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

Abstract

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 fi...

The Clinical, Philosophical, Evolutionary and Mathematical Machinery of Consciousness: An Analytic Dissection of the Field Theories and a Consilience of Ideas

“…The high entropy of the human brain with a potential abeyance of the area law of entropy concurs with Karl Pribram's holonomic or holographic theory of brain function [14].…”

“…Thermodynamic entropy, which is proportional to the number of states in a system, is also expressed as 670 2/1340 = 0.000149 10 14 4.7…”

“…In a review article on the biophysics of vision, the author summarized the concept of the formation of a visual percept. To form a visual percept, prior to cognizance, the brain convolves a stimulus, in this case, the function of light intensity, with a filter function, the selection of an ensemble of neurons [ 14 ]. The very essence of percept formation involves afferentation, a stimulus.…”

“…More accurately, one should ask what are the mechanisms that are operational for a privileged or strategic workspace to elicit or evince consciousness? We earlier alluded to the convolution of a visual stimulus with a Gabor filter function, or selective activity of a neuronal ensemble, as the minimal pre-requisite for setting up the perceptual mechanism of a visual scene [ 14 ]. We also explored the concept of synchrony of neuronal firing and gamma oscillations in evincing perception and awareness of an olfactory stimulus.…”