Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding

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1 Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding Kohji Kamejima Faculty of Engineering, Osaka Institute of Technology Omiya, Asahi, Osaka JAPAN Tel: ; Fax: kamejima@dim.oit.ac.jp Abstract: Newton potential is reformulated in terms of the Hausdorff distance to design reduced affine mappings associated fractal attractors. By applying maximum entropy analysis to observed patterns, stochastic features are extracted as well as boundary points where the fixed points of the mappings should be located. To linear segments of potential fixed points, feature points are nondeterministically attracted following the Hausdorff potential. Guided by this feature clusters, random patterns are partitioned to estimate mapping parameter. Keywords: Hausdorff Potentials; Nearest-Neigbour Aggregation; Pattern Clustering; Self- Similarity Table of Contents 1 Introductory Remarks 2 2 Contraction Structure 2 3 Hausdorff Potentials 3 4Clustering Scheme 4 5 Statistical Mapping Design 5 6 Experiments 6 7 Discussions 7 8 Concluding Remarks 7 1

2 Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding 2 1 Introductory Remarks Image features are organized by applying universal morphological rules. This implies that we have compact and robust description of geometric complexity of observed imagery in terms of pattern dynamics. Pattern dynamics induces interaction among pixels unstructurally distributed in image plane. The intensity of induced interaction attract pixels mutually to yield global structure to be organized. This implies image structure is identified with invariant set associated with introduced pattern dynamics. Through parallel relaxation of nonlinear constraints, the dynamical systems specifies complex patterns generatively. Due to such generativity, however, the identification of pattern dynamics easily falls into computational difficulty. Topologically, arbitrary contraction mapping induces 2D mechanism attracting constituents of patterns to specific fixed point: a simple pattern. The superimposition of 2D contraction mappings expands associated set of fixed points to a complex pattern of unexpectable form, called fractal attractor. The fractal attractor can be identified with self-similar aggregation of fixed points of finite composite of the contraction mappings [2]. Thus, self-similar patterns are considered as complete sets for identifying grammars of fractal attractors [5]. In such fractal attractors, furthermore, it has been pointed out that self-similarity rules can be visualized as invariant measure [1] and invariant feature [4] as well. Noting the invariant feature is a finite representation of the invariant measure of infinite dimension, in this paper, a method is presented for statistical estimation of contraction mappings. The mapping set is required to partitioning observed complex patterns into self-similar subsets: images of unknown mappings. Basic idea of the partitioning is to classify feature points in terms of not-yet-identified contraction mappings. For this purpose, the Newton potential is reformulated in terms of the Hausdorff distance. Within the framework of such nondeterministic kinetics, attractive forces are evaluated between feature points and estimates of fixed points of unknown mappings. The self-similar organization of features introduces a version of range estimates of unknown mappings. Based on statistical moments of the range estimates, reduced affine mappings are identified to transform the totality of invariant feature into structurally consistent feature clusters. 2 Contraction Structure Let Ω R 2 be a fixed image plane and consider random patterns generated within F[Ω]: the totality of subsets of Ω. The disparity between patterns in F[Ω] is assumed to be indexed in terms of the Hausdorff distance η[, ]. Suppose that Ξ is the fractal attractor generated by a set of contraction mappings ν = {µ i }. Assume that the attractor Ξ approximates observed pattern Λ F[Ω] in the following sense: η[λ, Ξ] 0. Then we have a fractal code ν for random pattern Λ. Noticing the following collage evaluation [1] [ η Λ, ] µ µ i ν i(λ) η[λ, Ξ] 1 max s, (1) µ i µ i ν where 0 <s µi < 1 denotes the contractivity factor of µ i ν, the fractal coding results in the design of mapping set satisfying Λ ν µ i(λ). Geometric and stochastic analysis of fractal patterns reveals two clues for mapping design. First, by the contractivity, any point ξ Ξ must be attracted to the fixed point of µ i ν, designated by ω f i. This implies that ξ Ξ should be mapped to µ i (ξ) Ξ satisfying µ(ξ) ω f i ξ ωf i. (2) Hence, ω f i should be approximately located on the boundary given as a level set Λ of the capturing probability ϕ(ω ν) [6]. Second, the introduction of the capturing probability yield the following recursive mapping representation [4]: { Θ = θ Θ } µ i ν : µ 1 i (θ) Θ, (3)

3 Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding 3 where Θ denotes the feature pattern given by Θ = { θ Λ ϕ =0, det [ T ϕ ] > 0, ϕ <0}. By definition, feature points Θ must be located interior of unknown attractor Ξ in a stochastic sense. Furthermore, by constraints (3), it follows that Θ i = µ i ( Θ) Θ, (4) for any µ i ν. In this paper, a dynamic clustering process is considered for aggregating Θ i satisfying deterministic constraint (2) as well as the following nondeterministic fixed point positioning 3Hausdorff Potentials ω f i ˆ (Λ ν). (5) Suppose that Θ i. Then, we can locate the fixed point ω f i within the following subset { Λ i = ω ˆ (Λ ν) } ω θ i ω θ j, (6) for some θ i Θ i and for any θ j Θ Θ i. In particular, for the maximal feature set satisfying {θ Θ Θ i µi (θ) Θ i } =, (7) the linear segment Λ i specifies admissible region for locating ωf i. Hence, we can identify the structure of imaging scheme without any assumption of explicit mapping forms. Consider dynamic generation of {Θ i }. Suppose that properly selected feature set Θ i µ i ( Θ) is expanded by the increment δθ i. It is natural to select the increment δθ i consisting of the nearest features to the fixed point ω f i associated with unknown µ i. The nearest increment δθ i is consistently merged into the cluster Θ i if δθ i is selectively attracted to not-yet-identified fixed point ω f i. Such 2D interaction can be directly evaluated in terms of the Newton potential induced on image plane [7]. Noting that the disparity between patterns in F[Ω] is exactly evaluated in terms of the Hausdorff distance, the consistency of the increment δθ i within a kinetic framework can be nondeterministically indexed as follows: φ(δθ i, Θ i ) = log η[δθ i,ω f i ]. (8) The introduction of this Hausdorff potential implies that the attractive force between δθ i and ω f i is evaluated by ψ(δθ i,ω f i ) = 1 η[δθ i,ω f (9) i ]. For not-yet-maximized cluster Θ i, suppose that structurally consistent estimates of the fixed point ω f i are generated as a subset of admissible boundary: ˆΩf i Λ i. Let θ { be in the margin of increment consisting of the Lagrange points with respect to attractive forces ψ(δθ i, ˆω f i ) ˆω f i ˆΩ } f i. Then we have the following quasi-static balance within observed pattern Λ: (ˆω f j θ) 0. (10) ˆω f j,θ 2 µ j ν ˆω f j ˆΩ f j

4 Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding 4 By the uncertainty ˆΩ f i, the attractive force to true fixed point is not ensured to directed to selected point ˆω f i ˆΩ f i. Noticing this, let effective attractive force within the distribution of fixed point estimates be represented by ψ(θ, ω f i ν) = 1 ˆω f i,θ φ ν, (11) where ˆω f i,θ φ ν is the kinetic distance within the framework of the Hausdorff potentials given by ˆω f i,θ φ ν = ψ(θ, ˆω f i µ j) ˆω f i,θ. µ j ν µ i Consider kinetic estimation of possible expansion of fixed points ˆΩ f i. By the maximality of pattern expansion at the fixed points (2), any point θ of consistent increment δθ i must be repelled by any possible fixed point, i.e., ˆΩ f i [θ] = { ω Λ i ω θ φ ν max }. By aggregating this a posteriori estimate, we have ˆΩ f i = ˆΩf i [θ]. (12) θ δθ i For not-yet-maximized clusters, on the other hand, the fixed point estimates are expected to support possible increments to be consistently merged in the maximal cluster. This a priori { consistency } can also be nondeterministically indexed based on the Hausdorff potential generated by Θ i, ˆΩ f i structure as follows: Ω f i = Ωf i [θ], θ δθ i (13) Ω f i [θ] = { ω Λ i ω θ φˆν max }. where the search area is expanded to kinetically possible segment given by: Λ i = {ω ˆ (Λ ν) } ψ(ω, ˆω f i ν) > 0. 4 Clustering Scheme By introducing the Hausdorff potentials, we can select feature sets with geometrically consistent increments. Noticing the minimal complexity generation in self-similarity processes [3], the risk of inconsistent growth should be indexed in terms of the variation of δθ i. The combination of these geometric and structural evaluations we have the following monotone clustering process: } Step-0: Set initial clusters as Θ i = { θi, θ i Θ. Step-1: If there exist clusters with nonempty a posteriori fixed points ˆΩ f i, then go to Step-2. If not, halt the procedure. Step-2: For each Θ i, select the increment δθ i in Θ Θ i so as to minimize the Hausdorff disparity from a posteriori fixed points.

5 Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding 5 Step-3: Select a priori fixed points Ω f i in Λ i and evaluate the fixed point disparity from Λ i. Step-4: Select clusters with the minimum fixed point disparity as consistent class for expansion. Step-5: Select expandable clusters with non-empty increment of the minimal variance. Step-6: If there exist expandable clusters {Θ i }, then update Θ i by Θ i δθ i and return to Step-1. The clustering process is intrinsically designed on a set of universal rule for stochastic and computational analysis of geometrically complex patterns: the maximum entropy capturing of unknown attractor points, contractivity reserving feature-boundary association and nearest increments selection. The compression of unknown attractor Ξ to stochastic features Θ results in the implementation of nondeterministic algorithm on tolerable size of symbol space. Induced kinetic { metaphor yields a monotone process for nondeterministic refining of discrete self-similarity structure Θ i, ˆΩ } f i. 5 Statistical Mapping Design The partitioning of feature pattern {Θ i } induces a geometrically consistent clustering of random pattern Λ as follows: Λ = {Λ i }, (14a) { Λ i = ω Λ } η[ω, Θi ] η[ω, Θ j ],, (14b) for any Θ j {Θ i }. By this pattern clustering, we have the following first and second moments: Λ = Σ Λ = 1 ω, Λ ω Λ (15a) 1 [ω Λ 1 Λ] T, (15b) where Λ i and Σ Λ i denote first and second moments of feature distributions in the cluster Λ i : Λ i = Σ Λ i = 1 Λ i ω Λ ω Λ i ω, 1 Λ i 1 In Eq. (15), means the size of the set ( ). contraction mappings satisfying ω Λ i [ω Λ i ][ω Λ i ] T. (15c) (15d) By these statistical moments, we can design a set of Ξ = ν µ i (Ξ), Ξ Λ. For instance, let the mappings be sought with the reduced affine mappings of the form: ˆµ i (ω) = Âiω + ˆb i, Âi < 1. For such mappings, the parameter (Âi, ˆb i ) is estimated as follows: [ ][ q  i = [e i 1 e i i 2 ] 1 /q1 Λ 0 e ΛT 1 0 q i 2 /q2 Λ e Λ T 2 ], (16a)

6 Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding 6 Figure 1: Simulation Results: On a fractal pettern sampled via Monte Carlo method, stochastic features are extracted for partitioning complex pattern into finite clusters to estimate mapping parameter. ˆbi = Λ i Âi Λ, (16b) where e i k (eλ k ) and qi k (qλ k ) are eigenvectors and associated eigenvalues of covariance matrix Σi (Σ Λ ), respectively. 6 Experiments The clustering scheme was verified via simulation studies. In these simulations, self-similar mappings are randomly selected to generate associated fractal attractor. For sampling selected attractor, a sequence of random points ω t,0 t 1000 is generated in a image plane Ω with resolution via Monte Carlo method. On digitized attractors Λ, the capturing probability ϕ was generated to indicate stochastic features Θ and boundary points in array. By applying steps-1 to -6 to this feature array, feature points are successively aggregated into discrete clusters. Examples of a clustering process are shown in Fig. 1. In this case, three affine mappings ν = {µ i,i=1, 2, 3} with the following (reduced rotation shift)-parameter set were selected as unknown generator of observed pattern Λ. ((REDUCTION ( 84% 84%))(ROTATION -3deg) (SHIFT ( 4% 17%))) ((REDUCTION (-33% 33%))(ROTATION 65deg) (SHIFT ( 30% 61%))) ((REDUCTION ( 33% 33%))(ROTATION 65deg) (SHIFT (-30% -52%))) In Fig. 1, simulated fractal attractor was sampled (Observables view) to estimate the capturing probability ϕ on which stochastic features Θ were extracted as shown in Features view. By invoking the Hausdorff potential, unstructured points in the set Θ were aggregated into feature cluster {Θ i }. For this sample attractor, Θ was partitioned into four disjoint point sets. In Features view, the center and the expansion of each feature cluster Θ i is indicated by the location and the size of cross mark as well as the location and the size of Θ.

7 Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding 7 Guided by the finite information {Θ i,i=1, 2, 3, 4}, observed pattern Λ was partitioned into four regions {Λ i,i=1, 2, 3, 4} as estimates of the ranges of observed pattern Λ through unknown mappings ν = {µ i,i=1, 2, 3}. By combining Eq. (16) with Eq. (15), iconic information {Λ i,i=1, 2, 3, 4} is coded into 4 reduced affine mappings ˆν = {ˆµ i,i=1, 2, 3, 4} that regenerate fractal attractor as indicated in Restoration View. In this case, ˆν is described in terms of the following (reduced rotation shift)-parameter list: #:CLUSTER_1807 ((REDUCTION (63% 59%))(ROTATION 21deg) (SHIFT ( 2% 32%))) #:CLUSTER_1820 ((REDUCTION (59% 58%))(ROTATION 3deg) (SHIFT ( 22% -40%))) #:CLUSTER_1817 ((REDUCTION (41% 41%))(ROTATION 33deg) (SHIFT (-39% -27%))) #:CLUSTER_1819 ((REDUCTION (36% 37%))(ROTATION 50deg) (SHIFT (-29% -61%))) 7 Discussions As shown in these results, proposed method provides finite description of complex patterns. The description is finitely coded in terms of (reduced rotation shift)-parameter so as to transfer complex patterns easily. The code is robust in the sense that mapping length is essentially dependent only on structural information, stochastic feature. Based on statistical adaptation, mapping descriptions are parametrically refined to regenerate observed complex pattern. Via random sampling in imaging process and nondeterministic scheme in clustering process, original and estimated code are completely separated. Despite this intrinsic unpredictability, extracted finite code can easily be verified through efficient Monte Carlo simulation. The combination of finite transferability and intrinsic unpredictability is expected to play a crucial role in efficient communication based on complex imagery. 8 Concluding Remarks A nondeterministic kinetics was introduced in feature space for self-similar clustering of random patterns. Within the field of induced force, feature points are kinetically aggregated towards fixed points of contraction mappings to be identified. Proposed method was verified through simulation studies. Simulation results demonstrate that the nondeterministic kinetics induces autonomous feature clustering mechanism to design consistent mapping set. References [1] M. F. Barnsley. Fractals Everywhere, Academic Press, [2] J. E. Hutchinson. Fractals and self similarity. Indiana University Mathematical Journal, 30: , [3] K. Kamejima. Complexity evaluation of imaging processes with applications to design and detection of selfsimilar patterns. In Proc. CISST 97, pages , [4] K. Kamejima. Multi-scale image analysis for stochastic detection of self-similarity in complex texture. In Proc. IEEE-SMC 97, pages , 1997.

8 Nondeterministic Kinetics Based Feature Clustering for Fractal Pattern Coding 8 [5] K. Kamejima. Propositional-grammatical observability of self-similar imaging processes for feature based pattern verification. In Proceedings of the 5th International Workshop on Parallel Image Analysis (IWPIA 97), pages , [6] K. Kamejima. Maximum entropy contouring and clustering for fractal attractors with application to selfsimilarity coding of complex texture. In Proc. EUSIPCO98, pages , [7] K. Kamejima, Y. C. Ogawa, and Y. Nakano. A fast algorithm for approximating 2D diffusion equation with application to pattern detection in random image fields. In Distributed Parameter Systems: Modeling and Simulation. North Holland, 1989.