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L162 Digitized by the Internet Archive in 2013 http://archive.org/details/computermethodsf601jaya Tjf hj uiucDcs-R-73-601 yuo. kdl C00-2118-00U8 top. <£-* -T Computer Methods for Analysis and Synthesis of Visual Texture by Sadali N. Jayaramamurthy September 1973 THE LIBRARY OF THE JAN 9 UNIVERSITY OF ILLINOIS DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS r uiucDcs-R-73-601 Computer Methods for Analysis and Synthesis of Visual Texture "by Sadali N. Jayaramamurthy September 1973 Department of Computer Science University of Illinois Urbana, Illinois 6l801 Supported in part by the Department of Computer Science and the Atomic Energy Commission under contract US AEC AT(ll-l)21l8 and submitted in partial fulfillment of the requirements of the Graduate College for the degree of Doctor of Philosophy in Computer Science. St 0.7 ? ■ V j-6>6l-(>6b iii ACKNOWLEDGMENT I acknowledge with gratitude the assistance, guidance and academic enlightenment received from my thesis advisor, Professor Bruce H. McCormick. His continuous support, illuminating discussions and abundant patience are mainly responsible for the successful completion of this research. I would like to thank Mr. John S. Read for his participation in the preparation of the paper 'Automatic Generation of Texture Feature Detectors', parts of which have been included in this report to preserve continuity. Thanks are also due to Professor R. S. Michalski for many interesting and useful discussions concerning interval covering theory and various other topics and also for allowing me to use his PL/1 programs implementing the A algorithm. Special thanks go to Mr. Stanly Zundo for preparing several neat drawings . IV TABLE OF CONTENTS Page 1. INTRODUCTION 1 1 . 1 Decision Theory Method 1 1. 2 Time Series Model for Texture 3 1. 3 Linguistic Approach 4 1 . 4 Summary of the Chapters 6 2. LITERATURE SURVEY 8 2 . 1 Visual Texture 8 2. 2 Classification of Textures 9 2.2.1 Structural Textures 10 2.2.2 Statistical Textures 12 3. DESCRIPTION OF THE DECISION THEORY METHOD 15 3.1 Texture Recognition as Statistical Decision Problem 15 3 . 2 Description of the Scheme 16 3.2.1 Local Patterns ( Events ) 16 3.2.2 Likelihood Ratio 17 3.2.3 Decision Goals: Optimal Decision Making 19 3.2.4 O.C. Curve 21 3.3 Coloring the Scene of Analysis and Classification Scheme 23 3.4 Multiple Textures 25 3. 5 Texture Discrimination 25 V 3.6 Analysis of Color Images 26 4. INTERVAL COVERING THEORY: GENERATION OF TEXTURAL FEATURE DETECTORS 28 4.1 Notation 29 4. 2 Generation of Interval Covers 30 4.3 Intervals as Textural Feature Detectors 37 5. APPLICATION OF THE DECISION THEORY METHOD 38 5. 1 Texture Recognition 38 5. 2 Border Extraction 42 5. 3 Extraction of Texture Regions 46 5.4 Iteration. 46 5. 5 Interval Complexes 57 6. TIME SERIES MODEL FOR TEXTURE SYNTHESIS 63 6.1 Texture Synthesis: Its Importance 63 6.2 Seasonal Time Series Model for Texture 63 6.3 Stochastic Models for Time Series 65 6.3.1 Non-Seasonal Time Series 66 6.3.2 Seasonal Time Series 67 6.4 Forecasting 69 6. 5 Generation of Time Series 70 6.6 Synthesis of 2-D Texture 72 6. 7 Redundancy Reduction 73 7. SYNTHESIS OF TEXTURE : A CASE STUDY 75 7.1 Selection of an Appropriate Model 75 7. 2 Estimation of the Parameters 79 7.2.1 A Method to Obtain Starting Values for Recursive Calculations of Residuals... 79 VI 7.2.2 Sum of Squares Function 80 7.3 Checking the Adequacy of the Fit 81 7.4 Generation of Texture 82 7.5 A Method for Determining the Size and Shape of Template Used in Decision Theory Method 84 7. 6 Texture Recognition 90 8. TWO-DIMENSIONAL FORMAL GRAMMARS FOR GENERATING TEXTURE SCENES 93 8.1 Linguistic Approach 93 8. 2 Array Grammars and Array Automata 93 8.2.1 Monotonic Array Grammar 94 8. 2. 2 Turing Array Automaton 94 8.2.3 Array Bounded Automaton . 95 8.3 Examples of MAG ' s Which Generate Texture Scenes... 96 8.4 Multilevel Array Grammars 101 8.4.1 Limitations of Conventional Array Grammars. 101 8.4.2 Introduction to Multilevel Array Grammars.. 102 8.4.3 Definition 103 8.4.4 Examples of MLAG Which Generate Texture Scenes 106 8.4.5 Restrictions on the 'Shape' of the Template 110 8.5 ' Interval Complexes ' and Grammars Ill 9. SUMMARY AND CONCLUSIONS 118 9.1 Discussion of Results 118 9.2 Suggestions for Future Research 119 LIST OF REFERENCES 121 VITA 129 lo INTRODUCTION There seems little doubt that texture plays an important role in the visual perception*, It has been convincingly argued [60] that textures carry information of use in object detection and recognition Many laboratory studies [18, 19, 29, 60, 80 1 have also shown that various types of textural information have measurable effects on perception of the depth, slant and shape of the surface. In the context of computer vision three problems have received wide attention recently, viz D texture recognition/ discrimination, texture analysis and texture synthesis In the present thesis we propose schemes to deal with the problems of visual texture These schemes, unlike many others which have been proposed in the literature, are sufficiently versatile in nature and are adequately adaptive to deal with different problems of texture. Id Decision Theory Method This proposed method is based on the principles of statistical decision theory Q For convenience we shall refer to this as the decision theory method (DTM). To serve as an introduction to this method, let us initially attack the problem of texture recognition. We shall consider a situation where we are presented with a pair of families of visual scenes (digitized images , of course) which differ in texture; say, those of malignant and nonmalignant tumor cells. The problem is to distinguish and appropriately label any member of these families. This is the problem of texture recognition. For this purpose we define a template to extract patterns from the scene of analysis, i.e., the n- tuple of pixels centered around each given positioning of the template. The occurrence of any local pattern can be considered as an 'event'. All possible patterns that can be extracted by a template define the universe of events. With the help of samples from both texture families we calculate the statistics of each event: including specifically the probability of its occurrence in any given family and the "likelihood ratio" which is the ratio of the probability of occurrence in one family to the other. Using criteria, which we shall discuss later in this thesis, we extract two disjoint sets of patterns (events). The patterns sharing a common set have the property that their occurrence in one family is more likely than in the other. The classification of 'unseen' sample (i.e., one that is not included in the training set) is achieved on the statistics of the occurrences of patterns belonging to the above-mentioned sets in the scene of analysis. The thresholds needed for the classification are determined with the help of training set. We have essentially constructed a filter by defining one set as the 'pass band' of events and the other as the 'stop band'. A scene of analysis is classified by the statistics of the events that fall in these bands. Here we treated one texture family as 'signal' and the other as 'non-signal' or 'noise' [binary case]. In this context texture recognition can be viewed as the popular problem encoun- tered in the signal detection theory. After the universe of events is defined by the choice of a template, in the decision theory method , the criteria for building up the 'pass band' and 'stop band' of events with the help of training set, is based on the statistical decision theory as applied to signal detection, In Chapter 5 we illustrate how the decision theory method can be successfully employed to extract texture borders even in the presence of noise, and to discriminate various textural regions in a complex scene. We also show how this method can be extended to deal with scenes which are both spatially and chromatically textured. Starting from this method we can generate a set of interval complexes which act as texture feature detectors with the application of Internal Covering Theory [49, 50]. This is explained in detail in Chapter 4 after a brief introduction to the Internal Covering Theory, as developed here at the University of Illinois by Professors Bruce H. McCormick and Richard S. Michalski. 1. 2 Time Series Model for Texture We propose here a method for the synthesis of texture. It is based on a model which treats the pixels (picture elements) of a digitized textural scene as a two-way seasonal time series. This method possesses the desirable characteristic that the parameters needed for synthesis are derived directly from the analysis of the "parent" texture (i.e., the texture to be imitated). With the help of well developed methods in the time series analysis we identify the process that generates the pixels of the parent texture. This process is essentially a white noise process that is allowed to pass through several filters. The seasonality of the time series represents the orderly repetitiveness of the patterns which is characteristic of all textures. The presence of white noise accounts for the minor variations in the repetitive patterns from one 'season' to another, e.g. , one video line to the next. Starting from certain boundary conditions we 'generate' the time series from the known process which in essence are the pixels of the synthesized textural scene. In Chapters 6 and 7 we illustrate this procedure of texture synthesis with examples. We also show how the parameters derived from the time series analysis can help select the appropriate size and shape of the template used in the decision theory method. With an example we show how it is possible to use this method also for texture discrimination. 1.3 Linguistic Approach The world of textures can be conveniently subdivided into two categories: statistical textures and structural textures. The nomenclature is self-explanatory. So far many authors have treated these cases separately. Present methods suggested to deal with textures have been more suitable to one class than the other. The methods we suggested above make no such distinctions and treat both cases equally efficiently. Textures can be viewed at many levels. For example a brick-wall texture consists of a fairly regular arrangement (structure) of bricks while individual bricks themselves display statistical texture. Though the existence of many levels in texture perception has been acknowledged, many attempts of researchers in the field have been limited to at most one level of textural order. We propose a linguistic model which is adequate to treat textures at many levels. At any level it can either be statistical or structural, that is to say we introduce a grammar that is capable of generating, say, brick-wall type textures. We at first attempt to generate some commonly occurring textural scenes by the array grammars, which have recently been introduced by Milgram and Rosenfeld [51, 52"! We then extend the concept of the stochastic linear grammars to the two-dimensional grammars in order to generate statistical textures. We note some inadequacies of the simple array grammars to generate complex textural scenes. By introducing the concept of 'shape' of a terminal symbol at any given level, we extend the simple array grammars to define 'multilevel array grammars'. Here the 'terminal symbols' in the grammar at a higher level are actually the 'sentences' of a given shape generated by grammars defined at lower levels, unless it is at the lowest level (i.e. , level 0) in which case grammar is a simple array grammar. We further introduce constraints on the choice of •shape 1 for the sake of grammatical simplicity and demonstrate the ability of the multilevel array grammars to generate multi- level textural scenes with some examples. Later in Chapter 8 we proceed to show that in some real cases, the 'interval complexes' derived from the decision theory method [49, 50, 64] help writing rewriting rules of the grammars that would describe these scenes. Thus we demonstrate that the methods we have suggested act to complement one another. For example, we start with time series analysis of some textural scenes in order to suggest an appropriate size and shape of the choice of template. Using this template, the decision theory method, in conjunction with covering theory, comes up with a set of interval complexes. With the help of these interval complexes we can then write an array grammar. Thus with these methods we can potentially discriminate, synthesize, and formally describe textural scenes. 1.4 Summary of the Chapters In Chapter 2 we survey the literature to review present strategies for the analysis and synthesis of visual texture. Chapters 3 and 4 are devoted to the introduction to the Decision Theory Method and Internal Covering Theory respectively. In Chapter 5 we study the application of the Decision Theory Method in practical cases with some examples. The time series model is explained in detail in Chapter 6. The results of the texture synthesis experiments are shown in Chapter 7 D Chapter 8 is devoted to the two-dimensional grammars in which we introduce the multilevel array grammars. We present the summary and conclusions in Chapter 9. 8 2. LITERATURE SURVEY 2. 1 Visual Texture Everyone seems to understand what "texture" means as we live in a world rich in textures. If viewed at an appropriate angle, texture can be seen in almost every scene. Yet it is one of those terms which has escaped a precise scientific definition. As a matter of fact, the host of visual scenes indicated by the term texture is so enormously large and varied in nature, it appears that it is a very difficult task to span the varied concepts of textures by a single definition. The definitions like "texture is that property of material which indicates what it feels like if touched" and adjectives like "rough," "smooth," etc. deal with tactile textures. We are here exclusively interested in visual textures. Pickett [591 observes that for any visual scene to be seen as texture, there should be "a large number of elements (spatial variations in intensity and/or wavelength)" and that "the elements and rules of spacing or arrangement may be arbitrarily manipulated provided a characteristic repetitiveness remains." He adds "provided there is sufficient detail shown in a small enough visual angle, a characteristic texture emerges even when the basic elements or spacings are randomly distributed." Thus the primary attributes of visual texture are "many variations" and "repetitive variations." Almost similar aspects are found in other available definitions. But even after a suitable choice of a working definition, there remain many difficulties inherent in the task of texture extraction technology. Hawkins [351 in his paper enumerates these in great detail and comes to the con- clusion that "texture classification may very well be one of the more difficult tasks in the field of image processing." Visual texture is really very sensitive to external conditions like lighting and angle of view, etc. The same scene may present a very different texture even for slight modification in the external conditions. Pictures in Brodatz's [7] book on "Textures" can be used to endorse the above statement. To simplify the already complicated problem, we assume that the scenes with which we are concerned are taken under similar conditions. With these problems and its prospective applications in important fields like biomedical sciences and remote sensing of the environment, texture analysis stands as an attractive as well as a challenging problem. 2.2 Classification of Textures Textures can be subdivided into at least two categories: statistical textures and structural textures. Statistical textures in a visual scene can be regarded as defined by a set of statistics extracted from a large ensemble of local measure- ments made on the scene. We need more information than this 10 to define a structural texture ; here the texture is considered to be defined by subpatterns which occur repeatedly within the overall pattern according to well-defined placement rules, as for example, wallpaper. Though we find many textures which can be classified adequately under one of these asymptotic categories, there are still many textures for which a strict classification may be questionable. 2.2.1 Structural Textures As its name suggests, the subpattern/placement model [72] appears to be the most appropriate model to deal with structural textures. This model is by far the most widely used model in the available literature. The "subpattern" is sometimes referred to as the "unit 0611." Usually, but not necessarily, the sub- pattern itself might contain sub-sub patterns and so on. As we accept this model, the description of texture is indeed simplified to developing a language to describe the unit cell(s) and spelling out the rules of its placement over a given region. Similarly, synthetic generation can be employed to develop the unit cell(s) and distribute it over the given area according to (appropriate curvilinear) placement rules. The works of Trout [89], Conroy [12], and Rosenfeld [72] reflect a similar treatment of these problems. Trout attempts a formal description of wallpaper- type patterns. The constraints of the wallpaper design are liberalized to provide a formal description of texture on a "generalized wallpaper" design. 11 He develops a syntax-like vehicle to describe primitive patterns. In terms of these patterns and appropriate syntactic rules, the unit cell is described. The wallpaper pattern, and eventually more complex textural patterns , can be described effectively in terms of these unit cells and their distribution. The success of this scheme obviously depends on the judicial choice of the unit cell, primitives, and appropriate syntactic rules. The crux of the problem lies in the choice of the above parameters based on the scene of analysis. This deserves more attention. Without practical simulation one cannot decide how appropriate the descriptive scheme is. Conroy [121 introduces the notion of "signs," defining these to be any collection of lines, usually in close spatial proximity to each other. These are similar to Trout's "primitives." Suitably manipulating the composition of various signs of different sizes, Conroy attempts to generate realistic textures. His method can be used to generate a unit cell and construct the whole scene by properly distributing the unit cell. His program requires as an input the choice of "signs" and their sizes, densities, and appropriate spatial distribu- tions. Rosenfeld and Lipkin [72 1 also start with the unit cell/placement rule model for the synthesis of texture. Various problems encountered in this process have been neatly explained. Results are encouraging only when dealing with man-made or artificial textures where the extraction of unit cell is not 12 a difficult problem. When dealing with the real world, for the description or generation of natural texture, the extensions of the above methods have not yielded satisfactory results. The reason for the failure of these methods when dealing with natural textures is obvious. The extraction of the holy unit cell(s), whose size and shape is unknown, is a formidable task. (So is the choice of primitives, "signs," placement rules, etc. on which the success of the schemes described are very much dependent. ) This is the present impediment when dealing with the analysis of texture. Further, when analyzing a piece of texture, even if one has successfully extracted a unit cell, it is yet very difficult to decode the "secret" of the placement rules — which demands a thorough analysis at the global level. 2.2.2 Statistical Textures Many investigators in this field have developed methods which are usually statistical in nature when working with natural textures. Rosenfeld et al have attempted to discriminate textures by merely detecting the differences in averages of local properties. They demonstrate their scheme by detecting textural edges using the gray level as the local property [71 ]. Muerle [53] also uses statistical analysis for the discrimination of textures. He divides the whole scene of analysis into many cells. Starting with a single cell, he expands the region by comparing the statistical distributions 13 of the cell with its neighbors and adding the new cells to fragments which have similar distributions. Bajcsy [3 1 also proceeds in a similar direction, that is by dividing the region of analysis into 'windows', extracting 'texture descriptors' in each window and patching windows which have 'similar' descriptors. Her work mainly deals with textural scenes that occur in nature (natural textures) like trees, clouds, water, grass, etc. 'Texture descriptors' are evaluated both in spatial domain and frequency domain. Directionality turns out to be one of many useful features that is detected easily in frequency domain. In fact texture descriptors like, 'homogeneous, random, blob-like, mono and bi-directional' are derived from fourier analysis of the scene. Analytic expressions of spacing, size and contrast of 'texture elements' are also derived. The process of region growing is represented by sheaf- theoretical model which formalizes the operation of pasting local structure (over a window) into global structure (over a region). Success of such methods largely depends on some crucial choices like: the size of the windows, determination of the similarity conditions, etc. Also there are many practical difficuties involved when dealing with natural textures, say, for example determining the texture element in a scene and computing its size and so on. Acknowledging these problems, Bajcsy claims that some higher level programs have been written which would help make these decisions without excessive user interaction. 14 There is yet another approach for dealing with problems of statistical textures as suggested by Bartels et al [4]. In this scheme a digitized textural scene is regarded as a two-way time series. Methods have been developed [5 1 to determine the 'order of dependence 1 in a two-way time series. That is to say, the number of neighboring pixels which influence the value of a given pixel can be estimated. This information and other relevant stochastic properties extracted from the time series have been successfully used for the proper diagnosis of tumor cells by Bartels [4], 15 3. DESCRIPTION OF THE DECISION THEORY METHOD When presented with a textural scene for analysis, we need to perform the following tasks. First we must discriminate the various textural regions. This is the task of scene segmentation using textural information. Later, if needed, we may have to identify each textural region with any one of the classes of textures presented before. This is the task of texture recognition. The next in line is the more difficult task : to analyze each homogenous texture and try to extract significant textural features from it. Ideally, of course, we would like to extract minimal information needed, perhaps to faithfully reconstruct or synthesize the original texture. This procedure, when fully developed, would constitute texture synthesis. 3.1 Texture Recognition as Statistical Decision Problem Texture Recognition can be basically viewed as a statistical decision problem where , as in a single trial of a psychophysical experiment, a cycle begins with the presentation of textural scenes and ends with the response of the decision maker. The type of responses expected of the decision maker varies with the problem. For example, in a typical recognition problem, given a textural scene T„ , a decision is to be made 16 if it belongs to T or T . Here T represents a family of visual scenes consisting of a particular texture, or, one of the hypotheses. The alternate is T which does not contain that particular texture. The anticipated response here would be either "1" or "0", representing "signal" and "noise" respectively. This is the binary case we considered in the previous chapter and the decision problem can be extended to a case with multiple hypotheses. 3.2 Description of the Scheme Ideally, each texture is composed of one single pattern, "the unit cell" and if one is able to extract it, the texture recognition problem would be reduced to comparing unit cells. But as we have already discussed, things need not be so simple: More than one unit cell may be present in a given scene, the placement rules may be difficult to extract, or no unit cell may be detected. We avoid these problems by defining our own universe of local patterns. We try to determine which of these local patterns characterize any given texture. At this point, we use the elements of statistical decision theory. In what follows, we define some of the basic terms needed for furtner presentation of the method. 3.2.1 Local Patterns (Events) We define a template centered around a point which samples the patterns from the scene of analysis. Each possible 17 pattern is an n-tuple of gray levels of the nearest n neighbors of the given point and is represented as the n-dimensional vector, e.g., e = (x, , x_ , ..., x ) and is regarded as an "event." All possible local patterns define the Universe of Events, E . For example, if the digital picture was quantized to "h" gray levels and the sampling template has n-pixels, then __ TT there are h elements in E . Figure 3.1 shows the number of elements in the universe for the templates shown. 3.2.2 Likelihood Ratio The "events" described just now serve as the local evidence for the decision maker at the local level. The "likelihood ratio" of an event is a single number which is an indicator of the strength of evidence that the occurrence of that particular event would provide for the presence of the signal. More precisely it is the ratio of the probability of the occurrence of the event in T to that in T . This is estimated from the "training samples" provided from both the families of T and T . Let E and E be the sets of events obtained from scanning T and T with a given template, respectively. Let n (e, ) = The number of occurrences of event e, in T n n (e, ) = The number of occurrences of event e, in T nT = The number of events in T nT = The number of events in T 18 EVENT NUMBER OF EVENTS IN THE UNIVERSE (E U ) TEMPLATE REGULAR HEXAGONAL NGL ■ # of GRAY LEVELS IN THE PICTURE NGL « 2 3 U 16 S in nr 1L ■ ■ * » f ■ — h 16 32 61+ 128 27 81 16 61+ 729 1+096 2187 l6,38U 16 256 U096 256 65,536 21+3 1021+ 1,068,576 17M 268M 512 19,683 262,11+1+ 68,600M Figure 3.1. Chart Showing Total Number of Events in the Universe for Various Templates 19 P (e, |T ) = The probability of occurrence of e, in T i.e. , probability of e, conditional on T . _ n l (e k ) nT 1 P (e, |T ) is defined similarly. Then, the "likelihood ratio" of the event e, is: ' k P(e, It 1 ) LR(e ) = 7 ^— — P(e |T U ) where "y" is a normalization factor which compensates for the intrinsic probabilities. The "a posterior probability," that is the probability of truth of the hypothesis at a local level conditional on the occurrence of the event , can be shown to be : . LR ( e, ) P(T X | e, ) = 'k' ' l+LR(e,) 3.2.3 Decision Goals: Optimal Decision Making There are four outcomes in the binary signal detection problem: two errors and two correct decisions. A "false alarm" is to choose T when it is T and a "miss" is vice-versa.. A "hit" is a correct choice of T , and the correct choice of T is known as a "correct rejection." There may be different values associated with the correct decisions and different costs associated with the errors. There can be many alternative decision goals. One of the goals, for instance, is to maximize the expected value. 20 The "expected value" is the sum of the four terms, each representing the value or cost associated with an outcome weighted by its probability. Other decision goals could be to: maximize a weighted combination, maximize the percentage of correct responses, satisfy the Neyman- Pearson objective, and so on. These are discussed in detail by Green and Swets [28] and they prove that whatever is the decision goal, the likelihood ratio criterion is the optimal decision rule, that is to choose T if LR(e, ) > p , where P is a positive number. P may vary for each decision goal. With this background, we are ready to partially define a local categorizer \ir on the basis of training set of information: R Let E = event space F 1P = [e| e e E 1 U E° and LR(e) > P} F° P = {e| e e E 1 U E° and LR(e) < P) F* = {e| e €E U \e 1 U E°)} Then, define^-, by its acceptance set R, i.e., \|r D (e) = 1 iff e e R Is. where F 1P < R < F 1P U F* and R fl F° P = Note that the determination of which events in F* are in R has not been made at this point. These represent the "don't-care" events and their assignment will be deferred until later. 21 3.2.4 O.C. Curve The Operating Characteristic (OC) curve (Figure 3.2) is a useful device for observing and predicting the behavior of these categorizers. To make the curve, each event e e E U E is regarded as a two-component vector with x = p(e|T ) and y = p(e|T ). An ordering can be imposed on these vectors by sorting them in descending order by the likelihood ratios of the e's. The curve is generated by placing the tail of the first vector at the origin and then concatenating the rest in order (Figure 3.2a). The OC displays several useful items of information in an easy-to-see form. For one thing, the training-set perfor- mance of a categorizer for each value of P is shown directly, since v for each threshold, the y-coordinate is equal to r i TT ,, w Q i ' J ^ ( e |LR(e )>p; i 1 Z i p(e T ) and the x-coordinate is equal to r i T „, Wft ->p(e T ). 1 le| LR(e ;>p j c ' The point on the OC corresponding to a given value of P is easy to find, since it is the head of the vector with slope = p* . (Note that p" only has a discrete number of values with different performance effects. ) The OC also provides a measure of the inherent separability of the textures in the training set. The area under the curve (A ) is equal to .5 if the textures are non- distinguishable (all events occur with equal probability in both textures) and is equal to 1.0 if the textures are perfectly distinguishable (all events occur in one or the other texture but not both). 22 x 0. C. * Figure 3.2a Operating Characteristic Curve 23 The OC curve can be displayed in another form, by plotting it on probability scales, that is, on axes scaled linearly for normal deviate (Figure 3.2.b). This type of representation will be useful when the density functions of the events extracted from T and T are of certain known forms, like Gaussian, exponential, etc. The OC curves will be straight lines for Gaussian and exponential distributions. In such cases the measure of separability can be specified by a single parameter [88]. This measure, unlike A, permits the recon- struction of the OC curve from which it is derived. But the value of A has another property useful for conceptual purposes , that it is equal to the percentage of correct choices that the system will make when attempting to select from a pair of events , one extracted at random from T and the other extracted at random from T , the one that belongs to T [28, 88]. 3. 3 Coloring the Scene of Analysis and Classification Scheme It is sometimes convenient to have the following trans- formation of the scene. A point in the transformed plane is marked "1" (or dark) if the corresponding point in the scene of analysis has an event in the acceptance set R. Otherwise, it is marked "0" (light). The resultant picture is the "colored" image of the scene of analysis. An ideal case: colored images of samples of T would be all "dark" and those from T would be all light. In practical cases classification of the scene would be achieved based on the global characteristics such as distribution x 0. c. * 24 ar" o en' o CM A en t— •— i 1 • - 1 * ■ t— CO •— i > UJ Q_ o • 21 e»" CO O Z V o • 1 o • C\J- 1 o CO- 1 1 1 1 1 1 \ -4.0 -3.0 •2.0 -1.0 0.0 1.0 2.0 3.0 Figure 3.2b Operating Characteristic Curve on Probability Scales 25 of l's and 0*s in the "colored" scene. The thresholds needed for the classification can be estimated from training samples. 3.4 Multiple Textures Thus far we have considered only binary signal detection (T and T ). The scheme can be extended to multiple textures in the same way that the binary signal detection is extended to that of M-ary signal detection. When there are M different classes of textures present (T n , T„, .„., T„ ) , we need to jr 1 ' 2 ' ' M ' calculate at least M-l "likelihood ratios" from which other likelihood ratios can be calculated. Here P(eJT, ) LR. . (e, ) = ki i ij v "k' 7 ij P ( e. I T . ) The universe of events will be partitioned into M-disjoint acceptance sets. An event e, will be included in the R. accep- tance set, iff, LR ij (e k ) > 1 for a11 i ^ ^ In general for satisfactory results, for larger M, we need a larger universe of events, which means more computation time. 3. 5 Texture Discrimination We can mark the various textural regions in a composite scene if we are given the samples of textures that might be present in it. The procedure is the same as before: extract 26 disjoint acceptance sets of events representing each texture, "color" the composite scene with each set in a different "color" (gray level). Ideally, we end up with each textural region marked with a different appropriate "color." 3. 6 Analysis of Color Images Thus far we have discussed only black and white pictures. In practice we come across many color images. The spectral (color) information appears to be very valuable and is used in many classificatory schemes involving natural biological images [95] , and also in remote sensing technology for photo interpretation. We can perform the analysis of color images using the decision theory method by modifying the interpretation of the 'event'. Here the event is allowed to extract the spectral information of each point defined in the template. One way is to codify the 'spectral signature' of each point, i.e., if S^ is the spectral distribution of a point, the event extracts the k- tuple of values { S^, , S^_ , ... S^,} where *■ . is a discrete frequency in the visible spectrum and k is finite. Thus the 'event' is an (nxk) dimensional vector for an n-point neighborhood as defined by the template. After defining the event in this manner, we can proceed with the decision theory method for the analysis of chromatically textured scenes. Normally we use a triplet of values of S^ for each point, corresponding to colors red, green and blue in order to match the performance of human eye. A set of intervals derived in 27 this method of analysis would specify the allowable band of intensities over selected set of frequencies in the visible spectrum. For extracting the set of intervals which act as texture feature detectors we need to introduce the elements of interval covering theory which is attempted in Chapter 4. 28 4. INTERVAL COVERING THEORY: GENERATION OF TEXTURAL FEATURE DETECTORS We have described a local categorizer in the previous chapter,, In principle it could be implemented by just looking up input events in a table of events and likelihood ratios. However, for real textures and useful neighborhood sizes, the number of events in the acceptance set R could be too large to make the process practical. Also, no categorization would be performed for events not in the training set. By applying some concepts generalized from switching theory, equivalent but much more efficient categorizers can be generated. This is accomplished by a technique which extends the switching- theoretic procedures for minimization of the disjunctive normal form of a switching function. In the case of binary signal detection, the events 13 OP from F can be considered a "true" set and those from F are considered a "false" set. The disjunctive normal form can be expressed as V | . (e. ) where |. is a predicate that has output "true" when the input is a particular event, e. , from F , and output "false" if it is from F . The symbol V above represents the logical "OR" of the predicates. Events from F* are considered as don't-care events. McCormick and Michalski have developed "interval covering theory" as a generalization 29 of switching theory [50] which permits the transplantation of much of the minimization machinery already in existence. In particular, Michalski ' s A algorithm [49] for generation of quasi-minimal covers can be used. 4.1 Notation To explain the method, it is necessary to introduce a few items of notation from [49] : E is the event space as before. That is, the set of all events e = (x, , x~ , .o., x ) where < x. < h - 1. 1 ' 2 ' ' n — i — A literal , i i , is the set of all events e e E i whose i-th component lies between a. and b. : a i x i = { ( x x , x 2 , . c o , x n ) | a ± < x ± < b ± . i An interval is a set- theoretic product of literals, L = i Q I a i x b i, I < [1, 2, .oo, n}. i The interval represents a "box" in hyperspace which includes all events between (a,, a», ..., a ) and (b. , b_ . .. , b ). ± Z ' n 12 ' n Note that components not specified by the interval are free to take on any integer value in [0, h-1] . "lft Oft An interval cover of the set F against F is defined as a union of intervals, L . , such that: ' 3 ' F 1P < U L . < F 1 ^ U F* . - 3 ~ 10 Thus an interval cover contains all the events in F plus some OP in F* , but none in F . However, the interval cover will 30 represent this partitioning of the space of possible events much more concisely than just enumerating all the events in the acceptance set for T . Also, the interval cover can classify events which were not in the training set, because of the inclusion of F* events in the "boxes." A quasi-minimal cover o can be generated via the A algorithm, which we can only sketch here. 4. 2 Generation of Interval Covers We can make this procedure clear by means of a simple example. For 1-D textures shown in Figure 4.1a, the F and OP F are shown in Table 1. For this example, an interval cover (a minimal one, as it turns out) can be generated manually by means of a visual aid, the Generalized Logic Diagram (GLD) , which was introduced by Michalski [49] (Figure 4.2). In the particular case when variables assume only two values, the GLD reduces to a diagram which resembles a Marquand-Veitch diagram. The GLD is a representation of the entire event space, E : 64 18 events in this case. To use it, the events of F are mapped 06 in as ones and those of F as zeros. The squares left over represent F* (don't cares). The cover is found by an iterative procedure which begins by picking the first "one" encountered in a TV-like scan of the GLD, and discovering all of the maximal intervals which include that "one," but no zeros (an interval "star"). One of these, the interval including the most "ones," is added to the covering set (initially empty). All of the ones included in the "star" are temporarily eliminated, 31 * IO CVI - ro CM » * * * * * ro * ro q: I- X Ld ro CM 02 CD U -P X CD En cci O •H w C 0) a •H Q CD C o CD bD ■H 32 p(hit) 0.60 0.45 0.30 0.15 RESPOND T t IF 1(e) > .25 RESPOND "T" IF 1(e) > 1.3 RESPOND "T" IF 1(e) >1.3 J L 0.15 0.30 0.45 0.60 0.75 0.90 1.00 p (false alarms) Figure 4.1b O.C. for Textures Shown in Figure 4.1a TEMPLATE DEFINING EVENTS = X< X 2 X 3 INTERVAL COVER 33 L t XK 1 3 2 2 3 X, X, X, X X NOTE: Lower limit for each variable is indicated in lower right hand corner and the upper limit in the upper left hand corner. Figure l+.lc. Interval Cover for Textures Shown in Figure U.la 34 on on cm H O CM o o H o H ID W 0) u ■P X 0) E-» E M (4-1 T) rH c o <4-i x: o to c o •H ■p u •H 4h •H W CO on on on CM CM o o o O o CM o CM CM A. w w CM CM on CM CM CM on O CM H o CM H ' CM H on CM 00 H O CM CM O CM H on O H CM H H CM CM H O on CM on CM H H O H o J 35 Figure k.2. Generalized Logic Diagram with Interval Covering 36 and the scan of the GLD is resumed. The first "one" encountered is selected, and the iteration repeats. Eventually all the "ones" have been eliminated. If all "ones" are included in the covering set, then the cover is minimal. Otherwise, the cover is patched up to include the neglected events, and may not be minimal. This procedure was followed for the example, and a minimal covering using three intervals resulted: (Figure 4 D lc) T 2 Y 3 Y 3 T M °Y 2 2 Y 3 T 2 V 3 2 V 3 If these are used to form a categorizer t where ^ (e) = 1 for R R e e R = L 1 U L_ U L_ , then the event categorization shown in Figure 4.1 results. The asterisks appearing above Texture 1 and Texture indicate events for which \|r had output = 1. The R subscripts on the asterisks denote the interval producing the "hit." Notice that the probability of a hit (19/20 = .95) and the probability of a false alarm (3/20 = .15) is as predicted by the OC (Figure 4.1b) for the likelihood ratio decision rule with P = 1. These textures could easily be discriminated by labeling regions with hit density over some averaging aperture greater than, say, 55% as Texture 1. In a digital parallel processor, like the Illiac Ill's Pattern Articulation Unit, \|r is an image filter. The input to the filter is a digitized picture in several gray values, and the output is a binary plane labeling each element in the input picture as to which texture the picture element most likely belongs. The application of simple smoothing or noise-removal algorithms 37 would then make segmentation into texture regions relatively easy. 4.3 Intervals as Textural Feature Detectors Intervals were achieved as "boxes" into which events from the acceptance set R were efficiently packed. As the events in the acceptance set occur more frequently in T than in T , the intervals which are nothing but groups of such events, have a tendency to define features which are more likely to be found in T than in T . Sometimes it has been found in a practical case that some intervals pick up features like vertical lines, horizontal lines, herringbone patterns, etc. which are perceived by human beings. Some of the features extracted by other choices of intervals may be very poorly matched for human perception. Strategies for the choice of "good" intervals remains to date largely unexplored. These "intervals" can be treated as 2-D filters which detect textural features more common to T than T . The normalized "output count" of the filters (normalized number of input "local patterns" from the scene of analysis that fall in the "pass band" of the filter) can be used as a feature vector. In this multidimensional space, we will be dealing with clusters of the scenes from T and T and for classification purposes we can resort to any popular cluster analysis methods. It may be noted that the generation of "interval covers" was possible for the binary case only. Extension of this procedure for the case of multiple textures is reserved for further investigation. 38 5. APPLICATION OF THE DECISION THEORY METHOD 5.1 Texture Recognition We illustrate the applicability of the decision theory method to the problem of texture recognition with an example. Here T represents a family of textures which consist of grids of different sizes with varying amounts of noise added (Figure 5.1a). Some random pictures shown in Figure 5.1b belong to the family, T . All these samples are digitized binary pictures containing 32 x 32 pixels. We selected a 3 x 2 template to extract sets of events E and E from these training samples. For convenience, we define the following disjoint sets of events which are frequently referred to in future. F 1 : {e|e e E 1 and e ft E°] F°: {e |e e E° and e ft E 1 } F^: {e|ee (E 1 U E°) \ (F 1 U F°)} The events in F are arranged in descending order according to their likelihood ratios. The OC curve for this example is shown in Figure 5.1c. We selected P = 1 in order to maximize the percentage of correct responses [28] , and extracted the IP OP sets F and F . 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9 99 89 999 9 99 9 8 999 8 9 9 99f 1 9 9 9 9 99 99 9 9 98 9 9 99 9 98 99 9 8 9 9 8 9 9 99 89 9 99 9 89 9 8 9 8 9 9 9 8991! 1 99 99 9 9 8 99 9 'a" 8" 88 "" 9 9 9 9 9 9 ' 9 9 8 8 999 9 9 99 98 9 99 9 8 9 98 8 9 99 9 8 89 9 8 8 9 8 9 8 9 999 9 999 8 09 8 9 99 89 " "8 "89" 99 ' 9 89 88 98 99 9 9 1 8 8 88 9 998 9 9999 9 9990 9 8 99 8 8 8 99 99 9 9 9 9 9 8 09 9 8 9 9 9 88 8 88 809899 99 9 89 8 999 9 98 98 99 8 999 9 99 9989999 9 999 9 99 9 99 ft 998 9 9 9 9899 8 9999998 9 9 99 9 fl"""' 80 "*8~" 9~ "9 9 8 ~ 099 998 9 88 8 9 8999999 9 9 9 8 9 88 99 8999 99 9 9 9999 9 9 9 8 8 9 9 8 8 8 89999 9 9 98 9 999 99999999999 9 9 8 99999 99998998 9 9 99888999999998999 98 9 9 999 99 9 8 888 99999 8999 99 9 9 9 999 8 _ 9 99 9 99 990 9 89 8 889 9 8 9 88899 99 98 "" 88 8 "88' 88 "9 99999 9998 88 899 8 8 8 n 9 8 9 99 9 9 99 9 9 9 99 8 99 999998 98999999 99 989 fl 9 898 . 898 8 9 8 9 9 99 999 99999 9 89999 9 9 9 9 9 9 99 88898998 89 99998 8 9 9 9998 88 8 89 9 99 9 9 9 9 9 999 99 9 9 « 99 9 9 9 99 9 _ _f| 0_9 _ 898 _ 808 8 9 899099 9 99998 89988 8 88 8 !«' """""" 88 "" 8 8" 8 988 889"' 9 9999 9 998 99 998 988 8 89 9998 99 899 9 99 89888 99 9 99 99 9999 999 89 888 888 89 999 99999999 99 898 9999 9 999 88 89 99 9 99 98 9 99 99 9 9999 9999999 8898 8 99 99 9 9999 9 9 99 99 9 9999 99989999999 9 99 998 81 9 9 9 8 9 89 9 99 9989 999 9 99 9 9 89 99 9 998"8""9TT«"«"'8" 8 8*9" — 8~~ " 988 8 99 8888 9 8 89888 8988 8 99 9 8 88 8 88 88 88888888 9 99 9 9 998 99 9 89 9 99 9999 9 99 9 9 99 99 98 88 89 99 8 89 9 9 8 8888 8 9998 8 999 9 99 9 8 8 89 9 999 99999 99 99 989 98 8 899988 88 888 9988 8 88 88 8 p 8 ft n 9 9 9 9898 88 88 888 89 89 99 898 98 8 8 99 9999 99 9 9 9 9 9 8999 9999 999 9 9 8 8 9 99 9 9 9 9988889 9 89 88 88 8 Figure 5.1b Samples of Texture T° (Random Pictures) 41 0. IS 0.30 O.MS 0.60 0.7S 0.90 1.0 1.05 Figure 5.1c O.C. for Textures Shown in Figure 5.1a and Figure 5.1b 42 number of training samples. Using these thresholds we attempt to classify the test samples shown in Figure 5. Id and 5.1e. The results of the classification (Figure 5. If) show that all samples in the first set of test samples have been correctly identified. In the second set, we used grids of larger sizes with added noise, as test samples. Here too, the classification is correct except in the presence of excessive noise. A similar experiment is conducted for recognition of textures of chromatin samples and artifact samples from Pap smears, which is described in detail in [64], 5. 2 Border Extraction When two textured regions meet , the events picked up along the border can be expected to be different from those that occur in the core of either regions, for the simple reason that they are made up from parts of both the regions. We may say that the "border texture" is different from the textures on either side. We can use this fact to force all the events in the border into a set. Later when this set is colored, the border will appear distinctly. Let t, and t n be the protosamples of textures present in a combined scene T . Let us consider the union of t.. and 10 t fi as T . We proceed as before and obtain F , F , and F sets. It is intuitively clear that F is a null set because t, and 1 10 t fi are part of T . F contains all events in both T and T . F contains only the events that occur in T exclusively, which 43 #1 i)l*w Aft * o ft ii a nun n fj a ft p _ _n c n r< « fi « *« nn "i a ft 8 3 A n fi n fi AA A n a n n ■> n ftftft i ft i fip n n n _ft _ n ft n h n a n n An n *» ft ft p n i rift •» n nro rn n « ft i nn o n n fton n ft ft n fi ft a >vt "in * ftftft A ■i o ft ft n ri n ft -i *i(in n ft n ft ft ft ft. ft ft do n ft n i n fin fir, no n ft r< ft a ft ft « nn ft n ft'i n ft i n Oft ft i") ft ft ftp ft ft ft'i ft ft H 1 Oft rii'n ft ri" fift n ft, f ft n ft ft ft. ft 1 Oft a* n r fin n m rft nn n ft on no #2 #3 99*9 nun 9 9 nn* a a a 99999 99998099008889998888888*88989988 _o_e n a 9 99 w 98« e * a a e a ? i«B« 990 989 e a a ana n aaa 088aaa9989998999eaee888999898899 M • • I • 999 • • 8 • 888 8 8 8 999999999n9aeaeaaaaaaa8aaaa899aa _ ner ^ _e_ej? _f_ a 998 iiM aaa 888 ^8««eBBa«88M88a88 eaeaeaaaeaa 888 88888888* *8S 888 8 88888888888888888888888888888888 8888 8 8 8 888 888 8 8 8 8 8 888 8888888888888n8888aa888S88888a88 8fl 8 8_?e8 88888 9 8 8 8 8_8 8 8 *«n8888a888888ea888e8rat7seia8888 MM 888 8 S 8 8 8 8 898 988 88888988888888888888888888888888 8 9 9 9 8 98899 98888888 99999899999999888888888888889989 _.!!_9-999*9. 9 9 9 8 9 9 8_8_ 9_9. 8. 8988888888889989*8988889999 S89~98 98 9 6 9 9 898 889 9 9 8 8 8 8 8 88899898889899988888e989»99 98888 8 9 9 9 980 9 9 9 999 9 8 89998 99999999899988888888888888888888 . o.q V6.888.e 9 8. Maes a 8 9 9 99999999989889999988999999889999 8 98998 89988898 888 8 8 89999999999898888999889888999999 9 9 9 9 999 9 9 99988 888 999 9 909999999999998888888(8888888999 9 8a 8 888898 8988 8 888 88 888988 8 8888 88 8 8 9899 9 8 888 tea 988888 8888 8 8 8 888 88 8 * 9 888a 8 8 8 aa 9a *s aa aa tana 88 8 aeaaaaasa saaa aaa a aaaa a 88 8 8 88 88 8 8 888888 8 889 8 888*9 * 8888 8 88 888 98 8 8 8 88*88889 888 8*8 8*8 88 8 889989 aa aa aaaa aaa aa aaa a * aaa aa aaaaaaa aaaaa a a aaa a aaa a* aawaa a 8 8888888 8889 888 888 8*888 899 98*99 88 9888 8 8 8 88889 889 8 999*8 8 8 8 88888 8*8 8 888 8 88888 888 8 88 88 89 8 8 8 88 8 88 88 88 8 8 8888 8 aaa aaa aaa* aaaa a aaa asa8 a 99 999 a a aa aaaaaa 898*88 988 8888 8 8 8888*8 8 8888 8888 8 888888 88 8888 8 888 99 8 888 88998 88 88888 8 888 * 8 888898 88 8 8 88*888888 98 88 88 a 888 88 99998 889999 99 998 8* 8 8888888 8 88 88888 8 8a 888 89 988 8 88 888 88 8 8 8 * 99 9 99 9 999 9 988888898 8 aaa aa aa a a aa as aaaaa aaa 888 8 99 88 8 88889888 88888 9 88 8 8 98 888 88888 8 888 8 aaaa 888 a a a aaaasaaaa #4 #5 #6 n If o h 09 9 o o o 9 fi 1 999 "9 9 I) o 98 9 n oonont 09999909pnn99"9oronoo98non_ 9 n n r rr 9 r >9 9 o 99 n i ri n oo o o o onn Oo9p9noo.in9onnoonror9rrooooo8aon 90 9' i >9rr»9 99 9 II 9 '1 '1 9 9 9 9 9 9 99on9oonnoooooo9nnrri 9990088 99099 n 99 9 r n" r r r n 9 9 n ft a ftft aaao ft ftiftft ftft ft n ft Oft ft wo ftni_ oo (in ft no n ft or ' ft ft ft ■■ift oft no ft n o a oft ft ft r n ftft ft ft ftftft ftft ftft n o ft 1 n r ft ft ft ft no ft ft ft n ft n ft n run nn o ft o ft Oft n ft ft r ift ft ftft t ftftft 'ft ftft ftnp nnnn ft ft o o n ftft n on n n ftft ft n ftft A n nft flu* * * ftp,** ftri ft** nft-i ft « #>n ftft • fi * nn ft #1 • ft ftft t • P A A ft. AAA ftft ft A AA A ftfift A AA A • A ft ft ftft AA A AA A A SA rr9889^ r rM9»«M^M^-rn-*'M* AA A A A A « ajftft A ft ftft A ft ftft AA A A ft ft A AAA A AAA A A Aft A AA AAA AAA AAA A AAA AA«AAft A Aft Aft Aft A AAA A AAA A AAA ft ftftft ftft A A A A A ftft AAAA AAA A A AA A A 8 8 888 8 9 9 99 9 9 9 888 8 9 9 9 8 888 8 8 f 9 9 9 9 8 89 9 a 8 r 9 8 aa 9 9 9 8 9 8 "a 8 8 " a '" 99 8 9 8 89 9 9 99 9 9 8 9 8 89 8 088 99 9 9 99 9 99 9 99 99 9 98 8 8 8 8 9 88 8 80 00 8 888 98 8 r989888»88*8e88898*8*9"*9H8a8"»88T( 88 8 8 9 88 8 88 8 8 9 8 8 8 8 8888 8 8 8 88 888*9 9 aaa a 888888 88 88 88 88888 a 888 89999998 89 888 8888 88 88 assa * aaasasa as s as s aaa an* • "aa a a aaaaaaa a •a «* aaaa aa a a asss a as a 998 88 8 888 88888888888 8 8 e 99989asaa*aaaaa*a aa as 88889 99989888 8 8 8 888 aaaa e aaa a a a aa aaaaa a* 899** 88*8 99 988" H III 99 9 99 999*9* ••••*••••■• 888 88 999 99988 8 •••*( 8 8 8 8 aa aaaaa aaa aaa8 aaa a aa a 9 9 asa as aaa aa a a a aa 9****« • ••••• ••••• •• •■•• 9999 9 8S8 89 9 888 888 8 8 08 99888 9888 88 88 88888888 0999*888 aaaa *•• aaaa a aaa 99998 9 aaaa aaaaaaa as** s 9888 999889999*8 88 SaS 888 88 ■8 99998 99* •• 89 999 •* *8 9 99999* 9* •*•• " * * aaaaa aaaa 9888 8 8 88 a 888 88 88888888 8888 8 8 898 88 8 8888 aa aaaa eaaa * aa aa asa 899 99 aa aaaaa aaa aaaa a 8899 9 988 a 99 a a* a aa * 98* D88988" KB 88 t** I*** •9*999 88 888 8888 8 88 88 8 a s aaa as aa aa asa aa 8888 8888 888 8 8 * 8*8 89* 99989988 8 88 888 88 8 Figure 5. Id First Set of Test Samples #1 #2 44 #3 aaaaaaa«aaaaaaaaaaaanaananaanaaa aanaaaaaaanaaaan.iannaaaaaaanaaaa aaaaaaannananaaanaaaannnana aaaaa n»ranaaa fi nr^f if i#v*i\r*Fj-ttii» Fiin^ ^ranaaaaraaaa inanano-tnnftaaanaana'Vianaanaanaaa naa^narmaanaaaf nannnaarnnanraann aMa**»>M inn* naai>n'iaa"^rariaa' 1 aaaaa innnn -in iaa^n*nnnnannananaaaanaan aaaa ee eee aaaaeeae a_ _a_B8 a * * a a_ a aaaaaBBeaeBaaeaeafBeTeaaaeeeeeei aaaa ee aaa aaaaaeeee nee aaa n a • a a • • aaaeBaBeeBaaBeBaeaeeeBeeaesBaaaa _ _•« ee e e e e aa a eaeeaeae aa a a a a aa a a anaaaaeaaaaaaaAaeaaaeaaaaeeaaeaa a aaa a a a a a • aaaaaaaaa o a_8a_ aa_a _e a a a aaaaaft'aaeaaen'anaeraeSgafterieaVffiM r a aaa a aaa a an an n a a a aaa aa a aaa a aa a a aanaaaaaaaeaeaaeeaeaaeeaaaeeaaas _ao rj_ g_ a _e_ a_ _a a aaa ~a aaaa aa a a a aa a a aa naneanaannaaaaaaaaanaeaaaaaaaaaa fi » n aaa a a a a a a a ■ aaaa aaa n a n a aaa a _ a _ a a atapaaaaaaaaaaaaaaaaaaeaaeaaeaaa ' ran aaa a a t aa aaa aa aaaa aaaaeaaa annaaaaannaaaaaaaaaaaaaaaaaaaeea a aa • a aaa a aa aa a a aa a a a aa aa a a a a a a aaa aaa a a a a a a aaamaaaaaaaaaaaaaaaaaaaaaaaaaaa a a a a aaa a aa a a a aa aa a a a a a aaa a a aa a aa eeaee a • • eaaemeaaeeaeeeaeeaaaeaeaeaaeaea aaaa aa a a aaa a aa a a a aa a aa a a aaa i a a a aa ae a a aaa aaa ee a eaeeeaeaaeaeaaeaeaaaaeeeeaeeeeae aaaa e a aa aa a a a ee aaa a aa aaa eee a ae aaaaaaa aaa aaa a a aaftaaaaftaAaAaaaaaaeaaaaaaaaaaaaa a a aaaa a aaa aaa aaa aa aa a aa aa a a aaa aaa a aa aa a a a a a aaa a a a aaaaaeaeeeaeaeeeaeeaeeeaeeeeeaae eae_ a aaa a a at a a aa aa a aa a aaa a • aa aaaa aaaaa » n » m » a aa eeaeeeaaeeaaeeeaaaeeeaeeeaeeeeee a aaa aaa aa a a aa aa a a aa a aa aaa eae aaa a aaa aaa a aaa a aaa aaaaaaeaeaaaaaaaaaeaaaaaaaaaa aaaa aaaaaaa aaaa a a aa a aa aa aa a a aaa a aa a a a aaa a aa aa eaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa #4 #5- #6 aaaaaanii'iaannnaaanaaaanannaaaan n a a n a a n r r< n * a n a n a a a a n a n _ . a n aan^ara^aaasaaaaaaaanaciaaaaannn u a n i a a n a a n a a a a a a a a a a a o a a aanaaaaaaaa aaaaaaa anar an naaaaoan a a a a a a a a a a a a a a a a a a a a a a a a aaaaaaaa laaaaaanaaaaaaaaaaaaaaaa ".. ._"._. ". a a _ a a a a a a a a a a a a a a a a a a aaaeanraaaaaaaaaaaaaaanaaai aaaaa aaaaaaanaaaaaaaaaaaaaaaaaaaaaaaa a a a a a " a aa a a aa a aa a a aa aaa aa a e _oaa _ a _ ae _ a _ a" a ' a "" a a "eee" aa aaaanaaaaeaeaaBaaeeaaaeaeeaaaeaa a aaa a a aa no a a aaaa a a a e a aaa a aaaa aaa _a _ a a _ a a_ __a aaaaaaaanaaaaaaaaaaaaaaaaaaaaaaa a a aa a a a a n ana a aaa aa aaa eae a a aaaaa aa a naaneanaaaaaaaeaaaaaaaaaaaaaaaae a aa a a aaa a a e a a aaa aaa a a aaa a a aa a a a a a eaaaaaaeaaaaaaeaaeeaaaeaeaaaaaaa a _ sag. _a.a _a_._e_e_ a_ _. a a aa a aaaa a a« aa a a aaa aa a a aaa a a anaaaaaaaaaaaaaaaaaaaanaeaaaaaaa a a a aa a a a a a a_ a a aaaa a a a a a aa aa a aaaaa a ana aaeaaaaaaaaaaaaaeaaaaaaaaaeaaaae a aa a aa a a aaa a a s a a aa aaa aa aa aaa aa a aa aaaa aa a a a aa a a aa aa aaa aa a aa aa a a a aaa a a a a aa a aa asaaasaaaaaaaeaaaaaaaaaaaaaaaaaa aaaaa a a aa aaa aa a a a aaaa aa aa aaaaa a a aaaa do aaaa aa aa _ee a a a a aa aa a a aa aaeeaeeeeeaaaeeaeaeaeeeeeee aaaaa a a aaa aaa aaaa aa a a a aa aa a aaaaa a aaa a a a aaa a a aaa a a aaa a a a eee a aaaaaaaaaaaaaaaaeaaaaaaaaaaaaaaa a a aaaa a a eaaaa a aaaa ae aaa aaa a aaa aaa aaa a a ee aaaa a aaa aaa naaaaaeeeeaeaeeeeeaeeeeeaaeeeeaa a aa aaa a aa % aaaaa ee aa a aaa a a aaaa ae ee a aaa a aaa a a aa a aaa a a a a aaa aaa a aaaaaaaaaaeaaaaaaaaaaaaaaaaaaaaa a a aaa aa aa aa aa a a a aaa a a a a a *m a a aa a ca aaaa aa aa a aaa aa aa a a aaaa a aaaeaaaaaaaeaaaaaaaaaaaaaaaaaaaa a aa a aa a ea ea a a a aa a a aa a #7 #8 #9 aaa aaaaaaaaraaaaaaaaaaaaaaa aaaaa r a a a a aaaaa aaaaanaaaaanaaaaaaaaraaaraaaoaaa aaaaaaaaaaaaaaaaaaaaraaraaaaaaaa aaaaanaaaa* aaaaaaaaa aaaaaaa aaaaa n a a n a aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaa a aaa aa aa a aaa a a a a aaa j a a_ aaa a aaa a a' aaaa aaaa a anaaaaaaaaaaaaaaaaaaaaaaaaaaeaaa aaa a a a aa a a a n a a aaa a _a a a aaa a a a 'a aa" a" a a a aaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aa a a a a aaa aaa a _ aa a a „e e a _ aaa aaa a a aa a aaa aa aaaaaaaaaaanaaaaaaaaeBaaeaaaeaaa aaa a a a n s aaaa a a b a _ ^ bb a aa _ a_ "">""" 5" '" a a a a""" m~* "a e a aa a aaa aaaaaaaaaaaaaaaaaaaaaaaeaeeaaeaa aaa aaaa ee a a a a aa' a _ aa a a n_a a e_a_ a_ a a" 8 a" a~~ 'a aa a a a a a aa aaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaa a a a ae a a a aa aaa aa a a a a a a a aa aa aa a aaa aaa aaa ae a a e_ ee j_ a_e aa a aaa a e a a a "a e> e a a aae eae a aaaaa aaaaeeaeeaeeeaeaeaaeaaeaMaaaaaa a aea e a aae aaa aa a a a ea a ae aee e ee a ae a aeee a ae e e e_e e a a aaaaa a a ee ae a a aatea ee a aaa aaeeeeaeMeeaeeaeeeeeeeaaaaeeeae aa aa a a a a a aea aa a ae aeea e c a a a e aaa e eee eae a a aea a aea aae aaaaaeeaaeeeaaaaaaaeaaa aaaaaaaaa ae a e a a ee ee a aa a aa aa aa aaaa a • a a a a _ _ee_ea aa _eeee aaa a a a a e ee ee aa a aeee e e ee e a aaa eeaaaeeaeeaeeaeetaeeeeeaeeaeetM aaa a eee e e e ee a aaa eeeee e ee aa _ae_ee_ eaeea_ a e_e_e __e a a a" a e" e ee aa e a a ee aee aaa aaaaeaaneeaaeaeaaaaeaeeaaaaaaaaa aeeeeeeea aeeaa aee eaee e e aa a Figure 5.1e Second Set of Test Samples 45 SA"?LF « • SA WD LE .« c . 1 w Pl E « S.""LE * • SAW?LF ■ 3.A w:, LE H •;A«Pl c m • SA"°L? » SA-PLC n • SA"PL (: 4 ;a vo ll 9 SA"?L r it • SA«'LE u Si-Pl^ H 5A"PL E • • SAW9LE • SA'OLE « SA"'L C • • ENTER P PC CLASSIFICATION 1 ? 3 •5 = 6 7 s A C Tr jrT'J 2 F T£XTJO.E_ TrxTUOF First set of test samples "\ Second set of test samples J Figure 5. If Results of Classification 46 are nothing but the border events; therefore, when we color the F set we obtain the location of the border (Figures 5.2a and 5. 2b) . 5. 3 Extraction of Texture Regions Given protosamples of textures present in a composite scene we should be able to mark each texture region separately. For this purpose we analyze the protosamples and extract disjoint acceptance sets for each texture as described in section 3.4. We then analyze the composite scene and mark the events belonging to each acceptance set differently. The result of such an experi- ment on a composite scene (Figure 5.3a) is shown in Figure 5.3b. Here we notice that each texture region is marked uniformly and differently. We repeat this experiment with natural textures, this time using the protosamples of the textures of nucleus , cytoplasm and the background of the brain cells. The result of 'coloring 1 the composite scene is shown in Figure 5.4. This is the raw output picture and it is possible to 'clean' it, by removing, for example, points which do not have a certain number of neighbors belonging to same class. 5.4 Iteration IP OP . . After obtaining F and F for a given pair of textures T and T , if we go back and color the original scenes as described in 3.3, we obtain T and T . This output pair is more easily distinguishable than the original pair. As a 47 MSMMMiiiMzMzMzSziSSMMiSMSBSM t»-4-M-9-a-a-88 99999 99-9-9-9 -9-9- fl ltl^:«ai»Wa>(ma«-li-8-»-8W8888888 M88 «-«-«-8-8-B-88 8889999-9-9-9-9-9- _WtH 8aa88888888-e-8-8-8ga^8»8M g8 *-*-*- a- a-8- aa M«aaas-8-«-8-«-a-~ _ flJadBflBeauaaa-a-B-g-egaauBBaaaaa »-■*-»-«-«- «-i« *j88»aa-8-4-8-8-e- aaaaaaaaaaaaa-a-a-a-aaaaaaaBaa aa a-a-a-a-a-a-eaaasaasB-a-a-s-a-e- si^^jf aaasaa-a-a-a-aaeaaaBBBaaa «-d-«-B-8-a-aaaee8a«a-#-*-#-e-»- wKumBnaeaaBa-a-B-i-aaaaaaBaaaaa a-a-a-a-a-B-aaaBaaaaa-a-j-j-i-i- _ jf«jft0»i)«i«wiiij0- a- s-a-Baaaaeeaaaaa *-•» -a-a-M-B-aaaaaaaaa-a-a-a-a-a- ,awjjiBjHjgjja a-a-8- < -g W>wa g a „ aaa tt-rt-d-s-a-B-seasaaaaa-e-a-e-a-a- ai»>iWjji«^ii(B M i8-8-8-a-aa8aii«aaB«f«a «-i»-tj-i«-9-«-88 89aaaea-a-a-8-e-a- _3UMMMMM §r trirgriifjgliili i » B-a-a-a-B- a-«8 aasaaes-a-a-e-a-s- _ Mj8j«aaa eB9aaa- 8 -B-B-Ba8aaaaaBa Ba «-a-9-i«-*-8-a8«aa*»aa-e-8-e-a-e- aeaaaeaaaaaaa-B-B-a-BaBBaaaaaaaa e-8-i-i-a-B-eaaeaa888-8-a-8-B-»- ^HiiMMiiaaara-t-a-aaaaaaaaaaaa fl-a-a-8-B-e-B8B8B888B-8-i-9-B-»- eaBwaaaa8Baaa-a-«-«-a. 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Let us define an operator C which operates on a pair of scenes : (T^T ) % (tV ) ' c ' c By repeated application of operator C , (T 1 ,! ) £> (T 1 ,T° } j ' n' n c c we can go on iterating until we obtain a satisfactory separation in the output pair (Figure 5.5a through Figure 5.5e). The improvement in OC curve towards the ideal case with each iteration can be seen in Figure 5.5f. It is clear that each point in the output picture is a function of neighbors around a given point in the original picture. As a result of iteration, the points in the output plane depend (maybe weakly) on a larger and larger area in the original scene. That is to say, we are operating on the original pictures at a more and more global level with each iteration. 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Figure 5.6e shows the union of the output of all eight filters for the scene of analysis. As expected, the herringbone texture is blocked. Figures 5.6c and 5.6d show the output of individual filters for the same input scene (Figure 5.6b). It can be seen that features like horizontal lines and vertical lines have been passed through—thus it is demonstrated that some of these filters can be used to extract/enhance some of the features which the human eye can perceive. 58 * Mr M M * * * * Mr Mr Mr Mr Mr M M Mr Mr Mr Mr -* {KM Mr M ///// m * M- M M M M M 1 M M yyy//, M M M M: M M u * M M M * Mr M M Mr M M Mr M M M M m □ W don't care Figure 5«6a Interval Complexes (.2-D Filters) for T = Random Texture and T = Herringbone Pattern 59 .■•.•.■.■..■..■•■I ■ i,mu,i.iu.i....i....i.i...iii.i.i.i i.n .88 B. a. 88... 88 1. 8. I I.. ■ ■ ■.■..■ I..8...8..8 8... 8.. 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Figure 5.6e Union of the Output of all the 8 Filters f 0r the Input Scene shown in Figure 5.6b 63 6. TIME SERIES MODEL FOR TEXTURE SYNTHESIS 6. 1 Texture Synthesis : Its Importance The synthesis of natural looking textures deserves considerably more attention than it has received. General methods are needed by which we can generate a scene that bears an acceptable resemblance to the texture to be imitated ('parent' texture ) o It is highly desirable that the parameters required for the synthesis are derived automatically from the analysis of the 'parent' texture. Texture synthesis procedures of this type can be evolved to solve the ever-pressing problem of image storage. For it requires considerably less storage to save a few parameters and boundary conditions than to store a complete digitized scene. This effectively means redundancy in the image has been reduced, leading to a more efficient analysis of the resultant scene. Using the values of the texture parameters as features for discrimination/recognition is also suggested. By specifying intervals on values of each parameter, we can potentially generate a family of textures. 6. 2 Seasonal Time Series Model for Texture Many authors have suggested methods for texture synthesis [12, 22, 24, 38, 57, 58, 72l . Among them only Rosenfeld et al [72] and Conroy [12] attempt to synthesize 64 natural looking textures while others merely generate scenes with prespecified statistical properties to be used in their experiments concerning the visual perception of texture. How- ever none of these methods possess a desirable quality, namely the choice of parameters needed for synthesis be based on the analysis of 'parent' texture. We suggest a scheme for the synthesis of texture that has this property. This scheme is based on a model that views the pixels of a digitized two- dimensional textural scene as a two-way seasonal time series. Any digitized visual scene can be viewed as a two-way time series. Bartels et al [4] were among the first to treat a textural scene as ordinary two-way time series. However the orderly repetitiveness of a subpattern which is an essential characteristic of texture strongly hints at the possible representation of a digitized textural scene by Seasonal Time Series . The choice seems quite natural and very appropriate because there is a striking correspondence between some of the problems that occur in textures and those that are considered in seasonal time series analysis. For example, in TV scan of a texture the repeating subpattern is not necessaraly identical from line to line though it retains similar characteristics. In the time series analysis the minor variations of the function from one season to another is accounted for, by the assumption of the presence of white noise. Well developed methods are 2 available to estimate the values of the parameters [u , o } of this noise. 65 It is possible that there can be more than one periodicity present in the textural scene, such as, the subpatterns consisting of sub- subpatterns and so on. This may be treated as multiple seasonality. In the case of statistical textures these latter periodicities may be absent and they may be treated as an ordinary time series with no seasonal effect. Spectral analysis, in the frequency-domain, comprises one class of techniques for time series analysis. Our interest here is to stay in the time-domain and build stochastic models for time series. This way we can gain a better insight into the nature of the system that generates the time series. The models developed then can be used to obtain forecasts of the future values of the time series. Our objective, from the engineering viewpoint is to obtain models that possess maximum simplicity and the minimum number of parameters consonant with representational adequacy. Precisely the same approach is taken by Box and Jenkins while dealing with this subject in their book "Time Series Analysis - Forecasting and Control" (Holden-Day, 1970) [6]. 6. 3 Stochastic Models for Time Series Let us introduce some notation for convenience in representation. Let . ., Z t _ 2 » Z t -1' Z t' Z t+1' '•• be the time series which can be denoted as [Z^ t 66 A series of values ' a . ' ("shocks") are assumed to be 2 generated from white noise process with mean '0' and variance cr , cL B is the 'Backward Shift Operator' such that BZ,. « Z. , ; hence B m Z.,_ = Z^ t t-1' t t-m V is the 'Backward Difference Operator' such that V Z t = Z t - Z t _ 1 = (1-B) Z t V 1 " Z t = (1-B) m Z t 6.3.1 Non-Seasonal Time Series Box and Jenkins [6] represent the process that generates the non-seasonal time series [ Z . ] by the following model: (B) V d Z. = 6 (B) a. (6-1) p t q t Where $ (B) and G (B) are polynomials in B of order p p q and q and are known as • autoregressive (AR) operator 1 of order p and 'moving average (MA) operator' of order q respectively. The process is known as ARIMA (autoregressive integrated moving average) process of order (p,d,q). This model is sufficiently powerful to represent time series which show both stationary* (d=0) and non-stationary** behavior. * The process that generates 'stationary' time series is assumed to be in equilibrium about a constant mean level. ** Only homogeneous nonstationary behavior which calls for the ^th difference of the series to be stationary has been considered. 67 There are (p+q+2) unknown parameters to be estimated from the data. It can be seen that the ARIMA process may be generated from white noise a. by means of three filtering operations as shown in Figure 6.1. 6.3.2 Seasonal Time Series In the same manner if [Z ] shows seasonal behavior (with period s), it can be represented by the following multiplicative model [6] : (B) (B S ) V d V° Z^ = 6 (B) 9 n (B S ) a. (6-2) p P s t q Q t s s s where (B ) and n (B ) are polynomials in B of order p and Q respectively and V is the seasonal backward difference operator (1-B S ). This modified version of ARIMA process is said to be of the order (p,d ,q)x(P ,D ,Q) . This multiplicative model is very useful in that it can be easily extended to take care of multiple seasonalities. Box and Jenkins [6] and Bacon [ 2] give detailed procedures for identifying, fitting and checking the adequacy of the fit of the appropriate model for the given time series. The programs developed by Bacon implementing these procedures have been modified to suit our needs and are described in detail in [481. These procedures will become clear when we make a detailed case study with an example in the next chapter. 68 co >i-H 5-1 CO 03 W M C 0) of Z . is the conditional expectation K u) = rw = E [ z t+ fl z t' Vi' -' ] The conditional expectations of the terms in (6.4.2) are evaluated by inserting actual Z's when these are known, fore- casted Z's for future values, actual a's when these are known, and zeroes for future a's (because E[a .] = for j > 0). 70 This forecast function is highly useful when the future values are needed for very short lead times as required in many business and industrial applications. For longer lead times the forecast error will be cumulatively larger. Also, as the future values of a*s are replaced by zeroes, the eventual fore- cast function takes on a deterministic mathematical form as dictated by the solution of the homogeneous difference equation containing autoregressive terms only. Our intention of fitting a time series model for the pixels from a textural scene is to be able to forecast the series for longer lead times and thus effect the texture synthesis from a set of boundary conditions. The forecast function will be unsuitable for this purpose for the reasons mentioned above. Especially the eventual disappearance of the stochastic effect is highly detrimental for the textural property of the scene. 6.5 Generation of Time Series As shown in Figure 6.2, the general ARIMA process can be generated from a white noise process with appropriate filtering operations. Figure 6.2 shows the rearrangement of the filters in Figure 6.1 for the generation of time series. Where $ (B) Z, = 1) (B) v Z. and a, is the output of the white noise generator. The present scheme requires (p+q+d) delays and registers to store the previous occurrences of Z's and a's which are required in the calculation of the present value. The multiplicative constants (<$> ' s and 0's) are the parameters of the model. To 71 WHITE NOISE GENERATOR Q t. q (B) >Ci 1 FILTER 1 1-(B) Z« FILTER 2 S-(B)a t FILTER 1 (l-$(B))Z t FILTER 2 A -UNIT DELAY Figure 6,2 Generation of Time Series [Z, ] 72 start with, the registers are loaded with the 'initial' conditions and the values of Z (i > 0) are successively regenerated. The above discussion can be extended to seasonal time series. 6. 6 Synthesis of 2-D Texture So far we have considered one-dimensional time series. By concatenating either successive rows or successive columns, a two-way series can be treated as a one-way time series. Of course, by doing this we are introducing one more (known) periodicity. The pixels from a digitized textural scene to be imitated ('parent* texture) are row (or column) concatenated to form a one-way time series [ Z , ]. With the help of programs USID* and LIKESURF**, an appropriate model is fitted and the values of the parameters are estimated. The generation process shown in Figure 6.2 is simulated in GENTEX , which synthesizes the Program USID: Univariate Stochastic Model Identification. This program inputs a time series [ Z t ] and plots the auto- correlation function (acf), r^ and the partial auto-correlation function (pacf), ^^^ which help in the identification of the series. For details see [2, 6, 48]. •k -k Program LIKESURF: This program plots the likelihood surface over the specified parameter space and outputs the maximum likelihood estimates of parameters for the selected model for a given time series. See [2, 6, 48]. + Program GENTEX: This program synthesizes the textural scene by generating the pixels for a set of boundary conditions from a given time series model. Using PAX [46] subroutines the output scene is printed as a grey-level picture. See [ 48] . 73 textural scene from a set of boundary conditions. This method is illustrated with an example in the next chapter. 6.7 Redundancy Reduction The digitized TV scan of a picture is a row-concatenated one-way time series. When an appropriate model is fitted for this series, we come up with an attractive scheme for redundancy reduction in the transmission of the TV scan of the picture. In the literature [82] we find a method known by the name 'optimum linear predictor' in which the next value in the series is pre- dicted by expressing it as a linear function of the previous occurrences : y t = z e i Y t-i i=l 1 r 1 t = y - y where y = the predicted value y = the actual value e = the error in prediction 2 0's are selected to minimize the mean square error [ cr (e )]. Data compression (redundancy reduction) is achieved by trans- mitting the error values only. At the receiving end the actual values (y's) can be reconstructed from the 'errors' (e's) once the system has been properly initialized. In the present scheme derived from the time series model, a one-step-ahead forecast is made by expressing the next value as a linear function of not only the previous values in the series, but also the previous 'errors': 74 Z. = S 0. z. . t . , 1 t-1 1=1 q s e . a. . , 1 t-1 1=1 (autoregressive terms) (moving average terms) [ z t - z t l where Z^ = one-step-ahead forecast of Z J t 1 a, = the 'residual' or 'error' The values of $ 's and e 's are selected to minimize the mean 2 square error [ a ]. We can see that this is an improved version GL of the 'optimum linear predictor' method for achieving redundancy reduction in the transmission of a TV scan of a picture. At the receiving end we can perform a similar 'filtering' operation as shown in Figure 6.2 to reconstruct Z's from a's, which are transmitted. 75 7. SYNTHESIS OF TEXTURE: A CASE STUDY 7.1 Selection of an Appropriate Model The textural scenes that are used in the following examples are taken from Brodatz [7], A digitized scene of cheesecloth texture (D-105, Brodatz [7] ) with 16 graylevels is considered as the 'parent texture' (Figure 7.1) in example 1. For the sake of analysis, pixels from a 32 x 16 window is treated as a two-way input time series. It is row-concatenated (s = 32) and is given as an input one-way time series ( [z ] , t = 1,512) to the program USID*. The output of this program consists of the estimated autocorrelations for [W 1 where w. = v n d V D z . t 1st (a) for the original series, Z. i.e., d = D = [ACF-OO] (b) for the series differenced once with respect to basic interval only, V, Z i.e. , d = 1 D = [ACF-10] (c) for the series differenced once with respect to seasonal interval only, v Z^ s t i.e.,d = D = 1 [ACF-01] (d) for the series differenced once with respect to both basic and seasonal intervals, V, Z^ '1st i.e. , d = 1 D = 1 [ACF-11] * See footnote in section 6.6. 76 Figure 7.1 Cheesecloth Texture ™S JTO 'ii'y * "■ Figure 7.2 Synthesized Texture 77 Figure 7.3 shows the plots of the autocorrelation functon (ACF) and partial autocorrelation function (PACF) for all the four cases mentioned above. The partial autocorrelation function is more useful in the identification of the non-seasonal models and hence is not further discussed. The autocorrelation values in case (a) are large and highly periodic as might be expected. In case (b) it is seen that by differencing with respect to the basic interval reducing the correlation in general while a heavy periodic component remains. We obtain stationarity by differencing with respect to the seasonal interval as seen in cases (c) and (d) where the correlation values diminish very rapidly. The values of d and D which produce stationarity in the present case are and 1 respectively. A prospective model is selected for further analysis based on the information furnished by the autocorrelation function by referring to Table 3.2 [2] and Appendix A9.1 [6]. In practice we can pick up many tentative models and even probably overparameterize them. The program LIKESURF* is used to estimate the parameters of a given model from analyzing the input data. This program is capable of checking many models at a time and its output indicates the adequacy of any model as well as any redundancy in the choice of number of parameters. By the procedure mentioned above, we select the follow- ing model to represent the given data: * See footnote in section 6.6 78 0.0 16.0 LAGS 3L\U M6.0 64.0 H 1- 80.0 36.0 112.0 126.0 H 1- RCF- 00 prcf \* RCF- 10 X^-/^, Iv^ ^^j^ - ^jAy^^ ' * v-^ wfy/W' PRCF ^-^-<^-cJ-^2^v7,- flCF- 01 V" .^..-., ,-^._ *-T\, ^A PRCF ^!?Tyv/2* ^ ■ c\r\ J _"- RCF- 11 Vv - ArV M/v - kaa^a/^- »A/ — " ■^•v\A/^ > v v^ PRCF "'dsAi'U^c' : Figure 7.3 Serial Correlations in Example 1 79 (1 - * B S ) W t = (l-\ B)(l-^ s B S ) a t 7.1.1 where W. = V Z. t s t s = length of season (length of row) 7. 2 Estimation of the Parameters We need to estimate the values of four parameters ($ , X -^ , X Ql a a ) by analyzing the input data. \ = (1+4) Z t _ s -* Z t _ 2s +a t - \ a t _ 1 _ \ a + X X 3l st-s lst-s-1 7.2.1 With the knowledge of a a +-_i > a +- o > etc * We can make a one- step-ahead forecast: Z t+1 = (H4j Z t+1 _ s -« Z t+1 _ 2s - \ a t - ^ a t+1 _ s + ^ X s a t _ s 7.2.2* /■v The 'residual' a , = z -»-4.i ~ z +- + i ' i* e */ tne residuals are con- sidered on the one-step-ahead forecast errors. With appropriate starting values we can see that the values of a can be recursively calculated for a given set of parameters. 7.2.1 A Method to Obtain Starting Values for Recursive Calculations of Residuals A procedure to obtain the starting values is described in detail and illustrated with an example in Chapter 9 of the book by Box and Jenkins [ 61 . It is as follows: For a given * While forecasting the unknown values of a are replaced by their expected value. In this case E(a ,) = o. 80 set of parameters the series may be forecast backwards starting somewhere in the middle of the series. To begin with the unknown a's are assumed to be zero. This introduces transients which will hopefully die down by the time Z n , Z_, , Z „ , etc. are estimated, provided the starting point is chosen sufficiently far along the series. The back- forecast continues till we obtain sufficient number of terms to be able to forward- forecast the first value (Z, ) : Z, = (1+$) Z, „ - <£ Z, - 1 1-s l-2s Here we need the values till Z n _ and the values of a. for l-2s t t < 1 is assumed to be zero. Now A. a i = Z l " Z l 7.2.2 Sum of Squares Function From now on we can recursively estimate the values of residuals a, , a_, ... a . We define a sum of squares function ' S" as : S(* f \ \ ) = L a* is t =l r We obtain the least squares estimate of the parameters by picking a set of values (0 , X , , X ) which provides the minimum sum of ' 1 ' s squares, S . . It is shown in [2, 6] that likelihood estimates n ' min ' are same as the least squares estimates if we assume that a's are normally distributed. Under the assumption program LIKESURF 81 actually plots the sum of squares function and obtains a set of values that make it minimal. Using the program LIKESURF the following model is fitted to example 1. (1 + 0-15 B s ) V z = (1 _ 0-25 B)(l - 0-5 B S ) a^ s t t where ° 2 = 1-03 a 7. 3 Checking the Adequacy of the Fit According to the model the residuals a are generated by a white noise process. If the fit were to be adequate, the calculated residuals should be uncorrelated. Figure 7.4 shows the correlation values of the residuals. It can be seen that the values are fairly small and there is no periodic component either. Thus, it offers no significant departures from randomness among residuals, confirming the adequacy of the fit. As mentioned in the previous chapter we can also check the adequacy by comparing the actual values to the forecast values from any origin. Forecasting is done on the same principle as before (i.e., using difference equation approach). The values of a for t > t is set to zero, origin Figure 7.5 shows the actual values (solid line) and forecast values (crosses) from the origin t = 128. Considering the limited number of parameters steering the model, the forecast function follows actual values rather closely even for t » t 1 origin, 81.5 ,?5.0 LRGS Figure 7 .k Serial Correlations of Residuals in Example 1 H50.Q '.,oo. n Figure 7.5 Forecast Values (shown by x) in Example 1 82 7.4 Generation of Texture We can create a white noise generator having a mean = 2 and the variance a , which generates the residuals a . To start the recursive procedure for the generation of future values we need at least 2s values of Z . These can be considered as the boundary conditions. Figure 7.2 shows the texture synthesized in this manner. Here using two rows of length 64 (s = 64) from the original scene as the starting values, the future values of the pixels have been generated by the procedure mentioned above. One of the ways of testing the effectiveness of the scheme of synthesis is by attempting to fill the holes in the 'parent' texture. The results are shown in Figure 7.6. We notice some edge effects here. This can be expected in the case of structural textures. It is possible to minimize these effects In the present case we have used pixels from two rows parallel to the top edge of the hole as the starting values. Instead we can incorporate pixels from a couple of rows and columns which are parellel to each edge of the hole as boundary conditions. Then we can proceed to patch the hole by forward forecasting from top edge and backward forecasting from bottom edge (as suggested in 7.2.1). In the center we can average out the values forecasted from either side, thus reducing the edge effects. Let us consider a different class of textures in the second example. The texture of handmade paper (Figure 7.7a) belongs to general class of statistical textures. It has no 83 (a) (b) Figure 7.6 Filling the Holes in a Textural Scene (Example 1) 84 structure. The seasonal time series model fitted for this texture is of the form: (1 - 0-8 B S ) Z = (1 + 0-25 B)(l - 0-5 B S ) a t where o 2 = 3«27 a Figure 7.7b and Figure 7.7c show the results of hole- filling operation using above model. Here the edge effects are hardly noticeable. We have shown that it is possible to synthesize natural- looking textures using the time series model. This method is particularly suitable in the case of statistical textures. Many biomedical textures of interest are statistical in nature. 7.5 A Method for Determinining the Size and Shape of Template Used in Decision Theory Method It is possible to determine the appropriate size and shape of template with the help of parameters derived from the time series analysis of the texture scenes. We illustrate the method with an example. Let the textures used in examples 1 and 2 belong to 1 2 families T and T respectively, which are represented by the following time series models: (1 + 0-15 B S ) V Z^ = (1 - 0-25 B)(l - 0-5 B S ) a,. ... 7.5.1 st t- (1 - 0-80 B S ) Z = (1 + 0-25 B)(l - 0-5 B S ) a ... 7.5.2 2 2 where cr = variance of a 's = 1-03 I t x 2 ff = variance of a ' s = 3*27 z 85 (a) (b) (c) Figure 7.7 a. Texture of a. Handmade Paper b. Holes c. Resultant Scene after Filling the Holes 86 We shall rewrite the above models as pure autoregressive models of the form: 1 (B)Z t =a t ; i=l, 2 ... 7.5.3 i ~1 ~2 The autoregressive operators § (B) and $ (B) are polynomials in B and their coefficients are arranged diagrammatically and shown in Figure 7.8. These values indicate the influence of the neighboring pixels in determining the values of Z , , apart from the residual, a whose mean value is zero. Figure 7.8d shows the template which picks up Z, and its neighboring pixels. The set of pixels picked up by the template constitutes an 'event 1 , as defined before. The autoregressive operator operating on Z can be viewed as if the 'event' picked up by the template is passed through the 'filter' formed by the coefficients. The contour of the smallest significant coefficient value determines the upper limit on the size and shape of the template. For, the selection of a larger template would not yield any new information on 'local properties' of the texture scene. Equation 7.5.3 indicates that the 'events' picked up by this template when passed through appropriate 'coefficient filters' the output consists of random shocks, ' a ' , which are normally distributed. From engineering point of view, we can considerably reduce the size and shape of template, and yet preserve the above-mentioned property (approximately) by selecting a suitable cut-off value of the coefficients. 1 2 In the case of texture families T and T , the template 6 (shown in Figure 7.9) appears to be suitable. The output of 87 CO (/) to (/) m *• to CM CD CD CD CD V) CD 00 N m o o o o q ««-* fO r»- W o • * • o o o «4 to o i o 1 o o 1 o t- 00 r~ lO o to o o to N (VI q o o o o o" 6 d o o o 1 o o o o o o 0) o o oo o (VI co o d o 1 o o 1 o 1 o 1 «-l (VI ^- IT) Nj RS o o o o d fcia ^ o o o d o o i o ro H p o o o y o 1 o . • • • • • to co GO CO m * to (VI CD CD CD CD CO CD CD (VI CD ro CD CD m CM cO U. 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Curves for T 1 and T 2 (Obtained using the set of templates shown. ) 90 have to check whether or not in its truncated form, the corre- sponding autoregressive operator would approximately satisfy the equation 7.5. 3. Figure 7.10 shows the results of texture discrimination by the decision theory method for the six templates shown in Figure 7.9. 7. 6 Texture Recognition We have seen that events extracted from a homogeneous texture region if passed through appropriate 'coefficient filter', the output consists of uncorrelated random noise. The auto correlation function can be employed to test this randomness. Using this property, we can recognize textures belonging to same family and discriminate others. We tested this hypothesis by extracting events from T 2 and T using template 6 and we filtered them using 'coefficient ~1 ~2 filter' C and C . We plotted the autocorrelation function of the output in all four cases, ACF11, ACF12, ACF21, ACF22* (Figure 7.11). For the sake of comparison we plotted the auto- correlation function of the output of the random-noise generators, a and a (ACF10 and ACF20). We notice that in spite of trunca- 1 2 tion of the template, quantization noise, etc., the output of filters is fairly random for the right input (ACF11 , ACF22) and not so in other cases (ACF21, ACF12). This process can be easily extended to treat the case of multiple textures. 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Pi - O ra •H Pi -p B } where A s s 8.2.1 Monotonic Array Grammar If, in the course of derivations the arrays never 'shrink' (i.e., the blank symbol is not generated); in other words, if the number of non-blank symbols is monotonically non-decreasing, the AG will be called 'monotonic array grammar' (MAG). 8.2.2 Turing Array Automaton A turing array automaton (TAA) is a quintuple: T = (Q, Z, 5, q , F) where Q is a finite set of states of the form Q' x A (where A= L, R, U, D)*. Any member of the set Q is of the form (q. , x) , x e A which indicates the present internal state (q. ) of TAA and the direction of its previous motion (x). Z is a finite set of symbols. (q , R) e Q is a start state. F c q is a set of final states. 5 i s a mapping from Q x Z into 9 Q x z x a * indicates the direction of motion of TAA. Assuming 4-point neighborhood, TAA can move in four directions: viz. left (l) right (R) , up (U) or down (D). 95 A transformation in 5 of the form ((p, x), A) =» ((q, Y) , B, Y) , x, Y e A is interpreted as follows: If T is in internal state p and has just moved in direction x and reads a symbol A, then it goes into an internal state q, writes a symbol B and moves in direction Y. Let g be an array on z which is finite and connected. If T starts at some point (i, J) in its start state and reads (and rewrites) the values of the points in the array as it moves around, it " accepts " G if it ever goes into a final state . 8.2.3 Array Bounded Automaton A TAA is considered as an "array bounded automaton" (ABA) if it always "bounces" off a blank symbol (#) , i.e., every triple in the image of any ((p, x) , #) under 5 is of the form ( (q, x" 1 ) , #, x _1 ) , x e A. The following two theorems are stated and proved by Milgram and Rosenfeld [51, 521 . Theorem 1 . Let £ be the language of an AG; then there exists a TAA that accepts just the arrays of £ . Conversely, let II be the input arrays accepted by a TAA; then there exists an AG whose language is II . Theorem 2 . Let .?, be the language of an MAG; then there exists an ABA that accepts just the arrays of £ . Conversely, let n be the input arrays accepted by an ABA; then there exists an MAG whose language is n , In the next section we write some monotonic array grammars that would generate texture scenes. We define the ABA that would recognize these texture scenes. 96 8. 3 Examples of MAG's Which Generate Texture Scenes If the terminal symbols (Z) in an array grammar represent gray levels, then the 'sentences' of the array grammar can be considered as the two-dimensional scenes. We write some MAG's which generate some commonly occurring texture scenes. I. Checkerboard pattern G : {V, z, R, #, S} R : (1) 1 *(2) [0 x #1 -* [0 x 1] (3) [1 #] -> [1 0] V = {#, S, 0, 1} 2 = (0, 1} x e A = (L, R, U, D} + X e A = (L, R, U, D} The starting symbol is rewritten as or 1 in the starting array (A ). Then every # symbol in the neighborhood of or 1 is rewritten as 1 or respectively. For this simple grammar G, we illustrate how to specify an ABA (T) that accepts L(G). Array Bounded Automaton, T: {Q,Z,8,q,F} Q': (internal states): { q g , q t , q p , q N , q s , q # , q Q , q^ Z : V u (V x V) where V = {0, 1, #} and V = {0, 1, #, S} For convenience in representation we use the following short- hand notation C R A for [ CA] ; C L A for [AC], CyA for [^~] and(_J is represented as [C_A] We assume a 4- point neighborhood. 97 and F : [ q p , x} , X e A = {L, R, U f D} 8: [In the following rules, xe a and Y e a] (1) ((q s , x), a) -> {((q s , Y) , (a, #) , Y) U ((q , Y) , (a, S) , Y)} , a = or 1 S The ABA can remain in the start internal state (q ) and ^s move around, rewriting a's as (a, #)'s; it changes its state to (q q ) only when it rewrites one of the a's as (a, S). (2) 5 ((q s , x), (0, S)) - ((q Q , Y) , (0, 0), Y) (2)' 5 ((q s , x), (1, S)) ~* ((q 1# Y) , (1, 1), Y) When in state q ABA rewrites (a, S) as (a, a) and goes into state q for a = or 1. (3) 5 ((q 1# x), (0, #)) - ((q , Y), (0, 0), Y) (3)' 6 ((q Q , x), (1, #)) - ((q lf Y), (1, 1), Y) When in state q n or q, if it reads a symbol (1, #) or (0, #) it rewrites them as (1, 1) or (0, 0) and goes into a state q 1 or q~ , respectively. (4) 5 ((q . x), #) - ((q x" 1 ), #, x -1 ) c # for c = (S, 1, 0, #) Being an ABA, as it should, it 'bounces' off a blank symbol (5) 5 ( ( q c , x ) , ( a , A ) ) -> ( ( q t , x" 1 ) , ( a , A ) , x' 1 ) for c = (S, 1, 0, #) a = (0, 1) A = (0, 1, #, S) 98 The ABA at any stage can go into a state q and reverse its latest move. On the first rule of 5 , a 'starting array* is created in the second term of each pair rewritten and in the following rules, ABA has been imitating the rules of MAG. Every non- blank symbol in the terminal array that is submitted for acceptance can be rewritten as a pair (0, 0) or (1, 1) only if it were the sentence generated by G. After the ABA has moved into the state q. , then some new commands can be given* to T , by which it moves around and goes into final state q„ accepting the 'sentence' if it finds that every connected non-blank symbol is a pair of the form (0, 0) or (1, 1). Otherwise it goes into q and rejects the sentence. In this example we have shown how to specify an automaton that has texture recognition property given a grammar that generates texture scenes. II . Horizontal lines G is same as before , except for R. R: (1) rr rcr r 1 " r °~] —> i L#J ^iJ Lo_j LOJ (2) 0x # J ~0x a J for a = {0, 1} and x = L , R (3) ro- r°i - L#J _lJ (3' ) r#" rl l —* I J l-OJ Milgram and Rosenfeld have proven that an ABA can determinis- tically decide whether or not a given array consists only of specified finite set of symbols [51, 52]. 99 (4) (5) 11 l -* #_ Lo. ol ro ih 1 #- _o (4' ) (5 1 ) r#i r° i _» i loJ Lo r# - r° -» llj -i After S is rewritten in the context as shown in rule 1, rules 3,4, 5 show how expansion is carried in the vertical direction. A '#' is rewritten as ' 1' if a pair of pixels just above or below are both '0's. If any one happens to be 1, then # is written as 0. Rule 2 shows the horizontal expansion. This grammar generates picture scenes with horizontal parallel lines with 2-unit-spacing. In the same way we can generate vertical lines, slanted lines, etc. Ill R: Herringbone pattern G same as above , except for R. (1) (2) (3) (4) (5) "S # # #" # # # # # # # #. 1 #" 1 1 # #_ "1 o #" 1 # .0 1 #. ~0 #" 1 1 # .0 1 #. r 1 #" 1 # 1 #. 1 01 1 1 1 0J "0 10 0' 10 10 1. 1 (T 10 1 L0 1 0.J ro o o r 10 10 10 0^ 10"^ 10 1 10 0J (2' ) (3' ) (4* ) (5' ) #10 0" #010 l# 1_ #000" #10 1 L# 1 0_ "#0 1' #010 L# 1 0. ~# 1 0' #10 1 #0 0- "0 10 0' 10 10 1. 1 °1 1 1 1 OJ 1 10 10 L0 1 0- ro oio" 10 1 10 100 (6) (7) (8) on ^"0 l — > 1 #J ,0 on r° -» #J Li 1~ r i — » #- ^0 (6- ) (V ) (8' ) _i_ — » ^°1 -1_ ^0^ ~* lO J i .o_> ~* i loJ As we can see it is becoming increasingly cumbersome to specify the rewriting rules even for very simple texture scenes, This is so because of some limitations in array grammars which will be discussed in more detail in the next section. IV. Statistical textures We have seen that structural textures can be generated by array grammars. For generating statistical textures we can extend the concepts of array grammars to introduce stochastic array grammars in the same way as in linear grammars. A stochastic array grammer can be defined as a weighted array grammar in which the rewriting rules R have an associated weight as shown: r ID j = 1 ... n ± i =1 ... k where k indicates the number of distinct sets of rules in R such that the left member is same in every rule belonging to the same set. < p. . (weight or probability) < 1 n k and 2 p. . = 1 j=l ID for all i 101 The following grammar illustrates how a statistical texture can be generated. G: {V, z, R, S, #} R: (1) [ S #] - [S si V = {S, #, 0, 1} for xe A £={0,1} (2) S °^ 6 (3) S ^> 4 1 The scene is first filled by the nonterminal 'S' and then S is rewritten as 1 with a probability 0*4 and as with probability 0*6. It is obvious that this grammar generates 'salt and pepper' scenes. 8.4 Multilevel Array Grammars Before introducing this special class of two-dimensional grammars, we shall study the limitations of the conventional array grammars as defined by Milgram and Rosenfeld. 8.4.1 Limitations of Conventional Array Grammars By virtue of their definition, there are some limitations in conventional array grammars which make them uneconomical in terms of rewriting rules in representing complex texture scenes as we have seen in the previous section. First of all, according to the definition, a terminal symbol is never rewritten. Hence, a nonterminal symbol is rewritten as a terminal symbol only once during the course of 102 derivations. Also each nonterminal will eventually occupy only one cell in the terminal array. These restrictions make nonterminal symbols much less powerful as far as expansion is concerned than their counterpart in linear grammars. The necessity for the presence of these restrictions can be explained in the following manner: In linear grammars the expansion takes place in only one dimension. There is no problem of juxtaposing the expansions of neighboring nonterminals, Whereas , the expansion of a nonterminal in an array grammar takes place in two-dimensions. To make a meaningful scene ("2-D sentence"), the "shape" of expansion of any nonterminal should be compatible to that of its neighboring nonterminal so that there are no 'holes' created while juxtaposing these expansions. It is a non- trivial problem to specify the shape of unrestricted expansion of every nonterminal that would satisfy this requirement. 8.4.2 Introduction to Multilevel Array Grammars As much as we would like to have the same powers of expansion for nonterminals as found in linear grammars, we are forced to make some compromises because of the problems involved. Generalizing the concept of conventional array grammars, we introduce what are known as "multilevel array grammars" (MLAG). As the name would suggest there are many levels and at each level several grammars are defined. These grammars at any level resemble the conventional array grammars in every sense except for one main difference. The so-called 'terminal symbols' at 103 level > are not exactly terminals in the conventional sense which is true only at the lowest level, 0. They are in fact •sentences' of particular 'shape' which are derived by a set of grammars defined at next lower level. It is assumed that all the terminals at any level have the same 'shape' and this shape information is embedded in the definition of all grammars at that level. Finally, all the 'sentences' of MIAG should consist entirely of a connected set of terminal symbols defined at level only. Grammars at intermediate levels act as subroutines that would expand the terminal symbols at highest level in terms of those defined at level 0. 8.4.3 Definition A multilevel array grammar (MLAG) can be formally defined as a set of 6- tuples: i =0, 1 ... p (Qi) J - 1 / 2 ... n . and n =1 1 P where G j = (V J' E j' Q± ' R j' S± ' #i} Here the superscript i indicates the level. The MLAG is said to be of order 'p' if there are p + 1 levels (0 thru p) V. = 'vocabulary' of j grammar at i level * j_ i_ 4-1^ Z 1 . = 'terminal symbols' of j grammar at i level S = 'start symbol 1 at i level: same for all j 104 # = 'blank symbol 1 at i level: same for all j Q = 'shape parameter' contains information as to how many cells each member of V^ for all j would eventually occupy in the 'final array' (the 'sentence' of MIAG). Q is said to define the size and shape of 'unit cell' at level i. It is generally expressed in terms of Q in the form of a template either pictorially or by a set of coordinates (see Figure 8.1). By definition Q is always 1. R.: 'rewriting rules' of j grammar at i level. Consists of pairs (r . T'). p is a finite connected P' P * subset of I x I and T t T ' are mappings from p into V.. Here every (k, £ ) e I x I refers to a unit cell defined by Q . At a given level terminal symbols are never rewritten. An ' initial array ' A 1 at level i , is a mapping from Q (if i = p, the highest level, it is a mapping from I x I) onto {# , S } . The initial array is all blanks except for one S . Similarly a ' terminal array ' A T at level i , is a mapping from I x I onto (Z 1 , ft 1 ) for all j such that {(k, £)|a£(1c,£ ) e £ .} is connected. As mentioned earlier any terminal symbol a , at level i+1 is a 'sentence' of shape Q , derived by a grammar at level i. That is to say, every a 1 can be expanded into an array of 105 shape Q 1 consisting of only terminal symbols of i level. This can be expressed as A — ^ a , i.e. , the 'array' can be derived from the initial array A (of shape Q , by definition) with the application of rewriting rules of j grammar at i level. We can derive A from A by expanding each and every terminal symbol in it, in the manner described above . Shape Parameter Set of Coordinates Template □ Q' (0,0), (-1,0), (-1,1), (0,1) (0,0), (0,-1) or (0,0), (-1,-1) and so on. i i i I i i i — ' 1 — | i Figure 8.1 The 'Shape' Parameter Q 1 and Corresponding Templates 106 The language of MIAG is defined as: L(MLAG) = {b|a =» B] A: Initial array at highest level, p B: Terminal array at lowest level, The terminal array at level is sometimes referred to as final array. The 'sentence' in the MIAG is generated in the following manner: A p ^ a p G ? a p-i A i G J A o s T n T 9 — A T — *•• A T . .— -+ n A T D-l/2...n 1 3=1,2. ,.n Starting with the initial array at highest level, a terminal array is generated at that level. Then with the help of grammars at next lower level, each 'terminal' is replaced by its image in terms of terminal symbols defined at lower level. This is analogous to laying floor 'tiles' on the. floor in the assigned space. This process is carried on until the 'final array' is derived which is a 'sentence' in the MIAG. The following examples along with the illustrations will clarify the concepts introduced above. 8.4.4 Examples of MIAG Which Generate Texture Scenes I. Herringbone pattern : In order to demonstrate the simplicity in specifications of an MLAG as compared to conventional array grammar, we 107 generate the same scenes as mentioned in example 3 of section 8.3 MLAG (of order 1) = { G 1 , G°) G 1 = {V 1 , Z 1 , Q 1 , R 1 , S 1 , #*} V 1 = [S 1 , tt 1 , of 1 } 1 r li z = (a J Q = (1) S" (4x3 template. Can also be expressed as a set of coordinates ) a lUi 1 1. (2) [a^# x ] _^[a x a] , x e A .0 and a R ^ and (G°: V° , E ° f Q° , R° , S # } v u = (S, #, 0, 1} z u = {0,1} Q° = 1 rs # # #" # # # # L# # # #. =* 10' 10 1 LI 0- [Note: Small unit cells can be directly defined as shown. More complex cells can be derived by a set of rules.] 108 Using G , the initial array A_ at level 1 is filled with or, or i the terminal using R (Figure 8.2). s then expanded into terminals at level II. Diamond pattern : With a slight modification, we can generate more complex patterns . We can introduce two terminals at level 1 , and two grammars at level 0. G : same as in the previous example except for £ and R Z 1 = (a 1 . P 1 }- and R 1 : (1) S 1 - aM P 1 (2) [°£ #*] -* [or a i x J , x = L, R (3) < * — ^ A , x = L, R (4) [al tt 1 ] — [o£ P 1 ] / x = U, D (5) [pi #'] — *> [£ a 1 ] , x = U, D With these rules we can generate alternating rows of en's and 3's in A (terminal array at level 1). R Oil where : A — ► a s R A TV 2 ol and A * p s G and G ? are same as G in previous example except: R°- R 1 . S # # #" # # # # 10 10 1 1 0J 109 #' #' #' #' # #' #' a a a' a' a' a Initial Array at Level 1 a a' a' a' a' t A° T Terminal Array at Level 1 Final Array (a) Herringbone Pattern a 0' a 0' cr' ft a' ef a a 1 t Terminal Array at Level 1 Final Array (b) Diamond Pattern (Dark Cell = 1 Light Cell = 0) Figure 8.2 Generation of Texture Scenes Using MLAG 110 and R 2 : S # # #1 # # # # ri o 0") 1 1 Lo o i oJ Figure 8.2 shows the generation of the diamond pattern. In these examples we have specified the 'unit cells' at level and the rules of placement at level 1. Thus MIAG is really suitable for generating texture scenes that can be represented by unit cell/placement model. We can also generate unit cells consisting of statistical textures (as described in example 4, section 8.3) at level and generate 'brickwall type' textures by a regular arrangement of these unit cells at level 1 and so on. 8.4.5 Restrictions on the 'Shape' of the Template To avoid the complicated problems that would arise in generating a meaningful scene with MLAG, we find it necessary to place the following restrictions on the 'shape' of the 'template' at every level. a) Every term in the vocabulary of all grammars at a given level expands into a pattern in the next terminal array whose size and shape is fixed as defined by the template at that level. b) In order to ensure that there are no 'holes' generated when neighboring terminals are expanded, we need to have (i) The left edge of a template should be the jig-saw-complement (jsc) of its own right edge and Ill (ii) The top edge of the template should be 'jsc' of the bottom edge. These restrictions are for four point neighborhood and for other neighborhoods, these are suitably modified. We have to exercise some caution while selecting some com- plex shapes. Unless the shape selected is simple like rectangular we may need to specify some additional rules for juxtaposing the cells of this shape in deriving terminal arrays of lower level from higher level. 8. 5 'Interval Complexes' and Grammars One of the basic needs in the 'linguistic approach' to pattern recognition is to be able to write a reasonably simple grammar by analyzing a set of 'sentences', in practical cases. Without this capability this approach is of limited use. Our concern here is to come up with 2-D grammars that would generate a family of textures, by analyzing them. We intend to show with examples that the 'interval complexes' in their capability as 'texture feature detectors' (section 4.3) supply sufficient information to specify a reasonably simple grammar. Example 1 : Let us first consider a linear case. The following 'interval complexes' have been derived for linear textures T and T shown in Figure 4.1 T 03233 _ 130223 T 232303 1 " x l X 2 X 3 2 " X l X 2 X 3 3 " X l X 2 X 3 We have seen that L , L and L act as 'filters' which have a 'passband' for 'events' (in this case a triplet of 112 pixels) from T . We can easily implement the characteristics of these filters in terms of rewriting rules of a grammar and generate only those triplets which fall in the 'passband* of these filters. Grammar - G = [V, Z, R, s) Vocabulary - V = (S, A, B, C, D, E, F, 0, 1, 2, 3} Terminals - = {0, 1, 2, 3} S is the start symbol R: S - SS [SSSj -* [CF3]|[EBF]|[FPC] A -» 0|l D- l| 2 B -A | 2 E- D| 3 C -* B|3 F- 2| 3 It can be easily seen that this nondeterministic context sensitive language, G is capable of generating scenes belonging to T family. An automaton that would accept the language of G is not only capable of recognizing textures in family T , it can also discriminate textures in T from those in T . This latter property is attributed to the nature of internal complexes, that they resulted from comparing a pair of scenes which specify 'passband 1 and 'stopband' of events. We should be careful in our choice of T (when it is not specified in the problem), in order to derive interval complexes that would extract prominent texture features in T . For example, if T is a structural texture, we select a scene filled with random noise (random texture) to be T . The interval complexes derived in this case, hopefully, tend to pick up structural features which make T different from the random 113 texture. We performed this experiment with the cheesecloth texture (Figure 7.1) using the template shown in Figure 8.3 to extract events. The interval complexes derived are shown in Figure 8.3. As before we can consider these interval complexes as 2-D filters. A part of the output of each filter with the cheesecloth texture at input is shown in Figures 8.4a through 8.4f. Figure 8.4g shows the union of the output of all six filters. We notice that Figure 8.4 accentuates the fact that the cheesecloth texture consists of many similar looking patterns (events) aligned in vertical direction. With the help of these interval complexes we can specify a 2-D grammar that is capable of generating the scenes belonging to the family of cheesecloth-type textures. MLAG (of order 1) = ( G 1 , G°} G 1 = {V 1 , Z 1 , Q 1 , R 1 , S 1 , # X ] V 1 = {S 1 , #\ a 1 } 1 r X i Q 1 = R 1 = (1) a (2) [a* tt 1 ] _^[«^ a 1 ] A .0 R ° «l and A — ». a s for x e A 114 X 3 X 4 X 5 X 2 X l X 6 x 9 X 8 X 7 Number of gray levels used = 4, i.e., 0< x . <_ 3 23 13 13 23 13 L l : x 2 X 3 X 7 x 8 X 9 01 13 13 02 02 13 x l x 2 X 3 X 4 X 8 X 9 13 01 13 02 12 13 1 x l x 2 X 4 X 5 X 6 8 9 L 4 : 01 12 13 13 13 13 0. 2 x 2 X 3 X 5 x 6 X 7 X 8 X 9 13 13 01 01 01 12 X l X 4 X 5 X 6 X 7 X 8 02 12 12 12 01 x l x 5 X 6 X 7 8 Figure 8.3 The template used and the interval complexes derived for cheesecloth texture (T 1 ) x random texture (T°) 115 •••• OUTPUT OF FILTF.P • 1 •••• • M • M • Ma * • • M • •M • • • M • M* t • M • Mt • • M • MC 1 M '• • M • Mt M • •' M • MM M • • M • • MM 1 M • M MM • • MM • M • M M M MMM • M ( Mt M • MM • • •( tl Mt M • MS • •M* • M t • MM M MM 1 M M • M* * • • • I* M* • M Mt Mt • • • It MM 1 M M* M • t M* t •MM • • M • MM t • MM MM* Mil t M* M M* MM tltt t • mi M( ( 1 Mt t • *M M* M MM t • *M 3 / 3/ 3/ /\ / / 3~7 V 7 V /2 / / T7 *7 3/ /l / 2 / I M tMI • M* FILTER « 1 • Mt* • Mt t • tM • M t • It* ■ M t ■ • •«• t tM t • t Mt t MMI Mt Mt MMM t M MM *M MM M t M MM t 1 Mt Mt M tM MM tIM tit • tMMM t III MM M MMtM •II Ml • MMttt •IM tl • t MMMt •mi • M MMtMM • M II t IMtlMI • •IMI 1 t t MtMM t MM* t •MNHIH • Ml I 1 t IMtMMt • MMM IMItMMt t • M It t t MltMIMI • 1 •M MMMtMt • ■1 I (0) 3 / 2 / 3/ /\ /O /0 T7\7 3/ /\ /o /q T7z7 3/ /\ /o /o FILTER »2 • ••• OUTPUT OF FILTH • I •••• • M Mil It *M*I II Mill I I Ml II IMII II III It • M M I I I II M Mt I It lit I I tIMIII MM Ml t II III U I I M MM t t It M Ml II III It t M I M • • • I Mt M • • til It I II tl tl II It Mil lltlt ■•••I* *IM •• II Ml Mill* II Mt I II II I Itltll I I tl I It It! Mill! IIM lltlt IMII I II It I II II lilt ■ Mil tMI t I ttltll • Mill! I It II Ittll tl**** • M ■ It I till lltl It! lit llltt •■•It I* a a « tilt I II I II I lilt lltlt III Ml till I II II Itl II lilt • Mil I II It I till lllltl *M (• ••■* I It It I IIM Mil tltt • I II I til lllltl t • till IM It It lilt HUH I* *•( a a a a lltl I It II IMtll tllltl IIM II It It I lllltl IM Mil Mtlll • III* ••• •••• tana lltlt IIM It I tit lit t IIIIM • •••• •• I IMI It • a a a aa • till M til II • till I ••• •• •••• ••••• • ••■ • t ititt lilt I til attitlt • till tl a a a lit titti till iiiii • tut a a a a • til • tun ttli t til t ti ti • ■•• • •• • till! tilt * * * **ittt • tit t tl • t ■• ti • II I I lit III t tilt till* •( itt t lltl • •••••••• ••• a aa aa lllltl*) •••• Mil I lltlt III lit itiat • •••• • ••• ••■ ••••• • •••••••• ••••••! Ittttllll ti it it • til ••••• •• •• •••• ••••• ••••• • • ••• • •• •••••••• • • ••• •• itititt (b) •••• OUTPUT OF FILTER I I I I I • I • • • • • • • % / 2 / 3 / / 1 FILTER #3 • • • t • • till • • • • ■ tit • lit • a t t tit • I itt • • t i • ti • it t t • t t t • t • t • t I t t • • t • t • t • t (c) OUTPUT OF f UTt* 2 / / 1 1/ /I 2 / 3 / FILTER *4 I • t I t I • I • t I t I • tit • l • I (d) Figure 8.4 A portion of the output of the filters 1-4 (a-d) with cheesecloth texture (Tl) as input. 116 •••* OUTPUT OF FILTER Ml Ml Ml t • • • • f t • • • ■■ I I • It • M ■ H • •■ • 3/ /o 3/ /o y% y : i 3/ / ° 2/ % naTPUT Of F1LT6« 3 / 3/ 2/ /I 3/ / 2/ 1 /o 2/ /I 3/ / ° l/ 2/ /I FILTER *6 Note : Lower limit for each variable is indicated in lower righthand corner and the upper limit in the upper lefthand corner. •»••►•••» • • •••• IIIIIIMUIIIIII imiiiiiiiiiiiiii • •• • HIMIIDn •• • MMIM •■•••••• ••••••••••••■•»• • •••••••••• ••••• •••••••■•••••••f f »•••••••• •• •• ••«•••• •• HIIII • • ■•••••• •• * ••••••• iiinmi • •••••••>•• •• ••••••tail •«•■•••• ■ *«•••••»• •• •• ■■•«•■••• •««• 1 • P« ••••••• •• • HIIIHII tisr • ••• fMMM •• • iminuuiti •• HIIIHHII ••• t» ••••••••••••« • •■■■mini « • • M ••••••••«•••« •mrett m • Mt •• •■••••••••••• « »•••••••• 1 HI ••• ••••••••••• • • • ••••••* • ••• ••••••••••••» • • f ••••«•• • •■• •••statiicttemi • ill «••• •■•■»■■•■*•*••••■■■■■ • « • » •••• iHiiiiiiiiinniHiii ■■• »•■■• •■■■ •• • •••••*•••••••••• • » «i»MM • Mil • •••»••• • • •••*•••••••••• • •MIMtlllMMI • ••••••••••••!••< • •••••■•• ••••••• • ••••«••>•••<•••• • I ■■••«•■•■•••«••• «• IIM1MIMI1MH • • IDIHIII • ••• • • IHMHM ■••■■• (g) Figure 8.4 (contd. ) A portion of the output of the filters 5, 6 (e, f) with cheesecloth texture Tl as input . Shows the union of the output of all six filters, 117 G° = IV°,Z°, Q°. R°, S, #) ,0 V" = (S, #, A, B, C, D, E, F, 0, 1, 2, 3) s " = {0, 1, 2, 3} R°: rs # #- # # # -* ^E C C FCC LE F E K -V 0|1 B -* AJ 2 C -± b| 3 D C E' ACE BEE. E B C~ E A C E B C^ C E A C E A C D A. D -> 1| 2 E -► D| 3 F -* 2| 3 C E B A E D LA E C C C D C B D C A D- In this example the rewriting rules at level '0' are written in such a way that every event generated falls in the passband of one of six filters ('interval complexes'). At level 1 we specified the global arrangements of these events. Thus we have shown how we can specify grammars that would generate textures that occur in practical cases with the help of the 'interval complexes'. 118 9. SUMMARY AND CONCLUSIONS 9. 1 Discussion of Results In this work we have presented methods suitable for computer implementation for the analysis and synthesis of visual texture. The Decision Theory Method, coupled with the application of interval covering theory is shown capable of automatically generating 2-D filters which act as texture feature detectors, from the analysis of input scene. We have demonstrated that this method is very versatile and can deal with the various problems concerning visual texture, such as scene segmentation using textural information, extraction of texture borders, discrimination/recognition of both spatially and chromatically textured scenes and so on. The time series model provides the basis for a method which is capable of synthesizing natural looking textures. Unlike many earlier methods, this one can derive the parameters needed for the synthesis automatically from the analysis of parent texture. Based on this model we also proposed scheme for redundancy reduction in the TV transmission of a digitized textural scene. The multilevel array grammars introduced are shown capable of generating brickwall-type textures. We also 119 illustrated a way of specifying these grammars for real textural scenes with the help of interval complexes derived from the analysis of those scenes. 9. 2 Suggestions for Future Research We suggest the following topics for further investigation. (a) By virtue of the present interpretation of the 'event 1 , the decision theory method of analysis of texture is based on the statistics of local patterns only. It is possible to perform analysis to global level by modifying the interpretation of event. Here each variable in the event is allowed to represent, instead of gray level of a point which is a local property, some measure of a global property, for example, number of edges per unit area, gray level distributions, spatial frequency content, etc. These measures are quantized and digitized. Each variable can assume different range of values. When a 'global event' is defined in this manner we can perform analysis of the scene at global level. The 'likelihood ratio' criterion provides a better 'similarity criterion' than those found in other methods performing similar analysis [3, 35, 53], There is a need for a scheme that would suggest minimum number of appropriate global properties to be used in dealing with the given set of texture families. Investigation should be carried in this direction. (b) In the time series analysis, in order to deal with multiple seasonalities that occur commonly in practice, we need efficient algorithms and faster programs for identifying, filtering and checking the adequacy of a time series model to represent real textures. 120 (c) The complete formulation of stochastic turing machines and studies of their properties have not yet been successfully accomplished even in linear case [16, 20]. 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"Veridical Perceptions of Cylindricality : a Problem of Depth Discrimination and Object Identification." J. Exp. Psychology , pp. 145-156, 1962. [81] Smith, A. R. III. "Two-Dimensional Formal Languages and Pattern Recognition by Cellular Automata." Proc. 12 SWAT, 1971 IEEE Conference (Switching and Automatic Theory) . [82] Special Issue on "Redundancy Reduction" - Proceedings of IEEE . [83] Special Issue on "Syntactic Pattern Recognition" (Parts 1 and 2). Pattern Recognition , 3, 1971, and 4, 1972. [84] Stoloff, P. H. "Detection and Scaling of Statistical Differences Between Visual Textures." Perception and PsychoPhysics , 6 (1969), 333-36. [85] Strong, J. P. III. Automatic Cloud Cover Mapping . TR-163/ July, 1971, University of Maryland Computer Science Center, College Park, Maryland. [86] Sutton, R. N. and E. L. Hall. "Texture Measures for Automatic Machine Recognition and Classification of Pulmonary Disease." Proceedings of Two-Dimensional Digital Signal Processing Conference , Columbia, Mo., 1971. [87] Swanlund, G. D. "Design Requirements for Texture Measure- ments." Proceedings of Two-Dimensional Digital Signal Processing Conference . Columbia, Mo., 1971. [88] Swets , J. A. "Effectiveness of Information Retrieval Methods." Scientific Report No. 8, Air Force Cambridge Research Laboratories, USAF, Bedford, Mass., June, 1967. [89] Trout H - R* G « "The Description of Texture." File No. 832, Department of Computer Science, University of Illinois, Urbana, 1970. [90] Troy, E. B. , E. S. Deutsch and A. Rosenfeld. Visual Texture Analysis , 3 TR-145, Computer Science Center, University of Maryland, Maryland, Feb. , 1971. [91] Two-Dimensional Digital Signal Processing Conference. Proceedings . Columbia, Mo., October, 1971. 128 [92] Watanabe, S. , ed. Methodologies of Pattern Recognition . New York: Academic Press, 1969. [93] . Knowing and Guessing . New York: John Wiley and Sons, Inc. , 1969. [94] Wied, G. L. and G. F. Bahr , eds. Automated Cell Identifi- cation and Cell Sorting . New York : Academic Press , = 1970. [95] Young, I. T. "Automated Leukocyte Recognition." Automated Cell Identification and Cell Sorting . Edited by Wied, G. L. , Bahr, G. F. New York: Academic Press, 1970. [96] Zusne , L. Visual Perception of Form . New York: Academic Press, 1970. 129 VITA Sadali Narasappa Jayaramamurthy was born in Anantapur, Andhra Pradesh (India), on July 15, 1946. He received the Bachelor's degree in Electronics and Communications from Andhra University, in 1967. He was given a special award by the Institution of Engineers (India) for obtaining highest percentage of marks among the graduating students of all disciplines from Andhra University in 1967. He obtained M.Tech. degree in Electrical Engineering from the Indian Institute of Technology, Kanpur (India) in 1969. He joined the University of Illinois , Urbana as a graduate research assistant in the Department of Computer Science in 1969. While associating with the Illinois Pattern Recognition Computer Project (ILLIAC III), he conducted research under the guidance of Professor Bruce H. McCormick. He is the coauthor with John S. Read of a paper entitled, "Automatic Generation of Texture Feature Detectors" which was published in IEEE Transactions on Computers, July 1972. He is also a coauthor with Professor Bruce H. McCormick of the DCS Technical Reports entitled, "Analysis of Texture" (July 1972) and "Synthesis of Texture" (July 1973). These publications evolved from his thesis research. Mr. Jayaramamurthy is a student member of Institute of Electrical and Electronic Engineers. JOGRAPHIC DATA ET 1. Report No. UTUCDCS-R-73-601 3. Recipient's Accession No. tie and Subtitle Computer Methods for Analysis and Synthesis of Visual Texture 5. Report Date September 1973 6. jthor(s ) Sadali N. Jayaramamurthy 8. Performing Organization Rept. No. rforming Organization Name and Address Department of Computer Science University of Illinois Urbana, Illinois 10. Project/Task/Work Unit No. US AEC AT(ll-l)21l8 11. Contract /Grant No. k6 -26 -15-303 ponsoring Organization Name and Address US AEC Chicago Operations Office 98OO South Cass Avenue Argonne, Illinois 60^39 13. Type of Report & Period Covered 14. upplementary Notes Abstracts Computer methods are presented which deal with the problems of visual texture, cifically Texture Analysis and Texture Synthesis. The Decision Theory Method, as its name indicates is based on the principles of tistical decision theory. This method in conjunction with the Internal Covering ory, automatically generates a set of interval complexes which act as 2-D filters t detect texture features in the scene of analysis. We demonstrate the versatility this method to deal with various problems of visual texture, such as, scene mentation using textural information, extraction of texture borders, discrimination/ ognition of both spatially and chromatically textured scenes, etc. We propose a model that views the pixels of a digitized textural scene as a two-way sonal time series. Based on this model, we develop a method for synthesis of ural looking textures. This method possesses a desirable quality that the ameters needed for the synthesis are derived from the analysis of 'parent' texture, ., the texture to be imitated. Extending the concepts of two-dimensional formal grammars, known as array grammars, introduce 'multilevel array grammars'. These are shown capable of generating plex texture scenes, specifically 'brickwall-type' texture, that is those textures ch are perceived at many levels and at each level, they can be either structural statistical. We also indicate a way of specifying these grammars for texture nes in real cases, with the help of 'interval complexes'. Finally we demonstrate the methods we suggested act to complement each other ling with the various problems of textured scenes occurring in practice,. Key Words texture analysis, texture synthesis, visual texture, texture likelihood ratio, operative characteristic curve, statistical decision theory, seasonal time series, autoregressive terms, moving average terms, array grammars COSATI Field/Group Availability Statement « NTIS-38 (10-70) 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 22. Price USCOMM-DC 40329-P7I )rm AEC-427 (6/68) AECM 3201 U.S. ATOMIC ENERGY COMMISSION UNIVERSITY-TYPE CONTRACTOR'S RECOMMENDATION FOR DISPOSITION OF SCIENTIFIC AND TECHNICAL DOCUMENT ( See Instructions on Reverse Side ) AEC REPORT NO. C00-2118-001+8 2. TITLE Computer Methods for Analysis and Synthesis of Visual Texture TYPE OF DOCUMENT (Check one): (3 a. Scientific and technical report Q3 b. Conference paper not to be published in a journal: Title of conference Date of conference Exact location of conference. Sponsoring organization □ c. Other (Specify) RECOMMENDED ANNOUNCEMENT AND DISTRIBUTION (Check one): P3 a. AEC's normal announcement and distribution procedures may be followed. ~2 b. Make available only within AEC and to AEC contractors and other U.S. Government agencies and their contractors. ]] c. Make no announcement or distrubution. REASON FOR RECOMMENDED RESTRICTIONS: SUBMITTED BY: NAME AND POSITION (Please print or type) Sadali N. Jayaramamurthy Research Assistant Organization Department of Computer Science University of Illinois Urbana^ Illinois 6l801 Signature ^fW#^*^r