Biol Cybern (2005) 93: 1–5 DOI 10.1007/s00422-005-0580-0 LETTER TO THE EDITOR K. Ghosh · S. Sarkar · K. Bhaumik A possible mechanism of zero-crossing detection using the concept of the extended classical receptive field of retinal ganglion cells Received: 2 November 2004 / Accepted: 26 April 2005 / Published online: 13 July 2005 © Springer-Verlag 2005 Abstract The extended classical receptive field (ECRF) of retinal ganglion cells has been modelled as a combination of three zero-mean Gaussians at three different scales that has been shown to be equivalent to a Biharmonic or Bi-Laplacian of Gaussian filter. It has also been shown that the ECRF can be approximated by a combination of Laplacian of Gaussian (LoG) and the Dirac-delta function. Zero-crossings detected with this operator are more informative than those detected by the traditional filters like LoG or Difference of Gaussians (DoG) that had been devised using the classical receptive field of the ganglion cells. We have also explained that such an additional information processing is not in contradiction with the recent experimental findings on the physiology of retinal ganglion cells. 1 Introduction As early as in 1868, Ernst Mach proposed a model for infor- mation retrieval from retinal illumination: The illumination of a retinal point will, in proportion to the difference between this illumination and the average of the illumination on neighboring points, appear brighter or darker, respectively depending on whether the illumination of it is above or below the average. The weight of the retinal points in this average is to be thought of as rapidly decreas- ing with distance from the particular point considered (Mead 1990). This model therefore points out the existence of a center- surround smoothing mechanism in the retina, in either polar- ity, a phenomenon that was to be verified much later through the discovery of the on and off-centered retinal ganglion cells (Kuffler 1953; Wiesel 1960) which motivated the develop- ment of a Difference of Gaussians (DoG) model of the clas- sical receptive field (CRF) of the ganglion cells (Rodieck K. Ghosh (B )· S. Sarkar · K. Bhaumik Microelectronics Division,Saha Institute of Nuclear Physics 1/AF Bid- hannagar, Kolkata-700064, India E-mail: kuntal.ghosh@saha.ac.in and Stone 1965; Enroth-Cugell and Robson 1966). The opti- cal illusion observed in Mach band can be explained by DoG model, a fact which shows that Mach’s model may be con- sidered to be the first psychophysical model of retinal signal processing. Mach further elucidated on this point: Let us call the intensity of illumination u = f(x,y). The brightness sensation v of the corresponding retinal point is given by v = u - m(d 2 u/dx 2 + d 2 u/dy 2 ), where m is a constant. If the expression in parentheses is positive, then the sensation of brightness is reduced; in the opposite case, it is increased. Thus, v is not only influenced by u, but also its second differential coefficients (Mead 1990). Marr and Hildreth (1980b) later looked upon the visual system as a complex information processing system that needs a zero-crossing detection for a raw primal sketch of the exter- nal world. They considered the Laplacian ∇ 2 , which is an ori- entation-independent second order differential operator and showed an equivalence of the DoG and ∇ 2 G (termed as the Laplacian of Gaussian or LoG) for a certain ratio of the scale parameters (i.e., the standard deviations) of the two Gaus- sians. According to this scheme, the zero-crossings are likely to be detected at the simple cell level of primary visual cortex, along with some orientation information and some measure of signaling amplitude. The latter is speculated to be obtained from the average local value of the slope of the zero-cross- ings, which may be determined from the sum of the inputs to the cell. It may be further pointed out that the realization of such a mechanism requires a large sampling density. This task of hyperacuity may be performed by the small cells, similar in nature to the cells found in layer IV of the monkey’s striate cortex. It was also shown with the help of Logan’s theorem that these zero-crossings alone could be very rich in infor- mation, although the ∇ 2 G filter itself falls outside the range of Logan’s condition (1977). Zero-crossing detection in pri- mary visual cortex was shown to be physiologically viable to a considerable extent (Hochstein and Spitzer 1984; Richter and Ullman 1986). On the other hand, from the observation of point weight function in the receptive field of a particular