A Theory of "Fuzzy" Edge Detection in the Light of Human Visual System

Kuntal Ghosh,1 Sandip Sarkar2 and Kamales Bhaumik3

'Center for Soft Computing Research, Indian Statistical Institute, 203 Β. T. Road, Kolkata-108;2 Microelectronics Division, Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata-64;3 West Bengal University of Technology, BF-142, Sector-1, Salt Lake, Kolkata-64; INDIA

ABSTRACT

In this paper, a new multi-scale Gaussian derivative filter is proposed, in which the higher order derivatives are expressed as a linear combination of the Gaussian function at various scales, one of which has been approximated to a Dirac-delta function to yield a filter which is in agreement with a very old psychophysical model of vision. This modification also finds support from neurophysiology of retina. The proposed filter can be optimized at any order of derivative in terms of the scale-ratios of the Gaussians. Such optimization yields further support to the approximation mentioned above. The proposed filter been shown to be very effective in extracting features from a noisy image in the form of a "fuzzy" derivative computation. Zero-crossing maps of any image filtered with the proposed model gives a half-toning effect to the retrieved image.

KEYWORDS vision, retina, receptive field, edge detection, primal sketch, halftoning

Correspondence to: Kuntal Ghosh, Center for Soft Computing Research, Indian Statistical Institute, 203 Β. T. Road, Kolkata-700108, INDIA; e-mail: [email protected]

229 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

1. INTRODUCTION

For over the last 50 years, a large number of experiments have been performed and consequently several models have been proposed on the response characteristics of the retinal ganglion cells, The output of the ganglion cells is what is utilized by the brain, possibly at the cortical level, to detect the edges and contours of the visual presentations. We are proposing here a modified model of the ganglion cell by taking into account the recent experimental observations about its receptive field. We shall show that the proposed model is in tune with the previous experimental results and the existent models of the ganglion cells. We shall further show that the algorithm derived out of the proposed model can lead to a better "raw primal sketch" of an image. The classical receptive field of retinal ganglion cells was modeled by the neurophysiologists as follows. Contributions received by a ganglion cell from photoreceptors show that the excitations in the ganglion cells are best described by a centre-surround effect. This effect assumes that the net input to a ganglion cell is obtained from a difference of two Gaussian inputs, the central one having a smaller variance than the surround. This prompted physiologists like Rodieck (1965) or Enroth-Cugell and Robson (1966) to develop a Difference of Gaussian or DoG model for the receptive field of the retinal ganglion cells. In one dimension,

2σ 2)= J- g ' - ' e > (1) •J2na\ V2πσ-ι

where,

certain value of σ]: σ2), Laplacian of Gaussian (LoG), a isotropic operator given by:

230 Vol. 17, No. 1-3, 2008 Theory of "Fuzzy" Edge Detection in Light of Human Visual System

1 r2 V2G(r,a) = 3- 1 re 2σ (2) πσ 2σ

r where, G(r,a) = —j=e 2σ (3)

is a good approximation to DoG in two dimensions. Two important points should be noticed here. Firstly, from the proposed equivalence of LoG with DoG, one may infer the possibility of involvement of higher derivatives of Gaussian in the filtering function in retinal image processing. Secondly, it may be noted that DoG is the pioneering model to indicate that a combination of Gaussian filters of different scales may reproduce the output of the ganglion cell. Such a view of filtering an image at different scales was later elaborated in the scale-space theories of vision by Witkin (1983), Yuille and Poggio (1986). The first aspect, namely the existence of filtering function as the higher derivative of a Gaussian, received a further boost in the works of Young (1985, 1987). Young demonstrated that the receptive fields of many neurons, both at the cortical and retinal level, in the mammalian visual system could be approximated by a combination of higher derivatives of Gaussian. For example, spatial frequency spectra of simple cells in monkey visual cortex showed a large variation in the bandwidth according to De Valois et al. (1982) and that variation could be accounted for by a Gaussian Derivative model. It is now known that although in rare cases one may have to include up to tenth order of differentiation, generally inclusion up to fourth order suffices. Whether these Gaussian Derivatives, appearing in the routine computation, are merely mathematical constructs or whether there is any physiological evidence in having networks or structures with such functions is not yet clear. Young (1987) proposed that retinal level processing can be nicely approximated by a linear combination of a Gaussian and a Laplacian of Gaussian (LoG) and attempted to trace this operator back to Mach's psychophysical model of retinal information processing which can be found in the works of Ratliff (1965).

231 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

The second aspect of the proposal, namely the scale space theories, achieved a rigour in the hand of Koenderink. With the development of the ideas of scale space by Witkin (1983), Yuille and Poggio (1986), the advantages of Gaussian smoothing kernel shown by Bracewell (2003) assumed a greater significance in the hands of Lindeberg (1994). Based on these ideas, Koenderink and van Doom (1984, 1987, 1990) developed a set of 'fuzzy derivatives' (which essentially belong to a family of Gaussian Derivatives) and established that members of this operator family can be transformed into each other through simple unitary transformations. This ensures significant computational advantages, because the higher derivatives and spatial offsets from pixel centers may be derived from a small canonical set of operators by linear combinations. Thus, the scale space analysis further iterated the necessity of studying the various orders of Gaussian Derivatives and their relation to the representation of image structure in visual system. Furthermore, Koenderink and van Doom (1982) have shown that the sense of three-dimensionality of an image cannot be conveyed only through the lines and edges. It also needs the knowledge of creases and folds within the image, which can be obtained through an analysis of higher derivatves. When Marr & Hildreth (1980) claimed the equivalence of LoG and DoG for a particular scale ratio between the two Gaussians, they could not provide any strong theoretical basis for the equivalence. That basis was provided much later in a paper by Ma and Li (1998), wherein they proved from very general consideration that any derivative filter could be expressed as a linear combination of functions with different scale parameters. Ma and Li (1998) have shown that any 2kth order derivative filter can be designed as the weighted sum of any k +1 even functions, every function having the same kernel, but different scales. Additionally, a (2k + l)th order derivative filter can be designed as the weighted sum of k +1 odd functions of different scales. In the present paper, we shall explain that multi-scale filtering, following the pre-scription of Ma and Li (1998), not only circumvents the problem of construction of higher order derivative filters but also actually brings in new advantages for performing more sophisticated edge detection. Our discussion will be confined only within even ordered derivative filters because we have chosen to construct filters at different scales by using two-dimensional Gaussian function, which happens to be an even function. In this paper, we

232 Vol. 17, No. 1-3, 2008 Theory of "Fuzzy" Edge Detection in Light of Human Visual System shall show that the proposed method of image analysis, by expressing such higher derivatives as weighted combination of Gaussians at various scales, is endowed with advantages like easy computability and may contain useful information in the sense of a half-toning or "fuzzy" effect in the zero- crossing-map. The present work in fact attempts to carry forward the "Theory of edge detection" which was propounded by Marr-Hildreth (1980) in the light of human visual system, towards a new theory of "fuzzy" edge detection.

2. THE PROPOSED METHODOLOGY

2.1 Zero-crossings and the Non-classical Receptive Field

Let us first introduce the concept of non-classical receptive field of ganglion cells. Contrary to the classical ideas, new experiments like for example, by Passaglia et al. (2001), indicate that the receptive field of a retinal ganglion cell consists of an extended surround well beyond the classical receptive field. Such an extended surround may be approximated by algebraically adding one or more Gaussians, having very large variances, with the DoG. According to Ma and Li (1998), such a linear combination of Gaussians would be equivalent to higher order derivatives. From such an argument it has been shown by Ghosh et al. (2005), that the non-classical receptive field of retinal ganglion cells can be modeled by a fourth order rotationally symmetric derivative of Gaussian, that is by V4G . The detailed expressions are given in Section 2.2 for the one-dimensional case. One may express V4G as DoG + where G\ is the widest Gaussian representing a disinhibitory surround beyond the classical receptive field or in other words the mean- increasing sub-units in the model of Passaglia et al. (2001). Similarly, V6G can be expressed as DoG + G\ -Gj, where G2 is another wide Gaussian representing the mean-decreasing sub-units in the same model. Zero-crossings, which represent the change in intensity at different points in the image, constitute an important tool for image processing. Marr (1982) stressed the need to study zero-crossings at various scales to understand the changes in the image caused by physical phenomena like change in reflectance, depth, surface orientation, and so on. Marr and others (1979, 1980, 1980a,

233 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

1982) looked upon visual perception as a complex information processing task, out of which a preliminary step might be performed by the zero crossing detection at the simple cell level in visual cortex, based on the information received from the LoG-based ganglion cells. It has also been shown by Marr- Hildreth (1980) as well as by Ghosh et al. (2005) with the help of Logan's theorem (1977), that the zero-crossing map obtained by convolution of the image with LoG does not satisfy Logan's condition. However, at the same scale σ = 2, the half power bandwidths forV4G and V6G filters are 0.75 and 0.7 octave respectively as compared to 1.25 octave for LoG filter. These bandwidths are less than 1 octave, satisfying Logan's condition, and hence are likely to be more informative according to Marr et al. (1979).

2.2 The Proposed Model

If the positive sub-units of the non-classical receptive field are primarily considered, which may be justified in view of the fact that reports of an extended disinhibitory surround have long been known from the works of Mcllwain (1966), Ikeda and Wright (1972), then in one dimension, following Ma and Li (1998), we can construct according to Ghosh et al. (2005), a fourth order derivative filter h^(x,a) as a linear combination of three Gaussians at scales σ0,σ] andcr2:

/ f \ f \ 1 X 1 X 1 X +a + a —g a0—g \—g 2 (4) σ σο σ2 1 2; where,

«0=^2-σι2)

«1 = ~k[°2 -°o) (5)

a2 = k{a1-al)

Here k is a constant.

So we essentially arrive at the DoG+Gy model described in the previous section. Now, in the limit of a large extended surround as shown in the works

234 Vol. 17, No. 1-3, 2008 Theory of "Fuzzy" Edge Detection in Light of Human Visual System of Passaglia (2001), Mcllwain (1966), Ikeda and Wright (1972), we can apply a condition σ0 : σ2 -> 0, so that from Eqs. (4) and (5), we arrive at:

/ // hA(χ,σ )-> mh2[x,a ) + h2[x,cr ) (6) where σ' and σ7/ are two arbitrary scales, m is an amplitude scale factor and

f \ f \ X X g (7) \σa J \Jb)j

Here c is a constant, and σα and ah are again two arbitrary scales as in the works of Ma and Li (1998). This means in two dimensions,

V4Gfr, σ) = mW2G(r,a/) + V2G(r, σ" ) (8)

Again, the limiting condition d 0 is imposed assuming that one of the two Gaussians is local while the other averages over a large area as shown by Ghosh et al. (2005), so that:

V4G(r)= mS(x,y) + V2G(r) (9)

where δ (χ, y) represents the Dirac-delta function in two dimensions.

In the same way, we can incorporate the negative sub-units of non- classical receptive field of Passaglia et al. (2001), reports of which have also been around for quite some time from the works of Enroth-Cugell and Jakiela (1980) and Kruger (1984). Then following the same procedure:

f \ ( \ f \ X a\ ' χ λ a2 X a3 X ho(x) = ^g +—g — +—g +—g (10) σ σ0 σι [ J 2 U2J σ3 l°3 J

235 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

1 Here \( \ * γ A 'I („)

where σ3 = σ, σ2 - ta, σι = ρσ, σ0 = qcr; and t, ρ, q are the scale ratios. So we essentially arrive at the DoG + Gi - G2 model described in the previous section. Following a similar procedure described above, with the help of Eqs. (8), (8) and (10), in two-dimensions we get:

V6G(r) = mV2G(r) + V4G(r) (12)

Again using the limiting condition, as above:

V6G(r) = mS(x,y) + V*G(r) (13)

Eqs. (9) and (13) bear interesting resemblance with Mach's operator as shown by Ratliff (1965), a point which Young's retinal model (1987) also tried to emphasize. Nevertheless, unlike the present case, in these works, Mach's model was restricted only to the Laplacian operator, of course as envisaged by Mach himself according to Ratliff (1965). On the other hand, the present model also combines the Dirac-delta function (the contribution of the original intensity profile according to Mach) with a BhLaplacian operator (V4).

2.3 Some Useful Properties of the Proposed Model

We are going to examine how good these higher order derivative filters like V4G or V6G would be as "edge-detectors". One should remember that zero crossings and edges are not synonymous. Not all zero crossings can be identified with real physical boundary according to Marr (1982). Moreover, edges are also to be distinguished from "lines" according to Iverson and Zucker (1995). Canny (1986) provided a set of criteria for good edge detection. It was pointed out that there should be three major criteria for a good edge detector. These are as follows:

236 Vol. 17, No. 1-3, 2008 Theory of "Fuzzy" Edge Detection in Light of Human Visual System

1. Good Detection: The filter should not miss real edge. It should not show false positives. In other words it should maximize Signal to Noise Ratio (SNR or in short Σ). 2. Good Localization: Points shown as edge by the filtering operator should be as close as possible to the midpoint of the actual edge in the image (this criterion being represented by Λ ). 3. Avoidance of Multiple Responses: Possibly, the filtering function may generate some spurious edges in addition to the real edge. The distance between the spurious edges should be maximized in order to minimize the number of spurious edges within the spatial spread of the filter (the Multiple Response Criterion or in short MRC).

Although these criteria were not accepted as the best set by Koplowitz and Greco (1994), we may take a prima facie test of our model using a modified expression of Sarkar and Boyer (1991). Following Sarkar and Boyer (1991), the criteria for a good edge- detecting filter with impulse response function f(x) for 1-D edge are given below. The edge is assumed to be at the origin and the noise is Gaussian. The best edge detector should maximize the two functions known as ΛΣ and MRC, which are given as:

\f(0)\ ι f(0)\ (14)

and

Ma and Li (1998) claimed that a good filter for image processing should maximize the value of the product of these two functions, given by

Ρ = ( ΛΣ )( MRC ) (16)

237 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

One can easily verify that it is not possible to optimize the performance of the higher-order isotropic Gaussian Derivatives, like LoG or Bi-LoG, with respect to their scales using this modified expression of Sarkar and Boyer (1991). Optimization is indeed possible, however, when the same Gaussian Derivatives are expressed as the weighted combination of multi-scale Gaussians. Thus, while for DoG the optimized ratio is 1:5.7, as shown by Ma and Li (1998), it can easily be shown (see Appendix) that the same for a three Gaussian representation [Eq. (4)] of V4G is approximately equal to 1.6:100 and similarly, for a four Gaussian representation [Eq. (10)] of V6G , it is 1:6:98. 100 . The last mentioned ratio clearly shows that, from such optimization also, we arrive at the possibility of two very wide Gaussians, as has been proposed in Section 2.1 .These two optimized ratios, for V4G and V6G , also serve to show that even from this point of view, it is possible to find justification behind the approximation σ0 : σ2 -> 0, which ultimately leads to Eqs. (9) and (13). Another advantage of representing higher order derivatives as multi-scale Gaussians may be noted. The increasing number of scale-ratios, as we move from a lower to a higher order derivative, can easily be shown, enables an additional freedom of altering the function parameters along with a visual inspection to arrive at much better results that cannot be otherwise obtained by simply optimizing the Sarkar-Boyer's (1991) expression, for the multi- scale derivatives. Such altering of the parameters is also vindicated from recent neurophysiological experiments by Sceniak et al. (2002), in which it has been shown that VI simple cells (existence of similar extra-classical receptive field has already been reported in cat LGN by Sun et al. (2004)) can be beautifully modeled with the help of a DoG with two flanking Gaussian sub-units. For all practical purposes, this is clearly equivalent to a Bi- Laplacian of Gaussian. In this case, what becomes altered is the height and spread of these sub-units.

3. RESULTS AND DISCUSSION

Iverson and Zucker (1995) pointed out the shortcomings of the edge- detectors that rely upon convolution followed by thresholding. The present

238 Vol. 17, No. 1-3, 2008 Theory of "Fuzzy " Edge Detection in Light of Human Visual System

Fig. 1: Zero-crossing map of (a) the famous image of Lena, using the filter mS(x,y) + V4G(r) at (b) σ = 1.2 and m = . 1 and at (c) σ = 1.2 and m = .04 . We can clearly see that m plays the role of threshold in the zero- crossing operation. model is advantageous in this respect, since it circumvents the problem of thresholding. Yet, at the same time, this model also retains a sort of flexibility with respect to the value of m in Eqs. (9) and (13) because m may be varied to bring in the effect of thresholding. This was shown in Figure 1 by changing only m. With a decrease in the value of m, there is a decrease in the detected zero-crossings. From the works of Weiss (1994), it is well known that depending on the geometrical complexity of the image contour, higher order derivative filters can be used to extract the local invariants of shape. Yet, because the problem of de-localization rapidly increases with the increase of scale as shown by Marr (1982), in choosing the scale of smoothening there is very little freedom in these algorithms. Previously it was not possible to optimize the higher order derivatives with respect to their scale. Hence, though important subtle local variations of the edge can be captured by the higher order derivative filters, the overall processing of a noisy image may worsen as one moves from lower to higher derivatives due to uncontrolled smoothing. The present algorithm circumvents this problem and may in fact take advantage of the noise present in the image, resulting in a pronounced "fuzzy" effect in the sense of a half-tone in the zero-crossing map, which we shall now see. The introduction of the Delta function in the original scheme of a linear combination of Gaussians, brings in an additional advantage. Such a possibility of the existence of Delta function in the visual system is also vindicated from the works of Marr et al. (1980a), who proved from

239 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

(a) (b)

Fig. 2: Zero-crossing map of the three dimensional box (a), with externally added noise, using the filter mS(x,y) + V4G(r) at σ = 1.2 and m = .01. The intensity variation in the image (between the background and the top and the sides of the box) is reflected in the map (b) in the sense of a half-toning effect or "fuzzy" derivative computation.

psychophysical experiments the existence of a very narrow (single cone) channel in the human visual system (HVS). The Dirac-delta function with its multiplicative constant imparts into the zero-crossing map, additional information about image intensity. This has first been examined, by externally added noise to the simple image of the three-dimensional box (Figure 2). In this image, we can see three distinct gray levels, one in the background and two more (the top and the sides) in the box. We can see that this intensity variation between the background and the two gray levels in the three-dimensional box has been reflected in the edge map. This is somewhat odd, because classically, the purpose of edge-extraction is to come up with a clean edge map, rid of the shallow shading gradients. Nevertheless, from the point of view of image reconstruction, like in HVS for example, important shading information is absent in such a clean map. The HVS will require higher-order perceptual processes, to retrieve this shading information. The present model can form the basis of such perceptual processes. To clarify this viewpoint further, we present the case of a more complex image (Figure 3) intrinsically containing noise, for which a comparison has

240 Vol. 17, No. 1-3, 2008 Theory of Fuzzy" Edge Detection in Light of Human Visual System

(a) (b) (c)

Fig. 3: Zero-crossing map of (a) the image of a child, with intrinsic noise, using (b) Canny's edge detector (c) the filter mS(x,y) + V4Gfr) at Canny edge detector.

been made with Canny edge detector. Unlike the proposed filter, Canny edge detector expectedly, only detects the intensity edges and not the intensity variation in the image itself. In contrast, the proposed filter in the zero- crossing map has, along with simple edges been also able to reflect the light and shade information in the original image through a density variation of the zero-crossing points. The most apparent edges, like those detected by Canny algorithm, are also detected quite prominently by the proposed filter alongside such shading information as can be seen from Figure 3. The effect is also visible in Figure 1 and Figure 2. The presence of the Dirac-delta function in Eq. (13) essentially results in computing the sixth order derivative image V6Gf r) ® I by adding the original image with proper sign and weight (m) with the fourth order derivative image V4Gf r)®I. Such addition of the original image modifies the zero-crossings of V4G(r)®I in such a way that the number of occurrence of zero-crossings varies in proportion to the intensity of the original image. The final zero-crossing image therefore resembles a half-toned image. The areas with higher zero-crossing density looks darker compared to the areas with lower density of zero-crossings. The net effect is "fuzzy" edge detection in contrast to or rather, in complement with classical edge detection. Such "fuzzy" edge detection by cortical simple cells, according to the edge detection scheme envisaged by Marr-Hildreth (1980), may form the basis of intensity interpolation in the "raw primal sketch" for the ultimate purpose of image retrieval in human visual system.

241 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

REFERENCES

Bracewell, R. N. 2003. Fourier Transform and Its Applications, Third Tata McGraw Hill Indian Edition, 151-97. Canny, J. 1986.A computational approach to edge detection, IEEE Trans- actions on Pattern Analysis and Machine Intelligence, 8, 679-98. De Valois, R.L., Albrecht, D.G. and Thoreil, L.G. 1982. Spatial frequency selectivity of cells in macaque visual cortex, Vision Research 22, 545-60. Enroth-Cugell, C. and Robson, J.G. 1966. The contrast sensitivity of the retinal ganglion cells of the cat, Journal of Physiology (London), 187, 517-52. Enroth-Cugell, C. and Jakiela, H.G. 1980. Suppression of cat retinal ganglion cell responses by moving patterns, Journal of Physiology (London), 302, 49-72. Ghosh, K., Sarkar S. and Bhaumik, K. 2005. A possible mechanism of zero- crossing detection using the concept of extended classical receptive field of retinal ganglion cells, Biological Cybernetics, 93, 1-5. Ikeda H. and Wright, M.J. 1972. Functional organization of the periphery effect in retinal ganglion cells, Vision Research, 12, 1857-79. Iverson, L.A. and Zucker, S.W. 1995. Logical/linear operators for image curves, IEEE Transactions on Pattern Analysis and Machine Intelligence, 17, 982-96. Koenderink, J.J. and van Doom, A.J. 1982.The shape of smooth objects and the way contours end, Perception, 11, 129-37. Koenderink, J.J. 1984. The Structure of images, Biological Cybernetics, 50, 363-70. Koenderink, J.J. and van Doom, A.J. 1987. Representation of local geometry in the visual system, Biological Cybernetics, 55, 367-75. Koenderink, J.J. and van Doom, A.J. 1990. Receptive field families, Biological Cybernetics, 63, 291-7. Koplowitz J. and Greco, V. 1994. On edge location error for local maximum and zero-crossing edge detectors, IEEE Transactions on Pattern Analysis and Machine Intelligence, 16, 1207-12. Kruger, J. 1984. The shift-effect enhances X- and suppresses Y-type response characteristics of cat retinal ganglion cells, Brain Research, 201, 71-84. Lindeberg, Τ. 1994. Scale-space theory in computer vision, Boston, MA, USA, Kluwer Academic Publishers, 31-60. Logan B.F. Jr. 1977. Information in the zero-crossings of bandpass signals, Bell Systems Technical Journal, 56,487-510. Ma, S.D. and Li, B. 1998. Derivative computation by multiscale filters, Image and Vision Computing, 16, 43-53. Marr, D., Poggio, T. and Ullman, S. 1979. Bandpass channels, zero-crossings, and early visual information processing, Journal of Optical Society of America, 69,914-6.

242 Vol. 17, No. 1-3, 2008 Theory of "Fuzzy" Edge Detection in Light of Human Visual System Marr, D. and Hildreth, E. 1980. Theory of edge detection, Proceedings of Royal Society of London B, 207, 187-217. Marr, D., Poggio, T. and Hildreth, E. 1980a. Smallest channel in early human vision, Journal of Optical Society of America, 70, 868-70. Marr, D. 1982. Vision: A computational investigation into the human repre- sentation and processing of visual information, New York, NY, USA, W.H. Freeman and Company. Mcllwain, J.T. 1966. Some evidence concerning the periphery effect in cat's retina, Experimental Brain Research, 1, 265-71. Passaglia, L., Enroth-Cugell C. and Troy, J.B. 2001. Effects of remote stimu- lation on the mean firing rate of cat retinal ganglion cells, Journal of Neuroscience, 21, 5794-803. Ratliff, F. 1965. Mach bands: quantitative studies on neural networks in the retina, San Francisco CA, USA, Holden-Day, 253-332. Rodieck, R.W. and Stone, J. 1965. Analysis of receptive fields of cat retinal ganglion cells, Journal of Neurophysiology, 28, 833-49. Sarkar, S. and Boyer, K.L. 1991. On optimal infinite impulse response edge detection filters, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13, 1154-70. Sceniak, M.P., Hawken, M.J. and Shapley, R. 2002. Contrast-dependent changes in spatial frequency tuning of macaque vl neurons: effects of a changing receptive field size, Journal of Neurophysiology, 88, 1363-73. Sun, C., Chen, X., Huang, L. and Shou, T. 2004. Orientation bias of the extra classical receptive field of the relay cells in the cat's dorsal lateral geniculate nucleus, Neuroscience, 125,495-505. Weiss, I. 1994. High-order differentiation filters that work, IEEE Trans- actions on Pattern Analysis and Machine Intelligence, 16, 734-739. Witkin, P. 1983. Scale-space filtering, in: Proceedings of International Joint Conferences on Artificial Intelligence, Karlsruhe, Germany, 1019-22. Young, R.A. 1985. The Gaussian derivative theory of spatial vision: analysis of cortical cell receptive field line weighting profiles, General Motors Research Publication, GMR-4920. Young, R A. 1987. The Gaussian derivative model for spatial vision: I. Retinal mechanisms, Spatial Vision, 2, 273-293. Yuille, L. and Poggio, T. A. 1986. Scaling theorems for zero-crossings. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8, 15-25.

243 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

APPENDIX

In Eqs. (4) and (5), let us take, σ2 = σ , at = ίσ and σ0 = ρσ . Then from Eqs. (14) and (15), we get

2 2 2 2 (ΐ-/ )/Γ·- (l(ζ -/) ΛΣ =. ι π

MRC = 2π {T2xTj where, Mf.M2,^ * afiMM ^ι-^νΙ,^-Α Η ρ * {^ϊ Μ2 ,Μ2 ^ 44ν) t * Μ2 kf2 MF2

r

. M1

Then if we plot Ρ (prod) from Eq. (16) versus / - ρ, we obtain Figure 4:

From this method, we get, Pmax = 1.2 and the corresponding f = .06 and /> = .01, so that:

σο :σ\ :σ2 = 1:6:100. We proceed similarly, from Eqs. (10) and (11). We put,

244 Vol. 17. No. 1-3, 2008 Theory of "Fuzzy" Edge Detection in Light of Human Visual System

prod

β 0

Fig. 4: The product of SNR, localization and multiple response criteria has been plotted against the two scale ratios for indicating the optimized domain for the multi-scale Bi-Laplacian of Gaussian filter.

Then from Eqs. (14) and (15), we get:

_ fija+ß + y + sf ΛΣ ν«·

MRC = 2π •Jhh where,

a q + β p + yt

aßpqjl ßyptjl yStJl '

= σ-[π + δ2+2

| Sccqjl | ayqtfj | βδρ^ϊ -Jli?

245 Κ. Ghosh, S. Sarkar and Κ. Bhaumik Journal of Intelligent Systems

az/q + β2/ρ + χ2/ί

f aßpqji ßrptji γδί^Ι2 λ 41 h = 2σ + δ +4 Saqyfl | aygt-Jl | βδρ^ϊ

2 3 2 2 3 2 3 K (y/Üq ) (ylq +t ) (Φ + Ρ ) }

a2q3+ß2p3+y2t3

' aßp^J_2_ βγρ3?42 γδί3^2 λ t-i, -„1= σ J* J 2 + δ2+4 3 3 | Saffi | aYg\ 4i | βδρ 42

(Jl2+t2)3 (^|Ϊ+P2J3)

Then proceeding similarly, as the previous case, we get, Pmax =1.19 and the corresponding t =. 98 , ρ =. 06 and q =. 01, so that:

σ0 . σ, :σ2 : σ3 = 1. 6. 98.100.

246