Image compression predicated on recurrent iterated function systems

Transcription

1 2n International Conference on Mathematics & Statistics June, 2008, Athens, Greece Image compression preicate on recurrent iterate function systems Chol-Hui Yun *, Metzler W. a an Barski M. a * Faculty of Mathematics, Kim Il Sung University, Pyongyang, D. P. R. Korea a Faculty of Mathematics, University of Kassel, Kassel, F. R. Germany May, 2008 Abstract. Recurrent iterate function systems (RIFSs) are improvements of iterate function systems (IFSs) using elements of the theory of Marcovian stochastic processes which can prouce more natural looking images. We construct new RIFSs consisting substantially of a vertical contraction factor function an nonlinear transformations. These RIFSs are applie to image compression. Keywors: Recurrent iterate function system (RIFS); Iterate function system (IFS); Fractal image compression; Fractal image coing. 1. Introuction Fractal image compression (FIC) was introuce by Barnsley an Sloan (1988, [2]) an after that has been wiely stuie by many scientists. FIC is base on the iea that any image (for example that of human face) contains selfsimilarities, that is, it consists of small parts similar to itself or to some big part in it. So in FIC iterate function systems are use for moeling. Jacquin (1992, [12]) presente a more flexible metho of FIC than Barnsley's, which is base on recurrent iterate function systems (RIFSs) introuce first by him. By his metho, the original image is partitione into small non-overlapping blocks calle regions (or ranges) an big overlapping blocks calle omains. For every region, a similar omain an some transformation are searche. Fisher (1994,[10]) improve the partition of Jacquin. Bouboulis etc.(2006, [7]) introuce an image compression scheme using fractal interpolation surfaces which are attractors of some RIFSs. RIFSs which have been use in image compression schemes consist of transformations which have a constant vertical contraction factor. This factor has a great influence on the shape of attractor. It is obvious that each point of a region has a ifferent contraction ratio. So, replacing the * Corresponing author. aress: metzler@mathematik.uni-kassel.e ** Presente at 2n International Conference on Mathematics & Statistics, June, 2008, Athens, Greece. KISU MATH 2008 E C 001

2 vertical contraction constant by a contraction function can give a more flexible construction so that the number of omains an regions neee for image coing can be reuce an the quality of the ecoe image can be better, that is, the FIC will be improve. We present a new construction of RIFSs which have vertical contraction factor functions an their application to image compression Recurrent bivariate IFSs on rectangular gris The iea of RIFSs was introuce by Jacquin ([12]). A recurrent iterate function system (RIFS) is efine as a pair of a collection,, of Lipschitz mappings in a complete metric space (i.e. an IFS) an a row irreucible stochastic matrix satisfying 1 for all 1,, an aitionally for any, 1,,, there exist,, such that 0 (Barnsley, Elton an Harin [5]). The attractor A of a RIFS is compute as follows: By a stochastic matrix, for an initial 1,, a sequence is given, which obeys Pr. With this, we get a sequence of transformations that generates an orbit for an initial R, that is, is chosen with the probability that equals an we have. In the matrix, gives the possibility of applying the transformation to the point in state, so that the system transits to state. The attractor A is efine as the limit set of these orbits, which consists of the points whose every neighbourhoo contains infinitely many for almost all orbits. By Barnsley, Elton an Harin ([5]) the existence, uniqueness an characterization of this limit set A was prove. We present a new construction of a RIFS with a given ata set on a gri. We construct a local IFS R ;,, 1,,, 1,, over the gri, whose general efinition is as follows ([3]). Definition 1 Let ( X, ) be a compact metric space. Let be a nonempty subset of X. Let : X an let s be a real number with 01. If,, for all, in, then is calle a local contraction mapping on ( X, ). The number is calle a contractivity factor for. Definition 2 Let ( X, ) be a compact metric space, an let : X be a local contraction mapping on ( X, ), with contractivity factor for 1, 2,,, where is a finite positive integer. Then : X : 1, 2,, is calle a local iterate function system (local IFS) (or partitione IFS ([10])). The number max : 1,2,, is calle the contractivity factor of the local IFS. Let the ata set on the rectangular gri be,, R ; 0,1,,, 0,1,,

3 3 such that,. Let enote N 1,, 1,,, I,, I,, I,, I,, EI I E I I (which we call the region), for, N, We choose big rectangulars (calle omains) E I I, where I, I I, I, I I an,, for 1,,. Then, there exist, 0,, 0, such that an ( an ), which we enote by an an respectively. We efine the contraction transformations :E E for, N by,,, where I I, I I are contraction transformations with contractivity factors, obeying for any, N : 1 : 1,,,,,. The functions :E R R, for, N are efine as follows:,,,,,, where, 1,, are continuous Lipschitz mappings on E, E with the Lipschitz constants L, L respectively satisfying,,,,, for, 1,, for, 1, 1,, where,,,,, that is, goes through 4 enpoints of the omain E, goes through 4 enpoints of the region E. Then, the functions satisfy 'join-up' conitions, for, 1,, where,,,,,,, 1, 1,. Thus, any maps the enpoints of the -th omain to the enpoints of the, -th region. These transformations are contractions for all, N

4 4 E E C E E E E 4 E E Figure 1 Connection matrix an it's irecte graph. 3 with respect to some metric which is equivalent to the Eucliean metric on R 3. This metric is efine on R 3 for,,,,, R 3 by where,,,,,, 1Max, ; 1,,, 1,, 2L F an L F Max L F ; 1,,, 1,,, where L F are Lipschitz constants of the functions F. The contractivity of the transformation is Max, L F, where 1Max, ; 1,,, 1,, 2 We enumerate the set N 1,, 1,, by a injective mapping : 1,, 1,, 1,, an enote MN. For simplicity, we enote, by an by. The following is an example of a connection matrix which can be efine as follows: 1 if E E 0 otherwise The connection matrix shows that the transformation can follow the transformation iff (See Fig. 1). Then, we have a row irreucible stochastic matrix given by: 1 iff 0 0 iff 0, Thus, the RIFS is given as a pair consisting of the above local IFS an the row irreucible stochastic matrix. The existence, uniqueness an characteri-

5 5 Figure 2 H on the omain. zation of the attractor A of this RIFS has been prove by Barnsley Elton an Harin ([5]). 3. Application: An algorithm for image compression using RIFSs Let us enote an image value at the position, E by,. 3.1 Construction of the vertical contractivity factor functions In this section, we present a metho of constructing the vertical contractivity factor function, on the region E by the image values on the region E an on the corresponing omain E, N, 1,,. The vertices,,,,,,, of the omain E are mappe to any one of the region E,,,,,,,, by the contraction transformation E E. Choose 1. For,,,,, 1,,, N, we calculate the absolute vertical istance between the image value, an the straight line through the enpoints,,, an,,,, an enote them by H,, H, 0, 1,, H,,, H,, respectively. In general, choose 1 for,, 1,,, N, H,,, are compute similarly. Now, choose. For 1, 1 1,,1, analogously calculate the absolute vertical istance between the image value

6 6, an the straight line through the enpoints,,, H, an, H,,, H,,,, respectively. By the same metho, H, an enote all these istances by, H,,, H, 0 (excepting H,, H,, H, are compute. In the case where, H,, H, ), changing the value or, we calculate that one again. Then, we get the set H H, : 0, 1,,, 0,1,, (See Fig. 2). Similarly, in the region E, for any point,, where,,,,,, 1,,, =, the set H r H r, : 0, 1,,, 0,1,, is constructe. With the sets H, H r, we get a set r D H, H, : 0, 1,,, 0,1,,, where 0. We construct the contractivity factor function, by an interpolation function for the ata set D. 3.2 Encoing algorithm Image encoing is a process to get reuce informations neee to preserve the ata of an image. We introuce a way for getting the reuce information by a RIFS. We partition the original image into non-overlapping blocks on a rectangular gri an construct a gri of non-overlapping omains which are (2 is integer) times as large as the regions. For each region, we look for the most similar omain to that as follows: By the above algorithm we calculate the contractivity factor function, on the region an the interpolation functions on the omain an on the region, respectively. Next we compute the istance between the image values on the region an the values mappe from the image values on the omain by a contraction transformation with regar to some metric. If the istance is less than a given tolerance, then we save the number of the omain an the ata set for the contractivity factor function. If there is no suitable omain, then we ivie equally the region into smaller rectangular (new regions), that is, repartition an repeat the above proceure for each smaller region. Finally, we get the reuce information, the enpoints of the regions, the number of the most similar omain for each region an the ata set for the contractivity factor function. 3.3 Decoing algorithm We calculate the interpolation functions,,,,, from the information of the encoe image an compute the transformations,, out of them, that is, we etermine the IFS. The ecoing algorithm is base on the Deterministic Iteration Algorithm (DIA) as in Barnsley [1] an it is similar to that in Bouboulis's [7], Section 3.3. The only ifference compare with [7] is the following simplification: For any

7 7 Table 1 Some results obtaine for Lena by the scheme in this paper. Compression Time(CT)(secon) PSNR(B) Compression Ratio(CR) : : : : : : 1 region, we fin out if its center is alreay rawn. If yes, then we go to the next region. If no, then we search for an appropriate transformation of the IFS to raw this center an aitionally the four mipoints of the eges of the region, an then we go to the next region. 4. Experiments We use a Lena image as original image in our experiment. Accoring to the above algorithm, while this image is encoe an ecoe, we measure the time neee to compress using a computer with a 1.86 GHz CPU clock an Winows XP. The experimental results show that our metho can compress the image very fast, holing high quality of the ecoe image (See Table 1). In Fig. 3-7, some images ecoe by our scheme are shown. 5. Results & Discussion In this paper we present a refine metho of constructing an RIFS an its application to the fractal image compression (FIC). Its ecoing scheme is simplifie compare with earlier FIC schemes. Encoing is improve by a more complicate interpolation metho base on the iea of using a vertical contractivity factor function instea of a constant contractivity factor. The performance of FIC is significantly improve using our schemes. Now the compression ratio (CR) is not higher. To improve CR we are planning to a some kin of lossless compression an entropy coing compression. References [1] Barnsley, M.F. (1988). Fractals Everywhere. New York: Acaemic Press.

8 8 ( a ) ( b ) Figure 3 ( a ) The original image of Lena. (b): The ecoe image of Lena using our scheme at CT=32 secons, PSNR=47.6B, CR=11.7:1. ( c ) ( ) Figure 4 (c): The ecoe image of Lena at CT=4 secons, PSNR=43.8, CR=11.7:1. (): The ecoe image of Lena at CT=13 secons, PSNR=46.3 B, CR=11.7:1.

9 9 ( a ) ( b ) Figure 5 ( a ) : The original image of the Boat. (b): The ecoe image at CT=34 secons, PSNR=43.6B, CR=11.7:1 ( a ) ( b ) Figure 6 ( a ) : The original image of the pile up benches. (b):the ecoe image at CT=33 secons, PSNR=36.5B, CR=18.75:1.

10 10 ( a ) ( b ) Figure 7 ( a ): The original image of the tower of chairs. (b): The ecoe image at CT=11 secons, PSNR=36.3B, CR=11.7:1. [2] Barnsley, M.F. & A.D. Sloan (1987). 'Chaostic compression.' Computer Graphics Worl Nov: [3] Barnsley, M.F. & L.P. Hur (1993). Fractal Image Compression. AK Peters, Wellesley. [4] Barnsley, M.F. (1986). 'Fractal function an interpolation.' Constr. Approx. 2: [5] Barnsley, M.F., J.H. Elton & D.P. Harin (1989). 'Recurrent iterate function systems.' Constr. Approx. 5: [6] Bouboulis, P., L. Dalla & V. Drakopoulos (2006). 'Construction of recurrent bivariate fractal interpolation surfaces an computation of their box-counting imension.' J. Approx. Theory 141: [7] Bouboulis, P., P.L. Dalla & V. Drakopoulos (2006). 'Image compression using recurrent bivariate fractal interpolation surfaces.' Internat. J. Bifur. Chaos 16(7): [8] Bouboulis, P. & L. Dalla (2007). 'Fractal interpolation surfaces erive from fractal interpolation functions.' J. Math. Anal. Appl. 336: [9] Falconer, K. (1990). Fractal Geometry Mathematical Founations an Applications. John Wiley & Sons. [10] Fisher, Y. (1994). Fractal Image Compression-Theory an Application. New York: Springer-Verlag.

11 11 [11] Geronimo, J.S. & D. Harin (1993). 'Fractal interpolation surfaces an a relate 2D multiresolutional analysis.' J. Math. Anal. Appl. 176: [12] Jacquin, A.E. (1992). 'Image coing base on a fractal theory of iterate contractive image transformations.' IEEE Trans. Image processing 1: [13] Massopust, P.R. (1990). 'Fractal surfaces.' J. Math. Anal. Appl. 151: [14] Metzler, W. & C.H. Yun (2010). 'Construction of fractal interpolation surfaces on rectangular gris.' Internat. J. Bifur. Chaos, 12, [15] Zhao, N. (1996). 'Construction an application of fractal interpolation surfaces.' The Visual Computer 12: