This explanation of light interaction with the retina of the eye makes it absolutely (actually geometrically!) clear that this surface evolved to detect, and does so detect, the 2-D Fourier transform of the perceived visual image. It does not directly detect the “camera photograph” that has been assumed for so long in the history of vision science. The retina is not an array of “red, green, and blue dots” as characteristic of color photographic film or the silicon imaging chips of contemporary digital cameras. The historically measured plan (Osterberg and following) of cone and rod receptors on the retinal surface doesn’t even look like this but the field persists in ignoring these data!
I have tried to explain in this work the nature of a 2-D Fourier transform in non-mathematical terms but have invariably found that the mere mention of the term “Fourier” causes the listener to nod off! I believe that this situation represents a fundamental problem for vision science. Matters “Fourier” are generally the province of the disciplines of electrical engineering and/or image processing who articulate the subject in mathematical terms not generally accessible to biologists, geneticists, etc. It seems that this is a “cross-disciplinary problem and that these EE groups must increase their involvement in the vision field.
The “Fourier equation” that describes the imaging process defines that light detection at the “Fourier plane” involves two terms that correspond to light intensity and phase detected at each point (or “pixel”) of the visual image. This plane corresponds to the focal plane of an imaging system and not the intensity-only sensitive image plane used in camera etc. Even our most advanced imaging technologies – both film and digital imaging – are capable of detecting and processing only the intensity of light falling on the imaging surface. We do not possess imaging technology that can simultaneously detect both light intensity and phase.
But the eye does!
I have attempted to describe in non-mathematical terms that detection of light phase at the retina corresponds to the direction of the impinging light ray with this factor, in addition to detection of light intensity at that point hoping that this would lead one’s mind to how the eye reconstructs the visual image. But, the concept of 2-D Fourier imaging remains difficult. A quote from Brian Hagan’s brilliant paper (“The Mathematical Transformation of Growth and Form”, Medical Hypotheses, 6, 559-609, 1980) under the heading Dissimilarity Between the Object and It’s Transform, “Under Fourier transformation, the big becomes small and vice versa…..”.
I suppose that the oft referenced attempt to portray the Fourier transformation of 2-D images is Kevin Cowtan’s Book of Fourier. These are beautiful illustrations of transformations that use color to encode light phase. They give some general idea of how Fourier transformations appear and should be studied.
A perhaps even better idea might be gained by viewing the images of a site authored by John Brayer of the University of New Mexico http://www.cs.unm.edu/~brayer/vision/fourier.html. If you wish, disregard the mathematical formulas at the beginning and scroll down to the images and their transforms. A quote from the site: “Now, with the above introduction, the best way to become familiar with Fourier Transforms is to see lots of images and lots of their FTs”. I would suggest reading this work. The analogy with a “radio station” is particularly good. It should become clear in viewing the images and their transforms how the transforms appear to relate to the retinal response posited in this work. The intensity of the central “spot” corresponds to the response of the fovea (solely at long wavelengths) and “sets” the overall brightness of the image for the visual system.
To conclude with two more thoughts: 1.) it seems clear that the retina processes color (in the manner described by Land) deriving this information from an “overlay” on the retinal surface of the spatial frequency 2-D Fourier transform. This is an exciting thought as it may portend entirely new technological imaging modalities that for the first time duplicate the process of vision, and 2.) I have described elsewhere in this work how the inter-receptor light detection devices of the retina, that are the basis of this explanation, simultaneously possess the ability to detect both the intensity and phase of incident light.
GCH,
Tucson, AZ
11/13/07
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