Institute of Telecommunications. We study applications of Gabor functions that are considered in computer vision systems, such as contour

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1 Applications of Gabor Filters in Image Processing Sstems Rszard S. CHORA S Institute of Telecommunications Universit of Technolog and Agriculture Bdgoszcz, ul.s. Kaliskiego 7 Abstract: - Gabor's theor implies that the information in image can represent in terms of the amplitudes of functions that are localized in both space and frequenc. We stud applications of Gabor functions that are considered in computer vision sstems, such as contour detection, stereo matching process and object recognition. It is known that structural information even in gre scale images resides in contours. The problem of extracting contours from images is approached with the use of Gabor lters. These lters have optimal properties relating to simultaneous localisation in space, spatial-frequenc and orientation. Matching algorithms using image properties maximize a similarit measure between two images. This similarit ma appl to features images derived from images. The main problem of stereoscopic vision is to nd corresponding parts of the left image and the right image. Gabor lters have been used to resolve the correspondence problem. The problem of recognition of free-form objects in real-world scenes is still a dicult. The magnitude of Gabor features can be used as objects features. The output of this lter is a vector that describes the structure around the image location. This vector are classied into N tpes form a set of regions. Each region is characterised b its index which describes image in terms of the Gabor lter response. Ke-Words: - Gabor functions, Gabor lters, stereo matching process, feature extraction. 1 Introduction In 1928 Hartle [5] concluded that "the total amount of information which ma be transmitted...is proportional to the product of frequencange which is transmitted and the time which is available for the transmission". In 1946 Gabor presented an approach tocharac- terize a time function in time and frequenc. Gabor showed how to represent time-varing signals in terms of functions that are localized in both time and frequenc. Information contained in a visual scene is pre-processed in the retina and represented in the brain in highl compressed form for further processing. Gabor's theor implies that the information in image can represent in terms of the amplitudes of functions that are localized in both space and frequenc. In a joint space/spatial-frequenc representation for images, frequenc is viewed as a local frequenc that can var with position throughout the image. Gabor functions are Gaussians modulated b complex sinusoids. In its general form, the twodimensional Gabor function and its Fourier transform can be written as [1]; [2] g(x; ; u0;v0) = exp(,[ x x 2 2 ]+2i[u0x+v0])(1) G(u; v) = exp(,2 2 ( 2 x(u, u0) (v, v0) 2 ))(2)

2 2 where x and dene the widths of the gaussian in the spatial domain and (u0;v0) is the frequenc of the complex sinusoid. We stud applications of Gabor functions that are considered in computer vision sstems, such as contour detection, stereo matching process and object recognition. It is known that structural information even in gre scale images resides in contours. The problem of extracting contours from images is approached with the use of Gabor lters. These lters have optimal properties relating to simultaneous localisation in space, spatial-frequenc and orientation. Matching algorithms using image properties maximize a similarit measure between two images. This similarit ma appl to features images derived from images. The main problem of stereoscopic vision is to nd corresponding parts of the left image and the right image. Gabor lters have been used to resolve the correspondence problem. The problem of recognition of free-form objects in real-world scenes is still a dicult. The magnitude of Gabor features can be used as objects features. The output of this lter is a vector that describes the structure around the image location. This vector are classied into N tpes form a set of regions. Each region is characterised b its index which describes image in terms of the Gabor lter response. 2 Stereo matching process We do not assume that the cameras are parallel. If (u l ;v l ) and (u r ;v r ) are the coordinates in the left and the right images respectivel, of the perspective projection of an object point P (X; Y; Z) then [7]: u l = f u r = f x l z l + f x r + f v l = f v r = f l z l + f + f () (4) The image coordinate sstems are translated b vector [a; b; c] T and rotated ( b the rotation matrix R determined b the three Euler angles ; ; ) such that 2 64 x r 75 =[R] 2 64 xl,a l,b z c,c 75 (5) Assuming that a =0 = 0 i.e. the cameras optic axes are coplanar, and = 0 ( cameras not rotate around the z - axis) 2 64 x r 0 cos sin 0, sin cos 75 = x l, a l, b z c, c 75 (6) We want to express z l in terms of v l and v r (horizontal disparities). For Z z l Z = f [(b, v l) (v r sin + f cos )+ +fv r +c(f cos, v r cos )] (7) where: f is the focal length of the lens, is the convergence angle, = sin (v l v r + f 2 )+fcos (v l, v r ). For c = =0 Z=f b,v l v l,v r (8) For computing stereo disparities we assume matching occurs onl there where edge point features exist. The matching begins b location feature points in each image f at multiple resolutions. Let W (x; ) denote the output of the feature detection lter at (x; ). W (x; ) is dened b the convolution of f(x; ) with Gabor elementar function h(x; ) i.e. W (x; ) = h(x; )f(x; ). The left feature point at L = (u l ;v l ) matches a right feature point at R =(u r ;v r ). The disparit of a match (L; R) is the horizontal pixel distance jv l, v r j. Feature points from image are extracted b uses Gabor wavelet decomposition and local scale interaction. Feature points are points of curvature maxima in the image. To determine a high spatial curvature point the response from a larger sized cell

3 6 P =(X; Y; Z) Gabor lters (u l ;v l ) f D =[a; b; c] 6 T focal point x l z l l x r right image plane left image plane focal point Figure 1: Stereo geometr (u r ;v r ) is subtracted from the smaller cell. Feature point detection utilizes a simple mechanism uses the responses of lter from dierent frequenc channels [], [10] I i;j (x; ; )=s(jjw i (x; ; ), W j (x; ; )jj) (9) where =,2(i,j) is the normalizing factor, and s is sigmoid nonlinearit function 1 s(x) = 1 + exp(,x) and W i (x; ; ) is done b [4]: W (x; ; ; )=f(x; ) (x; ; ) = f(x; ) ( j x; j ; ) (10) where denotes convolution, f(x; ) is an image, (x; ; ) is a wavelet representation of the Gabor function, and W is the lter output. The feature vector at each spatial location (x; ) is specied as W(x; ; )=fw(x; ; ; )g ; (11) The parameter values used in our experiment are = 2; = 4;i =,2;j =,5.We used twovector feature points I q i;j where q = l; r. The epipolar constraint limits the search space from 2D image plane to a single 1D horizontal line. The disparit vector is characterized b a cost function U d = X x ji l i;j(x; ; ), I r i;j(x + d x ;;)j 2 (12) Gabor lters are lters with Gabor functions as impulse response. The have been emploed in a number of applications, notabl in the area of the edge detection, texture segmentation and stereo matching process..1 Gabor lter for edge detection Gabor functions are Gaussians modulated b complex sinusoids. In its general form, the twodimensional Gabor function and its Fourier transform can be written as shown in eqn. 1 and eqn. 2. When the complex impulse response in eqn. 1is expanded: g(x; ; u0;v0)=g r (x; ; u0;v0)+ig i (x; ; u0;v0)(1) where g r (x; ; u0;v0) = exp[,( x x 2 2 )] cos[2(xu0 + v0)] (14) and g i (x; ; u0;v0) = exp[,( x x 2 2 )] sin[2(xu0 + v0)] (15) are the real part and the imaginar part referred to as cosine and sine, respectivel. Cosine part is even smmetric with respect to an oriented axis, dene b, and is strongl sensitive to features with smmetr propert along the same oriented axis, useful for detecting oriented blocs or roof edges. Sine part is odd smmetric version along the same oriented axis, useful for detecting oriented step edges. One of the attractive aspects of using Gabor lters are orientation selectivit. Structural information even in gra-scale images resides in lines and edges. Projections of the original image into a specied discrete basis set of 2D Gabor functions gives information for local line/edge position and direction. The parameters of the Gabor lters were studied: - the orientation of the grid, denoted b, with

4 4 = f0 0 ; 0 0 ; 45 0 ; 60 0 ; 90 0 g, - p the scale of the Gaussian denoted b, with p p p p p 2 = f 2; 4 2; 5 2; 6 2; 10 2g pixels, and = p 2 2, - the ratio between the scale and the grid wave length, denoted b with = f2; ; 4g for a constant scale ( p 2 =4 p 2) and a constant orientation ( =45 0 ). The edge representation was computed b convolving original image with the Gabor functions..2 Wavelet representation of Gabor functions The Gabor functions form a complete though nonorthogonal basis set. Like the Fourier series, a function g(x; ) can easil be expanded using the Gabor function. Consider the following wavelet representation of the Gabor function: (x; ; ) = exp[,( 2 x )+ix 0 ] (16) where x0 = x cos + sin 0 =,x sin + cos where is the spatial aspect ratio and is the preferred orientation. In the experiments, is the set to 1, and is discretized into four orientations. The orientation parameter determines the direction of the edges. Famil of wavelets can be generated b translations (x, x0;, 0;) and dilations ( j x; j ; ), where (x0;0) and j are the translation and scale parameters, respectivel. The resulting famil of wavelets is given b f[ j (x, x0); j (,0); k ]g (17) for 2 R ; j = f0;,1;,2;g, and k = k N, N =4,k=f0;1;2;g. We assume the following lter structure for analzing images [4] W (x; ; ; )=F(x; ) (x; ; )= =F(x; ) ( j x; j ; ) (18) where denotes convolution, F (x; ) is an image, (x; ; ) is a wavelet representation of the Gabor function, and W is the lter output. We call the ltering operator shown in eqn.(48) a Gabor lter. The feature vector at each spatial location (x; ) is specied as W(x; ; )=fw(x; ; ; )g ; (19) 4 Object detection In object recognition for each pixel location (X; Y ) we compute the magnitude Gabor features G mag (X; Y )= X x=w=2,1 =w=2,1 x=,w=2 X =,w=2 f(x + x; Y + )g(x; ; u0;v0) (20) The input image with 256 x 256 pixels and 256 grelevel was divided into a set 8 x 8 non-overlapping blocks and used Gabor lter we obtained 2 x 2 x 5(number of orientations) features for each image. Next these features are the input vectors for a k- nearest neighbour classier. The Gabor vector that describes the structure around the image location are quantised into N regions. Each region is characterised b code-book index which describes the corresponding image. References [1] M.J. Bastiaans, "Gabor's expansion of a signal into Gaussian elementar signals", Proc. IEEE, vol.68, pp.58-59, [2] R.S. Choras, "Gabor functions and applications to image processing and object recognition in computer vision sstems" Proc. of IASTED Int. Conf. Signal Proc. & Communicat., pp ,1998. [] J. Daugman, "Complete discrete 2-D Gabor transforms b neural networks for image analsis and compression", IEEE Trans.

5 Acoust. Speech, and Signal Processing, 6, pp.1169-1179, 1988. [4] D. Dunn, W.E. Higgins, J. Wakele, "Texture segmentation using 2-D Gabor elementar functions", IEEE Trans. Patt.Anal.Mach.Intell.

Kunt, "Video coding using a pramidal Gabor expansion", SPIE Visual Commun. and Image Proc., vol.160, pp.489-502, 1990. Figure 2: Input image [7] W.E.L.

Porat, Y.A.Zeevi, "The generalized Gabor scheme of image representation in biological and machine vision", IEEE Trans. Patt. Anal. Mach. Intell., PAMI-10, pp.452-468, 1988. [10] K.R. Namuduri, R.

5 5 Acoust. Speech, and Signal Processing, 6, pp , [4] D. Dunn, W.E. Higgins, J. Wakele, "Texture segmentation using 2-D Gabor elementar functions", IEEE Trans. Patt.Anal.Mach.Intell., PAMI- 16, pp , [5] D. Gabor, "Theor of communications", J. IEE, 9, pp , [6] T. Ebrahimi, T.R. Reed, M.Kunt, "Video coding using a pramidal Gabor expansion", SPIE Visual Commun. and Image Proc., vol.160, pp , Figure 2: Input image [7] W.E.L. Grimson, From Images to Surfaces, Cambridge,MA: MIT Press,1981. [8] R. Orr, "The order of computation for nite discrete Gabor transforms", IEEE Trans. Signal Proc., vol.41, pp , 199. [9] M. Porat, Y.A.Zeevi, "The generalized Gabor scheme of image representation in biological and machine vision", IEEE Trans. Patt. Anal. Mach. Intell., PAMI-10, pp , [10] K.R. Namuduri, R. Mehrotra, N. Ranganathan, "Ecient computation of Gabor lter based multiresolution responses", Pattern Recognition, 27, pp , Figure : Edge detection using Gabor ltering. [11] J. Wexler, S. Raz, "Discrete Gabor expansions", Signal Processing, vol.21, pp , Figure 4: Edge detection with another set of parameters.

Texture Segmentation Using Multichannel Gabor Filtering

IOSR Journal of Electronics and Communication Engineering (IOSRJECE) ISSN : 2278-2834 Volume 2, Issue 6 (Sep-Oct 2012), PP 22-26 Texture Segmentation Using Multichannel Gabor Filtering M. Sivalingamaiah