New method for feature extraction based on fractal behavior - IDRBT

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1074 Y.Y. Tang et al. / Pattern Recognition 35 (2002) 1071–1081 The image pixels (m; n) with distance less than one from (i; j) are taken to be the four immediate neighbors of (i; j). Similar expressions exist when the eight-neighborhood is desired. A point f(x; y) will be included in the blanket <strong>for</strong> when b (x; y) ¡f(x; y) ¡u (x; y). The blanket denition uses the fact that the blanket of the surface <strong>for</strong> radius includes all the points of the blanket <strong>for</strong> radius −1, together with all the points within radius 1 from the surfaces of that blanket. Eq. (7) ensures that the new upper surface u is higher by at least 1 from u −1 , andalso at a distance at least 1 from u −1 in the horizontal andvertical directions. The volume Vol of the blanket is computedfrom u and b : Vol = ∑ (u (i; j) − b (i; j)): (9) i;j As the volume Vol of the blanket is measuredwith radius , the area of a fractal surface can be deduced as A = Vol 2 or A = Vol − Vol −1 : (10) 2 • Let A() be the area of the surface of the blanket. Basedon the Minkowski dimension, we have A() ≈ 2−D ; if is suciently small; (11) where denotes a constant and D stands <strong>for</strong> the fractal dimension of the image. Since the dimension can be regarded as a slope on a log–log scale, only two points are needed to get the dimension. We use two values of to compute the fractal dimension, namely, we take = 1 and 2 , then A 1 ≈ 2−D 1 and A 2 ≈ 2−D 2 . Thus, we have A 1 =A 2 ≈ 2−D 1 = 2−D 2 . Taking the logarithm of both sides yields 2 − D ≈ log 2 A 1 − log 2 A 2 log 2 1 − log 2 2 ; D ≈ 2 − log 2 A 1 − log 2 A 2 : (12) log 2 1 − log 2 2 It is usedto approximate the fractal dimension of the signature images, i.e., D ≈ 2 − log 2 A() if is suciently small: (13) log 2 We call D as modied fractal signature of set F. It will be used to distinguish the dierent handwritten signatures. Several points are worth noting: 1. Why do the dierent document images have dierent fractal dimensions? The essential distinction of handwritten signature images is their values of A(). 2. The value of A() depends on the volume Vol 3 (F ) of thickenedblanket F only. 3. In summary, they can be representedas D ⇔ A() ⇔ Vol 3 (F ): Consequently, in this paper, the volume Vol 3 (F )ofthe thickenedblanket F is applied to identify dierent handwritten signature, insteadof using the fractal dimension. We call such technique of approximating the fractal dimension “Modied Fractal Signature”. 3. Applications From the specic point of view of shape characterization of patterns the main aim of the fractal approach is to nd a “measure” to distinguish between curves with different, often very complicated, contours. The main idea is to describe the complexity of the curve through a new parameter, the fractal dimension, so as to ll in the gap between one- andtwo-dimensional objects (<strong>for</strong> objects on a plane). Two applications of the fractal theory are presentedin this section, namely: (1) the fractal technique is usedto extract the <strong>feature</strong>s <strong>for</strong> two-dimensional objects; (2) the fractal signature is employedto identify dierent handwritten signatures. 3.1. Application of fractal technique <strong>for</strong> <strong>feature</strong> <strong>extraction</strong> Feature <strong>extraction</strong> is a crucial step in pattern recognition. It is responsible <strong>for</strong> measuring the <strong>feature</strong>s of the objects in an image. Pattern recognition requires the <strong>extraction</strong> of <strong>feature</strong>s from the regions of an image, andthe processing of these <strong>feature</strong>s with a pattern classication technique. We employeda central projection <strong>method</strong> to reduce the problem of two-dimensional patterns into that of one-dimensional ones, and thereafter, utilize the well-establishedone-dimensional wavelet trans<strong>for</strong>m coupledwith fractal theory to extract the one-dimensional pattern’s <strong>feature</strong> vectors <strong>for</strong> the purpose of pattern recognition. The key steps of the experimental procedure consist of the following: Step 1: Central projection of two-dimensional patterns: We denote each of the two-dimensional patterns in question by p(x; y). Thus, the central projection of p(x; y) can be expressedas follows: f(x k )= M∑ p(x k cos i ;x k sin i ): (14) i=0 Step 2: Wavelet trans<strong>for</strong>mation of the one-dimensional patterns: Let f(x k )=c j;k , where k =0; 1;:::;2N −1 and V 0 = {c j;0 ;c j;1 ;:::;c j;2N−1 }:
Y.Y. Tang et al. / Pattern Recognition 35 (2002) 1071–1081 1075 Fig. 2. Diagram of <strong>feature</strong> <strong>extraction</strong> by wavelet sub-patterns and divider dimensions <strong>for</strong> the Chinese character “Heart”. Thus, the expressions <strong>for</strong> the wavelet trans<strong>for</strong>mation of V 0 can be written as follows: 5∑ 5∑ c j+1;m = h k c j;k+2m ; d j+1;m = g k c j;k+2m (15) k=0 k=0 where m =0; 1;:::;N − 1. Hence, the wavelet trans<strong>for</strong>mation sub-patterns of V 0 obtainedusing Eq. (16) will become V 1 = {c j+1;0 ;c j+1;1 ;:::;c j+1;N−1 }; W 1 = {d j+1;0 ;d j+1;1 ;:::;d j+1;N−1 }: (16) Step 3. Computation of divider dimensions <strong>for</strong> wavelet sub-patterns: Since the divider dimensions of non-self-intersecting curves are asymptotic values, we can derive their approximations <strong>based</strong> on the following expression: log M (C) −log ; when is set to be small enough. M (C) is the compass dimension computed with a ruler of size along the curve C. In our experiments, we per<strong>for</strong>medthree consecutive wavelet trans<strong>for</strong>mations <strong>for</strong> each one-dimensional pattern. Hence, the one-dimensional pattern, such as the central projection of the Chinese character “Heart” (Fig. 2), will yieldthree non-self-intersecting curves; namely: V 0 ;V 1 ;W 1 . By comparing these two frequency spectra, we can ndthat V 0 contains all frequency components of f(x), while V 1 contains only one half of the frequency components of f(x). The reason is that V 0 is divided into two parts: V 0 = V 1 ⊕ W 1 . As <strong>for</strong> the rst part, V 1 , only the low-frequency components of V 0 are kept, andthe high-frequency components are lost. In addition, only the high-frequency components of V 0 are kept, andthe low-frequency components are lost in W 1 . For each of the three curves, we further computedits divider dimension, and there<strong>for</strong>e, related each pattern with a <strong>feature</strong> vector. These <strong>feature</strong> vectors are presented as follows: D vo =1:03911917; D V1 =1:03940148; D W1 =1:60938780: These <strong>feature</strong> vectors are selectedto be usedas the fractal <strong>feature</strong> <strong>for</strong> extracting the <strong>feature</strong>s of object. More details about the central projection trans<strong>for</strong>mation and the orthonormal wavelet decomposition can be found in Ref. [14]. In our experiments, we have testedour new approach <strong>for</strong> two types of images: 52 upper andlower case printed alphabets and3000 Chinese characters. We classify these <strong>feature</strong>s andper<strong>for</strong>m the recognition of dierent types of patterns by using the least distance classication <strong>method</strong>. For each type of the patterns, the center
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