Retinal DOG filters: high-pass or high-frequency enhancing filters?

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1 Retinal DOG filters: high-pass or high-frequency enhancing filters? Adrián Arias 1, Eduardo Sánchez 1, and Luis Martínez 2 1 Grupo de Sistemas Inteligentes (GSI) Centro Singular de Investigación en Tecnologías de la Información (CITIUS) Universidad de Santiago de Compostela Santiago de Compostela, 15782, Spain adrian.arias.abreu@usc.es,eduardo.sanchez.vila@usc.es 2 Instituto de Neurociencias de Alicante CSIC-Universidad Miguel Hernández Alicante, 03550, Spain l.martinez@umh.es Abstract. This paper analyzes the filtering operation carried out by the classical Difference-of-Gaussians model proposed by Rodieck to describe the receptive fields of retinal ganglion cells. Discrete DoG kernels of such functions were developed and compared with High-Pass and High- Frequency Enhancing filters. The results suggest that the DoG Kernels behave as High-Frequency Enhancing filters but in a limited band of frequencies. Key words: Retina, Difference of Gaussians, High-pass Filtering, High- Frequency Enhancing Filtering. 1 Introduction The term receptive field (RF) in the visual system was classically defined as a two-dimensional region in visual space where a luminous stimulus triggers a change in response on that neuron [1]. The concept was first applied to the retina to describe the area in which a stimulus drove responses of retinal ganglion cells (RGCs). Later on, Kuffler found that RGCs show RFs with a concentric shape made up of two antagonistic regions: a center, and a surround [2]. Thus, when a bright stimulus is applied to the center region, the RGCs are excited and generate a number of action potentials or spikes; and, conversely, when the same stimulus is applied to the surround, the neuron is inhibited and a weaker or no response is observed. Thereafter, the RFs of RGCs were characterized by this center-surround organization. The center-surround RF is an empirical model that is useful to understand the spatial organization of the afferent inputs to RGCs but lacks the ability to predict the neuron?s response to any given stimulus. An important contribution was later made by Rodieck (1965) by proposing a mathematical model to formally describe the function that maps the input-output relationship of the RFs

2 of RGCs. The proposed relationship was the sum of two Gaussian functions: a positive one, representing the center, and a wider negative one representing the surround, both centered at the same point. This model was called the Differenceof-Gaussians (DoG) model and has been used to represent the RFs of RGCs ever since. The parameters of the DoG model for RGCs were first estimated by Enroth- Cugell and Robson (1966, 1984). They recorded responses of ganglion cells to sinusoidal stimuli and fitted the model against contrast sensitivity curves that were obtained experimentally. For each recorded RGC, a DoG model was fitted and their parameters, the radius and maximum amplitudes of both the center and surround Gaussians, were estimated. As the individual estimated contrast sensitivity curves fitted nicely with the experimental ones, the results provided a strong support for the DoG model as a useful function to describe the centersurround RFs of RGCs. The issue about the information processing capabilities of a DoG function was subsequently assessed by Marr and Hildreth [6]. They proposed a very influential theory of edge detection on the basis of an analysis of intensity changes which occur in natural images as well as the appropriate filters to signal those changes. Intensity changes or edges could best be detected by finding the zero values of the second derivative, or the Laplacian, of a Gaussian. Moreover, they showed that an accurate approximation of the Laplacian of a Gaussian could be developed by using a DoG function with an appropriate ratio σ surround /σ center about 1.6. They concluded that neurons in the visual system with RFs being described by such DoG functions could indeed work as edge detectors. Since the landmark work of Marr and Hildreth, RGCs, which determine the output of the retina, have been understood to perform a kind of edge detection or sharpening filtering. In fact, different authors have reported evidence about the existence of RGCs of the type of Local Edge Detectors in Cats, Rabbits and even Primates [7 9]. Moreover, in the field of image processing, discrete kernels derived from DoG functions are usually characterized as high-pass filters [10]. However, it is interesting to note that there was no previous attempt to analyze what is the filtering carried out by the DoG models that were obtained experimentally by Enroth-Cugell and Robson. In this paper we aim at analyzing the information processing capabilities of different versions of the original DoG model and test to what extent it behaves as a canonical edge detector. 2 Methods 2.1 Parameters of the DoG model The Difference-of-Gaussians model is made up with two gaussians: the first one representing the excitatory center of the RF, the second one the inhibitory surround of the RF. The function as it was used by Enroth-Cugell and Robson [4, 5], was formalized as follows: DoG(r) = k c e (r/rc)2 k s e (r/rc)2 (1)

3 being the relevant parameters: the maximum amplitudes k c and k s, and the radius r c and r s. For each RGC, 17 cells reported in Enroth-Cugell and Robson (1966) and 6 in Enroth-Cugell and Robson (1984), the theoretical contrast sensitivity function derived from the DoG model was fitted to the empirical contrast sensitivity function measured at different spatial frequencies. As a result, a set of parameters (r c, r s, r s /r c, and k s r 2 s/k c r 2 c) were estimated for each cell. For brevity these parameters are not shown here but can be found in the original papers [4, 5]. 2.2 Discrete DoG kernels The continuous DoG functions fitted to experimental data has to be converted into discrete DoG kernels to operate with input images. In what follows, the four steps required in that procedure are described. As the continuous function can take infinite values, the first step consisted on truncating it in order to set a finite range of values. Figure 1 describes the process in one dimension. A variable T was defined to determine the width of the truncation, and an standard rule was followed in order to include the 99, 74% of the area under the curve of either the center or the surround gaussian. Therefore, two possible values were considered: T = 3 r c, or T = 3 r s. The second step consists on sampling the continuous function. By setting the size SxS of the kernel, the number of both the elements of the kernel matrix Fig. 1. Discretization of the continuous DoG model: truncation (upper inset), sampling (middle inset), and normalization (lower inset).

4 as well as the sampling points of the continuous DoG function are set. Both variables T and S define the step of the sampling process: Step = 2T S 1 After the convolution of the input image with the discrete kernel, the output image has to preserve the intensity ranges of the input. The third step therefore involves the normalization of the elements of the DoG kernel matrix. Each element or weight w ij of the matrix is normalized as follows: (2) w norm ij = w ij Σ i Σ j w ij (3) Finally, the goodness of the discretization procedure has been assessed by fitting back the normalized discrete DoG kernel to the original continuous DoG function. The Levenberg-Marquart algorithm was used to solve the non-linear least squares fitting problem. The results (not shown here) confirmed that the parameter set (r c, r s, r s /r c, and k s r 2 s/k c r 2 c) is preserved as well as other aspects of the DoG function. 2.3 Baseline filters Two filter kernels were chosen to analyze the information processing capabilities of the discrete DoG kernels described in the previous section. The first one is the kernel of a typical high-pass filter (HPF), which represents the discrete version of the second-order derivative or Laplacian operator. The kernel can be obtained by means of either the coefficients of the second-derivative operation or by substracting the low-pass filter kernel from the identity kernel (Fig. 2). HPFs are typically used for edge detection tasks as they signal the location of edges in the images. The second kernel belongs to the class of high-frequency enhancing filter (HFEF), which carries out local contrast enhancement operations. It is a popular HFEF technique known as unsharp masking widely used in the printing and photography industry. The kernel is obtained by adding a high-pass filter kernel to the identity kernel (Fig. 2). Fig. 2. Kernels of baseline filters: high-pass filter (left) and unsharp masking HFEF (right).

5 2.4 Image processing and kernel analysis A comprehensive python simulation environment was developed to carry out the discretization procedure as well as the analysis of the kernels presented in the Results section. The environment takes advantage of some powerful python libraries, such as: OpenCV, to convolve an image with a kernel; Numpy, to generate 2D kernels and compute 2D discrete Fourier transforms; Scipy, to solve the non-linear least squares fitting problem described in section 2.2; and Matplotlib to plot graphs and view images. 3 Results The first task was to analyze the behavior of the discrete DoG kernels obtained for the 23(= ) cells reported by Enroth-Cugell and Robson (see section 2.1). For each DoG kernel we generated: the kernel representation in both spatial and frequency domain, the Bode diagram, and the output after convolving the kernel with an standard test input (Lena image). It can be concluded that all kernels behave in a similar way regardless of the discretization parameters used (not shown here for the sake of brevity). On the basis of this result, only one of the discrete DoG kernels, which corresponds to cell number 1 of Enroth-Cugell and Robson (1966), was chosen to make the comparison with the baseline filters. The comparison with the high-pass filter (HPF) is shown in figure 3 for kernels of size 25x25. Both kernels in the spatial domain (red pixels indicating positive values and blue pixels, negative ones) show a positive center and a negative surround, but the extent as well as the structure of these regions are clearly different. These differences are made explicit when the kernels are represented in the frequency domain (Fig. 3, second row). The 2D Fourier spectrum of the DoG kernel in a db/log scale (red pixels indicating positive coefficients; blue pixels indicating negative ones) shows a region of positive coefficients starting from the center point (DC coefficient) that progressively change into negative coefficients at high frequencies. On the contrary, in the HPF kernel the value of coefficients is negative at low frequencies and positive after a certain threshold frequency. The effect of the kernel on the frequencies of the input image is better analyzed by means of the Bode diagrams (Fig. 3, third and fifth rows), which plots the filter gain for each spatial frequency. It can be seen that (1) the DoG kernel enhances those frequencies found in a band of frequencies; and (2) the HPF Kernel removes low-frequencies as well as keeps the gain of high-frequencies. The image outputs (Fig. 3, fourth and sixth rows), obtained after convolving the input with the kernels, confirm the different nature of DoG and HPF kernels. The first one enhances local edges as well as preserves the intensity levels of the rest of regions, while the second one detects the edges of the image but suppresses all intensity information of constant regions (black pixels). The transformation of the intensities at the output is clarified by looking at the intensity profile of one of the image rows (Fig. 3, fourth and sixth rows). The profile of the DoG kernel output (green line) follows the original profile of the input (black line) and stretches the values at the peaks of the curve. The behavior is different for

6 Fig. 3. Comparison of DOG kernel with HPF kernel. The DOG kernel corresponds to cell number 1 of Enroth-Cugell and Robson (1966), parameterized as follows: k c = 100, k s = 15.9, r c = 0.32, r s = 0.76, SxS = 25x25, T = 3 r c, and Step = The HPF kernel is of the same size as the DoG kernel. The kernels are represented in the spatial domain (first row) and the frequency domain (second row). The Bode diagrams (third and fifth rows) as well as the image outputs and intensity profiles at row = 500 (fourth and sixth rows) for both kernels are also plotted.

7 the HPF kernel output. The baseline is moved down to zero and the positive values of the curve indicate the location of edge points. The DoG kernel was also compared with the HFEF kernel, as shown in figure 4 for kernels of size 25x25. The main point here is that both the image outputs and intensity profiles (Fig. 4, fourth and sixth rows) are very similar, which might suggest that the behavior of the DoG kernel would belong to the class of HFEF filters. However, the 2D Fourier spectrum as well as the Bode diagram Fig. 4. Comparison of DOG kernel with HFEF kernel. The DoG kernel as well as the analysis plots are the same as shown in 3. The HFEF kernel is of the same size as the DoG kernel.

8 reveals that the kernel features in the frequency domain are somewhat different. The DoG kernel presents a band-pass behavior whereas the Unsharp HFEF does enhance the high frequencies in the same manner with no upper limit. 4 Discussion The results shown in section 3 suggests that the DoG kernels would be better classified as High-Frequency Enhancing filters rather than High-Pass filters. Our findings indicate that the retinal filters, as components of the first processing stage of the visual system, would preserve the information related to the intensity levels at each spatial location. However, the kernel representations at frequency domains indicate that the DoG kernels focus on some specific band of frequencies and do not operate on frequencies higher than a certain cut-off value. As the neurons and circuits of the visual system are fitted to the statistics of natural images, it could be interpreted that the retinal DoG functions have captured the optimal band of frequencies that can be found in natural images. This issue as well as the quantitative analysis of the DoG kernels will be assessed in a future work. Acknowledgements This research was sponsored by the Ministry of Science and Innovation of Spain under grant TIN References 1. Hartline HK: The response of single optic nerve fibers of the vertebrate eye to illumination of the retina. Am J Physiol Vol. 121 (1938) Kuffler SW: Discharge patterns and functional organization of mammalian retina. J Neurophysiol Vol. 16(1) (1953) Rodieck RW: Quantitative analysis of cat retinal ganglion cell response to visual stimuli. Vision Res Vol. 5 (1965) Enroth-Cugell C, Robson JG: The Contrast Sensitivity of Retinal Ganglion Cells of the Cat. J Physiol. Vol. 187 (1966) Enroth-Cugell C, Robson JG: Functional characteristics and diversity of cat retinal ganglion cells. Basic characteristics and quantitative description. IOVS Vol. 25 (1984) Marr D, Hildreth E: Theory of Edge Detection. Procs Royal Soc of London, Series B, Biological Sciences. Vol. 207 (1980) Cleland BG, Levick WR: Properties of rarely encountered types of ganglion cells in the cat s retina and an overall classification. J Physiol. Vol. 240: (1974) van Wyk M, Rowland Taylor W, Vaney DI: Local Edge Detectors: A Substrate for Fine Spatial Vision at Low Temporal Frequencies in Rabbit Retina J. Neurosci. Vol. 26(51) (2006) 13250?13263

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