Representational Strategy in the Visual Cortex

In this paper, we generalized Daubechies'[2] results on 1D wavelets to 2D, and elucidated the conditions in which a family of Gabor wavelets will provide a complete representation of an image. We found that the phase space (scale, orientation, x, y) sampling density of the striate cortical simple cells is sufficiently dense to form a tight wavelet frame. This tight frame approximates the continuous wavelet transform, and its redundancy allows high resolution information to be represented in the low-resolution firings of the simple cells. We presented some experimental results to demonstrate this idea. I. GABOR WAVELET REPRESENTATION Simple cells in the primary visual cortex are known to be self-similar in their receptive field structures, i.e. the ratio of frequency bandwidth of a cell and its tuned spatial frequency is more or less a constant. Each simple cell's receptive field can typically be characterized by a Gabor elementary function with a 2:l aspect ratio and 1.5 octave in frequency bandwidth. A family of these cells sample the frequency domain in a log-polar fashion [3], [l], [5]. Imposing these constraints, and adding the requirement that admissible wavelets must have zero d.c., we derive the following family of Gabor wavelets, which is a function of 4 continuous variables, .e$( ss in 8+y cos8)' , ~ e i ( w r c o s ~ + w y s i n 8 ) (1) where E is the spatial frequency along 1: and v is the spatial frequency along y. w = d w is the radial This work was supported in part by NSF grant DMS 9121266 to David Mumford. The author would like to thank David Mumford, John Daugman and Josef Segman for their helpful advice. frequency and 0 is the wavelet orientation, i.e. 0 = tan-'(./(). The Gabor wavelet is centered at (-2 = 0, y = 0) and the normalization factor is such that < $,$ >= 1 . This family of wavelets can be generated through scaling and rotation from the following mother Gabor wavelet, whose Fourier transform is q j ( , ~ , = & { ~ $ I ( C ~ ) ~ + ~ U ~ I (3) where k is a constant which depends on the frequency bandwidth of the wavelet. For 1.5 octave bandwidth, k = 2.5. For 1 octave bandwidth, k = T . Each member of a discrete family of the Gabor-wavelets models the spatial structure of the receptive field of a simple cell. A set of such wavelets of 1 scale is shown in figure 1. 2. FRAME AND COMPLETENESS For a discrete set of functions {+m,n; m, n E 2 ) to constitute a frame, there should exist A > 0 and B < 00 such that Allf ) I 2 I I < f, l h L , n , k , l > l 2 I Bllf 1 1 2 (4) m,n,kJ for all f E L 2 ( R ) . A and B are called frame bounds. When a discrete family of Gabor wavelets forms a frame, they can provide a complete representation of the input functions f [2]. A discrete family of admissible 2D wavelets can be generated from the mother wavelet by dilation and rotation, $m,n,k,~(x, Y) = G m $ e , ( a i m x nb,, ai"y kb , ) (5) where &,(-2, y) = $(z cos(l0,)+ysin(l0,), --2 sin(lO,)+ ysin(l0,)); $ is specified by equation 2; 0, is the sampling interval in orientation and I is the index for the 0-8186-6950-0/94 $ .00