Texture Segmentation Using Gabor Filters and LEGION

Erdogan Cesmeli

Department of Biomedical Engineering
The Ohio State University
Columbus, OH 43210, USA

cesmeli@chaos.bme.ohio-state.edu

1.0 Introduction

Image segmentation is a difficult yet very important task in many computer vision applica tions. Texture is one of the visual features believed to have role in segmentation. There have been several attempts to define texture in the literature, yet there is no satisfactory one currently available. Intuitively, texture is related to patterned variations in the image. It is, therefore, natural to deduce that texture is a region property, not a point property. Accordingly, all methods extract features from a region of certain size. Among several approaches, a method based on signal processing ideas is called a linear system theory approach. This method performs frequency analysis of textures by spatial or frequency domain filters [18]. A well known texture analysis method using signal processing approach is based on Gabor filters. This method is appealing due to its mathematical prop erties and its resemblance to the receptive fields of simple cells in the visual cortex which are believed to have role in segmentation [3, 10].

2.0 Gabor Functions in Texture Analysis

Most natural images are composed of patches of textures that we perceive as uniform within each zone. Although there is not yet a unique satisfactory definition, textures are made of a set of local structural elements that show pattern across the texture, producing a sensation of uniformity. Consequently, texture analysis requires combining local analysis for the basic textural elements, an then, larger region analysis looking for uniformity within a texture and discriminating between different textures. The joint localization of Gabor functions makes them a very interesting tool for a local description of textures. [1, 5, 6, 9, 11, 17, and 20]. In most of the above referenced models the phase information is ignored, even though it has been pointed out that Gabor phase may be important in texture discrimination [4].

Hubel and Wiesel [8] classified two different types of cells in the striate cortex; simple cells which showed receptive fields (RF) with clear excitatory and inhibitory areas, and linear response [14] and complex cells which lack distinct excitatory/inhibitory zones in their RFs and have a nonlinear behavior [15]. Simple cells directly receive the visual information from the lateral geniculate nucleus (LGN), and seem to constitute a first step in the image representation and processing in the visual cortex. RFs of simple cells show similarities to Gabor functions [13, 10]. Marcelja [13] reported good 1D fits of Gabor functions to experimentally measured RFs. Pollen and Ronner [16] added further support to this idea, finding cortical units whose RFs were identical, except for a 90 shift in phase, i.e. they were in phase quadrature. This strongly suggests several neurons could represent the real and imaginary parts of a complex Gabor function. They found pairs of cells without definite parity, indicating an arbitrary phase offset in the complex Gabor function, but still keeping the quadrature phase relationship. More recently, Jones and Palmer extensively studied 2D fits of Gabor functions to RFs of simple cells [10].

2.1 Mathematical Background

A complex 2D Gabor function has the general form

[equation1]

where

[equation1a]

are rotated coordinates and where

[equation2]

Thus, h(x,y) is a complex sinusoidal grating modulated by 2D Gaussian function with aspect ratio of l, scale parameter s, and major axis oriented at an angle f from x-axis. Its Fourier Transform is

[equation3]

where

[equation3a]

and (U',V') is a similar rotation of the center frequency (U,V). Thus, H(u,v) is a bandpass Gaussian with minor axis oriented at angle f from the u-axis, with aspect ratio of [equation3b] , radial center frequency of [equation3c] , and orientation, [equation3d] . Figure 1 depicts perspective plots of a Gabor filter and its Fourier Transform. Parameter B defines the radial bandwidth which is given as [equation3e] where [equation3f] [1].

[Gabor Function figure]

3.0 LEGION

LEGION (Locally Excitatory and Globally Inhibitory Oscillator Networks) is based on the idea of oscillatory correlation, where the phases of the neural oscillators encode the binding of the pixels [19]. In the correlation theory of von der Malsburg and Schneider [12] features are linked through temporal correlations in the firing patterns of neural oscil lators, where each oscillator represents neural activity. A single oscillator i of LEGION is defined as a feedback loop between an excitatory unit xi and an inhibitory unit yi:

[equation4]

[equation5]

where [equation5a] denotes the amplitude of a Gaussian noise term. [equation5b] represents external stimulation to the oscillator, and Si denotes coupling from other oscillators in the network:

[equation6]

where [equation6a] is the dynamic connection weight from unit k to unit i, H(z) is the Heaviside function which becomes 1 when its argument is positive and 0 otherwise. N(i) represents the neighborhood topology in the array which could be 4- or 8-nearest neighboring units around the unit i (Figure 2). The parameter [equation6b] is chosen to be small. [equation6c] is the weight of the inhibition from the global inhibitor z, defined as [equation6d] where [equation6e] if [equation6f] for at least one oscillator and [equation6g] otherwise. The neural network structure used in our image analysis is a two dimensional array and one global inhibitor (Fig. 2). The x-nullcline of equation 1 is a cubic curve while the y-nullcline is a sigmoid function, as shown in Figure 3. If I > 0, these curves intersect along the middle branch of the cubic nullcline and the system is oscillatory.

[LEGION Architecture figure]

The parameter [equation6h] is introduced to control the relative times that the solution spends in two phases. If I < 0, then the nullclines intersect at a stable fixed point along the left branch of the cubic. In this case the system does not oscillate. The parameter [equation6i] specifies the steep ness of the sigmoid [21].

[nullcline figure]

4.0 The Model: Bank of Gabor Filters and LEGION

Motivated by biological evidence and attractive mathematical features of both Gabor fil ters and LEGION, a system which combines the two is proposed. It is well known that texture perception is a local process rather than a global one. This feature can easily be incorporated into LEGION which is already a network functioning based on local interac tions through the local couplings. Therefore, the dynamic weights of LEGION are formed based on the filtered versions of the input image by a bank of Gabor functions (filters). For this purpose, a bank of Gabor filters with varying tuning frequency and orientation has been applied. Corresponding to each pixel in the input image a set of values resulting from different channels is obtained. This set of values is represented by a vector and called tex ton. The dynamic weight between the two neighboring units is defined as the angle between the two vectors belonging to the units. For this purpose, entries in each texton are first ranked in descending order, only m values are kept, the rest are set to zero. Then for given two neighboring units with textons, Ti and Tj, where T = [t1t2 ... tm],

[equation7]

where (*) denotes complex conjugate and |.| denotes magnitude of a complex number.

5.0 Results

For the demonstration of our analysis, synthetic and natural textures from Brodatz Photography Album [2] are used. Two images of size 128x128 containing different texture types are prepared. A bank of Gabor filters with three different frequencies (f=1/(k*1.41), [equation7a]

) and four different orientations [equation7b]

is used to obtain textons. Only the six largest entries in each texton are used.

In the first example, the image is composed of five regions (Figure 4). In the top right region of the input image (Figure 4.a), each pixel intensity is assigned to a randomly gen erated value independent from its neighbors. The other regions in the input are sinusoidal gratings with different frequency and orientation. Based on this input the weights in LEGION are calculated using (7). Following that, LEGION is simulated where the oscil lators in LEGION start with random phases (Figure 4b). As time evolves they interact and attain a repetitive pattern of activity. In this activity, the oscillators corresponding to the same texture region oscillate in synchrony. Different texture regions have different phases. Based on this phase difference, regions with different texture are easily segregated by observing the network activity at different time points (Figure 4c,d,e, and f).

In the second example, an image is composed of two natural textures [2]. Again, the weights of the network are determined and LEGION is run with random initial phases. Eventually, two major groups of oscillators are distinguished (Figure 5c and d). A third smaller group of oscillators which corresponds to a subregion with nonuniform texture of the image on the left is also segregated.

[results figure]

6.0 Conclusions and Future Study

An unsupervised texture segmentation method is suggested in this study. Two biologically motivated scheme, namely Gabor filters and LEGION, are combined to perform texture segmentation. The first stage of the model is composed of a bank of Gabor filters with dif ferent frequency and orientation. The second stage is LEGION with NxN many oscillators with local connections and one global inhibitor connected to all of them. Autonomous interaction of the oscillators in the network are defined through connection weights. A weight between two neighboring oscillators is defined by the angle between the textons corresponding to the oscillators. Success of the method in texture segmentation is demon strated on one synthetic image and one composed of two natural textures.

Currently, the suggested model has very few number of filters (12). The method will be more powerful if a large bank of Gabor filters is used to detect the differences among sev eral different texture types is provided. Better techniques to combine the filter outputs to calculate the weights are under investigation. It is the belief of the author that future suc cess of any method based on a bank of Gabor filters depends on the scheme for combining filter outputs.

Acknowledgments

This work was supported by Graduate Summer Fellowship Program from The Center for Cognitive Science, Department of Computer and Information Science, and The Center for Biomedical Engineering at The Ohio-State University, Columbus, Ohio.