The Biophysics of Visual Edge Detection: A Review of Basic Principles.

The mathematization of nature is an age-old concept. The Greeks sought harmony in the celestial spheres. The Arab geometers constructed a spherical geometry of the heavens. Later, Galileo Galilei arithmetized kinematics. As the centuries advanced, polymaths like the Dutchman Christiaan Huygens applied more advanced mathematics in order to understand natural phenomena. It was not until the turn of the twentieth century that a more comprehensive mathematical approach to understanding biological phenomena was sought by D'Arcy Wentworth Thompson. This leads us to our current review of the biophysics of visual edge detection. This is an unfolding saga of stunning experimental revelations in unison with an underlying mathematical edifice. The concept of visual contrast is a fundamental idea in order to understand the phenomenon of visual edge detection. We begin with contrast visual testing and the development of frequency tuning curves, which provided an insight into the multi-channel processing of selective spatial frequencies by the visual cortex. The single-cell recordings from the simple cells of the cat visual cortex unfolded the gamma distribution curves of different neuronal firing frequencies for different spatial frequencies. The theoretical construction of the convolution of Gabor wavelets with stimulus intensity goes hand-in-hand with the experimental observation of the separation of simple visual cortical cells into even and odd functions, a spectacular finding. In this review, we march the reader through the mathematical basics and the pathophysiologic correlates. Beginning with a simple Fourier analysis of a square wave, Weber's biophysical law, and the gamma distribution of contrast tuning curves, we gradually introduce Fourier transforms, the uncertainty principle of waveform analysis, the basics of wavelet theory, Gabor elementary signals and transforms, the concepts of coherence, and Weyl group representation theory. Group theory provides the symmetry operations necessary to preserve the fidelity of an image as it travels from the retina and cascades up the visual cortex. Unitary operators that allow a retinal displacement of an image to be reflected by a similar displacement in the visual cortex is also a fundamental principle. Along the way, we encounter the Convolution Theorem of Fourier transforms, which is critical in constructing a visual percept. We intermittently interject relevant clinical data as we unpack the mathematical complexities. The advanced mathematics deployed in the biophysics of vision makes for difficult reading. There is a paucity of step-by-step reviews of this subject. Our approach is heuristic and at the end of this review, one should be able to follow superficially the algorithmic steps in understanding visual edge detection using Gabor filters. Therefore, we will adopt the Socratic method of asking questions and providing answers to help us through the complex web of mathematics. In a nutshell, we will show that a Gabor filter is the inner product of a Gaussian distribution and the wave function. The Fourier transform of the convolution of the Gabor filter and the stimulus intensity function is what is recorded from simple visual cortical cells. This is not a coincidental observation, as nature economizes and utilizes a function that minimizes the uncertainty principle of signal extraction.