Joined: May 2002
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.
But did he actually read the paper? Nilsson and Pelger in fact spend a paragraph explaining the calculation of each graph.
For Figure 1a, they say:
We let the evolutionary sequence start with a patch of light-sensitive cells, which is backed and surrounded by dark pigment, and we expose this structure to selection favouring spatial resolution. We assume that the patch is circular, and that selection does not alter the total width of the structures. The latter assumption is necessary to isolate the design changes from general alterations of the size of the organ. There are two ways by which spatial resolution can be gradually introduced: (i) by forming a central depression in the light sensitive patch; and (ii) by a constriction of the surrounding pigment epithelium Both these morphological changes reduce the angle through which the individual light-sensitive cells receive light. The relative effects that depression and constriction have on the eye's optical resolution is compared in figure 1a. Initially, deepening of the pit is by far the most efficient strategy, but when the pit depth equals the width (P= 1 in figure 1a), aperture constriction becomes more efficient than continued deepening of the pit. We would thus expect selection first to favour depression and imagination of the light-sensitive patch, and then gradually change to favour constriction of the aperture During this process a pigmented-pit eye is first formed which continues gradually to turn into a pinhole eye (see Nilsson 1990).
Looks like Nilsson 1990 (see reference above) might be the place to see how this was calculated. Is Nilsson required to repeat all his previous work in every new article, for the benefit of people like Berlinski?
For Figure 1b, Nilsson and Pelger write,
|As the aperture constricts, the optical image becomes increasingly well resolved, but constriction of the aperture also causes the image to become gradually dimmer, and hence noisier. It is the random. nature of photon capture that causes a statistical noise in the image. When the image intensity decreases, the photon noise increases in relative magnitude, and the low contrast of fine image details gradually drowns in the noise. If we assume that the retinal receptive field, delta[ro]ret and the optical blur spot, delta[ro]lens, are identical Gaussians, with half-widths being the angle subtended by the' aperture at a central point in the retina (this effectively means that the retinal sampling density is assumed always to match the resolution of the optical image), then we can use the theory of Snyder (1979) and Warrant & McIntyre (1993) to obtain the maximum detectable spatial frequency, vmax as:|
vmax = (0.375P/A) [ln (0.746A2 /I)]^½
where A is the diameter of the aperture, P is the posterior nodal distance, or pit depth and I is the light intensity in normalized units of 'photons per nodal distance squared per second per steradian'. We can now use this relation to plot resolution against aperture diameter (figure 1b). For a given ambient intensity and eye size there is an optimum aperture size where noise and optical blur are balanced in the image. A large eye or high light intensity makes for an optimum aperture which is small compared with the nodal distance. When the aperture has reached the diameter which is optimal for the intensity at which the eye is used, there can be no further improvement of resolution unless a lens is introduced.
Look at that, an equation, with these references to the theory cited:
|Snyder, A. W. 1979 Physics of vision in compound eyes. In Handbook of sensory physiology, vol. vii/6A, led. H.-J. Autrum, pp. 225-313. Berlin: Springer. |
Warrant, E.J. & McIntyre, P. D. 1993 Arthropod eye design and the physical limits to spatial resolving power., Prog. Neurobiol. 40, 413-461.
Did Berlinski really look up these references and not find the relevant theory?
For Figure 1c, Nilsson and Pelger write,
In a lensless eye, a distant point source is imaged as a blurred spot which has the size of the imaging aperture. A positive lens in the aperture will converge light such that the blur spot shrinks, without decreasing the brightness of the image. Most biological lenses are not optically homogeneous, as man-made lenses normally are (Fernald 1990; Nilsson 1990; Land & Fernald 1992). In fact, a smooth gradient of refractive index, like that in fish or cephalopod lenses, offers a superior design principle for making lenses: the optical system can be made more compact, and aberrations can be reduced considerably (Pumphrey 1961). A graded-index lens can be introduced gradually as a local increase of refractive index. As the focal length becomes shorter, the blur spot on the retina will become smaller. The effect this has on resolution was calculated by, using the theory of Fletcher et al. 1954) for an ideal graded-index lens (figure 1c). Even the weakest lens is better than no lens at all, so we call be confident that selection for increased resolution will favour such a development all the way. from no lens at all to a lens powerful enough to focus a sharp image on the retina (figure 1c).
Hmm, they say the optical theory is in:
|Fletcher, A., Murphy, R. & Young, A. 1954. Solutions two optical problems. Proc. R. Soc. Lond. [Physical Series] A223, 216-222 |
...which would, I expect, have some equations in it. Did Berlinski really look this up? If not, how can he say,
|But no details are given either in their paper or in its bibliography.|
(leaving aside the equation that they do include, which sure seems like details to me)
Edited by niiicholas on April 05 2003,11:42