Showing papers in "Journal of The Optical Society of America A-optics Image Science and Vision in 1987"
TL;DR: A closed-form solution to the least-squares problem for three or more paints is presented, simplified by use of unit quaternions to represent rotation.
Abstract: Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task . It finds applications i n stereoph and in robotics . I present here a closed-form solution to the least-squares problem for three or more paints . Currently various empirical, graphical, and numerical iterative methods are in use . Derivation of the solution i s simplified by use of unit quaternions to represent rotation . I emphasize a symmetry property that a solution to thi s problem ought to possess . The best translational offset is the difference between the centroid of the coordinates i n one system and the rotated and scaled centroid of the coordinates in the other system . The best scale is equal to th e ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids . These exact results are to be preferred to approximate methods based on measurements of a few selected points . The unit quaternion representing the best rotation is the eigenvector associated with the most positive eigenvalue o f a symmetric 4 X 4 matrix . The elements of this matrix are combinations of sums of products of correspondin g coordinates of the points .
4,522 citations
TL;DR: The results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy into first- order redundancy.
Abstract: The relative efficiency of any particular image-coding scheme should be defined only in relation to the class of images that the code is likely to encounter. To understand the representation of images by the mammalian visual system, it might therefore be useful to consider the statistics of images from the natural environment (i.e., images with trees, rocks, bushes, etc). In this study, various coding schemes are compared in relation to how they represent the information in such natural images. The coefficients of such codes are represented by arrays of mechanisms that respond to local regions of space, spatial frequency, and orientation (Gabor-like transforms). For many classes of image, such codes will not be an efficient means of representing information. However, the results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy (e.g., correlation between the intensities of neighboring pixels) into first-order redundancy (i.e., the response distribution of the coefficients). Such coding produces a relatively high signal-to-noise ratio and permits information to be transmitted with only a subset of the total number of cells. These results support Barlow's theory that the goal of natural vision is to represent the information in the natural environment with minimal redundancy.
3,077 citations
TL;DR: In this paper, exact nonsingular solutions of the scalar-wave equation for beams that are non-diffracting were presented, which means that the intensity pattern in a transverse plane is unaltered by propagating in free space.
Abstract: We present exact, nonsingular solutions of the scalar-wave equation for beams that are nondiffracting. This means that the intensity pattern in a transverse plane is unaltered by propagating in free space. These beams can have extremely narrow intensity profiles with effective widths as small as several wavelengths and yet possess an infinite depth of field. We further show (by using numerical simulations based on scalar diffraction theory) that physically realizable finite-aperture approximations to the exact solutions can also possess an extremely large depth of field.
2,283 citations
TL;DR: In this article, a method for the representation of (pictures of) faces is presented, which results in the characterization of a face, to within an error bound, by a relatively low-dimensional vector.
Abstract: A method is presented for the representation of (pictures of) faces. Within a specified framework the representation is ideal. This results in the characterization of a face, to within an error bound, by a relatively low-dimensional vector. The method is illustrated in detail by the use of an ensemble of pictures taken for this purpose.
2,089 citations
TL;DR: The model appears to deal with the aperture problem as well as the human visual system since it extracts the correct velocity for patterns that have large differences in contrast at different spatial orientations, and it simulates psychophysical data on the coherence of sine-grating plaid patterns.
Abstract: A model is presented, consonant with current views regarding the neurophysiology and psychophysics of motion perception, that combines the outputs of a set of spatiotemporal motion-energy filters to extract optical flow. The output velocity is encoded as the peak in a distribution of velocity-tuned units that behave much like cells of the middle temporal area of the primate brain. The model appears to deal with the aperture problem as well as the human visual system since it extracts the correct velocity for patterns that have large differences in contrast at different spatial orientations, and it simulates psychophysical data on the coherence of sine-grating plaid patterns.
642 citations
TL;DR: In this article, it was shown that it is possible to reconstruct a complex-valued object from the modulus of its Fourier transform if one has a sufficiently strong support constraint.
Abstract: Previously it was shown that one can reconstruct an object from the modulus of its Fourier transform (solve the phase-retrieval problem) by using the iterative Fourier-transform algorithm if one has a nonnegativity constraint and a loose support constraint on the object. In this paper it is shown that it is possible to reconstruct a complex-valued object from the modulus of its Fourier transform if one has a sufficiently strong support constraint. Sufficiently strong support constraints include certain special shapes and separated supports. Reconstruction results are shown, including the effect of tapered edges on the object’s support.
529 citations
TL;DR: A frequency-selective mechanism is added to the ideal-observer model, similar to the channel mechanism that has been demonstrated through experiments that measure a subject's ability to detect grating stimuli.
Abstract: Several authors have measured the detection ability of human observers for objects in correlated (nonwhite) noise. These studies have shown that the human observer has approximately constant efficiency when compared with a nonprewhitening ideal observer. In this paper we add a frequency-selective mechanism to the ideal-observer model, similar to the channel mechanism that has been demonstrated through experiments that measure a subject's ability to detect grating stimuli. For a number of detection and discrimination tasks, the nonprewhitening ideal-observer model and the channelized ideal-observer model yield similar performance predictions. Thus both models seem equally capable of explaining a considerable body of psychophysical data, and it would be difficult to devise an experiment to determine which model is more nearly correct.
431 citations
TL;DR: In this article, an experimental investigation of light scattering from random rough surfaces is described, whose height fluctuations approximately follow Gaussian statistics, are fabricated in photoresist with a metal overcoating, and the effects of multiple scattering are shown to be consistent with the major features of the observations.
Abstract: An experimental investigation of light scattering from random rough surfaces is described. The surfaces, whose height fluctuations approximately follow Gaussian statistics, are fabricated in photoresist with a metal overcoating. When the lateral correlation length is larger than a wavelength and the surface slopes are mild, measurements of diffuse scattering are found to agree with the Beckmann theory, as long as the angle of incidence is not too large. For other surfaces that have stronger slopes, depolarization and enhanced backscattering may be observed in the diffuse scattering. Though we are unaware of theoretical calculations that compare with the measurements, the effects of multiple scattering are shown to be consistent with the major features of the observations.
368 citations
TL;DR: In this paper, first and second-order statistics of complex random signals are reviewed, and an example is taken from rf signal analysis of the backscattered echoes from diffuse scatterers.
Abstract: Both radio-frequency (rf) and envelope-detected signal analyses have lead to successful tissue discrimination in medical ultrasound The extrapolation from tissue discrimination to a description of the tissue structure requires an analysis of the statistics of complex signals To that end, first- and second-order statistics of complex random signals are reviewed, and an example is taken from rf signal analysis of the backscattered echoes from diffuse scatterers In this case the scattering form factor of small scatterers can be easily separated from long-range structure and corrected for the transducer characteristics, thereby yielding an instrument-independent tissue signature The statistics of the more economical envelope- and square-law-detected signals are derived next and found to be almost identical when normalized autocorrelation functions are used Of the two nonlinear methods of detection, the square-law or intensity scheme gives rise to statistics that are more transparent to physical insight Moreover, an analysis of the intensity-correlation structure indicates that the contributions to the total echo signal from the diffuse scatter and from the steady and variable components of coherent scatter can still be separated and used for tissue characterization However, this analysis is not system independent Finally, the statistical methods of this paper may be applied directly to envelope signals in nuclear-magnetic-resonance imaging because of the approximate equivalence of second-order statistics for magnitude and intensity
354 citations
TL;DR: It is shown that the depth dependence of the distribution of photon absorption events can be inferred from measured parameters of the surface emission profile, and illustrative experimental data are shown to be in accord with the theory.
Abstract: Various characteristics of photon diffusion in turbid biological media are examined. Applications include the interpretation of data acquired with laser Doppler blood-flow monitors and the design of protocols for therapeutic excitation of tissue chromophores. Incident radiation is assumed to be applied at an interface between a turbid tissue and a transparent medium, and the reemission of photons from that interface is analyzed. Making use of a discrete lattice model, we derive an expression for the joint probability Γ(n, ρ)d2ρ that a photon will be emitted in the infinitesimal area d2ρ centered at surface point ρ = (x, y), having made n collisions with the tissue. Mathematical expressions are obtained for the intensity distribution of diffuse surface emission, the probability of photon absorption in the interior as a function of depth, and the mean path length of detected photons as a function of the distance between the site of the incident radiation and the location of the detector. We show that the depth dependence of the distribution of photon absorption events can be inferred from measured parameters of the surface emission profile. Results of relevant computer simulations are presented, and illustrative experimental data are shown to be in accord with the theory.
335 citations
TL;DR: In this paper, a cellular automaton was discovered that can unwrap consistent phase data in n dimensions in a path-independent manner and can automatically accommodate noise-induced (point-like) inconsistencies and arbitrary boundary conditions (region partitioning).
Abstract: Research into two-dimensional phase unwrapping has uncovered interesting and troublesome inconsistencies that cause path-dependent results. Cellular automata, which are simple, discrete mathematical systems, offered promise of computation in a nondirectional, parallel manner. A cellular automaton was discovered that can unwrap consistent phase data in n dimensions in a path-independent manner and can automatically accommodate noise-induced (pointlike) inconsistencies and arbitrary boundary conditions (region partitioning). For data with regional (nonpointlike) inconsistencies, no phase-unwrapping algorithm will converge, including the cellular-automata approach. However, the automata method permits more simple visualization of the regional inconsistencies. Examples of its behavior on one- and two-dimensional data are presented.
TL;DR: The findings indicate that the same formulation can be applied to contrast discrimination and that changes in the estimated values of internal noise, rather than changes in sampling efficiency, play the major role in determining properties of contrast discrimination.
Abstract: Even the highest contrast sensitivities that humans can achieve for the detection of targets on uniform fields fall far short of ideal values. Recent theoretical formulations have attributed departures from ideal performance to two factors—the existence of internal noise within the observer and suboptimal stimulus information sampling by the observer. It has been postulated that the contributions of these two factors can be evaluated separately by measuring contrast-detection thresholds as a function of the level of externally added visual noise. We wished to determine whether a similar analysis could be applied to contrast discrimination and whether variation of the increment threshold with pedestal contrast is due to changes in internal noise or sampling efficiency. We measured contrast-increment thresholds as a function of noise spectral density for near-threshold and suprathreshold pedestal contrasts. The experiments were conducted separately for static and dynamic noise. Our findings indicate that the same formulation can be applied to contrast discrimination and that changes in the estimated values of internal noise, rather than changes in sampling efficiency, play the major role in determining properties of contrast discrimination. Implications for models of contrast coding in vision are discussed.
TL;DR: In this article, the generalized K model with a non-uniform distribution of phase by a fluctuating population of objects is derived on the bases of simpler and more general arguments.
Abstract: The generalized K distribution, introduced recently by Barakat [ J. Opt. Soc. Am. A3, 401 ( 1986)] as a model for the intensity of radiation scattered with a nonuniform distribution of phase by a fluctuating population of objects, is derived on the bases of simpler and more general arguments. In particular, Barakat’s analysis is based on a random walk in two dimensions and a specific model for the phase distribution; our analysis is presented in terms of an n-dimensional walk with a directional bias that, apart from its scaling property as a function of the mean number of scatters, is arbitrary. The relationship of the generalized K model with possible underlying scattering mechanisms is also discussed.
TL;DR: In this paper, the temporal processing of chromatic and luminance perturbations of a 600-nm field, measuring both modulation sensitivity (sinusoidal frequencies from 0.25 to 40 Hz) and pulse-detection thresholds (pulse durations from 5 to 2560 msec) was studied.
Abstract: We studied temporal processing of chromatic and luminance perturbations of a 600-nm field, measuring both modulation sensitivity (sinusoidal frequencies from 0.25 to 40 Hz) and pulse-detection thresholds (pulse durations from 5 to 2560 msec) for mean luminances of 0.9 to 900 Td and field sizes of 0.5 degrees to 8 degrees. Chromatic stimuli were produced by antiphase modulation of lights matched by heterochromatic flicker photometry. Both mean luminance and field size affected sensitivity, and the magnitude of field-size effects increased with mean luminance. We derived both luminance and chromatic impulse response functions for each set of experimental conditions, using the modulation-sensitivity data. At high mean luminances and large field sizes the chromatic impulse response functions are complex, suggesting contributions from both chromatic and luminance mechanisms. Pulse-detection data were fitted by a peak detector model based on these impulse response functions.
TL;DR: A method for the determination of the bidimensional optical transfer function (OTF) and the point-spread function of human eyes is presented and typical aerial and retinal images of a point source as well as the OTF's results are presented.
Abstract: A method for the determination of the bidimensional optical transfer function (OTF) and the point-spread function of human eyes is presented. Aerial short-term images of a point source are directly recorded with a TV camera and fed into a digital image-processing system that allows one to determine and display such functions. The method has been implemented in such a way that recording and computation can be carried out on a routine basis with minimum discomfort for the observer. A detailed description of the method and typical aerial and retinal images of a point source as well as the OTF's results are presented in this paper.
TL;DR: In this paper, the authors examined how the spatial characteristics of simple discrimination tasks change with time after the onset of a stimulus and concluded that the changes in spatial characteristics concern the range of spatial filters in operation.
Abstract: The manner in which the spatial characteristics of simple discrimination tasks change with time after the onset of a stimulus were examined. The experiments measured the improvements in sensitivity to the length, orientation, curvature, and stereoscopic depth of short lines that accrue with increased exposure durations. These improvements can be consistently interpreted in terms of a change of the spatial scale of analysis from coarse to fine over a period of at least 1000 msec. Variations in visual resolution acuity over the same period are negligible, and it is concluded that the changes in spatial characteristics concern the range of spatial filters in operation. This range progressively shrinks after stimulus presentation.
TL;DR: A code modeled on the simple cells of the primate striate cortex is explored, which maps a digital image into a set of subimages (layers) that are bandpass in spatial frequency and orientation and which is reconstructed from the code.
Abstract: Hypothetical schemes for neural representation of visual information can be expressed as explicit image codes. Here, a code modeled on the simple cells of the primate striate cortex is explored. The Cortex transform maps a digital image into a set of subimages (layers) that are bandpass in spatial frequency and orientation. The layers are sampled so as to minimize the number of samples and still avoid aliasing. Samples are quantized in a manner that exploits the bandpass contrast-masking properties of human vision. The entropy of the samples is computed to provide a lower bound on the code size. Finally, the image is reconstructed from the code. Psychophysical methods are derived for comparing the original and reconstructed images to evaluate the sufficiency of the code. When each resolution is coded at the threshold for detection artifacts, the image-code size is about 1 bit/pixel.
TL;DR: The maximum spatial frequency for the detection and resolution of sinusoidal gratings was determined as a function of stimulus location across the visual field as discussed by the authors, and the detection limit was about three times the resolution limit, and this factor increased to about 10 as the test stimulus was moved 35° into the periphery.
Abstract: The maximum spatial frequency for the detection and resolution of sinusoidal gratings was determined as a function of stimulus location across the visual field. Stimuli were produced directly on the retina as interference fringes, thus avoiding possible loss of image quality, which may occur when the optical system of the eye is used to form the retinal image. Contrary to earlier reports, we found that subjects could detect gratings with spatial frequencies much higher than the resolution limit. At 5° of eccentricity from the fovea, the detection limit was about three times the resolution limit, and this factor increased to about 10 as the test stimulus was moved 35° into the periphery. Quantitative comparison of the data with retinal anatomy and physiology suggests that pattern resolution is limited by the spacing of primate beta (midget) retinal ganglion cells, whereas pattern detection is limited by the size of individual cones.
TL;DR: In this article, a novel formulation of light beam propagation through any complex optical system that can be described by an ABCD ray-transfer matrix is presented. But this formulation does not consider the effects of finite-sized limiting apertures (i.e., diffractive screens) in the optical train, tilt and random jitter of the optical elements, and distributed random inhomogeneities along the optical path.
Abstract: 20We describe a novel formulation of light beam propagation through any complex optical system that can be described by an ABCD ray-transfer matrix. Within the paraxial approximation, optical propagation can be formulated in terms of a Huygens principle expressed in terms of the ray-transfer ABCD matrix elements of the optical system. We extend and generalize previous treatments to include the effects of finite-sized limiting apertures (i.e., diffractive screens) in the optical train, tilt and random jitter of the optical elements, and distributed random inhomogeneities along the optical path (e.g., clear air turbulence and aerosols). In the presence of limiting apertures the ABCD matrix elements of the optical system are complex. For the case of laser beam propagation and Gaussian-shaped limiting apertures in the optical train, we obtain analytical expressions for both the spot radius and the wave-front radius of curvature at an arbitrary observation plane and give illustrative examples of practical concern. In particular, analytical expressions for the fringe visibility obtained in a coherent laser interferometric system are presented. An analytical expression for the mean spot radius of a laser beam propagating through an optical system in the presence of tilt and random jitter is obtained. We also consider the propagation of partially coherent light through optical systems. In particular, we derive a generalized van Cittert–Zernike theorem that is valid for an arbitrary optical system that can be characterized by an ABCD ray-transfer matrix. Finally, the propagation of laser beams through a general optical system in the presence of distributed random inhomogeneities is considered. An explicit expression for the mean irradiance distribution of a Gaussian-shaped beam is derived that is valid for an arbitrary optical system. In addition, we derive an expression for the mutual-coherence function for wave propagation through an arbitrary optical system. In all cases the results are expressed in terms of the ABCD matrix elements of the complete optical system. The formulation of optical propagation presented here is a rather simple and straightforward way of determining the effects of finite-sized optical elements, tilt and random jitter, and distributed random inhomogeneities along the optical path. It is merely necessary first to multiply the relevant ray matrices together to find the complete system matrix and then to substitute this matrix into the expressions given in this paper.
TL;DR: It is commonly assumed that the visual resolution limit must be equal to or less than the Nyquist frequency of the cone mosaic, but under some conditions, observers can see fine patterns at the correct orientation when viewing interference fringes with spatial frequencies that are as much as about 1.5 times higher than the nominal Nyquist Frequency.
Abstract: It is commonly assumed that the visual resolution limit must be equal to or less than the Nyquist frequency of the cone mosaic. However, under some conditions, observers can see fine patterns at the correct orientation when viewing interference fringes with spatial frequencies that are as much as about 1.5 times higher than the nominal Nyquist frequency of the underlying cone mosaic. The existence of this visual ability demands a closer scrutiny of the sampling effects of the cone mosaic and the information that is sufficient for an observer to resolve a sinusoidal grating. The Nyquist frequency specifies which images can be reconstructed without aliasing by an imaging system that samples discretely. However, it is not a theoretical upper bound for psychophysical measures of visual resolution because the observer's criteria for resolving sinusoidal gratings are less stringent than the criteria specified by the sampling theorem for perfect, alias-free image reconstruction.
TL;DR: Stochastic amplification of photon noise by one stage of an imaging system is shown to constitute an effective signal to the next, while the underlying photon-noise component is unaffected by a subsequent scattering process, which leads to useful expressions for the noise power spectrum and detective quantum efficiency for multistage image systems.
Abstract: We have analyzed the influence of stochastic amplifying and scattering mechanisms on the transfer of signal and noise through multistage imaging systems in terms of multivariate moment-generating functions. Stochastic amplification of photon noise by one stage of an imaging system is shown to constitute an effective signal to the next, while the underlying photon-noise component is unaffected by a subsequent scattering process. In the case of stationary, photon-limited inputs, these considerations then lead to useful expressions for the noise power spectrum and detective quantum efficiency for multistage image systems. The application of these results to the analysis of radiographic screen–film imaging systems is discussed.
TL;DR: There was no change in equiluminance ratios, indicating no contribution of the blue-sensitive cones to the luminance channel even at low temporal and spatial frequencies.
Abstract: Equiluminance ratios for red/green, red/blue and green/blue sine-wave gratings were determined by using a minimum-motion heterochromatic matching technique that permitted reliable settings at temporal frequencies as low as 0.5 Hz. The red/green equiluminance ratio was influenced by temporal but not spatial frequency, the green/blue ratio was influenced by spatial but not temporal frequency, and the red/blue ratio was influenced by both. After bleaching of the blue-sensitive cones, there was no change in equiluminance ratios, indicating no contribution of the blue-sensitive cones to the luminance channel even at low temporal and spatial frequencies. The inhomogeneity of yellow pigmentation within the macular region was identified as the source of the spatial-frequency effect on the blue/green ratio.
TL;DR: In this article, the effect of a linear random medium on the state of polarization of the transmitted light is investigated, and the connection between the Stokes vector formalism and the coherence or polarization matrix formalism is discussed.
Abstract: The effect of a linear random medium on the state of polarization of the transmitted light is investigated, and the connection between the Stokes vector formalism and the coherence or polarization matrix formalism is discussed. It is shown that an ensemble of Jones matrices corresponds to the Mueller matrix in general.
TL;DR: An achromatic surface in a complex scene has both an apparent reflectance attribute (lightness) and an overall intensitive attribute (brightness).
Abstract: An achromatic surface in a complex scene has both an apparent reflectance attribute (lightness) and an overall intensitive attribute (brightness). We studied changes of these two attributes as a function of changes in illumination level and pattern complexity. Subjects observed simultaneously two arrays of simulated achromatic surfaces with identical reflectance distributions. The left-hand array (standard) was always illuminated at a moderate level. The right-hand array (test) had different illuminances from trial to trial. The subjects adjusted patches in the test array to match the corresponding patches in the standard array in either lightness or brightness. In complex patterns (32 grays) lightness constancy was nearly perfect; test reflectance settings were invariant over illuminance. In disk/annulus patterns (two grays), the lightness-match data confirmed previously published reports. At high illuminances, the standard patches could be matched with a smaller range of test-array reflectances than at low illuminances, i.e., lightness constancy was imperfect. Brightness matches varied substantially as a function of illuminance in all conditions.
TL;DR: In this paper, a rigorous electromagnetic theory of the diffraction of radiation by a circular aperture in a thick screen is developed, and the effects of varying the thickness of the screen and varying the wavelength, polarization, and angle of incidence of the incident wave on the reflection and transmission properties of a screen are investigated.
Abstract: A rigorous electromagnetic theory of the diffraction of radiation by a circular aperture in a thick screen is developed. In particular, the case of an incident plane wave is considered, and the effects of varying the thickness of the screen and of varying the wavelength, polarization, and angle of incidence of the incident wave on the reflection and transmission properties of the screen are investigated.
TL;DR: Fourier phase-retrieval algorithms are modified and applied to in-line holography, where phase is lost during the hologram recording process and the rate of convergence is enhanced.
Abstract: Fourier phase-retrieval algorithms are modified and applied to in-line holography, where phase is lost during the hologram recording process. Retrieval of phase permits separation of real-object distributions from the twin-image interference that accompanies conventional optical reconstruction. The rate of convergence is enhanced by the availability of a good initial guess based on the digital equivalent of conventional optical reconstruction.
TL;DR: In this paper, a unified formulation for the analysis of linear stable resonators containing a lens of variable focal length, which represents the rod of a solid-state laser, and other intracavity optical systems is presented.
Abstract: A unified formulation for the analysis of linear stable resonators containing a lens of variable focal length, which represents the rod of a solid-state laser, and other intracavity optical systems is presented. The stability, the mode spot sizes, the dynamical stability, and the misalignment sensitivity are investigated, and general properties that are valid for any resonator are derived. Some important practical consequences for resonator design are discussed.
TL;DR: In this article, a pseudoparaxial approximation of the axial variation in the complex amplitude was proposed to predict the effect of axial variations on optical imaging, including depth of focus, interference microscopy, and the imaging of objects with appreciable depth.
Abstract: Nonparaxial effects on optical imaging are considered. It is shown that the dominant effect is on the axial variation in the complex amplitude, affecting most strongly depth of focus, interference microscopy, and the imaging of objects with appreciable depth. A so-called pseudoparaxial approximation is introduced, which gives reasonable prediction of these effects up to quite large angular apertures.
TL;DR: In this article, the authors applied Mueller matrix ellipsometry to assess the change in the state of polarization of a light beam that has double passed the ocular media and is scattered at the fundus of the human eye in vivo.
Abstract: We have applied Mueller matrix ellipsometry to assess the change in the state of polarization of a light beam that has double passed the ocular media and is scattered at the fundus of the human eye in vivo. At several positions in the pupil plane, which together cover the area of the dilated pupil, Mueller matrices are assessed. From them the magnitude of the retardation and the orientation of the eigenvector are calculated. The properties of the retardation process are surveyed by measuring the retardation along a horizontal meridian as a function of wavelength, density of visual pigment, and location of retinal fixation. Furthermore, photographs are taken from the polarization patterns on the iris with circularly polarized light. We posit that the cornea behaves as a biaxial crystal with its fastest principal axis normal to its surface and its slowest nasally downward. The retardation of light by a model eye with such a cornea is calculated, and the results are compared with the data.
TL;DR: In this paper, the Fourier transform of a K-dimensional function is zero on continuous surfaces (here called zero sheets) of dimension (2K − 2) in a space that effectively has 2K dimensions.
Abstract: A multiple convolution (e.g., an image formed by convolving several individual components) is automatically deconvolvable, provided that its dimension (i.e., the number of variables of which it is a function) is greater than unity. This follows because the Fourier transform of a K-dimensional function (having compact support) is zero on continuous surfaces (here called zero sheets) of dimension (2K − 2) in a space that effectively has 2K dimensions. A number of important practical applications are transfigured by the concept of the zero sheet. Image restoration can be effected without prior knowledge of the point-spread function, i.e., blind deconvolution is possible even when only a single blurred image is given. It is in principle possible to remove some of the additive noise when the form of the point-spread function is known. Fourier phase can be retrieved directly, and, unlike for readily implementable iterative techniques, complex images can be handled as straightforwardly as real images.