Gaussian-Shaped Circularly-Symmetric 2D Filter Banks

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1 Gaussian-Shaped Circularly-Symmetric D Filter Bans ADU MATEI Faculty of Electronics and Telecommunications Technical University of Iasi Bldv. Carol I no.11, Iasi 756 OMAIA Abstract: - In this paper we propose a class of D circularly-symmetric filter bans based on a 1D II Gaussian prototype. We approach two cases, namely dyadic and uniform filter bans. The Gaussian prototype is obtained using an efficient rational approximation, namely the Chebyshev-Padé approximation. Then, using a frequency transformation, the corresponding D filters with circular symmetry are derived. These filters can be implemented both as digital filters, or in the analog version, using for instance the concept of cellular neural networs, which is detailed in the paper. Key-Words: - Gaussian filter bans, Chebyshev-Padé approximation, Cellular neural networs 1 Introduction Filter bans have been extensively used in various image processing tass []. Various different filtering approaches have been presented in the literature. For instance, an important application of filter bans is in texture discrimination [3], [6], i.e. feature extraction in order to mae possible their classification. A texture can be identified based on the energy distribution in the frequency domain [3]. Therefore, using appropriate filter bans, the frequency spectrum of the texture image is decomposed into a number of sub-bands [], [3], [7]. Analog implementation of such filter bans may be a noteworthy alternative in applications requiring fast processing, mainly using the computing power of systems such as cellular neural networs, as detailed below. Cellular eural etwor Filters The cellular neural networ (C is basically an array of identical analog processing elements called cells, which are identically connected [8]. When operating in the linear region, the C satisfies the following system of differential state equations [8], [9]: d x i, ( t x i, ( t + A l, x l, ( t + B pq, u pq, + I dt (1 l, pq, for all xi, ( t < 1; yi, ( t xi, ( t. One of the most important applications of Cs in image processing is linear filtering [8], [9]. Generally, in the -D case, the linear C filter is described by the spatial transfer function [9]: H ( ω1, B( ω1, A( ω1, ( where A ( ω 1, ω, B ( ω 1, ω are -D Discrete Space Fourier Transforms (DSFT of feedbac and control templates A, B. An important constraint is local connectivity, a ey feature of Cs. The design of -D high-order spatial filters requires large-size templates, which cannot be directly implemented. The templates currently implemented are 3 3 or 5 5 [8], [9]. Larger templates can only be implemented by decomposing them into elementary templates. Only separable templates can be written as a convolution of elementary templates. Correspondingly, the filtering function is a product of elementary functions. In II spatial filtering with H ( ω 1,, the final state will be: X( ω1, U( ω1, H1( ω1, H( ω1, H n ( ω1, (3 The image X1( ω1, U( ω1, H1( ω1, resulted in the first filtering is re-applied to C input, giving the second output image X( ω1, X1( ω1, H( ω1, and so on, until the whole filtering is achieved. We get: H( ω, ω H ( ω, ω B ( ω, ω A ( ω, ω (4 i i which implies a cascade interconnection of several II filters [13], [14]. The templates A and B can be expressed as discrete convolutions: B B B B A A A A (5 where B i, n A are 3 3 or 5 5 templates. In -D Cs, this successive filtering is possible only for separable templates. Thus, a given filtering operation can be achieved as a sequence of elementary filtering tass, directly implemented. 3 Circularly-Symmetric Filters In the following approach we propose an efficient design technique for -D circular-symmetric filters, based on 1- D filters, considered as prototypes. Given a 1-D prototype with transfer function H p (ω, the corresponding -D filter function H ω 1, ω results through the transformation ω ω 1 + ω : H ( ω + 1, ω H p ω1 ω (6 This can be interpreted as a rotation of the prototype function around its central axis, which generates the -D m ISS: ISB:

2 characteristic [13]. Here we will only discuss zero-phase C filters (FI and II, i.e. with real-valued transfer functions, which correspond to symmetric control and feedbac templates. It can be easily shown, using the fundamental theorem of algebra, that any symmetric 1-D template, for instance A ( 1, can be decomposed into elementary, symmetric templates of size 1 3 or 1 5, i.e. it can be written as a discrete convolution: A A1 A A (7 A currently used approximation of the -D cosine cos ω 1 + ω is given by the 3 3 template: (8 C such that we have the approximation: cos ω1 + ω C( ω1,.5 +.5(cosω1+ cos (9 +.5cosω cosω Let us consider the symmetric control template B p : [ b b b b b b ] B p (1 3 1 b3 and its frequency characteristic given by DSFT: p B ( ω b + b cosω (11 1 Using trigonometric identities for cos ω, 1..., we finally get a polynomial in powers of cos ω : p( c + c 1 B ω (cosω (1 Considering this 1-D characteristic as a prototype and using the frequency transformation given before, we get: 1, Bp ω1 + ω c + c C ( ω1, 1 B( ω ω (13 We used the notation: C ( ω + 1, cos ω1 ω. Therefore, in (13, the function B ( ω 1, is approximated by a polynomial of order, in powers of the function C ω 1, ω. Lie any real polynomial function, B ( ω 1, has real and complex-conugate roots and can be factorized into real polynomials of first and second order: n m 1, ω c ( C ri ( C α C + α + i 1 1 C C( ω 1, ω, r i are real roots ( i 1... n B( ω β (14 where and r α ± β are complex-conugate roots ( 1... m, with n + m ; c is the coefficient of the largest power term. Since the filtering function can be written as a product of elementary functions, the -D filter with circular symmetry, obtained from the 1-D prototype, is separable, which is a maor advantage in implementation. Consequently, the large-size template B corresponding to the filter B ω 1, ω (in our case a FI filter can be decomposed into elementary, small-size ( 3 3 or 5 5 templates, writing it as a discrete convolution: B c ( C1 Ci C n ( D1 D Dm (15 If we use the 3 3 template C given in (8 to approximate the function cos ω + 1 ω, each 3 3 template C i from (14 is obtained from C by subtracting the value r i from the central element. Each 5 5 template D from (14 results as: D 1 + ( α + C C α C β C (16 C is a 5 5 zero template, with central element one; C 1 is a 5 5 template obtained by bordering C ( 3 3 with zeros. Thus, starting from an arbitrary 1D prototype filter, a D filter with circular symmetry can be derived. 4 Filter Approximation Methods The Gaussian function in frequency has the form: G( ω exp( σ ω (17 A C Gaussian-shaped filter can be designed by determining the template parameters that approximate a Gaussian with a desired selectivity. The Gaussian function is real and symmetric, therefore so will be the templates. We loo for a rational approximation for the 1D Gaussian frequency response (17 in the form of a ratio of polynomials in cosω : ωσ M B( ω ( ω mcos( ω ncos( ω A( ω m n G e b m a n (18 where ω ( ππ, and a 1. The degrees of the numerator and denominator (M, may be not necessarily equal. 4.1 Chebyshev-Padé Approximation One of the most efficient rational approximation (best tradeoff between accuracy and approximation order is the Chebyshev-Padé rational approximation. It consists of a rational function which is determined from the Chebyshev series expansion of Gx ( over the interval [ a,1], similarly to the Padé approximation. Detailed references on this approximation can be found in [8], [9]. In the expression of G( ω, we will now mae the change of frequency variable of the form: x cos( ω ω arccos( x (19 and we obtain the Chebyshev-Padé approximation in x: σ (arccos x x ( a+ 1 Gx ( e ct, (1 a ( M x ( a+ 1 x ( a+ 1 qt m m, pt n n, (1 a (1 a m n This method gives better approximation accuracy than Padé for the same order. The main drawbac of the method is that the coefficients of the rational approximation can be only determined numerically using ISS: ISB:

3 a symbolic calculation software lie MAPLE. An explicit dependence of the coefficients on the standard deviation σ cannot be derived easily, especially for higher orders, as can be done in the Padé approximation. After finding the Chebyshev-Padé approximation in the form (, i.e. the coefficients q m ( m, M and p n ( n,, the explicit expressions for the Chebyshev polynomials may be substituted, finally obtaining a simple rational expression in variable x, similar to (18. We return to the original variable ω, then finally obtain a rational function in cosω, then a function of cos( nω. Examples:Forσ 1, the first-order Chebyshev-Padé approximation is: G ( cos 1( e ω ω ω (1 ( cos ω resulting in a pair of 1 3 templates: B [ ] ( A [ ] For σ, a second-order approximation is accurate enough: G cos cos ( e ω ω ω ω ( cos ω cosω The second-degree polynomials cannot be factorized, having complex roots. 4. Iterative Filter Approximation We loo first for a rational approximation of the Gaussian (17 in the range [ π, π], which represents the spatial transfer function H (ω of the corresponding 1-D filter: F( ω ( b + b1cosω ( a + a1cosω (4 This approximation is sufficiently accurate for σ 1, but inadequate for σ > 1 (the approximation error is too large. We can obtain it as a Chebyshev-Padé approximation, as before. At this point, the Gaussian (17 can be written as a power of a Gaussian with smaller dispersion: ( ω ( ω, where: G G G ( ω exp( σ ω exp( σ ω (5 and σ σ, with positive integer. The input image U is filtered times with the same "ernel" function G ( ω ; after each filtering step, the resulting output image is re-applied to the C input. We (a Fig.1: (a Kernel function G ( ω ( σ.9 ; (b Filter G( ω ( σ 1.8 obtained by iterating G ( ω for 4 times (b will call this operation an iterative filtering. Example: We want to implement a Gaussian with σ 1.8 using an iterative filtering; since we must have σ 1.8 1, we need a first-order filter iterated for 4 steps. We obtain: σ Using the Chebyshev-Padé approach as in section 4.1, we get: cosω H1,1 (6 1.98cosω The characteristics are plotted in Fig.1 5 Filter Bans We will next discuss two types of filter bans which are usually applied in multi-resolution image processing, namely dyadic and uniform. 5.1 Dyadic Filter Ban We propose a Gaussian filter ban with a dyadic structure, consisting in Gaussian-shaped filters with central frequencies at ω π, with integer The -th filter of the ban will have the frequency response of the form: H( ω G( ω ω + G( ω + ω (7 where G( ω ω and G( ω + ω are the Gaussians G ( ω shifted in frequency to the values ± ω. We will consider the first filter of the ban H 1 ( ω to have a standard deviation σ. The higher-order filters of the dyadic ban, i.e. H ( ω for will have the central frequencies ω π and the bandwidths decreasing in a geometrical progression, such that their quality factor is ept constant. The frequency response of a Gaussian filter ban comprising the first four filters H ( ω, with 1,,3, 4 is displayed in Fig.(a on the positive frequency range ω [, π ]. As we now, a Gaussian function with zero mean, of the form (17, can be considered almost zero in the frequency range ω > 3 σ, with an error less than 1%. The -3dB cutoff frequency of a Gaussian filter can be found imposing the condition: exp( σω c, such that we obtain the expression of the -3dB bandwidth: BW, ln ωc σ σ (8 If the filters of the ban are centered around the frequencies ω π, their frequency responses will have the same quality factor, given by the relation: ω.833 Q.65 (9 BW π For 1 we obtain the first filter: G1 ( ω exp( ω ( ω π ( ω+ π H1( ω G1ω + G1ω+ e + e (3 Using the Chebyshev-Padé approximation for G 1 ( ω given in (3, we easily obtain after some calculations: cos( ω cos(4 ω H1( ω ( cos( ω cos(4 ω ISS: ISB:

4 For the second filter with σ 4, we can find directly a Chebyshev-Padé approximation for the frequency response: where A 1 is given below as an example for template parameter design: A ( Unfortunately A 1 is not separable, i.e. it cannot be written as an outer product of two vectors, but rather as a sum of 3 outer products. (a (b (a (c Fig.: (a Ideal filters H1, H, H3, H 4 of the dyadic filter ban; (b, (c Filters H 1 ( ω and H ( ω of the filter ban ω ω+ 4 4 H( ω Gω + G1ω+ e + e 4 4 ( cosω.487cos( ω cos(3 ω.68481cos(4 ω H ( ω cos ω cos( ω.3478cos(3 ω+.6497cos(4 ω (33 A convenient factorized form is:.154 (cos ω+.959(cos ω (cos ω+.837(cosω 1.8 H ( ω (34 ( cosω+.5cos( ω ( cosω +.5cos( ω Using the method described in section 3, from the factorized 1D prototype transfer function H ( ω given in (34 it is straightforward to derive the templates for the corresponding circular filter. For instance, the feedbac template A results of size 9 9 and can be written as the discrete convolution of templates: A(9 9 A(5 5 A (5 5 (35 5. Uniform Filter Ban Unlie the filter bans with dyadic structure discussed before, in which the bandwidth decreases with a given ratio (in particular two as their center frequencies decrease towards lower values, in uniform filter bans the frequency responses are uniformly spread along the frequency range, as can be seen in Fig.4(a, for a filter ban with center frequencies ω π 8,,...,7. In this case all filters will have the same bandwidth. Design example: For instance we want to design the filter H 3,8 ( ω exp( 3( ω 5 π 8 + exp( 3( ω+ 5 π 8 (37 which has quite a high selectivity. Using the iterative method, we can realize this band-pass filter by first designing the filter: H 3,4 ( ω exp( 8( ω 5 π 8 + exp( 8( ω+ 5 π 8 (38 and then iterating this filtering for four times: 4 H ( ω H ( ω (39 3,8 3,4 Using the Chebyshev-Padé approximation we find the following rational realization for H 3,4 ( ω : (b Fig.3: (a, (b The first two filters of the circular Gaussian filter ban ISS: ISB:

5 cosω.545cos( ω cos(3 ω H3,4( ω cos ω cos( ω cos(3 ω+.1719cos(4 ω with the factorized form:.15(cos ω+.9764(cosω.655 H 3,4 (cosω.951 ( ω ( cosω +.5cos( ω (4 (41 ( cosω +.5cos( ω Using the same procedure we can design all the filters of the form: H,8 ( ω exp( 3( ω π 8 + exp( 3( ω + π 8 (4 such that we obtain an uniform filter ban, displayed in H ( ω exp 3 ω. Fig.4. It includes the low-pass filter,8 ( (a (b (c Fig.4: (a Uniform Gaussian filter ban with 8 filters; (b Ideal and approximated frequency response of the filter H 5,8 ( ω ; (c Frequency response and contour plot of the circular filter 6 Conclusion We proposed an efficient design method for both dyadic and uniform two-dimensional circularly-symmetric filter bans, useful in various image processing tass. The D filters are designed starting from 1D Gaussian prototypes with a desired selectivity. Using a rational approximation lie Chebyshev-Padé and a frequency transformation, the factorized prototype transfer function gives a direct and systematic realization of the D circular filter, by calculating the template parameters. We discussed here mainly filter design methods used in C, but they may be also applied to D digital filters which are commonly used in image filtering. eferences: [1] D.E.Dudgeon,.M.Mersereau, Multidimensional Digital Signal Processing, Englewood Cliffs, J: PrenticeHall, 1984 [].C.Gonzalez,.E.Woods, Digital Image Processing, Second Edition, Prentice Hall, [3] J.M.Coggins, A.K.Jain, A Spatial Filtering Approach to Texture Analysis, Pattern ecognition Letters, vol.3, no.3, pp.195-3, 1985 [4]M.Unser, Texture Classification and Segmentation Using Wavelet Frames, IEEE Trans. Image Processing, vol.4, pp. 1,549-1,56, ov.1995 [5] T.anden, J.H.Husoy, Multichannel Filtering for Image Texture Segmentation, Optical Eng., vol.33, pp.,617-,65, Aug.1994 [6]T.anden, J.H.Husoy, Filtering for Texture Classification: A Comparative Study, IEEE Trans. On Pattern Analysis and Machine Intelligence, vol.1, no.4, April 1999 [7] S. K. Mitra. Digital Signal Processing: A Computer- Based Approach, McGraw-Hill, [8] L. O. Chua, L. Yang, "Cellular eural etwors: Theory", IEEE Transactions CAS-I, Vol.35, 1988 [9] K.. Crounse, L. O. Chua, "Methods for Image Processing and Pattern Formation in Cellular eural etwors: A Tutorial", IEEE Trans. CAS-I, vol.4, no.1, pp [1] M. Abramowitz and I. A. Stegun, Handboo of Mathematical Functions, ew Yor: Dover, [11] H. Press, B. P. Flannery, S. A. Teuolsy, W. T. Vetterling, umerical recipes in C. The art of scientific computing., Second edition, Cambridge University Press, pp , 199. [1].E.Shi, Gabor-type image filtering in space and time with cellular neural networs, IEEE Trans. on Circuits and Systems, Vol.45, o., Febr [13] Matei, L.Goraş, Design Methods for C Spatial Filters with Circular Symmetry, Proc. of 7 th Seminar on eural etwor Appl. in Electrical Engineering, EUEL 4, Belgrade, 3-5 Sept. 4 [14] Matei, Design Method for Orientation-Selective C Filters, Proceedings of the IEEE International Symposium on Circuits and Systems ISCAS 4, May 3-6, 4, Vancouver, Canada ISS: ISB: