Mathematical Theory of Generalized Fractal. Transforms and Associated Inverse Problems. Edward R. Vrscay. Department of Applied Mathematics

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1 Mathematical Theory of Generalized Fractal Transforms and Associated Inverse Problems Edward R. Vrscay Department of Applied Mathematics Faculty of Mathematics University of Waterloo Waterloo, Ontario, Canada N2L 3G1 WWW: To appear in the Proceedings of the ImageTech Conference on the Mathematics of Imaging and its Applications, March 17-20, 1996, Atlanta, Georgia. (Hosted by Georgia Tech and sponsored by Iterated Systems.) 1 Introduction This work has been done in collaboration with Bruno Forte, University of Verona, Italy. Much of the work to be outlined below was presented at the NATO ASI on Fractal Image Coding and Analysis held in Trondheim, Norway, July 8-17, 1995 and will appear as two papers in the Proceedings, edited by Yuval Fisher. Preliminary versions of these papers are available by anonymous ftp from links.uwaterloo.ca:/pub/fractals/papers/waterloo/. The most popular fractal-based algorithms for both the representation as well as compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces, e.g. IFS with probabilities (IFSP), Iterated Fuzzy Set Systems (IFS), Fractal Transforms (FT), the Bath Fractal Transform (BFT) and IFS with grey-level maps (IFSM). (FT and BFT are special cases of IFSM.) There are two major aspects of our work which are summarized below. First is the eort to formulate a consistent mathematical theory of Generalized Fractal Transforms. This procedure leads to a unication of the various IFS-type methods over function spaces in the order IFS! IFS! IFSM. The above methods are then linked with the method of IFSP (which is formulated over the probability measure space M()) by means of a formulation of Fractal Transforms over D(), the space of distributions on. The IFSP and IFSM methods may then be considered as special cases of the Distributional Fractal Transform. The second major aspect concerns the inverse problem of function/image approximation. In earlier works, we have provided formal solutions the inverse problem for the approximation of (a) measures using IFSP and (b) functions/images using IFSM as well as algorithms to construct approximations to arbitrary accuracy. In both cases, 1

2 we employ a xed, innite set of (necessarily overlapping) IFS maps possessing suitable renement properties. The minimization of the squared collage distance leads to a quadratic programming problem in the transform coecients. As stated earlier, most practical algorithms for the representation and compression of computer images use nonoverlapping IFS maps, due to the relative simplicity of the problem. In addition, these methods are based on a direct method of approximation, i.e. the image or function itself is approximated by tiling or \collage" of copies of either itself or subsets of itself. In the case of measures or distributions, however, such direct methods are not applicable and one must resort to indirect methods involving, for example, moments or Fourier Transforms. These indirect methods are justied by Collage Theorems for Moments and Fourier Transforms. It is then natural to investigate the action of Fractal Transform operators on generalized orthogonal expansions of functions which, in turn, denes an \adjoint" mapping on the Fourier coecients. In the case of the usual dyadic-type \Local IFS"/Fractal Transform approach and associated wavelet expansions, this mapping is equivalent to an IFS-type method on the wavelet expansion coecients themselves, leading to a collage theorem for wavelet coecients. This provides the mathematical basis for \discrete fractal-wavelet compression", which was recently discovered independently by a number of investigators. The use of moments, Fourier transforms and orthogonal expansions may be considered as special cases of a more generalized formulation of the inverse problem in a \dual space", with an appropriate Collage Theorem. 2 Generalized Fractal Transforms In the various IFS-type schemes, an image or target is represented as a point y in an appropriate complete metric space (Y; d Y ). The spaces employed by various IFS-type methods are listed below: IFS [19, 3, 2]: H(), the set of nonempty compact subsets of. IFS [8]: F (), the set of all functions u :! [0; 1] which are (1) upper semicontinuous on (; d) and (2) normalized, i.e. for each u 2 F () there exists an x 0 2 for which u(x 0 ) = 1. Fractal Functions [22]: which also includes \Fractal Interpolation Functions" [1, 5], the space of k-times continuously dierentiable functions C k (). IFSM [16]: L p (; ), the space of p-integrable functions with respect to a measure, 1 p 1. Fractal Transforms [7, 12, 13] are a special case of IFSM as are Fractal Functions. The Bath Fractal Transform [23, 24] is an IFSM with place-dependent grey level maps. IFSP [19, 3]: M(), the set of probability measures on B(), the -algebra of Borel subsets of. 2

3 IFSD [17] the linear space D 0 () of distributions on (; d). Along with the IFS maps w (except in the case of IFS) there is an associated set of either (a) functions = f 1 ; : : : ; N g or (b) probabilities p = fp 1 ; : : : ; p N g that satisfy conditions which depend on the particular space (Y; d Y ) used. The pair of vectors (w; ) or (w; p) then determines a \fractal transform operator" T : Y! Y. (The conditions imposed on the w i and i or p i guarantee that T maps Y into itself.) It is also desirable that T be contractive on (Y; d Y ) so that, from the Banach Contraction Mapping Theorem, there exists a unique and globally attractive xed point y 2 Y such that T y = y. In most, if not all, fractal-based approximation methods, it is assumed that the sets w i () are nonoverlapping (or at least ignore any overlapping), i.e. that w i?1 (x) exists for only one value i 2 f1; 2; : : : Ng. As such, given an image function u, its so-called fractal transform (T u)(x) at a point x 2 w i () is given by the fractal component f i (x) = i (u(w?1 (x))) as discussed below. i However, we consider the more general overlapping case where a point x 2 possibly has more than one preimage, i.e. w?1 i (x); 1 < k n(x). The question is then how to k combine the n(x) fractal components f i (x) to form what we refer to as the generalized k fractal transform (T u)(x). In what follows, we outline a formulation of generalized fractal transforms of image functions. The mathematical setting is as follows: 1. The base space: denoted, as above, by (; d). The space representing the pixels; a compact subset of R n. Without loss of generality, = [0; 1] n with Euclidean metric. 2. The IFS component: For an N 2 N, let w = fw 1 ; w 2 ; : : : ; w N g. In many cases we can relax the condition that the w i be contraction maps on. Note that the sets w i () may overlap. 3. The image function space: F() = fu :! R g Rg, the functions which will represent our images. The grey level range R g will denote the range of a particular class of image functions used in a given fractal transform method. (In practical applications, R g is a bounded subset of R +.) 4. The grey level component: Associated with the IFS maps w will be a vector of N functions = f 1 ; 2 ; : : : ; N g, i : R g! R g. We may also consider i : R g! R g, i.e. \place-dependent" grey level maps. See, for example, Ref. [24, 23, 16]. 5. The fractal components of u will be given by f i :! R g ; 1 i N, where ( i (u(w i?1 (x))); x 2 w f i (x) = i (); (1) 0; x =2 w i (): In other words, the fractal component f i (x) represents a modied value of the grey level of u at the ith preimage of x (if it exists). 3

4 6. The generalized fractal transform of u 2 F(): F k : [R g ] k! R g, 1 k N, where F k (t) = F k (t 1 ; t 2 ; : : : ; t k ); t i 2 R g ; 1 i k: (2) The transform F N denes an operator T : F()! F() that associates to each image function u 2 F() the image function v = T u. The F k operators combine the k distinct fractal components t i = f i (x) subject to a prescribed set of conditions [8, 17]. These conditions include (1) permutation symmetry and (2) recursivity which imply that F 2 is an associative binary operation S on R + R +. The choice of the binary operation S depends on the image function space used. The grey level ranges and fractal transforms for IFS and IFSM are given below: IFS: R g = [0; 1]. F N (x) = sup 1iN ff 1 (x); f 2 (x); : : : ; f N (x)g. IFSM: R g = R. F N (x) = P N f i (x). Remarks on the Nonoverlapping Property: 1. From an image compression viewpoint, nonoverlapping may be viewed as \expensive" and wasteful: The coding of a region of an image with more than one fractal component may be considered redundant and contrary to the goals of fractal image compression. However, techniques of overlapping have been used to overcome the problem of \blockiness" in fractally encoded images. There is still much to be explored in this area. 2. The nonoverlapping property is useful, in fact necessary, in our formal solutions to the various inverse problems of approximation using IFS-type methods. Of course, increasing the renement of IFS maps will produce better approximations. A standard way to do this is to to simply divide the base space into smaller regions by using more nonoverlapping IFS maps. For example, on [0,1], construct the IFS maps w i = 1 (x + i? 1), i = 1; 2; : : : N and just let N in- N crease. In our formal solutions, we employ an innite set of xed IFS maps w = fw 1 ; w 2 ; : : :g. This set is considered to be a kind of basis set which contains IFS maps of arbitrarily high degrees of renement. We then employ truncations of this set w N = fw 1 ; : : : ; w N g from which N-map IFSM or IFSP are constructed along with the associated generalized fractal transform operators T N. 3 Unication of IFS-type Methods The main ideas of this unication scheme are outlined below. Details are given in Chapter 3 of [17]. First, let us recall the basic features of IFS [19, 3]. Given a base space (; d) (typically [0,1] or [0; 1] 2 with Euclidean metric), the complete metric space of \images" Y is H(), the set of nonempty compact subsets of. The metric 4

5 d Y is the Hausdor metric h. Now let w = fw 1 ; : : : ; w N g denote a set of N contraction maps on. Dene the contraction factor of each map w i to be c i := sup d(w i (x); w i (y))=d(x; y): (3) x;y2;x6=y Associated with each contraction map w i is a set-valued mapping ^w i : H()! H() dened by ^w i (S) = fw i (x)jx 2 Sg for S 2 H(). Then the usual IFS operator ^w associated with the N-map IFS w is dened as follows: ^w(s) = N[ The IFS operator ^w is contractive on (H(); h) [19]: ^w i (S); S 2 H(): (4) h( ^w(a); ^w(b) ch(a; B); 8A; B 2 H(); (5) where c = max 1iN fc i g < 1. The completeness of (H(); h) guarantees the existence of a unique xed point y = A 2 H(). The set A, also called the attractor, is the IFS representation of an image. From Eq. (4), it satises the following self-tiling property, 3.1 From IFS to IFS A = N[ ^w i (A): (6) We now formulate the IFS method over an appropriate function space. First, we restrict the IFS maps w i to be contraction maps on which are one-to-one. Let I S (x) denote the characteristic function of a set S 2 H(), i.e. I S (x) = 1 if x 2 S and 0 otherwise. Now let A; B 2 H() and C = A [ B 2 H(). Then I C (x) = supfi A (x); I B (x)g: (7) From the property that I ^wi (S)(x) = I S (w i?1 (x)), it follows that I ^w(s) (x) = sup fi S (w i?1 (x))g: (8) 1iN Now consider the function space F BW () (for \black and white") dened by F BW() = fu :! f0; 1g j supp(u) 2 H()g: (9) In this case, the support of u 2 F BW() is given by supp(u) = fx 2 j u(x) = 1g =: [u] 1 ; (10) where we have introduced the IFS notation [u] 1 to denote the \1-level" set of u. Note that the grey level range R g = f0; 1g: The function u 2 F BW() represents a bitmap 5

6 (black and white) image whose black (or perhaps white) region [u] 1 is nonempty and closed. It is thus natural to consider the following metric on F BW(): d BW (u; v) = h([u] 1 ; [v] 1 ); 8u; v 2 F BW : (11) Completeness of (FBW(); d BW ) follows from the completeness of (H(); h). From Eq. (8), the fractal transform operator T : F BW ()! F BW () associated with the N-map IFS w is given by (T u)(x) = sup fu(w i?1 (x))g; 8x 2 : (12) 1iN The contractivity of T on (F BW (); d BW ) follows immediately from the contractivity of the IFS operator ^w on (H(); h). Thus there exists a unique xed point u 2 F BW() of the operator T, i.e. T u = u. Moreover, [u] 1 = A, the attractor of the IFS w so that [u] 1 = N[ ^w i ([u] 1 ): (13) In this formulation, pixels can assume only two grey level values, namely 0 and 1, or black and white (or vice versa). As such, only the geometry of an attractor is revealed. \Real" images, however, are not only black and white. Instead, their pixels can assume a range of nonnegative grey level values, e.g. u(x) 2 [0; 1]. For this reason, it would be desirable to modify the above IFS method so that such a range of grey level values could be produced. This is easily accomplished by modifying the fractal components in Eq. (12) as follows: (T u)(x) = sup f i (u(w i?1 (x)))g; 8x 2 ; (14) 1iN where the i : [0; 1]! [0; 1] are grey level maps. Subject to some conditions on these maps which guarantee that T : F ()! F () we then arrive at the method of Iterated Fuzzy Set Systems (IFS) [8]. Here, images are represented by functions u 2 F (). The grey level range is R g = [0; 1]. The metric d Y for the space Y = F () is given by d 1 (u; v) = sup h([u] ; [v] ); 8u; v 2 F (); (15) 01 where the -level sets of u 2 F () for 2 [0; 1] are dened as follows: [u] := fx 2 : u(x) g; 2 (0; 1]; (16) [u] 0 := closure(fx 2 : u(x) > 0g): The metric space (F (); d 1 ) is complete [11]. Associated with a set of IFS contraction maps w = fw 1 ; : : : ; w N g is a set of grey level maps = f 1 ; 2 ; :::; N g, i : [0; 1]! [0; 1], such that for all i 2 f1; 2; : : : ; Ng: (i) i is nondecreasing on [0,1], 6

7 (ii) i is right continuous on [0,1), (iii) i (0) = 0. In addition, (iv) i (1) = 1 for at least one i 2 f1; 2; :::; Ng. The pair of vectors (w; ) comprise an IFS with associated fractal transform operator T : F ()! F (). For a u 2 F (), its image T u is dened in Eq. (14) (up to a minor technicality which will be ignored here, cf. [8]). The relation between level sets of a function u and those of its image T u is given by [T u] = N[ ^w i ([ i u] ); 2 [0; 1]: (17) The contractivity of the IFS maps w i implies that that T is a contraction map on (F (); d 1 ) since [8] d 1 (T u; T v) cd 1 (u; v); 8u; v 2 F (): (18) The completeness of this space guarantees the existence of a unique xed point u 2 F () of the operator T. From Eq. (18), the -level sets of u obey the following generalized self-tiling property: [u] = N[ w i ([ i u] ); 2 [0; 1]; (19) which can be compared to the self tiling of = 1 level sets for the IFS case in Eq. (13). 3.2 From IFS to IFSM The Hausdor metric d 1 is very restrictive, however, from both practical (i.e. image processing) as well as theoretical perspectives. In [16], two fundamental modications were made to the IFS approach: 1. For a 2 M() and u; v 2 F (), dene 8 2 [0; 1], G(u; v; ) = ([u] 4[v] ) = ji [u] (x)? I [v] (x)jd(x); (20) where 4 denotes the symmetric dierence operator: For A; B, A4B = (A [ B)? (A \ B). 2. Now let be a nite measure on B(R g ) and dene g(u; v; ) = R g G(u; v; )d(): (21) 7

8 An application of Fubini's Theorem yields g(u; v; ) = (f0g)([u] 0 4[v] 0 ) + + u ((v(x); u(x)])d(x) v ((u(x); v(x)])d(x); (22) where u = fx 2 j u(x) > v(x)g and v = fx 2 j v(x) > u(x)g. It can be shown that g(u; v; ) is a pseudometric on L 1 (; ). In the particular case that = m, the Lebesgue measure on the grey level range R g, Eq. (22) becomes g(u; v; m) = ju(x)? v(x)jd(x); (23) the L 1 (; ) distance between u and v. The restrictive Hausdor metric d 1 over - level sets has been replaced by a weaker pseudometric (metric on the measure algebra) involving integrations over and R g. The result is a fractal transform method on the function space L 1 (; ). While it appears that only the L 1 distance can be generated by a measure on B(R g ), it is still natural to consider fractal transforms over the general function spaces L p (; ). 3.3 IFSM Let be a measure on B() and for any integer p 1, let L p (; ) denote the linear space of all real{valued functions u such that u p is integrable on (B(); ). We choose Y = L p (; ). The metric d Y is dened by the usual L p {norm, i.e. d Y (u; v) := d p (u; v), where 1=p d p (u; v) =k u? v k p = ju(x)? v(x)j d(x) p : (24) The IFSM is then dened by the following [16]: 1. The IFS component: w = fw 1 ; : : : ; w N g, w i 2 Aff 1 (), 1 i N, where Aff 1 () denotes the set of one-to-one ane transformations on. (Note that for the moment, we have relaxed the constraint that the w i be contractions.) 2. The grey level component: = f 1 ; 2 ; :::; N g, i 2 Lip(R; R), where Lip(R; R) = f : R! R j j(t 1 )? (t 2 )j Kjt 1? t 2 j; 8 t 1 ; t 2 2 R; for some K 2 [0; 1)g: (25) Since our function space involves integrations, it is natural to dene the following fractal transform operator T corresponding to an N-map IFSM (w; ): (T u)(x) := 8 N k=1 f k (x); (26)

9 where the fractal components f k are dened in Eq. (1). The above conditions on the w i and i guarantee that T : L p (; m)! L p (; m) for all p 2 [1; 1], where m denotes Lebesgue measure on. Now let = [0; 1] D, = m and 1 p 1. Then for u; v 2 L p (; m), d p (T u; T v) C p d p (u; v); C p = N k=1 jj k j 1=p K k ; (27) where jj k j is the Jacobian associated with the ane transformation x = w k (y). In the special -nonoverlapping case, i.e. where the sets w i () overlap only on sets of zero -measure (the standard assumption in practical applications in the literature), d p (T u; T v) Cp d p (u; v); Cp = " N k=1 jj k jk p k# 1=p : (28) Note that C p C p K; In the nonoverlapping case, with p = 1, K = max 1kN K k: (29) k T u? T v k 1 K k u? v k 1 ; 8u; v 2 L 1 (; m): (30) This is the usual bound presented in the literature on fractal transforms [7, 13]. In applications, we shall be using ane IFSM, i.e. w k 2 Aff 1 () and k (t) = k t + k ; t 2 R; 1 k N: (31) If the associated operator T is contractive on L p (; ), then its xed point u satises the equation u(x) = N k=1 [ k k (x) + k k (x)]; (32) where i (x) = u(w k?1 )(x) and k (x) = I wk (). In other words, u may be written as a linear combination of both functions k (x) and piecewise constant functions k (x) which are obtained by dilatations and translations of u and I (x), respectively \Place-Dependent" IFSM The above IFSM method may be easily generalized to \place-dependent" IFSM (PDIFSM), that is, IFSM with grey level maps having the form k : R! R, 1 k N. In other words, the i are dependent both on the grey-level value at a preimage as well as the location of the preimage itself. (This is analogous to IFS with place-dependent probabilities [4].) Much of the theory developed above for IFSM easily extends to place-dependent IFSM. This is the basis of the \Bath Fractal Transform", the eectiveness of which in coding images has been discussed in the literature [23, 24, 29, 16]. 9

10 3.4 IFS on the Space of Distributions D0( ) In what follows = [0; 1] although the extension to [0; 1] n is straightforward. Distributions [26] are dened as linear functionals over a suitable space of \test functions", to be denoted (following the standard notation in the literature) as D(). Here we choose D() = C 1 (), the space of innitely dierentiable real-valued functions on. (Note: In the literature, D() is normally taken to be C 1 0 (), the set of C 1 () functions with compact support on. With this choice, the expressions for distributional derivatives simplify due to the vanishing of boundary terms.) The space of distributions on, to be denoted as D 0 (), is the set of all bounded linear functionals on D(), that is, F : D()! R, such that 1. jf ( )j < 1 for all 2 D(), 2. F (c c 2 2 ) = c 1 F ( 1 ) + c 2 F ( 2 ), c 1 ; c 2 2 R, 1 ; 2 2 D(). The space D 0 () will include the following as special cases: 1. Functions f 2 L p (; m), 1 p 1, for which the corresponding distributions are given by F ( ) = f(x) (x)dx; 8 2 D(); (33) 2. Probability measures 2 M(), for which the corresponding distributions are given by F ( ) = (x)d(x); 8 2 D(); (34) 3. The \Dirac delta function", (x?a), which may be dened in the distributional sense as follows: For a point a 2, F ( ) = (a) for all 2 D(). This is often written symbolically as F ( ) = (x)(x? a)dx: (35) The goal is to construct an IFS-type fractal transform operator T : D 0 ()! D 0 () which, under suitable conditions, will be contractive with respect to a given metric on D 0 (). From the above list, such a formulation will place the various IFStype methods over function spaces and the IFSP method on probability measures under a common scheme. Associated with a set of N IFS maps w i will be a set of grey level maps i which will have to satisfy the following condition: Denition 1 A function g : R! R will be said to satisfy a weak Lipschitz condition on D 0 () if there exists a K 0 such that for all 2 D(), [(g f 1 )(x)? (g f 2 )(x)] (x)dx K [f 1 (x)? f 2 (x)] (x)dx 10 8f 1 ; f 2 2 D 0 (): (36)

11 Note: If g is ane on R, then it satises a weak Lipschitz condition on D 0 (). We now dene an N-map IFS on Distributions (IFSD) (w; ) as follows: 1. The IFS component: w = fw 1 ; w 2 ; : : : ; w N g, w i 2 Aff 1 (), with Jacobian jj i j 6= 0, 2. The grey level component: = f 1 ; 2 ; : : : ; N g, i : R! R satises a weak Lipschitz condition on D 0 () (with Lipschitz constant K i ). A fractal transform operator T : D 0 ()! D 0 () will now be constructed for this N-map IFSD. For any f 2 D 0 (), the distribution g = T f will be dened by the linear functional G( ) = = = N g(x) (x)dx (T f)(x) (x)dx We now dene the following metric on D 0 (): d D 0(f; g) = sup 2D 1 () f(x) (x)dx? where D 1 () = f 2 C 1 () j k k 1 1g. complete [17]. Then for any f; g 2 D 0 (), ( i f w?1 i )(x) (x)dx: (37) g(x) (x)dx d D 0(T f; T g) C D d D 0(f; g); C D = ; 8f; g 2 D0 (): (38) The metric space (D 0 (); d D 0) is N jj i jk i : (39) If C D < 1 then, from Banach's Contraction Mapping Theorem, there exists a unique distribution u 2 D 0 () such that T u = u. Furthermore, d D 0(T n u; u)! 0 as n! 1 for all u 2 D 0 (). 3.5 IFSP Here we briey review the formulation of IFSP over probability measures. Associated with an N-map IFS w with contraction maps w i is a set of probabilities p = fp 1 ; p 2 ; :::; p N g, p i 0, with P N p 1 = 1. Let B() denote the -algebra of Borel subsets of generated by all the elements of H(). Then Y = M(), the set of all probability measures on B(). Here, the (Markov) operator associated with the IFSP (w; p) is dened as follows: For a 2 M() and each S 2 H(), (T )(S) = (M)(S) = N p i (w?1 i (S)): (40) 11

12 The metric d Y on Y = M() is the Monge-Kantorovich metric (often referred to in the IFS literature as the Hutchinson metric) d H (; ): where d H (; ) = sup f2lip 1 (;R) fd? fd ; 8; 2 M(); (41) Lip 1 (; R) = ff :! R j jf(x 1 )? f(x 2 )j d(x 1 ; x 2 ); (42) 8x 1 ; x 2 2 g: The contractivity of the IFS maps w i implies the contractivity of M on (M(); d H ) [19]: d H (M; M) cd H (; ); 8; 2 M(): (43) There exists a unique 2 H() such that (1) M = and (2) d H (M n ; )! 0 as n! 1. Moreover, supp() A, with the equality when all p i > 0. From Eq. (40) it follows that (S) = N p i (w?1 i (S)); 8S 2 H(); (44) which leads to the following \change of variables" result for integration: For f 2 C() (or simple functions), f(x)d(x) = N (f w i )(x)d(x): (45) It is well known that by setting f(x) = x n, n = 1; 2; : : :, one can obtain recursion relations for the moments g n = R xn d of the invariant measure. 4 Inverse Problems for Generalized Fractal Transforms Recall that we consider target functions or images to be elements of an appropriate complete metric space (Y; d Y ). The underlying idea in fractal compression is the approximation, to some suitable accuracy, of a target y 2 Y by the xed point y of a contraction mapping f : Y! Y, i.e. f(y) = y. It is then f which is stored in computer memory. By Banach's Fixed Point Theorem or Contraction Mapping Principle (CMP), the unique xed point y may be readily generated by iteration of f, using an arbitrary \seed" image y 0 2 Y. Much of the recent and intense eort in this area has focussed on the \compression" aspect: For some suitable accuracy, as far as discretized images in pixel space are concerned, try to nd a suitable f which is dened in terms of the fewest possible parameters (or the coding of which occupies the smallest amount of computer memory). However, it is important that a rigorous mathematical theory of fractal-based approximation also be developed. Just as in more classical methods of analysis, it is necessary that the following aspects, among others, be understood: 12

13 1. Is it guaranteed that a given target y 2 Y can be approximated to a given accuracy in the d Y metric? Can this accuracy be be made arbitrarily small (or even \visually small") as we add more IFS maps? (This is essentially the question: \Is the set of all attractors of a given IFS-type method dense in the space (Y; d Y )?") 2. If the answer to 1. is \yes", then how fast is the convergence? 3. Is it guaranteed that a given algorithm will provide desired approximations to the target y? We have been concerned with developing a systematic theory of approximation of functions, measures and distributions using fractal-based methods [16, 15, 17, 18]. The central theme is the mathematical solution to the following formal inverse problem of approximation by xed points of contractive operators: Dene the set of contraction maps on (Y; d Y ) as follows: Con(Y ) = ff : Y! Y j d Y (f(x); f(y)) c f d Y (x; y) 8x; y 2 Y; c f 2 [0; 1)g (46) Given a \target" y 2 Y and an > 0, nd a map f d Y (y; y ) <, where f (y ) = y. 2 Con(Y ) such that There are three important mathematical results which provide the basis for fractal transform methods and fractal-based compression. It is worthwhile to list them here. 1. Banach Fixed Point Theorem/Contraction Mapping Principle: Theorem 1 Let (Y; d Y ) be a complete metric space. Suppose there exists a mapping f 2 Con(Y ) with contractivity factor c 2 [0; 1). Then there exists a unique y 2 Y such that f(y) = y. Moreover, for any y 2 Y, d Y (f n (y); y)! 0 as n! Continuity of Fixed Points With Respect to Contraction Maps: [9] Theorem 2 Let (Y; d Y ) be a metric space and f; g 2 Con(Y ) with xed points y f and y g, respectively. Then d Y (y f ; y g ) < 1 1? min(c f ; c g ) d Con(Y )(f; g); (47) where c f ; c g denote the contractivity factors of f and g, respectively. This result is a generalization of Barnsley's \continuity with respect to a parameter" result [2]. It was used to derive continuity properties of IFS attractors and IFSP invariant measures [9] as well as IFS attractors [14]. Although never stated explicitly, most if not all, fractal compression algorithms depend on this property since an optimization of the approximation of a target involves the variation of parameters which dene fractal transform operators. 13

14 3. \Collage Theorem"[6], essentially a corollary of the Banach CMP: Theorem 3 Let (Y; d Y ) be a complete metric space. Given a y 2 Y, suppose that there exists a map f 2 Con(Y ) with contractivity factor c f 2 [0; 1) such that d Y (y; f(y)) <. If y is the xed point of f, i.e. f(y) = y, then d Y (y; y) < =(1? c f ). The Collage Theorem permits a reformulation of the above formal inverse problem as follows: Given a \target" y 2 Y and a > 0, nd a map f 2 Con(Y ) such that = d Y (y; f (y)) <. The term is often referred to as the \collage distance". From the Collage Theorem, the xed point y of f will lie within a multiple of. By providing sucient conditions for the construction of approximations y for arbitrary values of, we have answered the rst question posed earlier. 4.1 Solutions to Inverse Problems for Measures and Functions Our basic strategy in solving the inverse problem is to work with an innite set of xed, ane contraction IFS maps: W = fw 1 ; w 2 ; : : : ; g; w i 2 Aff 1 (); (48) that satisfy renement conditions for the particular metric space (Y; d Y ) concerned. (The reader is referred to the appropriate references for details regarding these re- nement conditions.) A very useful set of contraction maps on [0,1] which satisfy the renement conditions for both measure and function approximation are the following \dyadic wavelet" maps: w ij (x) = 1 2 i (x + j? 1); i = 1; 2; : : : ; j = 1; 2; : : : ; 2i : (49) (The extension to [0; 1] 2 is straightforward.) The two major inverse problems which have been considered are 1. Measures: on the space (Y; d Y ) = (M(); d H ) [15], 2. Functions: on the space (Y; d Y ) = (L p (; m); d p ) [16]. We now summarize the solutions to these problems. In both cases, we choose N-map trunctations of W, w N = fw 1 ; : : : ; w N g in order to construct either 1. (Measures) N-map ane IFSP (w N ; p N ), where p N 2 N = f(p N ; : : : ; 1 pn N ) j N pn i 0; p N i = 1g R 2N (compact): (50) 14

15 N is the feasible set of probability vectors for N-map IFSP. Each point in N denes a fractal transform (Markov) operator T N in Con(M()). 2. (Functions) N-map ane IFSM (w N ; N ), where N = f N 1 ; : : : ; N Ng, N i (t) = N i t + N i ; ( N ; N ) 2 2N R 2N (compact): (51) 2N is the feasible set of grey level map parameters for N-map ane IFSM. Each point in 2N denes a fractal transform operator T N 2 Con(L p (; m)). In both the measure and function approximation problems we then proceed as follows. Given a target y 2 Y, then for an N > 0, nd the minimum collage distance N min = min N or 2N d Y (y; T N y): (52) (The minimum exists due to the compactness of the feasible sets N and 2N.) In both cases, we have the important result: N min! 0 as N! 1: (53) This guarantees the existence of solutions to the formal inverse problems for measure and function approximation. The solution of these problems is equivalent to the following result: Corollary 1 Given a xed set of ane IFS maps W = fw 1 ; w 2 ; : : :g satisfying the appropriate renement conditions for (Y; d Y ) being either (a) (M(); d H ) (measures) or (b) (L p (; m); d p ) (functions), then the set of all xed point attractors for (a) (w N ; p N ) or (b) (w N ; N ), respectively, N = 1; 2; : : :, is dense in (Y; d Y ). In both the measure and function approximation problems, the minimization of the squared collage distance ( N ) 2 in the appropriate metric becomes a quadratic programming problem in the probability/grey level map vector with constraints. Some numerical results of these algorithms are presented in [15, 16, 17, 18]. 4.2 Direct and Indirect Methods in the Inverse Problem The above formal solutions guarantee the existence of xed point IFSP/IFSM attractors which lie arbitarily close to a given target measure/function in the d Y metric. The practical construction of such solutions may proceed in two ways: 1. \Direct" methods, in which you work directly with the target, e.g. function approximation using IFSM: Given a target function/image y 2 L 2, minimize the collage distance k y? T N y k \Indirect" methods, in which you formulate the inverse problem in (Y; d Y ) as an equivalent inverse problem in a \dual space" (; d ). Example: formulating the inverse problem for measure approximation using IFSP as an inverse problem in \moment space" [15]. 15

16 4.2.1 Indirect Methods for Measure Approximation using IFSP Moment Matching: Much of the early work on the approximation of measures using IFSP was based on a knowledge of the moments of a target measure. Some form of \moment matching" was applied: Given a target measure 2 M() (let = [0; 1] for simplicity) with moments g n = R xn dx; n = 0; 1; 2; : : :, nd an IFS invariant measure whose moments g n = R x n dx are \close" to the g n. Early investigations sought to minimize a sum of squared distances between target and IFS moments. In [15] a formal solution to this problem was given. As mentioned above, we work with a xed and innite set of ane IFS maps W which satisfy a density condition. The moment matching is accomplished by means of a Collage Theorem for Moments on the following space D() of moment vectors for probability measures in M(): D() = fg = (g 0 ; g 1 ; g 2 ; : : :) j g n = x n dx; n = 0; 1; 2; : : : ; 8 2 M()g: (54) (Note that g 0 = 1.) Now dene the following metric on D(): For u; v 2 D(), 1 1 d 2 (u; v) = k (u 2 k? v k ) 2 : (55) k=1 Then (D(); d 2 ) is a complete metric space [15]. Now let (w; p) be an N-map IFSP with associated Markov operator M : M! M(). There corresponds an associated linear operator A : D()! D(). Moreover, A is contractive in (D(); d 2 ). As a result, there exists a unique element g 2 D() such that Ag = g. (The elements of g are the moments of the invariant measure of the IFSP.) The Collage Theorem for Moments follows. Fourier Transforms: [18] Equivalently, we may consider the inverse problem of measure approximation as one of \Fourier transform matching". Dene the following space of Fourier transforms of probability measures on M(): F T () = f^ : R! C j ^(!) = e?i!x d(x);! 2 R; 8 2 M()g: (56) (Note that ^(0) = 1 and that j^(!)j 1; 8! 2 R.) Now dene the following metric on F T (): For ^; ^ 2 F T (), let 1 1=2 d F T (^; ^) = j^(!)? ^(!)j 2! d!?2 ; (57)?1 The metric space (F T (); d F T ) is complete. Let (w; p) be an N-map ane IFS with Markov operator M : M()! M(). Corresponding to M, there is a linear operator B : F T ()! F T () which is contractive in (F T (); d F T ). The operator B has a unique and attractive xed point ^ 2 F T (). (^ is the FT of the invariant measure of the ane IFSP (w; p).) A Collage Theorem for Fourier Transforms then follows. 16

17 4.2.2 Indirect Methods for Function Approximation using IFSM It might seem that an indirect method for function approximation is unnecessary since the direct method of minimizing the collage distance is the standard approach in fractal compression algorithms. However, as is shown in [18], the indirect method corresponding to Local IFSM methods (e.g. the Jacquin block encoding approach) is precisely the \discrete fractal wavelet transform" method which was reported independently by a number of workers [10, 18, 21, 25, 27, 28]. We rst consider the more general case of IFSM and standard orthogonal function expansions. For simplicity we restrict our attention to the one-dimensional case, i.e. = [0; 1]. Let fq n g 1 n=0, with q 0 (x) = 1, denote a complete set of orthonormal basis functions on. Then for a given u 2 L 2 (; m), where u(x) = c k =< u; q k >= 1 k=0 c k q k (x); (58) u(x)q k (x)dx: (59) Now let (w; ) denote an N-map ane IFSM on with associated operator T and let v = T u. Then where The coecients a kl = N v(x) = d k = 1 l=1 1 k=0 i < q k ; q l w?1 i >; e k = d k q k (x); (60) a kl c l + e k ; (61) N i < q k ; I wi () > : (62) dene the ane mapping A : c! d associated with the IFSM operator T. If T is contractive in (L p (; m); d p ) then the mapping A : l 2 (N)! l 2 (N) is contractive in the l 2 (N) metric. (The xed point c = Ac is the vector of Fourier coecients of u in the q i basis.) In general, the matrix elements in Eq. (62) do not vanish, leading to a rather full (i.e. not sparse) matrix representation of A. This would be the case for the Discrete Cosine Transform, for example. However, in the special case that the q k are localized in space, e.g. wavelets, many of these matrix elements vanish. The relations between the d k and the c l simplify even further when the IFS maps w i are local and map domain blocks which support wavelets of lower resolution/frequency to range blocks which support wavelets of higher resolution/frequency. This is the basis of what has been referred to in one form or another as \wavelet-based fractal compression" [10, 21, 25]. It is possible to subject the wavelet expansion coecients to a fractaltype compression scheme. The details of this transformation are given in [18]. It would be interesting to investigate other possible \dual" formulations of the inverse problem. 17

18 References [1] M.F. Barnsley, Fractal interpolation functions, Constr. Approx. 2, (1986). [2] M.F. Barnsley, Fractals Everywhere, Academic Press, New York (1988). [3] M.F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London A399, (1985). [4] M.F. Barnsley, S.G. Demko, J. Elton and J.S. Geronimo, Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities, Ann. Inst. H. Poincare 24, (1988). [5] M.F. Barnsley, J. Elton, D.P. Hardin, P.R. Massopust, Hidden variable fractal interpolation functions, SIAM J. Math. Anal. 20, (1989). [6] M.F. Barnsley, V. Ervin, D. Hardin and J. Lancaster, Solution of an inverse problem for fractals and other sets, Proc. Nat. Acad. Sci. USA 83, (1985). [7] M.F. Barnsley and L.P. Hurd, Fractal Image Compression, A.K. Peters, Wellesley, Mass. (1993). [8] C.A. Cabrelli, B. Forte, U.M. Molter and E.R. Vrscay, Iterated Fuzzy Set Systems: a new approach to the inverse problem for fractals and other sets, J. Math. Anal. Appl. 171, (1992). [9] P. Centore and E.R. Vrscay, Continuity properties for attractors and invariant measures for iterated function systems, Canadian Math. Bull (1994). [10] G. Davis, Self-quantized wavelet subtrees: A wavelet-based theory for fractal image compression, 1995 Data Compression Conference Proceedings (preprint). [11] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35, (1990). [12] Y. Fisher, A discussion of fractal image compression, in Chaos and Fractals, New Frontiers of Science, H.-O. Peitgen, H. Jurgens and D. Saupe, Springer-Verlag, Heidelberg (1994). [13] Y. Fisher, Fractal Image Compression, Theory and Application, Springer-Verlag, New York (1995). [14] B. Forte, M. LoSchiavo and E.R. Vrscay, Continuity properties of attractors for iterated fuzzy set systems, J. Aust. Math. Soc. B 36, (1994). [15] B. Forte and E.R. Vrscay, Solving the inverse problem for measures using iterated function systems: A new approach, Adv. Appl. Prob. 27, (1995). 18

19 [16] B. Forte and E.R. Vrscay, Solving the inverse problem for functions and image approximation using iterated function systems, Dyn. Cont. Impul. Sys (1995). [17] B. Forte and E.R. Vrscay, Theory of generalized fractal transforms, to appear in Fractal Image Encoding and Analysis, edited by Y. Fisher, Proceedings of the NATO Advanced Study Institute, July 8-17, 1995, Trondheim, Norway. Springer Verlag, New York (1996). [18] B. Forte and E.R. Vrscay, Inverse Problem Methods for Generalized Fractal Transforms, to appear in Fractal Image Encoding and Analysis, edited by Y. Fisher, Proceedings of the NATO Advanced Study Institute, July 8-17, 1995, Trondheim, Norway. Springer Verlag, New York (1996). [19] J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. Math. 30, (1981). [20] A. Jacquin, Image coding based on a fractal theory of iterated contractive image transformations, IEEE Trans. Image Proc (1992). [21] H. Krupnik, D. Malah and E. Karnin, Fractal representation of images via the discrete wavelet transform, Proc. IEEE Conv. EE (Tel-Aviv, 7-8 March 1995). [22] P.R. Massopust, Fractal Functions, Fractal Surfaces and Wavelets, (Academic Press, San Diego 1994). [23] D.M. Monro, A hybrid fractal transform, Proc. ICASSP 5 (1993), pp [24] D.M. Monro and F. Dudbridge, Fractal Block Coding of Images, Electron. Lett. 28, (1992). [25] B. Simon, Explicit link between local fractal transform and multiresolution transform, Proc. IEEE Conf. Image Processing (Oct. 95). [26] R. Strichartz, A Guide to Distribution Theory and Fourier Transforms, CRC Press, Boca Raton (1994). [27] A. van de Walle, Relating fractal compression to transform methods, Master of Mathematics Thesis, Department of Applied Mathematics, University of Waterloo (1995). [28] G. Vines, Orthogonal basis IFS, pp in Fractal Image Compression, Theory and Application by Y. Fisher, Springer Verlag, New York (1995). [29] S.J. Woolley and D.M. Monro, Rate/distortion performance of fractal transforms for image compression, Fractals 2 (1994), pp