ANALYSIS DIRECTIONAL FEATURES IN IMAGES USING GABOR FILTERS

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1 ANALYSIS DIRECTIONAL FEATURES IN IMAGES USING GABOR FILTERS Z.-Q. Liu*, R. M. Rangayyan*, and C. B. Frank** Departments of Electrical Engineering* and Surgery**, The University of Calgary Calgary, Alberta, Canada T2N 1N4 ABSTRACT A computational technique for directional analysis of piece-wise linear patterns in images has been developed using Gabor filters. A given image is preprocessed by a sequence of Gabor filters, which are Gaussian-modulated sinusoids in the spatial domain. Linear segments at specific orientations are detected by Gabor filters tuned to the corresponding directions. Information regarding orientation of the linear patterns in the image and the area covered by the patterns in specific directions is then computed from the filtered images. The performance of the method is illustrated through applications to synthetic images and scanning electron microscope images of collagen fibrils in rabbit ligaments. The method has significant applications in quantitative analysis of ligament healing and comparison of treatment methods for ligament injuries. I. INTRODUCTION Recent research in early computer vision has focused on extracting, from intensity images, such information as edges, depth, reflectance, and illuminance. An assumption underlying some of these approaches is that the properties mentioned above carry essential information describing physical characteristics of objects in the intensity images. Organization of these properties in the image into more abstract terms is an early and important process, which provides useful intermediate information for recognition, analysis, and other interpretation processes. Research based on this methodology is supported by studies on animal visual systems. For example, the finding that there exist special mechanisms sensitive (selective) to oriented linear patterns [6] has inspired significant interest in the development of techniques for oriented pattern analysis in machine vision [ll, 121. Zucker [12] investigated the estimation of orientation by combining the outputs of linear operators (difference of Gaussian operators). Kass and Witkin [7] proposed a method using bandpass filters to decompose an oriented pattem image into two parts: a flow field describing the direction of anisotropy, and a residue pattern in a flow coordinate system. Their research is aimed at understanding oriented pattems 68 CH2845-6/90/0000/0068$01.OO IEEE

2 Track 1 : Communications and Image ProcessinglSession 5 69 through the extracted components. Ligaments are highly organized connective tissues that stabilize joints, normally consisting of nearly parallel arrangements of collagen fibers. Serious injuries may result in ruptured ligaments, where the collagen fiber organization is lost. Collagen remodeling during healing returns the alignment gradually over a period of 6-12 weeks [9]. In an effort to develop techniques for quantitative analysis of collagen arrangement in ligaments, we proposed a Fourier domain directional filtering method [l-3,101. A drawback of this method is that it requires a proper threshold chosen manually to extract directional information from a set of component images generated from the original image. Recently, we reported an another method for linear pattern analysis using scale-space methods [8]. In this paper, we present a directional analysis method based on Gabor filters. 11. METHODS A. Gabor functions and properties Two-dimensional (2D) Gabor filters are expressed as Gaussian-modulated sinusoids in the spatial domain, and as shifted Gaussians in the frequency domain. In general, the 2D Gabor functions take the following form [5]: where g (x.y ) is a Gaussian function (aperture) given by and m (x y ) is a complex modulating function: In the above equations, xo and yo specify the center of the Gaussian, U,, = - fo N' h0 vo = -, N= image size, and fo and ho are the frequencies of the filter along the N x and y axes, respectively. The constants ox and ay in the Gaussian function determine the scale, and the width/length aspect ratio a = oxlcry which can be used to adjust the orientation sensitivity of the Gabor filter. Figure 1 shows typical

3 IO Third Annual IEEE Symposium on Computer-Based Medical Systems perspective views of the Gabor filter in the spatial domain. The overall modulation frequency for the Gabor filter is ~ u ~, + and v the ~ axis of the modulation in the frequency domain is oriented at an angle of tan-'(vo/uo) from the U axis. It is easy to show that, in the frequency domain, the Gabor filter is a Gaussian function centered at (uo,v0): which is a 2D bandpass filter. In his original work [5], Gabor, by applying arguments from quantum mechanics, proved that a signal's specificity in time and frequency is limited by a lower bound on the product of its bandwidth and duration. This phenomenon is referred to as the uncertainty principle. Further, he proposed a set of functions that are optimal in the sense that the product of effective widths in both time (At) and frequency (Am) domains reaches the fundamental lower bound, i.e. AtAm = 1/4x. Gabor functions in their one-dimensional form have been used fo? modeling simple-cell receptive-field profiles. Arguing that 2D Gabor filters will be more suitable for vision studies, Daugman proposed 2D Gabor filters, and extended the uncertainty principle to 2D situations as [4] 1 AXAYAUAV 2-16x2 where Ax and Ay are effective spatial widths, and AM and AV are effective frequency widths. As with ID Gabor functions, 2D Gabor functions are also a family of optimal functions which achieve the theoretical limit of 1/16x2. Gabor filters have the following important properties: (1) The response of the filter is spectrally localized allowing differentiation of linear patterns occurring over a range of spatial scales or channels. (2) The response of the filter is spatially localized, which allows accurate identification of linear patterns within specific orientations. (3) It has been shown that the complex Gabor filter achieves the lower bound of the uncertainty principle [4]. These properties have important implications in our analysis procedure, as described next. B. The algorithm The directional analysis algorithm developed using Gabor filters consists of the following steps:

4 Track 1 : Communications and Image ProcessinglSession (1) Generate a set of Gabor filters (in the spatial domain) tuned to a set of angle bands. (2) Convolve the Gabor filters with the given images, which results in a set of component images, one for each angle band. (3) Compute statistics (such as entropy, moments, linear pattern-covered area, etc. [l-3,101 ) from the component images RESULTS Experiments using both computer-generated patterns and ligament images were conducted with this method. It was found that the Cabor filter, tuned to a specific direction, can highlight those linear patterns oriented in the same direction. The images used in the experiments are in 128x128 pixels, 8-bit format. 16 Gabor filters spanning the range of 0'-180' were generated. For the purpose of illustration, only the results from three orientations, namely, O", 45', and 90' will be presented in the following sections. Figure 2 shows the Gabor filters at Oo, 45O, and go', with a = 1. A. Synthetic patterns The synthetic pattern used in this experiment was designed to illustrate the results produced by a sharp intensity change between regions: an S-shaped edge in the pattem, shown in Figure 3a, separates two regions. Figure 3b shows linear segments oriented at 0' extracted using the Gabor filter in Figure 2a. As seen in Figure 3c, some linear segments are extracted using the Gabor filter shown in Figure 2b, since the S pattem contains segments at 45'. Figure 3d shows the results after convolving with the Gabor filter at 90' (Figure 2c). No linear pattern is detected at this direction. B. Ligament images In the second example, a typical ligament image (Figure 4a) is used to demonstrate the effectiveness of the Gabor filter. It is seen that the majority of linear pattems in Figure 4a are at 0'. Figure 4b shows the results by using the filter in Figure 2a. As expected, the dominant linear patterns that are at 0' are extracted. In Figure 4c, a few fibers oriented at about 45' are picked up by the Gabor filter in Figure 2b. Figure 4d shows the results obtained by convolving the image in Figure 4a with the Gabor filter in Figure 2c. The image in Figure 4d is basically flat since the ligament image contains almost no fiber at 90'.

5 72 Third Annual IEEE Symposium on Computer-Based Medical System IV. CONCLUSIONS In this paper, we proposed a new directional analysis method using the Gabor filter, which is shown to be optimal in the sense that the product of the effective widths in the spatial and frequency domains reaches the fundamental lower bound. Gabor filters are particularly sensitive to the presence of collinear and elongated segments. This selectivity allows the use of Gabor filters to detect linear segments in ligament images. The method can be efficiently implemented using parallel computing machines. The procedure is being used for quantitative analysis of ligament healing and for objective comparison of ligament treatment methods. ACKNOWLEDGEMENTS We express our appreciation to the Alberta Heritage Foundation for Medical Research, and the Natural Sciences and Engineering Research Council of Canada for the research grants awarded. REFERENCES S. Chaudhuri, H. Nguyen, R.M. Rangayyan, S. Walsh, and C.B. Frank, A Fourier domain directional filtering method for analysis of collagen alignment in ligaments, IEEE Trans. Biomed. Engineering BME-34 (7) pp (July, 1987). S. Chaudhuri, R.M. Rangayyan, S. Walsh, and C.B. Frank, Quantitative analysis of ligament healing via directional filtering, pp in Proc. 13th Canadian Medical and Biological Engineering Conference, Halifax, Nova Scotia, Canada (June 9-12, 1987). S. Chaudhuri, R.M. Rangayyan, S. Walsh, and C.B. Frank, The Role of the supercomputer in image processing: Quantitative analysis of collagen alignment in ligaments, in Supercomputer Symposium 87, Calgary, Alberta, Canada (June 15-16, 1987). J.G. Daugman, Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters, J. Opt. Soc. Am. A. Vol. 2(7) pp (July 1985). D. Gabor, Theory of communication, J. Inst. Elect. Engr. 93 pp (1946). D. Hubel and T. Wiesel, Receptive fields, binocular interaction and functional architecture in the cat s visual cortex, J. Physiol. (London) 166 pp (1962). M. Kass and A. Witkin, Analyzing oriented patterns, pp in Proc. the 9th International Joint Conference On Artifrciul Intelligence, Los Angeles,

6 Track 1 : Communications and Image ProcessinglSession 5 73 Calif., U.S.A. (1985). 8. Z.Q. Liu, R.M. Rangayyan, and C.B. Frank, Linear pattern analysis using scale-space techniques, pp in Proc. Supercomputing Symposium 89, Toronto (June 19-21, 1989). 9. B.J. MacFarlane, P. Edwards, C.B. Frank, R.M. Rangayyan, and Z.Q. Liu, Quantification of collagen remodelling in healing nonimmobilized and immobilized ligaments, pp. 300 in Proc. 35th Annual Meeting of Orthopaedic Research Society, Las Vegas, Nevada (Feb. 6-9, 1989). 10. R.M. Rangayyan, Z.Q. Liu, B.J. MacFarlane, P. Edwards, and C.B. Frank, Computerized quantification of collagen alignment in nonimmobilized and immobilized healing ligaments, pp in Proc. IEEE Engineering in Medicine and Biology Soc. 10th Annual International Conference, New Orleans, Louisiana (Nov. 1988). 11. K.A. Stevens, Computation of locally parallel structure, Bio. Cybernetics 29 pp (1978). 12. S.W. Zucker, Early orientation selection: Tangent fields and the dimensionality of their support, Computer Vision, Graphics, and Image Processing 32(1) pp (1985). Figure 1. Typical profiles of the Gabor filter in the spatial domain: (a) the imaginary part, (b) the real part. (a) (b) (C) Figure 2. The Gabor filters used in this paper. The filters are tuned to different orientations with a = 1, and spatial frequency = 3 cycledframe: (a) at O, (b) at 45, (c) at 90.

7 14 Third Annual IEEE Symposium on Computer-Based Medical Systems Fi&e 3. (a) Synthetic pattern'with two regiozof different intensities separated by an S-shaped edge; (b) the result with the Gabor filter at 00; fc'l with the Gabor filter at 45O; (d) with the Gabor filter at 90. J) C) d) Figure 4. (a) A 128x128 ligament Image; (1 ".. (b) with the Gabor filter at 45O; (d) with the Gabor filter at ;