The Spatio-Temporal Fourier Transforms Of The Eye
July 17th, 2005 | No Comments »At the risk of losing (further!) the interest of the biologically oriented vision scientist I must continue with physics. I noted on 7/14 a temporal interpretation of longitudinal chromatic aberration leading to a concept of three separate “time differentiated snapshots” corresponding to the three primary colors”. I further proposed that these corresponded to Land’s separate “Retinex’s”. In simplest terms, these represent three separate focal points for the RGB colors with the longest wavelength (red) coming to focus on the fovea and the other two within the eye. These correspond to three separate spatial Fourier transforms recalling Feynman’s interpretation that it is a fundamental property of a condensing lens system to bring light rays into time coincidence on the Fourier or focal plane (”the thicker portion of the lens in the middle slowing down light rays so that those coming through the thinner periphery can catch up”), All well and good..with three discrete “Fourier color separations” of the image defined.
But now…I must go back a bit and remind that in image processing a fundamental axiom is that the “Fourier transform of a Fourier transform reconstitutes the original image”. The second transform is termed an “inverse transform”. But… image processing has always considered that both of these transforms were of the same type, i.e., both spatial. It may not be so in the eye. The second inverse transform in the eye may actually be temporal. The image then in the eye may be a spatial temporal sequence of Fourier transforms. The first being spatial resulting from the condensing lens system of the eye and the second temporal possibly resulting from a simple geometrical construction to be demonstrated. This would mean that both space and time would be simultaneously brought into coincidence on the retinal surface.. a result from pure geometrical considerations! This would be interpreted as meaning that a “spatial instant” and a “temporal instant” come together on the retina.
If this were true it would seem that the eye brings three geometrically defined Fourier transforms into a single spatial-temporal coherence on the retinal surface.
The second temporal transform might follow from a geometrical construction similar to this:
Simply stated, the short wavelength transforms “take different paths” to reach the light detecting retina…possibly bringing all three tranforms into temporal coincindence?
For those who have trouble understanding the concept of a Fourier transform please go back and read the sidebar on this subject on the index (first) page of this site. It is truly easy to understand!
On another note,
The three geometrically defined light detection structures on the retina (i.e., cone/cone, cone/rod, and rod/rod), in addition to defining the three primary colors, define both long and short wavelength endpoints and exact center of the visual band. Thus there are actually only two “regions” defined - on either side of the geometric mid-band point to the ends of the band. These correspond to the long and short wavelength “records” that Land correctly deduced must be present. The “long record” extends from the fovea to the mid-band point and the “short record” from there to the rod/rod region ( ~ 20 degrees of retinal angle). I think that this is the genesis of the misunderstanding about Land’s theory and the charge that he was proposing a “two color” theory.
