Aberration Chromatique Explication Essay

For years, camera-makers have sought ways to avoid chromatic aberration—the color fringes that occur when various wavelengths of light focus at different distances behind a lens.

But where photographers see a problem, some sea creatures see possibility.

A new study, co-authored by the father-and-son team of Christopher and Alexander Stubbs, suggests that chromatic aberration may explain how cephalopods—the class of animals that includes squid, octopi and cuttlefish—can demonstrate such remarkable camouflage abilities despite only being able to see in black and white. The study is described in a July 4, 2016 paper in the Proceedings of the National Academy of Sciences.

"There's been a long-standing paradox that (cephalopods) manifest these vivid chromatic behaviors," Christopher Stubbs, the Samuel C. Moncher Professor of Physics and of Astronomy, said. "That would lead any observer, even a lay person, to conclude that they must be able to deduce things about coloration."

"I have always been fascinated by these animals, and have had the opportunity to watch them perform their camouflage act while conducting field work in Indonesia," Alexander Stubbs, a Berkley graduate student and lead author of the study, said. "We believe we have found an elegant mechanism that could allow these cephalopods to determine the color of their surroundings, despite having a single visual pigment in their retina."

But what would possess a Harvard physicist to devote time and energy to one of the most persistent mysteries in biology? For Stubbs, the answer is simple—his son.

"He chased me down with an idea he'd come up with, and the more we talked about it, the more sense it made," he said. "I credit my co-author with having the a-ha moment here."

That a-ha moment, Christopher Stubbs said, was the realization that cephalopods could potentially detect color by adjusting the focal position of their eyes to detect different wavelengths of light, and then composite each into a "color" image of their world.

"You can think about it like a digital camera dithering back and forth to find the crispest image," he said. "To me, what's really persuasive about this argument is...the pupils in these animals are an off-axis U shape, and that actually maximizes this chromatic signature at the expense of image sharpness. So it actually looks like there's been selective evolutionary pressure for their pupil shape to maximize this phenomenon."

To understand just how cephalopods might take advantage of chromatic aberration, Christopher Stubbs turned to code he's earlier written for astrophysics research and created a computer model of how the animals' eyes work.

"People have done a lot of physiological research on the optical properties of lenses in these animals," he explained. "We wrote some computer code that essentially takes test patterns and moves the retina back and forth, and superimposes that on the image and then measures the contrast."

Though it's not definitive evidence of how cephalopods understand color, Christopher Stubbs said the mechanism described in the study does agree with earlier studies of cephalopod eyes.

"I'm not a life scientist, but I think in some ways, this is such an elegant mechanism that it would be a shame if nature didn't capitalize on it," he said.

Ultimately, Alexander Stubbs said, the hope is that the study will offer other researchers a direction for study in the search for a conclusive answer to how squid and octopi became masters of camouflage.

"This is an entirely different scheme than the multi-color visual pigments that are common in humans and many other animals. High-acuity "camera style" lens eyes in octopus, squid and cuttlefish represent a completely independent evolution of complex eyes from vertebrates so in some sense we shouldn't necessarily expect that this lineage would solve problems like color vision in the same way. These organisms seem to have the machinery for color vision, just not in a way we had previously imagined."

Alexander Stubbs said. "We also conducted an in-depth review of prior literature evaluating conflicting evidence for color vision, and found prior behavioral studies suggesting a lack of color vision represent special cases and are consistent with our model. We hope this study will spur additional behavioral experiments by cephalopod community."

Explore further:Studying 'squid skin' to create new camouflage patterns

More information: Spectral discrimination in color blind animals via chromatic aberration and pupil shape, PNAS, www.pnas.org/cgi/doi/10.1073/pnas.1524578113

Significance

We describe a means of obtaining spectral information using the principles of physical optics and an off-axis pupil shape without requiring spectrally distinct photoreceptor classes. The mechanism described here offers a possible solution to a long-standing puzzle in marine animals: cephalopods dramatically change color for both producing chromatically matched camouflage and signaling to conspecifics, despite having a single photoreceptor channel. The ability of these animals to achieve such excellent color matching to their surroundings, despite being “color blind” in the traditional sense, can be understood if they exploit chromatic aberration to deduce spectral information. The bizarre off-axis pupils of these animals can be understood as an adaptation that maximizes spectral information, even at the expense of image acuity.

Abstract

We present a mechanism by which organisms with only a single photoreceptor, which have a monochromatic view of the world, can achieve color discrimination. An off-axis pupil and the principle of chromatic aberration (where different wavelengths come to focus at different distances behind a lens) can combine to provide “color-blind” animals with a way to distinguish colors. As a specific example, we constructed a computer model of the visual system of cephalopods (octopus, squid, and cuttlefish) that have a single unfiltered photoreceptor type. We compute a quantitative image quality budget for this visual system and show how chromatic blurring dominates the visual acuity in these animals in shallow water. We quantitatively show, through numerical simulations, how chromatic aberration can be exploited to obtain spectral information, especially through nonaxial pupils that are characteristic of coleoid cephalopods. We have also assessed the inherent ambiguity between range and color that is a consequence of the chromatic variation of best focus with wavelength. This proposed mechanism is consistent with the extensive suite of visual/behavioral and physiological data that has been obtained from cephalopod studies and offers a possible solution to the apparent paradox of vivid chromatic behaviors in color blind animals. Moreover, this proposed mechanism has potential applicability in organisms with limited photoreceptor complements, such as spiders and dolphins.

We show in this paper that, under certain conditions, organisms can determine the spectral composition of objects, even with a single photoreceptor type. Through computational modeling, we show a mechanism that provides spectral information using an important relationship: the position of sharpest focus depends on the spectral peak of detected photons. Mapping out contrast vs. focal setting (accommodation) amounts to obtaining a coarse spectrum of objects in the field of view, much as a digital camera attains best focus by maximizing contrast vs. focal length. We note that a similar phenomenon has been advanced as a possible explanation for color percepts in red/green color-blind primates (1); however, primates have not evolved the off-axis pupil shape found in nearly all shallow water cephalopods that enhances this effect.

The only other known mechanism of color discrimination in organisms involves determining the spectrum of electromagnetic radiation using differential comparisons between simultaneous neural signals arising from photoreceptor channels with differing spectral acceptances. Color vision using multiple classes of photoreceptors on a 2D retinal surface comes at a cost: reduced signal to noise ratio in low-light conditions and degraded angular resolution in each spectral channel. Thus, many lineages that are or were active in low-light conditions have lost spectral channels to increase sensitivity (2).

Octopus, squid, and cuttlefish (coleoid cephalopods) have long been known to be among the most colorfully active organisms, vividly changing color to signal conspecifics and camouflage. In 350 BCE, Aristotle (3) remarked that the octopus “seeks its prey by so changing its color as to render it like the color of the stones adjacent to it; it does so also when alarmed.”

Cephalopods use their control of skin coloration to become (i) inconspicuous by camouflaging against local backgrounds (Fig. 1, Fig. S1, and Movie S1) or (ii) highly conspicuous during colorful mating and threat displays (Fig. 1, Fig. S2, and Movie S2). Despite this chromatically active behavior, genetic and physiological studies (4⇓⇓–7) show that (with one exception) cephalopods lack multiple photoreceptor types. Cephalopods also fail certain behavioral trials (7⇓⇓⇓–11) designed to test for color vision by opponent spectral channels.

Fig. 1.

Cephalopod behavior and pupil shapes. Figs. S1 and S2 and Movies S1 and S2 show additional examples. Many shallow water cephalopods produce colorful displays [(A) Australian giant cuttlefish Sepia apama] to conspecifics and accurately color-match natural environments to camouflage [(B and C) broadclub cuttlefish Sepia latimanus]. Their pupil shapes [(D) S. bandensis] maximize chromatic blur. Images courtesy of (A) Klaus Stiefel, (B) Flickr/Lakshmi Sawitri, (C) Ken Marks, and (D) Roy Caldwell.

Fig. S1.

Color matching and pupil shape in cephalopods. Fig. 1 and Movie S1 show more examples. (AD) The coral reef broadclub cuttlefish Sepia latimanus lives in one of the most chromatically complex ocean environments and accurately color-matches natural environments to camouflage. More examples of color matching in cephalopods are found in Movie S1. (E) The pupil in shallow water squids (such as this Sepioteuthis) maximizes off-axis light when imaging in the horizontal plane. (F) Shallow water octopus species, such as this Octopus vulgaris, maximize off-axis light when imaging the bottom. Images courtesy of (A) Ken Marks, (BD) Flickr/Lakshmi Sawitri, (E) Klaus Stiefel, and (F) Alexander Stubbs.

Fig. S2.

Examples of highly colorful signaling in cephalopods. Fig. 1 and Movie S2 show more examples. (A–C) The big-fin reef squid Sepioteuthis lessoniana vividly changes color while signaling to members of its own species. (D) The Australian giant cuttlefish Sepia apama also uses a colorful display, and the fine network of black lines allows for an easy determination of chromatically induced defocus and thus, chromatic information. During displays, the cuttlefish pupil is typically maximally contracted as a semiannulus, suppressing other sources of image degradation while maximizing chromatically induced blurring. Images courtesy of (A and C) © Gary Bell/OceanwideImages.com, (B) Ria Tan/Wildsingapore.com, and (D) Richard Ling.

We are faced with two distinct but related paradoxes: (i) how can these animals with a single photoreceptor achieve good background color matching, and (ii) why would they break camouflage to produce risky colorful mating displays (readily visible to predators with color vision) unless this chromatic information was visible to conspecifics and carried some selective advantage?

Previous attempts to reconcile these apparent paradoxes include suggestions that (i) the animals do not actually match natural background colors (12) or (ii) multiple photoreceptor types could exist (13, 14) in the animal’s skin. Neither of these explanations resolves the puzzle of “color-blind camouflage,” and researchers remain in search of a mechanism that allows for this ability (7, 14⇓–16). We are unaware of a proposal for how natural selection would drive the evolution and maintenance of colorful intraspecific displays in these soft-bodied mollusks if this information was not available to the animals themselves.

Contradictory Evidence: Chromatic Behavior but a Single Opsin

The extent of color matching in cephalopods remains somewhat controversial in some circles, but we assert that shallow water cephalopods often match the coloration of natural backgrounds (Fig. 1, Fig. S1, and Movie S1), and we encourage readers to examine Movie S1, which shows cephalopod camouflage in their natural habitat, and reach their own conclusions. Some had claimed (12) that these organisms simply match the brightness and spatial scale of patterns in their environment, tricking the human visual system without actually requiring a color match. Numerous studies (14, 17⇓⇓⇓–21) show, however, that cuttlefish and octopus actively vary their spectral reflectance in response to background color rather than simply modulating their luminance.

Kühn (21) conducted a series of behavioral experiments comparing the octopus and cuttlefish camouflage responses when placed on a series of greyscale and colored substrates. His data show statistically significant evidence that these organisms expand their long wavelength-reflecting chromatophores when on spatially variable red or yellow backgrounds but that they primarily expand black chromatophores when on corresponding greyscale backgrounds (21). Kühn (21) concluded that these organisms must have the ability to discriminate spectral content.

Contemporary laboratory and field observations (17⇓⇓–20) show that octopus and cuttlefish produce high-fidelity color matches to natural backgrounds (Fig. 1). The most definitive recent evidence for color matching in a laboratory setting used (14) a hyperspectral imager in conjunction with spectral angle mapping to show that cuttlefish varied their spectral reflectance (chromatic properties) to maintain excellent spectral matches to a diversity of natural backgrounds and interestingly, maintained poorer matches in brightness (luminance). These studies (14, 17⇓⇓–20) corroborate the earlier result by Kühn (21): cephalopods vary their spectral reflectance by active control over their chromatophores in response to natural backgrounds rather than simply varying their luminance.

Some have suggested that cephalopod skin might contribute to spectral discrimination through either undiscovered additional opsins (14) or filtering the single known opsin response. The recently published octopus genome (22) did not identify any additional opsins using both whole-genome sequencing and transcriptome sequencing of skin tissue, despite a focus on identifying G protein-coupled receptors. Across a diversity of taxa, all cephalopod studies to date have found rhodopsin transcripts in the skin identical to those in the eye (23), and the skin’s spectral response to light is nearly identical to that of the retina (24). Given multiple strong lines of evidence against additional undiscovered skin opsins and no described mechanism for spectral discrimination arising from rhodopsin alone, this competing hypothesis is not currently viable. Additionally, absent a focusing element, detectors on the skin act as wide-angle nonimaging light sensors and cannot provide useful information regarding background coloration or signals produced by conspecifics.

Chromatic Blurring and the Importance of Pupil Shape

Fig. 2 shows the mechanism that we are proposing for how chromatic aberration can be exploited to achieve spectral sensitivity. As we show below, the off-axis pupils of cephalopods combine with the wavelength dependence of the lens index of refraction to generate chromatic blur; different wavelengths come into focus at different distances from the lens. The spectral content of a structured scene can be deduced by sweeping through focus (i.e., changing the lens to retina distance) and seeing how the image blurring varies. A key element in our argument is the observation that the off-axis pupils common in cephalopods actually maximize the chromatic blurring in their visual system (Table S1). These animals would have better acuity if they had evolved a small, on-axis pupil, such as the one in the eye of the reader. Instead, they seem to have sacrificed overall acuity in favor of chromatic blurring, which we suggest here as a mechanism for spectral discrimination. This mechanism is similar to that in the recent observation (25) that vertical and horizontal pupils produce astigmatic blurring.

Fig. 2.

Chromatic blur and pupil geometry. The (A) full and (C) annular aperture pupils produce more chromatic blurring (CB) than (B) the small on-axis pupil, because they transmit rays with a larger ray height h. Vertical lines show best focus positions for blue, green, and red light.

SI Experimental Procedures provides a detailed description of the numerical modeling that we performed to assess the quantitative variation of blurring vs. spectral structure and focal spacing. These calculations were based on the measured optical properties of cephalopods, and we show that (despite claims to the contrary in previous works by others) chromatic blurring dominates the image quality for these animals. This chromatic aberration is what affords them the opportunity to exploit this mechanism for achieving color sensitivity.

Table S1.

Cephalopod retinal image quality budget

SI Experimental Procedures

Chromatic Aberration Computation.

Chromatic blurring (Fig. 4) was computed with an MATLAB code adapted from a program initially written by C.W.S. to investigate the out of focus properties of the large synoptic survey telescope. The point spread function (PSF) and encircled energy diagrams (Fig. S4) shown were computed with a related program. Both are available on request.

The detected light intensity I(i,j) in each pixel (i,j) of the simulated retinal image is given bywhere Φsolar(λ) is the solar photon irradiance spectrum; the exponential term accounts for the reduction in down-welling photon flux at a water depth D with an attenuation length z(λ); R(x,y,λ) is the spectral reflectance of the portion of the scene at a location x,y; PSF(x,y,i,j,λ) is the wavelength-dependent PSF of the visual system at pixel (i,j) for light arriving from location (x,y) in the scene; and O(λ) is the sensitivity function of the photoreceptor opsin. The limits of integration span the spatial extent of the scene within the field of view and the spectral range of interest [in this case, band-limited by the opsin response O(λ)].

We explored the performance of the cephalopod visual system for different pupil shapes, colored test patterns, and accommodation (lens to retina) distances with a numerical simulation that

  • i) Computed the spectrum of solar illumination after taking into account the filtering properties of seawater at the chosen depth of 3 m,

  • ii) Constructed simulated test patterns with reflectance spectra characteristic of the marine environment to provide a standardized framework for quantitative comparison of image sharpness,

  • iii) Determined the amount of light at each of a set of discrete wavelengths that was out of focus for each choice of pupil, test pattern coloration, and accommodation value,

  • iv) Computed the fraction of light detected because of the limited spectral response of the single opsin in the retina,

  • v) Summed up the wavelength by wavelength blurred images to arrive at a final simulated retinal image, and

  • vi) Computed an image sharpness metric for each situation analyzed.

We then assessed the extent to which variation in image contrast with accommodation can be used to extract information about the spectral characteristics of the scene. Of particular interest was the comparison of spectral information for different pupil shapes to explore the evolutionary advantage of the peculiar off-axis annular pupils that are common in cephalopods.

We describe each of these aspects of the image simulation and analysis in more detail below.

Illumination.

We used a ground-level spectrum of solar irradiance from https://www2.pvlighthouse.com.au/resources/optics/spectrum%20library/spectrum%20library.aspx multiplied by λ to convert to relative photon irradiance. To account for illumination attenuation at a 3-m depth D, we used the Pacific seawater optical attenuation length data from the work by Morel and Maritonrena (51). We used the attenuation lengths appropriate for 0- to 2-m depths, which correspond to 0.043-mg m−3 chlorophyll density. The resulting photon irradiance spectrum is presented in Fig. S3A.

Reflectance Spectra and Test Image Generation.

Each pixel in the simulated test image was assigned a reflectance spectrum that was a weighted superposition of template reflectance spectra. To simulate the colors encountered in marine settings, we use measured (50) reflectance spectra of blue and yellow tropical fish.

The reflectance spectra are shown in Fig. S3C. We produced a variety of dual-color bar charts (shown in Fig. 4 A–E) with varying spatial frequency to assess the acuity of the resulting images. The spatial scale of these test images was set to 5 μm/pixel, which corresponds to the typical diameter of the photosensitive structures (rhabdomes) that pave the retinal surface of cephalopods. We produced bar chart test patterns with 112- to 2-pixel bright bar widths. At our sampling of 5 μm/pixel, this resolution corresponds to 165- and 10-μm widths, respectively, on the retina.

PSF Computation.

The PSF at each wavelength determines the sharpness of the image formed on the retina. The PSF, in turn, is determined by the combination of the pupil shape, the refractive properties and configuration of the optical components, the spectrum of light reflected by the scene, and the shape and location of the retinal surface. Rays first propagate through the pupil, which determines both the collecting area and the off-axis distances h of the rays that are imaged onto the retinal surface. The cephalopod lens is well-approximated by a sphere with a radial gradient in the index of refraction that produces a remarkably effective correction for spherical aberration (26). The wavelength dependence of the index of refraction does, however, produce chromatic aberration. Different wavelengths have different effective focal lengths. The blur induced by this depends on the angle at which the rays intersect the optical axis, which in turn, scales with the ray’s distance off axis. Pupils that transmit a large proportion of off-axis rays produce more chromatic blurring than pupils that are predominantly on axis.

When chromatically out of focus rays emanating from a point source at infinity of a given wavelength intersect the retina, they produce a PSF that is a scaled image of the pupil (Fig. 2).

Our MATLAB program uses the wavelength dependence of the focal length of the Octopus australis eye as reported by Jagger and Sands (26). Their laboratory measurements show a subpercentage perturbation in focal length caused by residual spherical aberration but a chromatic fractional shift in focal length, δf/f, of 4.1% between 450 and 700 nm (26). A second-order polynomial fit to the data in the work by Jagger and Sands (26) yielded δf/f = −5.4676 × 10−5λ2 + 0.0794λ − 27.1047, with δf/f = 0 at 550 nm.

We modeled a cephalopod lens with a 10-mm diameter d and a focal length of f = 12 mm at 550 nm measured from the center of the lens. The spectral range used in the computer model was restricted to 450 nm < λ < 650 nm to avoid making an extrapolation from the measured chromatic focal changes reported by Jagger and Sands (26). We ran through this spectral range with a step size of Δλ = 5 nm. Substantial amounts of illumination (15% of the photons) can be between 350 and 450 nm, depending on the transmission through the water column. The chromatic focus perturbations are enhanced at short wavelengths; therefore, for our PSF estimates (but not for the image contrast calculations), we do extrapolate the data from Jagger and Sands (26) down to 350 nm using the expression given above in conjunction with the attenuated photon spectrum and the opsin response. We used the opsin photon sensitivity curve as a function of wavelength from Chung (6).

We performed PSF calculations for three different planar pupil masks. One corresponds to the full useful aperture of the lens with an 8-mm pupil diameter. The second mask is an axially centered pupil with a 1-mm diameter, the size at which diffraction and chromatic effects are comparable. The third mask approximates the U-shaped semiannular pupil seen in many free-swimming diurnal cephalopods under bright illumination (shallow water squid and cuttlefish), with a 6-mm i.d., a 6.66-nm o.d., and a polar angle extent of 180°.

Representative PSFs for the semiannular pupil and a point-like white reflector, R(x,y,λ) = δ(x,y), are shown in Fig. S4 A and B. The PSF that we computed when 400-nm light is brought to a sharp focus (Fig. S4A) shows the spike from the point source at 400 nm superimposed on the out of focus pupil images from the other wavelengths from either side of focus. The PSF obtained when the lens is far out of focus (Fig. S4B) is, in effect, a radially dispersed image of the point source. These intensity distributions are not well-represented by Gaussian PSFs, and therefore, we used encircled energy (52) (Fig. S4D) computations to determine a Gaussian-equivalent FWHM for these PSFs.

The PSF produced by a large circular pupil (Fig. S4C) is axisymmetric and breaks the relationship between radial position and wavelength. The light at the center of the PSF contains all wavelengths but only from the rays that pass close to the optical axis.

CTF Analysis.

We simulated this visual system by computing the retinal image that would be produced using various combinations of pupils (and their corresponding PSFs) and test pattern scenes. The focal plane image at each wavelength is a convolution of the appropriate wavelength-dependent PSF with the test pattern image produced on the retina. We computed this convolution for a discrete set of wavelengths (each 1 nm across the opsin response) and summed the resulting hyperspectral synthetic image along the spectral direction to arrive at a final full spectrum simulated image. We then computed an image quality metric (a quantitative indicator of image sharpness) by measuring the contrast in the simulated image.

Our image simulation program used an outermost loop that stepped through a sequence of lens to retina separations. For each of these accommodation values (i.e., lens to retina separation), we then iterated through a set of discrete wavelengths integrated over illumination wavelengths and the opsin response function for 450 nm < λ < 650 nm and computed the appropriate focus offset and corresponding blur for that wavelength. The sum, in the wavelength direction, of the blurred hyperspectral image stack produced a 2D simulation of the test pattern image on the retina,. We took care to introduce appropriate parity flips of the annular pupil (apparent in Fig. S4 A and B) according to the sign of the focal length offset at each wavelength. The wavelength-summed images were normalized so that the intensity value in a resolved test bar was unity.

The blurred test pattern images were each analyzed to assess the sharpness of the image using line profile plots across the images (Fig. 4). We defined a CTF metric that has the merit of being simple and that suppresses aliasing artifacts from the bar pattern and the pupil shape; we computed two times the SD of the pixel values in each image. A crisp image has a bimodal normalized intensity histogram (predominantly ones and zeros) and a high SD. A highly blurred image has an intensity histogram peaked at the mean pixel value and a low SD. By mapping out this CTF metric vs. lens to retina spacing, we can quantitatively assess the extent to which image sharpness can be used to deduce scene spectral content. These results are presented as CTF vs. accommodation plots for various pupil shapes and simulated scene spectral content in Fig. 4.

The animation in Movie S3 shows how the contrast of the simulated image depends on focal setting for the black to yellow test pattern shown in Fig. 4.

Determination of the Image Quality Budget.

To assess whether the chromatic aberration effects really do dominate the sharpness of the images, we produced an image quality budget that includes other sources of potential image blurring.

We evaluated the various terms in the image quality budget (Table S1) using geometrical or diffractive optics principles as appropriate. Each entry is provided as Gaussian-equivalent FWHM in the focal plane in units of micrometers for the f/1.2 spherical lens and a 12-mm focal length, with a radial gradient in index of refraction that compensates for spherical aberration. To convert to an equivalent angular resolution (which is independent of lens diameter for those terms that are in the geometrical optics regime), FWHM∠ = 2arctan(FWHM/(2FL)).

Adding all of the blur contribution in quadrature for the annular pupil case yields an angular resolution of FWHM∠ = 0.3°. This estimate is in broad agreement with behavioral experiments (53) that indicated a dynamic minimum separable angle (MSA) measured for 80-mm mantle-length animals in bright broadband light in the cephalopod Sepia officinalis of MSA = 0.6 ± 0.2°. A determination of cephalopod acuity on O. australis was performed by Muntz and Gwither (54) using static resolution targets. If we take the minimum detectable static contrast for cephalopods to be (7) 15%, then their results correspond to an angular FWHM of 0.2°, again in basic agreement with our image quality budget estimates, showing that there is no other term greatly compromising the image quality perceived by these organisms.

The geometric optics approximation (and the diameter independence of FWHM∠) breaks down if a physical length scale becomes important. There are two limiting cases where that occurs. (i) For sufficiently small (d < 1 mm) pupils, the wavelength of light becomes important, and diffraction dominates the image quality budget. (ii) The fixed photoreceptor size limits angular resolution for lens diameter d < 1 mm.

Despite the single photoreceptor type, chromatic aberration dominates the image quality budget, except for a small, on-axis pupil shape.

Photoreceptor size.

The typical diameter for the rhabdomes tiled across the cephalopod retina is reported (12) to be 5 μm. This length scale sets a limit on spatial sampling in the focal plane. The propagation of light rays across adjacent rhabdomes (1) (“rhabdome cross-talk”) would induce additional (and potentially chromatic) image degradation, but studies of cuttlefish retina concluded (27, 55) that their rhabdomes are clad in pigmented sheathing that may suppress this potential source of image degradation. We, therefore, elected to not include any potential image degradation from rhabdome cross-talk. If rhabdome cross-talk were a significant contributor to image blur, this effect would not favor the annular pupil shape seen in these animals.

Retinal displacement.

Cross-sectional light micrograph images of cephalopod retinal structure indicate (55) an rms axial displacement of, at most, a few micrometers over spatial scales of tens of micrometers. This displacement translates into a defocus blur of order 1 μm for the full aperture pupil. The retinal displacement along the optical axis would have to be comparable with the chromatic focus shift (300 μm) (Fig. 3) to produce image degradation comparable with the chromatic blur. The “retinal bump” in cephalopods could provide spectral information at fixed accommodation if the line of sight is varied (37) so as to shift the scene across this perturbation in effective focal length or if the object of interest’s image on the retina spans the retinal bump.

Residual spherical aberration.

Although a spherical lens of uniform index of refraction produces pronounced spherical aberration, numerous studies have shown that the lenses of fishes and cephalopods have a radial variation in the index that largely compensates for spherical aberration. Jagger and Sands (26) show a typical FWHM from on-axis residual spherical aberration in octopus of less than 5 μm at full aperture. That measurement was for lenses a factor of two smaller than our 12-mm focal length model, and therefore, we have scaled this up to 10 μm for the entry in the image quality budget. We note also that this is for full aperture imaging and that the annular pupil greatly reduces the radial span of rays in the system. This calculation is, therefore, a conservative overestimate for the annular pupil geometry, because the residual spherical aberration scales (56) as Δh.

Chromatic aberration.

The experimental data (26) clearly indicate a wavelength-dependent focal shift in the lens of the octopus. We used our quadratic fit to the fractional chromatic focal length shift measured for octopus lenses from the work by Jagger and Sands (26) to perform a numerical computation of the 80% encircled energy radius for a point source at infinity, with best focus accommodations corresponding to wavelengths between 350 and 650 nm, for the three different pupil geometries that we studied. For this computation, we were interested in the entire wavelength range of interest, and therefore, we extended the focal length dependence on wavelength down to 350-nm wavelengths. Representative PSFs for accommodation settings that correspond to 400 and 700 nm are shown in Fig. S4 A–C. The encircled energy as a function of distance from the centroid is shown in Fig. S4D. We converted from the 80% encircled energy radius, R80, to a Gaussian-equivalent FWHM = 1.3 × R80. This calculation produced FWHM Gaussian equivalents of 6, 48, and 61 μm for the small, full, and annular pupils, respectively, at the best focus wavelength of 500 nm, the peak of the opsin curve.

Diffraction.

The diffraction limit on the focal plane has a spatial FWHM given by FWHMdiff = (f/#)(λ). For our full pupil with d = 8 mm, at the wavelength of peak opsin sensitivity, this calculation gives FWHMdiff = 1.5 × 0.5 μm = 0.75 μm. Stopping down the pupil to a smaller diameter d increases this term by a dimensionless multiplicative factor of 8 mm/d. Our smallest circular pupil diameter of d = 1 mm produces a diffraction limit of 6 μm FWHM, which is equal to the chromatic aberration term at that small aperture.

Other achromatic aberrations.

To constrain the magnitude of aberrations other than those listed above, we turned to the narrowband measurements of cephalopod PSFs performed by Gagnon et al. (31). They measured the FWHM produced in collimated 550-nm light (with a 10-nm bandwidth) at full aperture (31). Their data show a strong correlation between FWHM and f number (31). For their results on f/1.2 lenses, such as the one modeled here, when scaled to a 12-mm focal length, they observe an FWHM of 20 μm. We have, therefore, entered this value in Table S1 as other achromatic aberrations. We do not know the ray height dependence of these other aberrations, but unless the blurring induced by these other sources of wave-front error is independent of ray height h, they will remain subdominant for all pupil geometries. These other contributions add order of 10% (when taken in quadrature) to the total FWHM, and therefore, the image quality budget is dominated by the chromatic term.

Translation of the Paper by Kühn (21) from the Original German.

A 2015 translation by J. Schoeneberg of the relevant section of the 1950 paper by Kühn (21) states

Results

Ideally, a set of monochromatic measurements of the point spread function produced by a cephalopod lens for different pupil sizes and lens to retina spacing would establish an empirical determination of the chromatic blur seen by these creatures. We are unaware of an appropriate comprehensive dataset, and therefore we have used the available laboratory measurements to produce a computer model of the chromatic properties for a representative cephalopod. Because the primary eye design features (complex pupil shape, spherical gradient index lens, and single-opsin retina) are common across coleoid cephalopods, we will use this model as representative of this class of animals.

Using measured (26) optical properties of Octopus australis, we performed a simulation by constructing a hyperspectral image cube [at 5 μm/pixel in the spatial directions, corresponding to a typical cephalopod rhabdome diameter (12, 27), and 200 planes spanning 450 nm < λ < 650 nm in the spectral direction at Δλ = 1 nm]. We modeled an f/1.2 spherical lens with a 10-mm diameter, but our computed chromatic blurring results are independent of this choice of length scale. For each lens to retina focal distance, which brings a single wavelength into crisp focus, we computed the pupil-dependent chromatic image blur at the other wavelengths. We summed up the blurred image cube along the wavelength direction [weighted by the product of the seawater-filtered solar photon illumination, the reflectance spectrum, and the opsin response curve (Fig. S3)] to arrive at a final simulated chromatically blurred image on the retina. This procedure was repeated for three different pupil shapes for a sequence of accommodation values.

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