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 REPORT NO. 283 
 
 yov^a 
 
 FUNCTIONAL APPROXIMATION OF GRAPHICAL DISPLAYS 
 
 by 
 
 MICHAEL NOEL PAYNE 
 
 September, 1968 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/functionalapprox283payn 
 
Report No. 283 
 
 FUNCTIONAL APPROXIMATION OF GRAPHICAL DISPLAYS* 
 
 by 
 
 MICHAEL NOEL PAYNE 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois, 6l801 
 
 Submitted in partial fulfillment of the requirements for the degree of 
 Master of Science in Computer Science in the Graduate College of the 
 University of Illinois, September, 1968. 
 
Ill 
 
 ACKNOWLEDGEMENT 
 
 The author is very grateful to his advisor, Professor W. J. 
 Poppelbaum for suggesting this problem and for much guidance and 
 encouragement . 
 
 He is also indebted to Professor Louis van Biljon and the 
 members of the Circuits and Systems Research Group for their suggestions. 
 He would also like to thank Miss Carla Donaldson and Mark Goebel for 
 typing the manuscript and producing the drawings. 
 
XV 
 
 TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION 1 
 
 2. MATHEMATICAL ANALYSIS '. 3 
 
 2.1 Approximation Problem . 3 
 
 2.2 Choice of Criterion k 
 
 2.3 Minimax Principle 5 
 
 2.k Orthogonality and Recurrence Relations 10 
 
 3. TCHEBYSHEV POLYNOMIALS 12 
 
 3.1 Definition and Properties 12 
 
 3.2 Main Theorem Ik 
 
 k. COMPUTER IMPLEMENTATION 17 
 
 k.l Curve-Fitting Program 17 
 
 k.2 Pictures 20 
 
 5. CONCLUSIONS 26 
 
 5.1 Limitations on Degree and Coefficients 26 
 
 5.2 Scanning Techniques 27 
 
 5.3 Hardware Feasibility 28 
 
 LIST OF REFERENCES 30 
 
LIST OF FIGURES 
 
 Figure Page 
 
 1 Function with Minimum Deviation 6 
 
 2 Alternating Sign Function 9 
 
 3 Geometrical Representation of Tchebyshev 
 Polynomials 13 
 
 k Linear Transformation 16 
 
 5 Original Car 21 
 
 6 Car Using Approximating Functions 22 
 
 7 Original Fish 2k 
 
 8 Fish Using Approximating Functions 25 
 
 9 Electronic Scanning 29 
 
-1- 
 
 1. INTRODUCTION 
 
 In the field of computers, graphical displays are becoming 
 increasingly important: Displays are being used to input and output 
 information, the fields of programming debugging, business, data retrieval, 
 computer-aided design, and data reductions make extensive use of graphical 
 displays. Unfortunately, wide bandwidth has been necessary in the trans- 
 mission of graphical displays. The goal of this thesis is to present 
 possible techniques to achieve bandwidth reduction. 
 
 One field of great importance is that of functional approxi- 
 mation and functional encoding of graphical displays by an on-line computer 
 system. The purpose of such a system is to take a given graphical display 
 and decompose the display, by means of some scanning technique, into sets 
 of continuous mathematical functions. These functions are approximated by 
 some other mathematical function, such as polynomials. In choosing the 
 approximating functions, limitations of the hardware of the system, degree 
 of accuracy, and other restrictions must be taken into account. Once 
 each function has been approximated by a polynomial, it will be represented 
 by the coefficients of that polynomial. After the encoding of the display 
 has been achieved, reconstruction of the display may begin. In order to 
 reconstruct the display elsewhere, the coefficients are sent to the 
 receiving place and are used to regenerate the approximating polynomial. 
 Finally, the entire graphical display is constructed from these approxi- 
 mating polynomials. 
 
 This thesis will concentrate on the area of the approximating 
 functions, in particular, on the mathematical analysis of the choice of 
 the approximating function. Also, there will be remarks concerning the 
 
•2- 
 
 choice of scanning techniques and the limitations imposed by an on-line 
 computer system on the choice of the approximating functions. Many of 
 the decisions were made on the basis of actually approximating simple 
 curves and reproducing simple pictures with the aid of the 799*+ and 
 Illiac II computers. 
 
 The application of such graphical processing seem unlimited. 
 The Circuit and System Research Group of the University of Illinois has 
 and is doing extensive work in this area and the present outlook is very 
 promising. 
 
■3- 
 
 2. MATHEMATICAL ANALYSIS 
 
 2.1 Approximation Problem 
 
 One of the most important problems in applied mathematics is 
 the approximation of a real continuous function f (x) by an approximating 
 function F(A,x) which has a fixed finite number of parameters, A. 
 A = (a, , a , . . . a ) the parameters of the approximation function F. 
 
 There are two principle- aspects to this problem. First, the 
 type of approximating function used, and second, the "exactness" of an 
 approximation. There is no mathematical method of determining which of 
 many approximating functions normally available will lead to the most 
 efficient approximation of f(x). A great deal of the time, such approx- 
 imations are made on the basis of intuition and experience. 
 
 Let d represent our distance function. Thus d[f(x), F(A,x)] 
 will represent the distance of F(A,x), our approximating function, from 
 f(x). These distance functions for approximations are called norms. 
 They are denoted by ' and satisfy the following properties: 
 
 (1) | |f(x) | | > and | |f (x) | | - if and only if f(x) = 
 
 (2) ||cf(x)|| = |c| ||f(x)| for any real number c 
 
 (3) ||h(x) + f(x)|| < ||h(x)|| + ||f(x)|| 
 
 Approximation Problem 
 
 Let f(x) be a given real- valued continuous function defined on 
 a set X, and let F(A,x) be a real- valued approximating function depending 
 continuously on xeX and n parameters A. Given the distance function d, 
 
-4- 
 
 determine the parameters A*eP, set of polynomials, such that for gl 1 
 
 AeP 
 
 d[F(A*,x), f(x)] < d[F(A,x), f(x)] 
 
 A solution to this problem is said to be a best approximation. The 
 general approach to the approximation problem is as follows : 
 
 (1) The choice of approximating function and the distance 
 function. 
 
 (2) The existence of a solution. 
 
 (3) Uniqueness of a solution. 
 
 (h) Characteristic and other special properties of the 
 
 solution. 
 (5) The computation of the solution. 
 
 2.2 Choice of Criterion 
 
 As mentioned in the previous section, the first step to approx- 
 imating a function f(x) is the choice of the approximating function F(A,x) 
 and the distance function d. Polynomials will be used as the approximating 
 functions and the distance function shall be determined by the Minimax 
 Principle (sometimes known as the Tchebyshev Criterion). 
 
 If one wished to find the "best" approximation to a function 
 over an interval [a,b] from the set of all polynomials, it can be shown 
 that no such polynomial exist. "Best" must be defined and restrictions 
 placed on the set of polynomials. One possible definition of "best" 
 
could be that the deviations 
 
 2. [f(x.) - F(A,x.)] 
 i=l 
 
 be a minimum. This can be shown not to be a very satisfactory criterion 
 
 of "best" fit by looking at Figure 1. The dotted line could satisfy the 
 
 criterion but is not a satisfactory fit. This difficulty could possibly 
 
 be avoided by specifying absolute values, minimize 
 
 E |f(x ) - F(A,x )| 
 i=l ± 
 
 Even here one runs into trouble trying to find a minimum because the 
 
 absolute -value function has no derivative at its minimum. This problem 
 
 leads to two different choices. 
 
 First, we could ask for a minimum of the square of the deviation 
 
 «£ [f(x ) - F(A,x.)] 2 
 i=l x 
 
 This can be differentiated, and moreover, the square of a variable does 
 not have a maximum, only a minimum. This approach leads to the Least- 
 Squares polynomials. 
 
 Secondly, we ask for the maximum error to be a minimum. 
 
 min max |f(x) - F(A,x) | 
 
 , . . < x < b a < 
 
 n 
 
 This is known as the Minimax Principle, 
 
 a < x, < . . . < x <b a<x<b 
 
 1 n 
 
 2.3 Minimax Principle 
 
 Definition : A polynomial P (x) of degree n is a "best" 
 
 approximation to a function f (x) in a < x < b 
 in the Minimax sense if and only if 
 
f(X) 
 
 -6- 
 
 F(A,X) 
 
 Figure 1. Function with Minimum Deviations 
 
-7- 
 
 max |f(x) - P (x) | < max |f(x) - Q (x) | 
 a < x < b n a < x < b ' n 
 
 for all polynomials Q of degree n. 
 
 Main Theorem 
 
 Given a continuous function f (x) in a < x < b and an integer 
 n > then: 
 
 (A) There exist a polynomial P (x) of degree n -which is a 
 best minimax approximation to f(x). 
 
 (B) The polynomial P (x) is unique. 
 
 (C) P (x) is the best minimax approximation to f(x) on [a,b] 
 
 if and only if there are n + 2 points a < x, < x < . . . 
 
 < x _ < b such that If (x. ) - P(x. ) I = max 
 n + 2- ! v i' v i y| 
 
 a < x < b 
 
 |f(x) - P (x) I and f (x. ) - P (x. ) alternate in sign 
 1 n ' l n i 
 
 for i = 1, 2, . . .n + 2. 
 Proof: 
 
 (A) The proof of the existence of an external element in a 
 space, in particular the space of all polynomials of degree 
 n, is very difficult. There the proof of (A) will not be 
 given. [For a rigorous proof of part (A) the reader is 
 referred to Ralston, A First Course in Numerical Analysis ; 
 p. 316 (No. 37, 38)] 
 
 (B) P (x) is unique. Assume Q (x) and P (x) are both 'best" 
 
 approximations to f(x) on [a,b]. Let R (x) = 
 
 1/2 [P (x) + Q (x)] which is also a 'best" polynomial. 
 . n n 
 
-8- 
 
 An extrema of f (x) - R (x) can occur only at a point 
 
 where f(x) - P (x) and f(x) - Q-(x) have extrema of equal 
 
 sign. Hence, P (x) and Q^(x) pass through n + 2 points, 
 
 which are too many points for polynomials of degree n. 
 
 Therefore, P (x) ■ Q^(x). 
 
 (C) P (x) is the best minimax approximation to f(x) on [a,b] 
 
 if and only if there are n + 2 points a < x < x < . . . 
 
 < x < b such that If (x. ) - P (x. ) I = max If (x) - P (x) I 
 n— ' i' n i ' i x ' n ' ' 
 
 and f (x. ) - P (x. ) alternate in signs for i = 1, 2, . . . 
 l n l ' ' 
 
 n + 2. 
 
 (i) Assume there are n + 2 points x, , . . . x _ at which 
 
 1 n+2 
 
 f (x) - P (x) attains its maximum absolute value 
 n 
 
 E = max |f(x) - P (x) | with alternating signs, 
 a < x < b n 
 
 If Q,(x) is any function with max |f(x) - Q(x) | < E, 
 
 a < x < b 
 
 then P (x) - Q(x) i = 1, . . .n+2 has alternating signs, 
 
 hence P (x) - Q(x) has n + 1 zeroes. If Q(x) were a 
 
 polynomial of degree n, then P (x) - Q(x), which is also 
 
 a polynomial of degree n, should be =0, or P (x) = Q(x) 
 
 which contradicts the assumption max |f(x) - Q(x) | < E. 
 
 a < x < b 
 
 (ii) Assume there are fewer than n+2 consecutive extrema with 
 
 alternating signs of the function f(x) - P (x) on [a,b]. 
 
 Divide [a,b] into, at most n + 1, subintervals each of 
 
 which contains either only positive extrema or only negative 
 
 extrema of f (x) - P (x) . This can be shown by the example 
 
 in Figure 2. 
 
-9- 
 
 -E -f 
 
 Figure 2. Alternating Sign Function 
 
-10- 
 
 Let the boundaries of these subintervals be a, y , y , 
 . . . y , b k < n, and notice that the polynomial 
 
 K. 
 
 (x - y^ • (x - y ) . . . (x - y ) has constant sign 
 within each subinterval, and opposite in neighboring sub- 
 intervals. For an appropriate choice of n (small) we 
 
 can have 
 
 |f(x) - P (x) - n(x-y ) . . . (x-y )| <E 
 
 in a < x < b, hence P (x) + n(x-y ) . . . (x-y ) is a nth 
 
 ft JL K 
 
 degree polynomial which is a better minimax approximation 
 
 than P (x) . 
 n 
 
 Q.E.D. 
 
 2.k Orthogonality and Recurrence Relations 
 
 Definition : A set of polynomials p (x), p, (x) . . . P,(x) 
 
 O _L 1C 
 
 where p, (x) is a polynomial of exact degree k is 
 
 orthogonal over the set of points x, , . . . x 
 
 with respect to a weight function w(x) > if and 
 
 n 
 only if "51 w(x)p (x )p (x ) = for j ^ k. 
 i=l J 
 
 The reader is probably more acquainted with the following 
 definition. 
 
 Alternate Definition : A set of polynomials p (x), p (x) . . . 
 is orthogonal over an interval [a,b] with respect 
 to a weight function w(x) > if and only if 
 ,.b 
 
 (Pj>P k ) = / w(x)p^.(x)p k (x)dx = for j ^ k 
 
 where (p.,P w ) is "the scalar product of p. and p, . 
 
■11- 
 
 Theorem 
 
 Every set of orthogonal polynomials p (x) , p (x) . . . satisfies 
 a 3-term recurrence relation of the type 
 
 P , (x) = (Kac + K») + K» 'P ( x ) 
 
 n+1 n-1 
 
 where K, K', K'' are constants that depend on, but not on x, 
 
 Proof: 
 
 Highest coef . of P , (x) 
 n+1 
 For K = we can construct a polynomial 
 
 Highest coef. of P (x) 
 & n 
 
 of degree n, say q^x), by ^(x) = P n+1 (x) - Kxp n (x). Expanding q^(x) 
 
 in terms of our orthogonal polynomials, we have 
 
 n 
 4 (x) = V c -P .(x) 
 n jtb J J 
 
 which yields 
 
 n 
 (A) P n+1 (x) - Kxp n (x) = <zZ c p (x) 
 
 Now take the scalar product of both sides of equation (A) with 
 
 n 
 p k (x), k = 0, 1, . . . n -2 (P n+1 , P k ) - K(xp n , p R ) = H c (p , p R ) = 
 
 j =o 
 
 c, (p, 5 Pn ) • Since (p. , p, ) 1 0, then c, = for K = 0, . . . n - 2. 
 k vx k k k' k i ' k 
 
 Therefore, only c. which may not be zero are c , and c , hence equation (A) 
 j n-1 n 
 
 becomes 
 
 p , (x) = (xK + c )p (x) + c p (x] 
 n+1 n n n-1 n-1 
 
 n 
 K 1 K' ' 
 
 Q.E.D. 
 
-12- 
 
 3. TCHEBYSHEV POLYNOMIALS 
 
 3.1 Definition and Properties 
 
 After several types of approximation were tried, it was 
 decided that either Least Squares or Tchebyshev Polynomial approxima- 
 tions would be used. The properties of both Least Squares and Tchebyshev 
 were studied, and Tchebyshev were chosen as the "best" approximation 
 for functional encoding. The following properties contributed to the 
 selection of Tchebyshev polynomials. 
 
 (1) They satisfy a three-term recurrence relation. 
 
 (2) They are easy to compute and convert to a power-series 
 form. 
 
 (3) They are orthogonal. 
 
 {h) Their zeroes interlace each other. 
 
 (5) They satisfy the Minimax Principle for a certain set of 
 polynomials. 
 
 (6) They lend themselves ideally to bandwidth compression, a 
 factor of prime importance in functional encoding. 
 
 Tchebyshev polynomials are defined as follows : 
 
 T (x) = cosn9 
 n 
 
 where x = cos0. Therefore, T (x) = cos(n arccosx) . 
 
 ' n 
 
 A geometrical representation of x, see Figure 3> is that the 
 semicircle of radius 1 is divided into n ejjual parts and the points 
 projected down on the x axis. The equidistant distribution in the angle 
 variable 6 causes a strongly none^uidistant distribution of the points 
 in the projections. 
 
-13- 
 
 -1 
 
 Figure 3« Geometrical Representation of Tchebyshev Polynomials 
 
.14- 
 
 Every orthogonal sets of polynomials satisfy a 3-term 
 relations and this is sometimes given as the definition of Tchebyshev 
 
 polynomials. 
 
 T (x) = 1 
 o 
 
 T x (x) = x 
 T n+l (x) = ^M " T n-l (x ^ 
 
 3.2 Main Theorem 
 
 The most important property of Tchebyshev polynomials can be 
 written in the form of a theorem. 
 
 Theorem 
 
 Among all polynomials P (x) of degree n > 1 with highest co- 
 
 n 
 efficient 2 , Tchebyshev polynomial T (x) has the smallest upper 
 
 bound for its absolute value in the interval -1 < x < 1. Since the 
 
 upper bound of |T (x) | is 1, then this upper bound is (l/2) 
 
 Proof: 
 
 Define R , (x) = T (x) - P (x) 
 n-1 ' n s ' n x ' 
 
 R (x) is a polynomial of degree n-1 and also we know that 
 
 T (x) attains extrema at n + 1 points which are alternately 
 
 positive and negative. If the extrema of P (x) is less than 
 
 that of T (x) , then at these n + 1 extrema R . (x) is 
 n " n-1 
 
 alternately positive and negative; hence must have n real 
 
 zeroes between these extrema. But R , is of degree n-1 
 
 n-1 
 
 and has at most n-1 zeroes, therefore R , (x) = and 
 
 n-1 
 
 T (x) = P (x). 
 
-15- 
 
 Corollary 
 
 Let f(x) = 5" c.T.(x). Then the nth partial sum S (x) = 
 .*tl 11 n 
 
 i=0 
 
 n 
 
 ^T c.T.(x) is the best nth degree minimax approximation to 
 i=0 X X 
 
 Proof: 
 
 f(x) on [-1, 1]. 
 
 f (x) - S (x) = c _T , (x) has n + 2 extrema + c . with 
 v ; n n+1 n+1 - n+1 
 
 alternating in signs on [-1, 1], hence S (x) is the best nth 
 degree minimax approximation. 
 
 Q.E.D. 
 
 It was necessary to devise a linear transformation such that 
 these theorems could be extended to intervals of arbitrary length. See 
 Figure h. 
 
-16- 
 
 X = 
 
 2X'-o-b 
 b-a 
 
 X'=a=s>X=-l 
 X'=b=>X= 1 
 
 Figure k. Linear Transformation 
 
-17- 
 k. COMPUTER IMPLEMENTATION 
 
 k.l Curve-Fitting Program 
 
 Reconstruction of several sample pictures was achieved with the 
 aid of a program developed by Steve Chase of the University of Illinois 
 Department of Computer Science and modified by the author. After de- 
 composing these pictures into sets of arbitrary curves, each curve was 
 approximated, in the Tchebyshev sense, by choosing selected points and 
 using these points as the function values and arguments in the following 
 program. 
 
 University of Illinois 
 Graduate College 
 Department of Computer Science 
 Illiac II Library Routine 
 E2-UOI-CF1T1-65-FR 
 May 27, 1966 
 
 IDENTIFICATION Chebyshev Curve - Fit for FORTRAN 
 
 PURPOSE Given an array of function values Y(l), Y(2), . . . Y (M) , and 
 their arguments X(l), X(2), . . . X (M) this subroutine will determine the 
 array A(l), A(2), . . . A(n+l) where the A's are the polynomial approxi- 
 mation 
 
 P (x) = a., + a^x + . . . + a n x (a. = A(i)) 
 n 12 n+1 1 
 
 to this tabulated function. 
 
 The criterion for this determination is the Tchebyshev criterion; 
 namely 
 
• 18- 
 
 max |P n (x j _) - y i |; (x. s x (l), y. = Y(l)) i = 1, . . '. M 
 
 is a minimum for all possible choices of the a. 's. 
 
 1 
 
 RESTRICTIONS 1) M > N + 2 
 
 2) X's must be distinct 
 
 (Overflow results if X(i) = X(j) for i ^ j and i, j < M) 
 
 3) M < 500 
 k) N < 20 
 
 ENTRY _____ 
 
 DIMENSION X(500), Y(500), A(2l) 
 
 read in (or compute) X, Y 
 
 CALL CFIT1 (M, N, X, Y, A, EPS) 
 
 where 
 
 M number of tabulated points of the function. Note M < 500. 
 
 N degree of the desired polynomial fit to the function N < 20. 
 
 X name of the array of the independent variables . The first M entries 
 
 in the array X must be distinct; i.e. X(l) =)= X(2) ^ . . . % X(M) . 
 Y name of the array of the independent variables; i.e. Y(i) = f(X(i)), 
 
 i = 1, . . '. M. 
 A name of the array in which the subroutine is to place the N + 1 
 
 coefficients. See EXIT. 
 
 -9 
 ESP See METHOD. Usually ESP =0.0 will work, if not try 10 . 
 
-19- 
 
 EXIT 
 
 METHOD 
 
 . . + a .x 
 n+1 
 
 Given a set of M points (x. , y. ) ; i =1, 2, 
 
 a. is stored in A(i) where P (x) = a. + a„x + 
 i n , 1 2 
 
 is the Tchebyshev polynomial approximation. 
 
 The basic idea of the method may be described as follows. 
 
 . . M, a subset 
 of N + 2 points is selected. This subset is called the 
 reference set; the rule for selecting it can be arbitrary since 
 any subset of N + 2 points is satisfactory. Now a reference 
 function f*(x) is calculated. This function is a polynomial 
 of degree N, and its coefficients are fixed by the condition 
 that 
 
 |f*(x.) - y.)| = h 
 
 for all x. in the reference set and 
 
 i 
 
 f*(x.) - y. 
 
 alternates in sign on the reference set 
 Then 
 
 max f*(x. ) - y. 
 
 i = 1, . . . M 
 
 is determined. If this maximum is located at a point, say 
 x., not in the reference set, a point in the reference set is 
 discarded and x. is added to the reference set and a new 
 reference function f*(x) is calculated. Then the search for 
 the maximum is repeated, etc. Finally, when the maximum falls 
 on a member of the reference set, this process is terminated. 
 
-20- 
 
 In comparison of value of 
 
 Z. = |f*(x.) - y. | 
 for the location of a maximum it is necessary, due to rounding 
 error, to tolerate a certain margin of error. Thus, in 
 comparing Z. , Z. we accept Z. is greater than Z. only if 
 
 Z . > Z . + EPS 
 
 where EPS is twice the maximum rounding error in Z. Although 
 this routine will often work successfully when EPS = 0, it 
 
 sometimes will go into an infinite loop in this case. To avoid 
 
 -9 
 this a value of ESP = 10 is usually sufficient. 
 
 ACCURACY The solution obtained by this program may deviate from the 
 
 exact condition of equal (magnitude) error at each of the 
 
 N + 2 reference points, due to round-off errors. 
 
 k.2 Pictures 
 
 Figure 5 represents the original sample and Figure 6 represents 
 the reproduction of the original drawing using Tchebyshev approximations 
 on the two arbitrarily chosen curves A and B. A number of selected points 
 were used as arguments in the approximation. The basis for the selected 
 points were : 
 
 (1) The initial point. This is the point where the curve 
 begins. 
 
 (2) Intermediate points. These are the intermediate points 
 along the curve. They ought to be connected smoothly with the points 
 preceding and following them. 
 
-21- 
 
 CURVE A 
 
 CURVE B 
 
 Figure 5. Original Car 
 
•22- 
 
 Figure 6. Car using Approximating Functions 
 
-23- 
 
 (3) Break point. This is an intermediate point on a curve 
 at which the curve bends in a different direction. 
 
 (k) Terminal point. This is the point where a curve ends. 
 
 Similar representations were done on Figure 7? yielding 
 Figure 8. 
 
-24- 
 
 CURVE C 
 
 CURVE D 
 
 Figure 7. Original Fish 
 
-25- 
 
 Figure 8. Fish Using Approximating Functions 
 
-26- 
 
 5. CONCLUSIONS 
 
 $.1 Limitations on Degree and Coefficients 
 
 The relatively narrow bandwidths of on-line communication 
 channels place a limitation on the degree and the number of coefficients 
 of the polynomials. The bandwidth representing the number of cycles 
 per second required to convey the information being transmitted. Not 
 only will bandwidth limit the degree of the approximating polynomial, 
 but also the realizability of the circuits necessary to generate the 
 approximating polynomials and their coefficients . 
 
 A polynomial of the fourth degree and its five coefficients 
 could be realized by a hardware system. Although the class of fifth- 
 degree polynomials and higher-degree polynomials may be desirable from 
 the point of view of approximation, they present problems with regards 
 to the hardware. This is not to discount the possibility of fifth- 
 degree and higher-degree polynomials. It would be possible, for instance, 
 to have approximating polynomials of higher degrees and eliminate the 
 transmission of certain coefficients. Such procedures must be entirely 
 dependent upon the approximating polynomial. 
 
 In transmitting the coefficients, consideration of the degree 
 of accuracy must be taken into account. The amount of error for the 
 entire approximating Tchebyshev polynomial is of prime importance. The 
 choice of Tchebyshev polynomials as the approximating functions eliminates 
 the necessity of distributing the error, equally or weighted, among the 
 coefficients . 
 
-27- 
 
 5.2 Scanning Techniques 
 
 The actual decomposition of the graphical display can he 
 achieved by different types of scanning techniques. It should be stated 
 here that among the problems associated with functional approximation 
 of graphical displays, scanning seems to be the most difficult. In 
 particular, electronic scanning techniques require a great deal of study. 
 The purpose of any scanning technique will be to decompose the graphical 
 display into a set of "relatively" continuous functions. "Relatively" 
 implying very few points of discontinuity. 
 
 One possible method of scanning the graphical display is with a 
 special curve reader. The display is read into the digital computer by 
 means of a series of points, and is then memorized by the computer. Each 
 point includes information concerning the position (the coordinates) and 
 type of point. The reader could consist of a pen for position detection 
 and a set of function switches. As the pen is moved along the curve, 
 the coordinates of the points are detected electromagnetically and are 
 transformed into digital quantities. The point coordinates are read 
 into the computer by the operating switch. The type of point is dis- 
 tinguished by selecting the appropriate function switch. 
 
 A second possible scanning technique is by electronic means. 
 Such a scanning technique would work very similarly to that in a tele- 
 vision. The given graphical display would be scanned hortizontally from 
 left to right and then drop down to the next line and scan hortizontally 
 once again, continuing this process until the entire display has been 
 scanned. Decomposition of the display into functions can be achieved by 
 having the first function consisting of the upper most set of points on 
 
-28- 
 
 the display. The second function being the next set of upper most 
 points. Figure 9 shows an example of a display before and after using 
 the electronic scanning technique. There are inherent problems associated 
 with such a scanning technique. Relatively simple displays could result 
 in sets of highly discontinuous curves of using this technique. Much of 
 this problem could be avoided by rotating the scanning. As an example, 
 scanning the top of the display, then the bottom and continuing this 
 alternating method. Other variations are possible. 
 
 3.3 Hardware Feasibility 
 
 The design and construction of a hardware system to achieve 
 functional approximation and functional encoding will entail a great 
 deal of research and ingenuity. However, considering the recent 
 advancements in on-line computers, particularly those in the Circuits and 
 Systems Research Group of the University of Illinois Digital Computer 
 Laboratory, functional approximation and functional encoding of graphical 
 displays seem to be definitely feasible. 
 
-29- 
 
 CURVE #1 
 
 z 
 
 CURVE #2 
 
 CURVE #3 
 
 Figure 9- Electronic Scanning 
 
-30- 
 
 LIST OF REFERENCES 
 
 1. Boas, Ralph Jr. and Buck, R. Creighton, Polynomial Expansions of 
 Analytic Functions . New York, N.Y. : Academic Press Inc., 1^6h. 
 
 2. Hamming, R. W. , Numerical Methods for Scientists and Engineers . 
 York, Pa. : The Maple Press, McGraw-Hill, 1962. 
 
 3. Lanczos, Cornelius, Applied Analysis . Englewood Cliffs, N. J. : 
 Prentice Hall Inc., 1961. 
 
 h. Nievergelt, Jury, "Classnotes : Math 387," Summer, 1967. University 
 of Illinois. 
 
 5. Ralston, A., A First Course in Numerical Analysis . New York, N.Y. : 
 McGraw-Hill, 1965. 
 
 6. Snyder, Martin Avery, Chebyshev Methods in Numerical Approximation . 
 Englewood Cliffs, N.J.I Prentice Hall Inc., I966. 
 
 7. Stewart, J. L. , Fundamentals of Signal Theory . New York, N.Y. : 
 McGraw-Hill, i960. 
 
 8. Van Valkenburg, M. E., Introduction to Modern Network Synthesis . 
 New York, N.Y. : John Wiley and Sons, Inc., i960. 
 
UNCLASSIFIED 
 
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 (Security classification ot title, body of abstract and indexing annotation mumt be entered when the overall report ia c lassitied) 
 
 1 ORIGINATING ACTIVITY (Corporate author) 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 2a. REPORT SECURITY CLASSIFICATION 
 
 Unclassified 
 
 26 GROUP 
 
 3 REPORT TITLE 
 
 Functional Approximation of Graphical Displays 
 
 4 DESCRIPTIVE NOTES (Type of report and inclusive dates) 
 
 Technical Report; Master's Thesis 
 
 5 AUTHORS (Last name, tint name, Initial) 
 
 Payne, Michael N. 
 
 6 REPO RT DATE 
 
 September, 1 
 
 7a- TOTAL NO. OF PASES 
 
 3^ 
 
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 12. SPONSORING MILITARY ACTIVITY 
 
 Office of Naval Research 
 Washington, D.C. 20360 
 
 13. abstract j n the field of computers, graphical displays are becoming increasingly 
 important: Displays are being used to input and output information, the fields of 
 programming debugging, business, data retrieval, computer-aided design, and data 
 'eductions make extensive use of graphical displays. Unfortunately, wide bandwidth 
 las been necessary in the transmission of graphical displays. The goal of this 
 thesis is to present possible techniques to achieve bandwidth reduction. One field 
 )f great importance is that of functional approximation and functional encoding of 
 raphical displays by an on-line computer system. The purpose of such a system is to 
 ;ake a given graphical display and decompose the display, by means of some scanning 
 echnique, into sets of continuous mathematical functions. These functions are ap- 
 rroximatedby some other mathematical function, such as polynomials. In choosing the 
 ipproximating functions, limitations of the hardware of the system, degree of accur- 
 icy and other restrictions must be taken into account. Once each function has been 
 ipproximated by a polynomial, it will be represented by the coefficients of that 
 polynomial. After the encoding of the display has been achieved, reconstruction of 
 i ;he display may begin. In order to reconstruct the display elsewhere, the coeffici- 
 ents are sent to the receiving place and are used to regenerate the approximating 
 polynomial. Finally, the entire graphical display is constructed from these approxi 
 aating polynomials. This thesis will concentrate on the area of the approximating 
 functions, in particular, on the mathematical analysis of the choice of the approxi- 
 aating function. Also, there will be remarks concerning the choise of scanning tech 
 liques and the limitations imposed by an on-line computer system on the choice of the 
 ipproximating functions. Many of the decisions were made on the basis of actually 
 
 &B*^TR7f 
 
 TT 
 
 iLiHjsli!. pi-ofrui ' eo with tho ai d ®£ »J ae 
 
 709^ and Illiac II computers 
 
 UNCLASSIFIED 
 
 Security Classification 
 
UNCLASSIFIED 
 
 Security Classification 
 
 14 
 
 KEY WORDS 
 
 bandwidth compression 
 graphical encoding 
 Tchebyshev polynomials 
 
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