Joined: May 2002
Here's another gem from Berlinski 2003. Berlinski lists a larger number of problems that he has with the paper. One of them is that Nilsson and Pelger don't give any details about how they calculated optical acuity:
Nilsson and Pelger treat a biological organ as a physical system, one that is subject to the laws of theoretical optics. There is nothing amiss in that. But while theoretical optics justifies a qualitative relationship between visual acuity on the one hand and invagination, aperture constriction, and lens formation on the other, the relationships that Nilsson and Pelger specify are tightly quantitative. Numbers make an appearance in each of their graphs: the result, it is claimed, of certain elaborate calculations. But no details are given either in their paper or in its bibliography. The calculations to which they allude remain out of sight, if not out of mind.
The graphics he is referring to are Figure 1a, 1b, and 1c (attached).
Figure 1. Strategies for improving spatial resolution in an evolving eye. (a) An originally flat light-sensitive patch, or retina, is gradually invaginated (solid line) to form a pit whose distal aperture keeps the size of the original patch. The optical resolution is calculated as the inverse of the field of view of a point in the centre of the retina. At various points on the curve, the deepening of the pit is interrupted and all morphological change is instead spent on constriction of the aperture (broken lines). Calculations are made for aperture constriction to start when the pit depth, P. is 0.1, 0.5, 1.0 and 1.5 times the original width of the patch. (b) Optimisation of lenses aperture. Continuations of the dashed P = 1 curve in (a), but with photon noise taken into account with equation (1). The three curves are calculated for ambient intensities (I) separated by two log units. The upper curve is thus for an intensity 10000 times higher than that for the lower curve. The intensity is in units normalised to the nodal distance (pit depth): photons per nodal distance squared per second per steradian. The unconventional use of nodal distance instead of micrometres in the unit allows the three curves to be interpreted as eyes differing in size by a factor of 10. assuming constant intensity, the upper curve is thus for an eye with h is 100 times larger than that for the lower curve.
There is an optimum aperture size, indicated vertical lines, beyond which resolution (maximum resolvable spatial frequency) cannot be improved without a lens. © The optical resolution plotted as a function of the gradual appearance of a graded-index lens. The spherical lens is assumed to fill the aperture, and to be 2.55 lens radii away from the retina (e.g. as in fish eyes (Fernald 1990). The central refractive index of the lens is plotted on the horizontal axis. From this value the refractive index is assumed to follow a smooth gradient down to 1.35 at the peripheral margin. The calculation demonstrates that optical resolution continuously improves prom no lens at all to the focused condition where the central refractive index is 1.59. The maximum resolution in the focused Condition will be limited by both photon noise, as in (b), and by diffraction in the lens aperture, but none of these limitations are significant within the range plotted. The vertical axis of all three graphs (a) © was made logarithmic to allow for comparisons of relative improvement. A doubling of performance is thus always given by the same vertical distance.
Edited by niiicholas on April 05 2003,13:03