IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. E82-A, No. 8, pp , August, 1999.

Transcription

1 5 PAPER Special Section on Digital Signal Processing An Encoding Algorithm for IFS Coding of Homogeneous Fractal Images Using Univariate Polynomial Manipulation Toshimizu ABIKO y, Student Member and Masayuki KAWAMATA y, Member SUMMARY This paper proposes a fast encoding algorithm for iterated function system (IFS) coding of gray-level homogeneous fractal images In order to realize IFS coding of high order fractal images, it is necessary to solve a set of simultaneous equations with many unknowns Solving the simultaneous equations using a multi-dimensional, numerical root-finding method is however very time consuming As preprocessing of numerical computation, the proposed algorithm employs univariate polynomial manipulation, which requires less computation time than multivariate polynomial manipulation Moreover, the symmetry of the simultaneous equations with respect to the displacement coefficients enables us to derive an equation with a single unknown from the simultaneous equations using univariate polynomial manipulation An experimental result is presented to illustrate that the encoding time of the proposed algorithm is about 5 seconds on a personal computer with a 00 MHz Pentium II processor key words: fractal, iterated function system, inverse problem, polynomial manipulation, intelligent signal processing Introduction Fractals have been used for modeling natural images, such as clouds, coastlines, plants, and so on [] These natural images have common properties such that the magnified local subsets look identical to the whole set This property is referred to as self-similarity and provides a new mathematical technique for image coding [2], [] Barnsley has studied the use of fractals for data compression in image coding [2] His compression technique is based on fractals generated by iterated function systems (IFS's), which have the following features: () An IFS expresses an image in terms of its self-similarities; (2) An IFS consists of contractive affine transformations which describe the selfsimilarities These features enable IFS's to encode natural images only with a few parameters which are the coefficients of affine transformations [] This paper is devoted to an inverse problem of fractal images, which is the determination of the IFS generating a given fractal image So far, many fast algorithms have been developed to generate a fractal image from a given IFS, such as the minimal plotting algorithm [5] and the parallel decoding algorithm [6] On the other hand, no fast algorithm has been developed Manuscript received December, 998 Manuscript revised March 2, 999 y The authors are with the Graduate School of Engineering, Tohoku University, Sendai-shi, Japan to derive an IFS from a given image Thus, it is important todevelop a fast algorithm for determining the IFS's, that is, for solving the inverse problems of fractal images The advantage of our approach in comparison with the conventional Jacquin's method [] is the possibility of achieving a high compression ratio His method cannot fully exploit the image redundancy viewed in the global self-similarity ofan image, because his method relies on the block-wise self-similarity of an image In order to solve the inverse problem of fractal images, we adopt the moments of a given image The moment method was first proposed by Barnsley and Demko [8] They succeeded in encoding a simple fractal image, which is called twin-dragon" The method, however, has not been used for encoding high order fractal images, because the solution of simultaneous equations involved in the moment method generally constitutes a difficult numerical problem [9], which requires a long computation time The difficulties in solving the simultaneous equations arise because of the large number of unknowns that are usually involved, and because of the uncertainty of the initial guess for a root-finding method such as the Newton-Raphson method Thus, our algorithm employs univariate polynomial manipulation as preprocessing of the numerical computation so that an equation with a single unknown can be derived from the simultaneous equations with many unknowns Using this approach, the simultaneous equations can be solved extremely fast, because the equation with a single unknown, which is derived using univariate polynomial manipulation, can easily be solved by the DKA method [0] Applying univariate polynomial manipulation to IFS coding requires the special property of the simultaneous equations for homogeneous fractals, that is, the symmetry of the simultaneous equations with respect to the displacement coefficients On the other hand, the simultaneous equations for inhomogeneous fractals have nosuch symmetry Thus, IFS coding of inhomogeneous fractals requires multivariate polynomial manipulation, which is relatively complicated and timeconsuming This is the reason why we focus on homogeneous fractals The outline of this paper is as follows Section

2 6 2 summarizes the IFS theory and the moment theory Section proposes a new algorithm for solving the inverse problems Section includes an experimental result verifying the algorithm of Sect Section 5 draws some conclusions 2 Preliminaries In this section, we first summarize the existing knowledge which is necessary for understanding this paper 2 Definition of Homogeneous IFS's In this paper, we limit our consideration to an IFS generated by affine maps in the complex plane Within this important class, we will discuss a particular IFS with uniform (homogeneous) contraction mappings and equal probabilities The IFS is called a homogeneous IFS with equal probabilities, or homogeneous IFS for short, and defined by fc; fiz+ fl n ; pg; jfij < ; fi; fl n 2 C; p ==N; (n =0;:::;N ); fl i j= fl j ; (i j= j; i; j =0;:::;N ); () where N denotes the order of the IFS; C denotes the set of all complex numbers; fi and fl n denote the deformation and displacement coefficients of the IFS, respectively; p denotes the associated probability, which is equal to =N in our definition of homogeneous IFS's A set of coefficients ffi; fl 0 ;:::;fl N g is referred to as an IFS code A fractal image generated by a homogeneous IFS of order N is called a homogeneous fractal image of order N 22 Review of Basic IFS Properties Let H(C) denote a set of images in the complex plane Put w n (z) = fiz + fl n (n = 0;:::;N ) Then w n : H(C)!H(C) defined by w(b) = fw(x) j x 2 Bg; 8B 2H(C) is a contraction mapping on the Hausdorff distance Since w n (B) (n =0;:::;N ) are contraction mappings, the Hutchinson operator W defined by W (B) = [ N w n (B); for all B 2 H(C), is a contraction mapping on the Hausdorff distance Thus, W has a unique fixed point F 2H(C); which obeys F = W (F ); and is given by F = lim i! W i (B) for any B 2H(C), where W i denotes i iteration of W The fixed point F is called the attractor of the IFS A more general definition of IFS is that of IFS with probabilities In this context, the gray-tone fractal image is modeled as a probability measure ν 2P(C) over its attractor in the complex plane C Namely, the total darkness over an image B 2H(C) corresponds to a probability ν(b) Then the fractal image is defined as the fixed pointofacontraction mapping on the space of the probability measures P(C) Specifically, if a set of probabilities is associated with the homogeneous IFS, then the Markov operator M is defined by (Mν)(B) = N ν(w n (B)); (2) where ν 2 P(C) is a probability measure and B 2 H(C) is an image Since the Markov operator M is a contraction mapping, successive application of the Markov operator M to an arbitrary initial distribution ν converges in distribution to the invariant measure μ 2P(C); which obeys μ(b) =(Mμ)(B); () and is given by μ = lim i!+ Mi (ν): () Furthermore, it is possible to show [2] that for any continuous function f, we have the integration-type invariance condition: f(z)dμ(z) = z2c f (fiz+ fl n ) dμ(z): N z2c Equation (5) is a direct consequence of the invariance of the measure μ as shown in Eq () and the definition of the Markov operator defined in Eq (2) 2 Review of the Moment Method (5) In this paper, we adopt the complex moments of an invariant measure μ, which corresponds to a gray-tone fractal image, in order to solve the inverse problem First, we define the kth order complex moment of the invariant measure μ as follows: g k = z2c z k dμ(z); g 0 =: (6) Substituting z k (k is an integer) for f(z) on Eq (5) yields z k dμ(z) = z2c (fiz + fl n ) k dμ(z): N z2c

3 ABIKO and KAWAMATA: AN ENCODING ALGORITHM FOR IFS CODING OF HOMOGENEOUS FRACTAL IMAGES Thus, we have the complex moment g k in terms of deformation and displacement coefficients as X kx kc i z k i fi k i fl i n dμ(z) N g k = N z2c i=0 kx = kc i g k i fi k i fln i N i=0 = kx ψkc i g k i fi k i fln i N i=0 By taking the term corresponding to i = 0 out of the summation, we can obtain the relations between the moments and the IFS code:! ( fi k )g k = kx ψkc i g k i fi k i fln i ; N i= (k =; 2;:::): () Solving the simultaneous equations () for fi, fl 0 ;:::;fl N yields several candidates for the IFS code Proposed Algorithm for the Inverse Problem In this section, we describe an algorithm of recovering IFS code of homogeneous fractal images The solution of the simultaneous equations () constitutes, in general, a difficult numerical problem Fortunately, there exists symmetry of the simultaneous equations with respect to the displacement coefficients fl 0 ;:::;fl N Thus, our algorithm exploits the symmetry of the simultaneous equations for solving them The procedure of our algorithm is based on polynomial manipulation The algorithm employs the following technique in order to fast solve the simultaneous equations Derive an equation with a single unknown fi from the simultaneous equations () using univariate polynomial manipulation In the derivation, the symmetry of the simultaneous equations with respect to the displacement coefficients is used for eliminating variables The algorithm also employs the following technique in order to certainly solve the inverse problems of all kinds of homogeneous fractals Select the suitable simultaneous equations for an IFS code according to whether the moment train of a given image is periodically increasing or not Due to the selection, the procedure for determining homogeneous IFS is divided into two procedures, one is for rotationally symmetric homogeneous IFS's and the other is for irrotationally symmetric homogeneous IFS's! : Classification of Homogeneous IFS's In this section, we describe in detail the classification of homogeneous IFS's We can assume that the origin of the coordinate axes is the centroid of a given image without loss of generality ffl Rotationally Symmetric Homogeneous IFS's This class of IFS's is characterized by the arrangement of the fixed points of the affine transformations The modifier rotationally symmetric" is derived from the fact that the fixed points are rotationally symmetric about the centroid of the fractal image This class of IFS's is moreover divided into two groups in order to solve the inverse problem Rotationally Symmetric Homogeneous IFS's with No Central Fixed Point This IFS consists of affine transformations all of which have no central fixed point z = 0, and can be described as follows: Φ C; fiz + flm exp(j 2ßn N c ); =N Ψ ; N = N c M; n =0;:::;N c ; m =0;:::;M : (8) Rotationally Symmetric Homogeneous IFS's with a Central Fixed Point This IFS consists of affine transformations one of which has a central fixed point z = 0, and can be described as follows: Φ C; fiz; fiz + flm exp(j 2ßn N c ); =N Ψ ; N = N c M +; n =0;:::;N c ; m =0;:::;M : (9) In this class of IFS's, M is the multiplicity, which controls the complexity of an IFS, and N c is the interval of the moments Figure (b) shows the moment train of a rotationally symmetric homogeneous IFS A fractal image generated by a rotationally symmetric homogeneous IFS is called a rotationally symmetric homogeneous fractal image" ffl Irrotationally Symmetric Homogeneous IFS's This class of IFS's is that of all homogeneous IFS's except for rotationally symmetric homogeneous IFS's, and can be described as follows: fc; fiz+ fl n ; =Ng; n =0;:::;N : (0) Figure 2(b) shows the moment train of an irrotationally symmetric homogeneous IFS A fractal image generated by a irrotationally symmetric homogeneous IFS is called a irrotationally symmetric homogeneous fractal image" Theorem in Appendix indicates that the moment train of a rotationally symmetric homogeneous fractal

4 8 g k Nc Select the optimal IFS with minimal Hausdorff [2] distance between the reconstructed image and the original image, from the solutions of the simultaneous equations The optimal IFS is the output of the algorithm (a) Fractal image Nc 2Nc Nc (b) Moments of (a) Fig Rotationally symmetric type (N c =) (a) Fractal image Fig 2 g k (b) Moments of (a) Irrotationally symmetric type is periodically increasing, as shown in Fig (b) If the moment train is periodically increasing, then the simultaneous equations () with k = ;:::;N are indeterminate Thus, a method which can deal with the periodically increasing moments is needed in order to avoid indeterminate simultaneous equations This fact affects the procedure for solving the inverse problem, so that we must classify the homogeneous IFS's 2 Overview of the Proposed Algorithm This algorithm can encode a homogeneous fractal image Its brief overview is as follows: Calculate the complex moments of the image Determine the interval N c of the moments using the discrete Fourier transform with zero-padding Select the procedure according to whether the moment train of the image is periodically increasing or not 2 Set up a set of relations between the moments and the IFS code The relations are simultaneous equations with many unknowns Solve the simultaneous equations using univariate polynomial manipulations k k The following sections describe in detail the above procedures Calculations of the Complex Moments In this paper, we adopt the complex moments of an invariant measure μ, which corresponds to a gray-tone image Without loss of generality, we assume the total darkness of the image to be unity, and take the centroid of the image as the origin of the coordinate system Therefore, the zeroth moment g 0 =, and the first moment g =0 Solving the simultaneous equations for irrotationally symmetric homogeneous IFS's of order N requires N + moments, because the number of unknowns is N + Thus, we simply use lower moments g ;:::;g N+ so that the simultaneous equations can be low degree and easily solved On the other hand, solving the simultaneous equations for rotationally symmetric homogeneous IFS's of multiplicity M requires M + moments, because the number of unknowns is M + The moments of the image generated by this type of IFS's are periodically increasing as described before Thus, we use lower periodically increasing moments g Nc ;g 2Nc ;:::;g (M+)Nc at intervals of N c so that the simultaneous equations can be determinate Relations between the Moments and the IFS Code In the case of irrotationally symmetric homogeneous IFS's, the relations between the moments and the IFS code are as follows: ( N( P P fi k k ) g k = i= kc i g k i fi k i N fli n ; (k =;:::;N +): () The candidates for the IFS code are obtained from solving Eqs () for fi; fl 0 ;:::;fl N In the case of rotationally symmetric homogeneous IFS's, we use the periodically increasing moments g Nc ;g 2Nc ;:::;g (M+)Nc Thus, the relations between the moments and the IFS code of a rotationally symmetric homogeneous IFS with no central fixed point are as follows: 8 >< >: M( fi Nck ) g Nck = P k i= P N ckc Nci g Nc(k i) fi Nc(k i) M m=0 flnci m (k =;:::;M +): (2) ;

5 ABIKO and KAWAMATA: AN ENCODING ALGORITHM FOR IFS CODING OF HOMOGENEOUS FRACTAL IMAGES 9 By substituting (MN c +)=N c for M of the left-hand side of (2), the relations between the moments and the IFS code of a rotationally symmetric homogeneous IFS with a central fixed point are obtained as follows: 8 >< >: MN c + ( fi Nck P ) g Nck N c k = i= P N ckc Nci g Nc(k i) fi Nc(k i) M m=0 flnci m (k =;:::;M +): () 5 Solution of the Simultaneous Equations Using Univariate Polynomial Manipulation ; Using the power-sums, Eqs () can be written in matrix form as 2 6 N( fi )g N( fi 2 )g2 N( fi N )g N = 0 2C gfi N C g N fi N N C2 g N 2 fi N s s2 s N (5) 5 ; The fast solution of the simultaneous equations using univariate polynomial manipulation is the most important part of the proposed algorithm In the following explanation, we concentrate only on the solution of irrotationally symmetric homogeneous IFS's The solution of rotationally symmetric homogeneous IFS's can also be obtained by the following replacements: fi! fi Nc, N! M, and fl n! fl m Nc The general solution of simultaneous equations is based on a multi-dimensional, numerical rootfinding method such as the multi-dimensional Newton- Raphson method However, dealing with the large number of unknowns requires much computation time and finding the good initial guess is difficult, so that the numerical root-finding method cannot be directly used for fast encoding algorithms In this paper, our limitation to the homogeneous IFS's enables us to employ the symmetry of the simultaneous equations for deriving an equation with a single unknown fi The equation can be easily solved by a one-dimensional, numerical root-finding method such as the DKA method [0] The derivation process is invoked at run time after calculating the moments of a given image, so that the moments g k (k =0;:::;N)in Eqs () can be predetermined Therefore, the undetermined variables in Eqs () are fl 0 ;:::;fl N and fi Thus, symbolic manipulation of univariate polynomials in fi is feasible using the symmetry of the simultaneous equations with respect to fl 0 ;:::;fl N Univariate polynomial manipulation is much faster than multivariate polynomial manipulation, which is adopted in general-purpose computer algebra systems, so that the equation for fi can be derived very fast 5 Derivation of the Equation for fi The following derivation procedure using the symbolic manipulation of univariate polynomials in fi is invoked at run time after calculating the moments of a given image Consider the power-sums of fl n (n =0;:::;N ) defined by s i = fl i 0 + fl i + + fl i N : () N( fi N+ )g N+ = N+C g N fi N N+C N gfi Λ 2 6 s s2 s N 5 + s N+ : Newton's formula [] can be also written in matrix form as 2 2 s + 2 ff 6 s2 0 +s 2 5 = 6 6 ff2 5 s N +s N s N 2 ( ) N N s N+ = s N s N ( ) N s 2 Λ6 ff ff2 ff N 5 ; ff N () (8) where ff ;:::;ff N are the fundamental symmetric expressions of fl 0 ;:::;fl N and defined by ff = fl 0 + fl + + fl N ff 2 = fl 0 fl + fl 0 fl fl fl fl N 2 fl N ff N = fl 0 fl fl N : (9) The derivation procedure for obtaining the equation with a single unknown fi is as follows: Solve Eq (5) for s ;:::;s N, so that the sequence s ;:::;s N is expressed in terms of fi 2 Substitute the sequence obtained above for s ;:::;s N in Eq () and solve Eq () for ff ;:::;ff N, so that the sequence ff ;:::;ff N is expressed in terms of fi Substitute the sequences obtained above for s ;:::;s N and ff ;:::;ff N in Eq (8), and express s N+ in terms of fi (6) 5 ;

6 0 Substitute the sequence obtained above for s ;:::;s N ;s N+ in Eq (6), so that the (N + )th degree equation with a single unknown fi is obtained 5 Divide both sides of the (N + )th degree equation by the obvious factor ( fi) that is always included in the equation, so that the Nth degree equation with a single unknown fi can be obtained We note that both Eq (5) and Eq () are lower triangular matrices This implies that the solving procedures are merely forward substitutions, which require very little computation time 52 Solving the Equation with a Single Unknown In our algorithm, the equation with a single unknown fi is solved by the DKA method [0], which isanumerical root-finding method for polynomials with complex coefficients and has global convergence In order to obtain the displacement coefficients fl 0 ;:::;fl N,wehave to solve the following equation with a single unknown fl because of the relations between roots and coefficients NX fl N + ( ) i ff i fl N i =0; (20) i= where ff ;:::;ff N can be expressed in terms of fi during the derivation of the equation for fi Therefore, we can evaluate the sequence ff ;:::;ff N beforehand by substituting the obtained value of fi Solving Eq (20) for fl yields N roots, which become the displacement coefficients fl 0 ;:::;fl N After obtaining fi; fl 0 ;:::;fl N, our algorithm constructs a set of coefficients: ffi; fl 0 ;:::;fl N g, which is called a candidate for the reconstructed IFS When the order of the reconstructed IFS is assumed to be N, N candidates are constructed and added into the list of the candidates for the reconstructed IFS 6 Selecting the Optimal Candidate for the IFS Code After creating the list of the candidates for the reconstructed IFS in this way, our algorithm selects the optimal IFS with minimal Hausdorff distance, as the solution of the inverse problem The selection procedures are summarized as follows: Remove IFS's which generate no attractor from the list of candidates 2 Select the optimal IFS from the list of candidates by comparing the original image with the reconstructed image generated by each candidate In the procedure, the absolute value of the deformation coefficient of the removed IFS's is greater than or equal to one In the procedure 2, the reconstructed images are generated using the random iteration algorithm [2] Table The control parameters for images Threshold value for rotational type R th Threshold value for irrotational type I th 9 Maximum order Nmax Maximum multiplicity Mmax 5 The Whole Process to Solve the Inverse Problems Figure illustrates the whole process to solve the inverse problems of fractal images The peak value P c of the moments on the frequency domain is determined using the discrete Fourier transform with zero-padding The threshold value R th is determined under the condition that the peak value P c of any rotationally symmetric fractal is greater than R th The threshold value I th is determined under the condition that the peak value P c of any irrotationally symmetric fractal is less than I th The maximum order N max and the maximum multiplicity M max are determined by computational precision of the moments Experiments The procedure outlined above has been applied to the reconstruction of many IFS images obtained using the model of Eq () As an illustration of the results, we consider the case of the Thunder-cloud The Thundercloud in Fig (a) is obtained from an IFS with six affine transformations whose parameters are shown in Fig (b) The Thunder-cloud is therefore a homogeneous fractal image of order six Figure 5(a) shows the reconstructed Thunder-cloud We have determined the control parameters as shown in Table, because of computational precision of the moments which are calculated on the images of size and 256 gray levels The Hausdorff distance between the original image and the reconstructed image is 2:9 dots This difference is caused by the truncation errors in the decoding process and by the round-off errors on the moments of the image The computational time for solving the inverse problem is :86 sec It has been measured on a personal computer with a 00 MHz Pentium II processor running the Linux operating system To the authors' best knowledge, no algorithm that can encode a high order fractal image within about 5 seconds has been reported for IFS coding 5 Conclusions In this paper, we have proposed an analytic method for IFS coding of fractal images using univariate polynomial manipulation In order to solve the inverse problem analytically, one has to solve the high degree simultaneous equations with many unknowns Thus, the proposed algorithm derives an equation only with a single

7 ABIKO and KAWAMATA: AN ENCODING ALGORITHM FOR IFS CODING OF HOMOGENEOUS FRACTAL IMAGES Input S : screen; (Λ An original fractal image Λ) R th ; I th : real; Nmax; Mmax : integer; Output fi; fl0;:::;fl N : complex; (Λ Parameters of a reconstructed IFS Λ) procedure SolveInverseProblem; var N; M : integer; begin Calculate the complex moments of the image S; Determine the interval N c of the moments; Determine the peak value P c of the moments on the frequency domain; if Pc >R th then (Λ Rotationally symmetric Λ) for M := to Mmax do The IFS is assumed to be a rotationally symmetric IFS of multiplicity M with no central fixed point; Derive the Mth degree equation of fi Nc ; Solve the above equation; Solve the Mth degree equation of fl Nc ; The IFS is assumed to be a rotationally symmetric IFS of multiplicity M with a central fixed point; Derive the Mth degree equation of fi Nc ; Solve the above equation; Solve the Mth degree equation of fl Nc ; endfor; endif; if P c <Ith then (Λ Irrotationally symmetric Λ) for N := to Nmax do The IFS is assumed to be an irrotationally symmetric IFS of order N; Derive the Nth degree equation of fi; Solve the above equation; Solve the Nth degree equation of fl; endfor; endif; Remove IFS's which generate no attractor from the list of candidates for the reconstructed IFS; Select the optimal IFS from the list by comparing the original image with the reconstructed image generated by each candidate; end: Fig The whole process to solve the inverse problems unknown from the simultaneous equations with many unknowns using the symbolic manipulation which exploits the symmetry of the simultaneous equations with respect to the displacement coefficients Owing to this derivation of the equation, the algorithm can encode a high order fractal image within about 5 seconds on a personal computer Moreover, if the given image is rotationally symmetric, the method which adopts the periodically increasing moments is needed for solving the inverse problem We have therefore proposed a practical method for encoding the rotationally symmetric homogeneous fractals References [] B B Mandelbrot, The fractal geometry of nature, W H Freeman and Company, New York, 9 [2] M F Barnsley, Fractals everywhere, Academic Press Inc, San Diego, 988 [] Y Fisher, ed, Fractal image compression (theory and application), Springer-Verlag, New York, 995 [] L P Hurd and M F Barnsley, Fractal image compression, AK Peters, Ltd, Wellesley, MA, 992 [5] D M Monro and F Dudbridge, Rendering algorithms for deterministic fractals," IEEE Computer Graphics and Applications, pp 2, Jan 995 [6] S-C Pei, C-C Tseng, and C-Y Lin, A parallel decoding algorithm for IFS codes without transient behavior," IEEE Trans Image Processing, vol 5, no, pp 5, March 996 [] A E Jacquin, A novel fractal block-coding technique for digital images," Proc IEEE ICASSP, pp , April 990 [8] M F Barnsley and S Demko, Iterated function systems and the global construction of fractals," Proc R Soc Lond, vol A 99, pp 2 25, 985 [9] M Peruggia, Discrete iterated function systems, AK Peters, Ltd, Wellesley, MA, 99 [0] O Aberth, Iteration methods for finding all zeros of a polynomial simultaneously," Math Comput, vol 2, pp 9, 9 [] Mathematical Society of Japan, Encyclopedic dictionary of mathematics," 2nd ed, p 28, The MIT press, Cambridge, MA, 99 [2] T Abiko, M Kawamata, and T Higuchi, An efficient algorithm for solving inverse problems of fractal images using the complex moment method," 5th IEEE International Workshop on Intelligent Signal Processing and Communication Systems : ISPACS'9, vol, pp S2 S26, Nov 99 Appendix: The Moments of a Rotationally Symmetric Fractal This section proves a theorem for the moments of a rotationally symmetric fractal The theorem states that the moments of a rotationally symmetric fractal is periodically increasing We can assume that the origin of the coordinate axes is the centroid of a given image without loss of generality

8 2 Definition : Let μ 2 P(C) be the invariant measure of a fractal A fractal is called an N c rotationally symmetric fractal if dμ(ze j2ßn=nc )=dμ(z); n =0;:::;N c ; z 2 C; where N c is a positive integer The following theorem indicates that the moments of an N c rotationally symmetric fractal is periodically increasing at intervals of N c Theorem : The kth order moment g k of an N c rotationally symmetric fractal equals 0 if k is not a multiple of N c, namely g k = 0; (k mod N c j=0): (a) Fractal image Proof: See [2] N = 6 fl2 = :8 :j fi = 0:2+0:2j fl = :9+2:j fl0 = 0 fl = :8 :j fl = :5+:5j fl5 = :9j (b) Fig IFS code for (a) Original Thunder-cloud Toshimizu Abiko received the Bachelor of Engineering and Master of Information Sciences degrees from Tohoku University, Sendai, Japan, in 996 and 998, respectively He is currently working towards the DE degree at Tohoku University His main interests and activities are in fractal theory, polynomial solving, Gröbner basis, and image coding He is a student member of the IEEE (a) Fractal image N = 6 fl2 = :85 :0j fi = 0:50 + 0:20j fl = : :0908j fl0 = 0:0256 0:050j fl = :85 :008j fl = :89 + :5050j fl5 = 0:0095 :905j (b) IFS code for (a) Masayuki Kawamata received BE, ME, and DE degrees in electronic engineering from Tohoku University, Sendai, Japan, in 9, 99, and 982, respectively He was an Associate Professor in the Graduate School of Information Sciences at Tohoku University and is currently a Professor in the Graduate School of Engineering at Tohoku University His research interests include -D and multi-d digital signal processing, intelligent signal processing, fractals and chaos in signal processing, control theory, and linear system theory Dr Kawamata received the Outstanding Transactions Award from the Society of Instrument and Control Engineers of Japan in 98 (with T Higuchi), the Outstanding Literary Work Award from the Society of Instrument and Control Engineering of Japan in 996 (with T Higuchi) and the th IBM-Japan Scientific Award in Electronics in 99 He is a member of the IEEE, the Society of Instrument and Control Engineers of Japan, the Information Processing Society of Japan, and Robotics Society of Japan He is an IEEE Senior Member Fig 5 Reconstructed Thunder-cloud