1 Corresondence Between Fractal-Wavelet Transforms and Iterated Function Systems With Grey Level Mas F. Mendivil and E.R. Vrscay Deartment of Alied Mathematics Faculty of Mathematics University of Waterloo Waterloo, Ontario, Canada NL 3G1 mendivil@augusta.math.uwaterloo.ca ervrscay@links.uwaterloo.ca WWW: htt://links.uwaterloo.ca (December 19, 1996) Abstract It is well known that the action of a \Fractal Transform" or (Local) Iterated Function System with Grey Level Mas (IFSM) on a function f(x) induces a very simle maing on its exansion coecients c ij in the Haar wavelet basis. This is the basis of the \discrete fractal-wavelet transform": subtrees of the wavelet coecient tree are scaled and coied to lower subtrees. Such transforms, which we shall also refer to as IFSW - IFS on wavelet coecients - have been introduced into image rocessing with other (comactly suorted) wavelet basis sets in an attemt to remove the blocking artifacts in the standard IFS block encoding algorithms. Although not as straightforward as in the Haar case, we show that there is a relationshi between such wavelet transforms and IFSM. In fact, for most such transforms, there is an equivalent IFSM, which rovides a further mathematical basis for their use in image rocessing. We also resent results for the case of eriodized wavelets, a common imlementation in image rocessing. Finally, we rove some results on the fractal dimension of the grah of an attractor of IFSM or IFSW oerators.
Corresondance between Fractal-Wavelet 1 Introduction Recently, a number of workers [D, FV, KMK, S, vw] (to name a few) have indeendently devised \discrete fractal-wavelet transforms" (DFWT) which involve an IFS-tye transformation on the wavelet exansion coecients of signals and images. In the case of Haar wavelet exansions, as was shown in [FV], there is a very simle connection between these wavelet transforms and (Local) Iterated Function Systems with Grey Level Mas (IFSM). Due to the nonoverlaing nature of the Haar wavelet functions, the action of the IFSM automatically translates into a simle scaling and coying of wavelet coecient subtrees onto lower subtrees. With suitable restrictions on the scaling coecients i, the transformation is contractive in l (IN). The wavelet exansion coecients c ij of a signal or image are then stored to a rescribed level k of renement, along with a set of codes describing the \arent-child" subtree airs and their corresonding \otimal" arameters i. Iteration of the fractal-wavelet transform then generates higher resolution coecients c ij, i > k. To reeat: In the case of Haar wavelets, this rocedure in wavelet coecient sace is equivalent to the action of an IFSM in \ixel sace". It also illustrates the fact that fractals and wavelet bases share the fundamental roerty of scaling. Such discrete wavelet transforms were then alied to exansion coecients for generalized (non-haar) systems of comactly suorted wavelets. In these cases, the suorts of contiguous wavelets overla. A major motivation for the generalization of this method was the attemt to reduce the blockiness exhibited by the usual fractal ( = Haar) block-encoding (see [D, KMK, vw]). The urose of this aer is to show that there is a connection between wavelet transforms for generalized wavelets and IFS-tye methods on function saces. However, the connection is not quite as straightforward as in the Haar case. In Section, we consider the case of noneriodized wavelets and show that the discrete wavelet transform is equivalent to a recurrent IFSM. In Section 3, we consider eriodized wavelets which are used extensively in image rocessing. The situation is slightly more comlicated but a corresondence to IFSM is shown. In Section 4, we briey resent some results on the fractal dimension of attractors for IFSM and IFSW. We now conclude this section with a brief review of the basic ideas of IFSM. An N-ma Iterated Function System with Grey Level Mas (IFSM) is dened by [FV1]: 1. The IFS comonent: w = fw 1 ; w ; : : :; w N g where each w i : X! X is a contraction and. The grey level comonent: = f 1 ; ; : : :; N g where each i : IR! IR is Lischitz. Here (X; d) is a comlete metric sace, the \ixel sace", tyically [0; 1] or [0; 1]
Corresondance between Fractal-Wavelet 3 with Euclidean metric. Associated with an IFSM (w; ) is an oerator T, the fractal transform, whose action on a suitable sace of functions F(X) is given by i f(w 1 i (x)) ; f F(X): (1) X T (f)(x) = i For simlicity, the both the w i and the i are often assumed to be ane so that w i (x) = s i x + a i and i (t) = i t + i. Another interesting and secial case of an IFSM oerator is the following: T (f)(x) = (x) + X i i f(w 1 i (x)): () Here, the function (x) acts as a \condensation" function for the IFSM. Under suitable conditions on the w i and the P i, the oerator T is contractive in F(X) (for examle, in L (X) it suces that ( i js i i j)1= < 1) and so has a unique xed oint u = T u (see [FV1]). Let fq i (x)g be an orthonormal basis of L (X). Given an IFSM oerator T there exists a corresonding oerator M on the sequence sace l (IN) (viewed as sequences of coecients in exansions of functions with resect to the q i ). The following gure illustrates this rocedure: L (X) = l (IN) - T M L (X)? = -? l (IN) In the event that the IFSM T is ane or linear, then the oerator M will also be ane or linear. If T is contractive in L (X), then M is contractive in l (IN). (See [FV] for further discussion.) As mentioned above, when the orthonormal basis is the Haar wavelet basis ij, then the oerator M becomes a maing of wavelet coecient subtrees to lower subtrees. Such maings will be introduced in the next section. Fractal-Wavelet Transforms (IFSW) and Their Corresondence with IFSM For simlicity, we restrict our attention to the one-dimensional case X = [0; 1]. The results extend in a straightforward manner to the imortant case X = [0; 1]
Corresondance between Fractal-Wavelet 4 (images). Let be a scaling function with which is associated a multiresolution analysis of L (R), and let be the associated \mother wavelet" function from which are derived the functions mn(x) = m= ( m x n); m; n Z; (3) which form an orthonormal basis of L (R). In articular we shall be concerned with functions f L (R) which admit wavelet exansions of the form (in the L sense) f(x) = b 00 X (x) + c ij ij (x); (4) i0;0j i 1 where b 00 =< f; (x) > and c ij =< f; ij >. We are interested in wavelets having comact suort on R (imlying, in turn, that f has comact suort). The exansion coecients are conveniently dislayed in the form of an innite wavelet tree: b 00 c 00 c 10 c 11 c 0 c 1 c c 3 B 30 B 31 B 3 B 33 B 34 B 35 B 36 B 37 Note that each entry B ij reresents a tree of innite length: we shall also refer to such a tree with aex c ij as the block B ij. We now consider some IFS-tye oerations on this wavelet tree. Examle 1: Using the above notation, consider the following transformation: c M : B 00! 00 ; j 0 B 00 1 B i j < 1 : (5) 00 The restrictions on the i follow from the condition that the wavelet coecient sequence c ij belong to l (Z). This is a simle examle of a \discrete fractalwavelet transform" to which, for simlicity, we shall refer as an IFS on wavelet coecients or IFSW. The goal is to nd an IFSM oerator T (as dened by Eq. ()) acting in \hysical sace", i.e. on functions suorted on R, which corresonds to an IFSW oerator M acting on the wavelet coecients. The dilation/translation relations within the wavelet basis rovide the key to this goal. Let w 1 (x) = x= and w (x) = x= + 1=. Then B00 w 1 1 = B 10 and B00 w 1 = B 11 (6)
Corresondance between Fractal-Wavelet 5 since i;j w 1 1 (x) = i+1;j (x) and i;j w 1 (x) = i+1;j+ i(x): (7) Thus, this simle IFSW oerator will corresond to the following two-ma IFSM with condensation function (x): T (f)(x) = c 00 (x) + 0 f w 1 1 (x) + 1 f w 1 (x) : (8) Note that the IFSM oerator deends on the articular wavelet basis chosen. In the simle case of Haar wavelets, the mother wavelet (x) decomoses into nonoverlaing comonents: (x) = I [0;1=) (x) I [1=;1) (x): (9) As such, the IFSM oerator T in Eq. (8) corresonds to a simle two-ma IFSM with IFS mas w 1 and w and grey-level mas 1 (t) = 1 t + 1 and (t) = t 1. If T is contractive, then its xed oint attractor function u has [0,1] as suort. However, in the case of other comactly suorted wavelets, no such satial decomosition into searate grey-level mas is ossible. As well, the suort of the attractor function u is necessarily larger than [0,1]. To illustrate, consider the articular IFSW in which 1 = 0:4 and = 0:6. The IFSM oerator T in Eq. (8) is contractive. Figures 1(a) and 1(b) show the IFSM attractor functions for, resectively, the Haar wavelet and \Coifman-6" cases. In both cases, we have chosen b 00 = 0 and c 00 = 1. Examle : Consider the fractal wavelet transform with four block mas as follows: W 1 : B 10! B 0 ; W : B 11! B 1 ; W 3 : B 10! B ; W 4 : B 11! B 3 ; (10) with associated multiliers i, 1 i 4. Diagramatically, M : B 00! c 00 c 10 c 11 : (11) 1 B 10 B 11 3 B 10 4 B 11 (The reader may susect that this has the aroma of a local IFSM.) Now iterate this rocess, assuming it converges to a limit B00 which reresents the wavelet exansion of a function u. Then u = c 00 00 + v: (1) We need only focus on the function v which admits the wavelet exansion c 10 c 11 1 B 10 B 11 3 B 10 4 B 11 : (13)
Corresondance between Fractal-Wavelet 6 4 3 1 u(x) 0-1 - -3-4 0 0.5 0.5 0.75 1 X 5 4 3 u(x) 1 0-1 - -3-1 -0.5 0 0.5 1 1.5 X Figure 1. Attractor functions u for Examle 1: (a) Haar wavelet basis, (b) Coifman-6 wavelet basis. Since < 10 ; 11 >= 0, etc., we may write where the comonents v i satisfy the relations v = v 1 + v ; (14) v 1 (x) = c 10 10 (x) + 1 v1 (x) + v (x) v (x) = c 11 11 (x) + 3 v1 (x 1) + 4 v (x 1): (15) We may consider these equations as dening a kind of vector IFSM with condensation. The vector v is comosed of the orthogonal comonents v 1 and v that satisfy the above xed oint relations in Eq. (15). These equations may be written more comactly as: where v i (x) = b i (x) + X j=1 ij (v j (w 1 ij (x))); i = 1; ; (16) w 11 (x) = w 1 (x) = 1 x; w 1(x) = w (x) = 1 x + 1 ; (17) b 1 (x) = c 10 10 (x); b (x) = c 11 11 (x) ; (18)
Corresondance between Fractal-Wavelet 7 and 11 (t) = 1 t; 1 (t) = t; 1 (t) = 3 t; (t) = 4 t : (19) Note that the contractive IFS mas w ij are maings from the entire base sace X into itself. As such, this IFSM is not a local IFSM in general. What aeared to be a \local" transform in wavelet coecient sace is a normal IFSM in the base sace. (Again, in the secial case of the nonoverlaing Haar wavelets, the above IFSM may be written as a local IFSM.) The \locality" of the block transform has been assed on to the orthogonal comonents v 1 and v of the function v. These comonents may be considered as \nonoverlaing" elements of a vector. This \vector IFSM" is really nothing more than a recurrent IFSM on B 00 where we slit B 00 as B 10 B 11 and have the IFSM act \between" the comonents of this slitting. Examle 3: Finally, consider the following modication of the fractal wavelet transform in Examle : W 1 : B 10! B 0 ; W : B 11! B 1 ; W 3 : B 11! B ; W 4 : B 10! B 3 ; (0) with associated multiliers i, 1 i 4. Grahically, M : B 00! c 00 c 10 c 11 : (1) 1 B 10 B 11 3 B 11 4 B 10 As above, iterate this rocess, assuming it converges to a limit B00 which reresents the wavelet exansion of a function u. Then where v admits the wavelet exansion As before, we write u = c 00 00 + v: () c 10 c 11 1 B 10 B 11 3 B 11 4 B 10 : (3) v = v 1 + v : (4) and determine the relations satised by the comonents v i. It is necessary to determine the ane changes of variables involved in maing B 10 to B 3 and B 11 to B. The result is v 1 (x) = c 10 10 (x) + 1 v1 (x) + v (x) v (x) = c 11 11 (x) + 3 v (x 1 ) + 4 v1 (x 3 ): (5) This is a vector or recurrent IFSM with condensation functions which could be written in the comact form of Eq. (16).
Corresondance between Fractal-Wavelet 8 3 IFSW for Periodic Wavelets on [0; 1] In many alications one needs wavelet bases adated to a comact interval. For this urose, eriodizing the wavelet basis functions is useful. Furthermore, in ractical comutations, one often deals with nite data sets and eriodizing the discrete transform is simle and clean to imlement. For a function f L (IR), dene the eriodized version of f to be f (x) = X i f(x i) (6) where the sum is over all integer i. This rocess \wras" the function f around over the interval [0; 1] and sums the various contributions. Let be a scaling function for an MRA on IR and be the associated \mother" wavelet. Then the eriodized wavelets ij (x) form an orthonormal basis of L [0; 1] and we have a nested MRA structure as in the case of wavelets on IR [Dau]. Unfortunately, the nice translation/scaling relations (reresented by Eq. (7) ) no longer hold for these eriodized wavelets. To see this, consider the following simle examle. By denition, 1j(x) = X k = X k 1j(x k) (x k j); (7) where the latter equation follows from the scaling roerty of the wavelets. However, the function (x j) = X k (x j k) 6= 1j (x): (8) As a result, the rather simle scaling analysis of Section is not alicable here. The IFSM associated with an IFSW will have to be dierent in the eriodic case than in the case of functions on IR. Suose that we are given an IFS on the wavelet coecient tree for some eriodized wavelets on the interval [0; 1]. This oerator induces an oerator T : L [0; 1]! L [0; 1]: Since we are interested in eriodic functions, we think of the unit circle with [0; 1] as the \natural" fundamental domain for the eriodization. Let be the mother wavelet for some wavelet decomosition. Suose that the suort of lies in the interval [0; N]. Then su( j ) su( ( n x j)) n [j; j + N] = [ j n; j + N n ]
Corresondance between Fractal-Wavelet 9 with j running from 0 to n 1. So, for suciently large n we have that their suorts have length less than 1 and that all the suorts lie in the interval [0; 3=]. We now dene the IFS mas w 1 ; w ; ^w 1 and ^w on the circle. These mas will be induced by corresonding mas W 1 (x) = x= and W (x) = x=+1= on IR. For the mas w 1 and w, we use the coordinate system on the circle induced by the fundamental domain [0; 1]. Notice that the image of [0; 1] under w 1 is [0; 1=] { this ma would not be a continuous ma on the circle, so we must make a \cut" at the oint x = 0 = 1. Thus, these mas are, in some sense, only \local" mas. For the mas ^w 1 and ^w, we use the coordinate system on the circle induced by the domain [1=; 3=]. Here the \cut" oint is x = 1= (the oint antiodal to the oint 0). Thus, the image of the circle under ^w 1 is [1=4; 3=4] { the \middle" of the circle { and under ^w is [3=4; 5=4]. Again, these are \local" mas. With these denitions, the usual scaling and translation relations among the wavelets are modied as follows. For i n, i;j (w 1 1 (x)) = i+1;j(x) and i;j (w 1 (x)) = i+1; i +j (x) (9) for 0 j < i 1 and i;j ( ^w 1 1 (x)) = i+1;j(x) and i;j ( ^w 1 (x)) = i+1; i +j (x) (30) for j i 1. The functions i;j are suorted on the interval [0; 1] for j < i 1 and on the interval [1=; 3=] for j i 1. Thus, we use the w i 's for the rst ones and the ^w i 's for the second ones. In our block notation, these scaling and translation relations become Bi;j w 1 1 = B i+1;j and for 0 j < i 1 and Bi;j ^w 1 1 = B i+1;j and Bi;j w 1 = B i+1; i +j (31) Bi;j ^w 1 = B i+1; i +j (3) for j i 1. These relations are the crucial observation needed to show that the IFS on the wavelet coecients induces an IFSM oerator on L [0; 1] (in the usual sense of an IFSM). In order to do this, we break the sace L [0; 1] into two orthogonal arts. The rst art is the sace sanned by the rst n levels of the wavelet basis: V 1 = sanf i;j j 0 i < n; 0 j < i g and the second art is the orthogonal comlement of V 1 in L [0; 1] so that V 1 V = L [0; 1]:
Corresondance between Fractal-Wavelet 10 Notice that T mas V 1 into both V 1 and V. However, T (V ) V. Thus, we have a sort of \recurrent" IFSM between V 1 and V. We reresent the oerator T as T = (T 1 + T ) T 3 (33) with T 1 : V 1! V 1 and T : V 1! V and T 3 : V! V (34) Let v be the xed oint of the oerator T, thus T (v) = v. Writing v = v 1 v, we obtain T 1 (v 1 ) (T (v 1 ) + T 3 (v )) = v 1 v (35) and thus T 1 (v 1 ) = v 1 (36) T (v 1 ) + T 3 (v ) = v : (37) So the IFS on V 1 is a simle IFS and the IFS on V is an IFS with condensation function T (v 1 ). The vector v 1 is relatively easy to calculate since V 1 is nite dimensional. One needs only to comute the coecients of v 1 with resect to the basis i;j for aroriate i; j. These coecients are given by the IFSW oerator. From v 1 one obtains the function T (v 1 ) { the condensation function for the IFSM on V. The IFSM on V basically corresonds to the case of non-eriodized wavelets (i.e. wavelets on IR). Finally, we return to the fractal-wavelet transform of Examle 1 alied to coecient trees for eriodized wavelets. In the Haar case, the result is trivially the same as in Figure 1(a). The eriodic attractor function u for the Coifman-6 eriodized wavelets is shown in Figure. (Note that the wavelet coecients c ij for all three grahs are identical.) 5 4 3 u(x) 1 0-1 - -3 0 0.5 0.5 0.75 1 X Figure. Attractor function u for IFSW in Examle 1 using Coifman-6 eriodized wavelet functions.
Corresondance between Fractal-Wavelet 11 4 Fractal Dimension of IFSM and IFSW In this section we discuss the dimension of the attractor of an IFSM or IFSW (which we think of as an IFSM with condensation). We will rimarily be interested in the box counting dimension, although in many cases this is the same as the Hausdor dimension. Our methods are direct adatations of the methods in [B, F], thus we give only the sketches of the roofs. Recall that our usual IFSM oerator has the form T (f)(x) = X i i f(w 1 i (x)) + i i ; (38) where i is the characteristic function of X i = w i ([0; 1]) and w i (x) = s i x + b i. We can think of the iecewise constant arts i i as being a iecewisedened \condensation function" for the IFSM. Thus, the more general form of the IFSM is given by T (f)(x) = X i i f(w 1 i (x)) + (x) (39) We obtain the following theorem on the fractal dimension of the grah of the attractor for the IFSM (39). Notice that if is C 1, then the grah of will have dimension 1. Theorem 1 Suose that the IFSM oerator T as dened in equation (39) is non-overlaing and that the dimension of the grah of (x) is 1. Then the dimension of the grah of the xed oint of T is the unique solution D to if P i j ij > 1 and 1 otherwise. X i j i jjs i j D 1 = 1 (40) Proof: As in ([Bar],. 6), we give just a heuristic reasoning. Denote by N() the number of boxes of side length that intersect the grah of the xed oint function. Then by the self-similarity, N() X i j i s i jn( js i j ) + C 1 where the last term aears because of the \condensation function". Notice that if we substitute N() = C D into this equation, we obtain 1 = X i j i jjs i j D 1 which is the desired relation (the C 1 term dros out as! 0).
Corresondance between Fractal-Wavelet 1 The following Corollary is also a secial case of results in [B, F] (the secial case where there is no shear in the IFS mas). Corollary For the ane IFSM oerator T as dened in equation (38), the dimension of the xed oint of T is the unique solution D to equation (40) if Pi j ij > 1 and 1 otherwise. Now, in the case of an IFSM corresonding to an IFSW, the maings are certainly not non-overlaing. Thus, the revious results do not aly. As in the case of a usual IFS, we obtain an uer bound on the dimension. Theorem 3 Suose that, the \mother wavelet" for the wavelet MRA, is dierentiable. Then for the ane IFSW oerator S dened as in equation (5) the dimension of the xed oint of S is bounded above by the unique solution D to X j i jjs i j D 1 = 1 (41) if P i j ij > 1 and 1 otherwise. i Proof: Let N(f; ) be the number of boxes in a \column" that intersect the grah of f. Then N(f + g; ) N(f; ) + N(g; ): (4) Using this observation one can obtain the uer bound in the case where the images in the IFSM are overlaing in a similar manner as one obtains the uer bound for the similarity dimension for geometric IFS. Acknowledgments This research has been suorted by the following grants from the Natural Sciences and Engineering Research Council of Canada (NSERC) which are gratefully acknowledged: an Oerating Grant (ERV) as well as an NSERC Collaborative Projects Grant (ERV, with C. Tricot, Ecole Polytechnique, U. de Montreal, B. Forte, U. of Verona, Italy and J. Levy-Vehel, INRIA, Rocquencourt, France). References [B] M.F. Barnsley, Fractal Functions and Interolation, Constructive Aroximation :303-39 (1986). [Bar] M.F. Barnsley Fractals Everywhere, Academic Press, New York, 1988.
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