SELF-AFFINE SNAKE: A NEW PARAMETRIC ACTIVE CONTOUR M. Saadatmand-Tarzjan H. Ghassemian saadatmand@ modares.ac.ir ghassemi@modares.ac.ir Department of Electrical and Computer Engineering, Tarbiat Modares University, P.O. Box 14115-194 Tehran, Iran ABSTRACT In this paper, a new active contour model called self-affine snake is proposed which integrates the self-affine mapping system (SAMS), wavelet transform, and snake model. It inherits wide capture range from the wavelet transform, both accurate fit to weak edges and effective reconstruction of boundary concavities from SAMS, and topological consistency from the snake model while avoiding their weak points. In self-affine snake, first, a force vector is computed for every pixel in each wavelet LL matrix using SAMS with disk domains. Then the obtained force fields of different wavelet scales are effectively combined to make the self-affine force filed. Finally, the snake is deformed using the resultant forces based on the snake dynamic formulation. Experiment results demonstrate good performance for self-affine snake compared to the balloon for a number of synthetic and biomedical benchmark images. Index Terms— Image shape analysis, wavelet transforms 1. INTRODUCTION The parametric active contours or snakes are parametric curves that can move toward desired features, usually edges, within an image domain under the influence of internal forces coming from within the curve itself and external forces derived from the image data [1]. The external forces usually draw the curve toward desired features while the internal forces hold the curve together (elasticity forces) and keep it from bending too much (bending forces). The external forces are divided into two categories: potential forces defined as the negative gradient of a potential function and dynamic forces formulated directly using a force balance formulation [2]. Snake models suffer three key difficulties: i) they should be usually initialized by an initial contour close to the object boundary [3], ii) they have difficulties to reconstruct the edge openings and progress into boundary concavities [4], and iii) they may likely converge to wrong results for weak edges, especially, when they lie beside strong edges [5]. However, most of methods introduced to address the above problems solve one problem while creating new difficulties. For example, multiresolution approaches increase the capture range of the external forces while deciding about how the contour should move across different resolutions is a challenging difficulty in such algorithms [4]. The balloon model addresses both the capturing range and boundary concavity problems using pressure forces, but they can not be too strong or weak edges will be overwhelmed [5]. Furthermore, the pressure forces are not bidirectional, a condition that mandates careful initialization. Another example is gradient vector flow and its variations [6-9] which effectively tackle the capturing range and boundary concavity problems though they may poorly perform for weak edges [10]. Additionally, they are computationally intensive due to computation of forces for almost all image pixels [11]. Considering the above difficulties, some researchers proposed several contour extraction algorithms that inherently differ from snakes. Ida and Yoko have been introduced a highly accurate method to approach and fit a roughly drawn line to the object contour using self-affine mapping systems [12]. The contractive self-affine mapping system has been typically used to produce fractal figures [13]. In an earlier work [14], Ida et al showed that edges attract mapping points during iterations of the map when they are initially set near them. They have utilized this attraction phenomenon to extract self-similar curves instead of a smooth curve like that in snakes. Ida’s approach, in spite of significant strengths including accurate fit to boundaries and wide capture range, has some weak points. It sometimes abnormally deforms the contour due to fractal behaviors [12]. The authors addressed this drawback by defining the contour as a topologically-consistent parametric curve in an earlier work [15]. This work is an attempt to tackle the difficulties of both snake and Ida’s algorithm by keeping their strengths while avoiding their weak points. In this paper, we propose a new parametric active contour model called self-affine snake which integrates the self affine mapping system (SAMS), wavelet transform, and snake model. Self-affine snake inherits wide capture range from the wavelet transform, both accurate fit to weak edges and effective reconstruction of boundary concavities from SAMS, and topological consistency from snakes. This paper is organized as follows: in sections 2 and 3, the self-affine mapping system and snake model are reviewed, respectively. Section 4 presents the proposed self-affine snake. Experiment results are given in section 5. Finally, section 6 is devoted to concluding remarks. 2. SELF-AFFINE MAPPING SYSTEM 2.1 Contractive self-affine maps Consider an image having the support G⊂R 2 with intensity g(x) for all x∈G. The contractive map m i with domain M i ⊂G (i=1,…,I) is defined as follows: 1 ), ~ ( ) ~ ( ) ( < + + - = i i i i i i r r m x τ x x x (1) where i x ~ is the center point of M i . The above equation translates M i by vector τ i =[s i ,t i ] and expands it by r i to form domain W i =m i (M i ). A contractive self-affine model is defined as {M i ,m i ,u i } where: 1 0 ), ( , ) ( ≤ ≤ = + = i i i i i p m q p u x z z z (2)