Showing papers in "Journal of The Optical Society of America A-optics Image Science and Vision in 1993"

Image rotation, Wigner rotation, and the fractional Fourier transform

Abstract: In this study the degree p = 1 is assigned to the ordinary Fourier transform The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt Commun (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms I approach the subject of fractional Fourier transforms in two other ways First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming Second, I propose two optical setups that are able to perform a fractional Fourier transform

Fractional Fourier transforms and their optical implementation. II

Abstract: The linear transform kernel for fractional Fourier transforms is derived. The spatial resolution and the space–bandwidth product for propagation in graded-index media are discussed in direct relation to fractional Fourier transforms, and numerical examples are presented. It is shown how fractional Fourier transforms can be made the basis of generalized spatial filtering systems: Several filters are interleaved between several fractional transform stages, thereby increasing the number of degrees of freedom available in filter synthesis.

Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating

Abstract: We recently introduced a new technique, frequency-resolved optical gating (FROG), for directly determining the full intensity I(t) and phase φ(t) of a single femtosecond pulse. By using almost any instantaneous nonlinear-optical interaction of two replicas of the ultrashort pulse to be measured, FROG involves measuring the spectrum of the signal pulse as a function of the delay between the replicas. The resulting trace of intensity versus frequency and delay yields an intuitive display of the pulse that is similar to the pulse spectrogram, except that the gate is a function of the pulse to be measured. The problem of inverting the FROG trace to obtain the pulse intensity and phase can also be considered a complex two-dimensional phase-retrieval problem. As a result, the FROG trace yields, in principle, an essentially unique pulse intensity and phase. We show that this is also the case in practice. We present an iterative-Fourier-transform algorithm for inverting the FROG trace. The algorithm is unusual in its use of a novel constraint: the mathematical form of the signal field. Without the use of a support constraint, the algorithm performs quite well in practice, even for pulses with serious phase distortions and for experimental data with noise, although it occasionally stagnates when pulses with large intensity fluctuations are used.

Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge

Abstract: Light propagation in strongly scattering media can be described by the diffusion approximation to the Boltzmann transport equation. We have derived analytical expressions based on the diffusion approximation that describe the photon density in a uniform, infinite, strongly scattering medium that contains a sinusoidally intensity-modulated point source of light. These expressions predict that the photon density will propagate outward from the light source as a spherical wave of constant phase velocity with an amplitude that attenuates with distance r from the source as exp(-alpha r)/r. The properties of the photon-density wave are given in terms of the spectral properties of the scattering medium. We have used the Green's function obtained from the diffusion approximation to the Boltzmann transport equation with a sinusoidally modulated point source to derive analytic expressions describing the diffraction and the reflection of photon-density waves from an absorbing and/or reflecting semi-infinite plane bounded by a straight edge immersed in a strongly scattering medium. The analytic expressions given are in agreement with the results of frequency-domain experiments performed in skim-milk media and with Monte Carlo simulations. These studies provide a basis for the understanding of photon diffusion in strongly scattering media in the presence of absorbing and reflecting objects and allow for a determination of the conditions for obtaining maximum resolution and penetration for applications to optical tomography.

Image recovery from data acquired with a charge-coupled-device camera

Abstract: A model for data acquired with the use of a charge-coupled-device camera is given and is then used for developing a new iterative method for restoring intensities of objects observed with such a camera. The model includes the effects of point spread, photoconversion noise, readout noise, nonuniform flat-field response, nonuniform spectral response, and extraneous charge carriers resulting from bias, dark current, and both internal and external background radiation. An iterative algorithm is identified that produces a sequence of estimates converging toward a constrained maximum-likelihood estimate of the intensity distribution of an imaged object. An example is given for restoring images from data acquired with the use of the Hubble Space Telescope.

Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity : addendum

Abstract: A numerically stable method is presented for the analysis of diffraction gratings of arbitrary profile, depth, and in conical mountings. It is based on the classical modal method and uses a stack of lamellar gratpermittivity to approximate an arbitrary profile. A numerical procedure known as the R-matrix propagation aling layers gorithm is used to propagate the modal fields through the layers. This procedure renders the implementation of this new method completely immune to the numerical instability that is associated with the conventional algorithm. Numerical examples including diffraction efficiencies of both dielectric and metallic propagation gratings of depths that range from subwavelength to hundreds of wavelengths are presented. Information about the convergence and the computation time of the method is also included.

Spectral sensitivities of the human cones

Abstract: Transient chromatic adaptation produced by an abrupt change of background color permits an easier and closer approach to cone isolation than does steady-state adaptation. Using this technique, we measured middle-wave-sensitive (M)-cone spectral sensitivities in 11 normals and 2 protanopes and long-wavelength-sensitive (L-) cone spectral sensitivities in 12 normals and 4 deuteranopes. Although there is great individual variation in the adapting intensity required for effective isolation, there is little variation in the shape of the M- and L-cone spectral-sensitivity functions across subjects. At middle and long wavelengths, our mean spectral sensitivities agree extremely well with dichromatic spectral sensitivities and with the M- and L-cone fundamentals of Smith and Pokorny [Vision Res. 15, 161 (1975)] and of Vos and Walraven [Vision Res. 11, 799 (1971)], both of which are based on the CIE (Judd-revised) 2 degrees color-matching functions (CMF's). But the agreement with the M-cone fundamentals of Estevez [Ph.D. dissertation, Amsterdam University (1979)] and of Vos et al. [Vision Res. 30, 936 (1990)], which are based on the Stiles-Burch 2 degrees CMF's, is poor. Using our spectral-sensitivity data, tritanopic color-matching data, and Stile's pi 3, we derive new sets of cone fundamentals. The consistency of the proposed fundamentals based on either the Stiles-Burch 2 degrees CMF's or the CIE 10 degrees large-field CMF's with each other, with protanopic and deuteranopic spectral sensitivities, with tritanopic color-matching data, and with short-wavelength-sensitive (S-) cone spectral-sensitivity data suggests that they are to be preferred over fundamentals based on the CIE 2 degrees CMF's.

Phase problem in crystallography

Abstract: Phase recovery from unphased data is the central problem in the interpretation of crystal diffraction data. There are a number of experimental methods for determining phases, and for small molecules computational methods work very well. The problem of developing robust computational methods for large molecules is not yet solved. Most of the approaches for large molecules are variants of the Grechburg–Saxton algorithm [ Optik35, 237– 246 ( 1972)]; with sufficiently good initial phase estimates these converge, but in general they fail with phase stagnation similar to that seen in image processing [ DaintyJ. C.FienupJ. R., in Image Recovery: Theory and Application, StarkH., ed. ( Academic, Orlando, Fla., 1987); FienupJ. R.WackermanC. C., J. Opt. Soc. Am. A3, 1897– 1907 ( 1986)].

Multiframe blind deconvolution of astronomical images

Abstract: Maximum-likelihood estimation techniques are presented for the problem of forming object estimates from turbulence-degraded images when the point-spread functions are unknown. The inability of unconstrained maximum-likelihood methods to form meaningful estimates is acknowledged, and iterative algorithms are derived for estimating the object by using both a penalized maximum-likelihood method and a physically meaningful parameterization of the point-spread functions by phase errors distributed over an aperture.

Twisted Gaussian Schell-model beams

Abstract: We introduce a new class of partially coherent axially symmetric Gaussian Schell-model (GSM) beams incorporating a new twist phase quadratic in configuration variables. This phase twists the beam about its axis during propagation and is shown to be bounded in strength because of the positive semidefiniteness of the cross-spectral density. Propagation characteristics and invariants for such beams are derived and interpreted, and two different geometric representations are developed. Direct effects of the twist phase on free propagation as well as on parabolic index fibers are demonstrated. Production of such twisted GSM beams, starting with Li-Wolf anisotropic GSM beams, is described.

Theory of morphology-dependent resonances : shape resonances and width formulas

Abstract: The theory of morphology-dependent resonances of a spherical particle is developed in analogy with the theory of quantum-mechanical shape resonances. Exact analytic formulas for predicting the widths of the resonances for both real and complex indices of refraction are developed.

Convergence of the coupled-wave method for metallic lamellar diffraction gratings

Abstract: Numerical evidence is presented that shows that, for metallic lamellar gratings in TM polarization, the coupled-wave method formulated by Moharam and Gaylord [ J. Opt. Soc. Am. A3, 1780 ( 1986)] converges slowly. (In some cases, for achieving a relative error of less than 1% in diffraction efficiencies, the number of spatial harmonics retained in the computation must be much greater than 100.) By classification of the modal methods for analyzing diffraction gratings into two distinct categories, the cause for the slow convergence is analyzed and attributed to the use of Fourier expansions to represent the permittivity and the electromagnetic fields in the grating region. The eigenvalues and the eigenfunctions of the modal fields in the grating region, whose accurate determination is crucial to the success of the coupled-wave method, are shown to converge slowly as a result of the use of these Fourier expansions. Despite its versatility and simplicity, the coupled-wave method should be used with caution for metallic surface-relief gratings in TM polarization.

Low-dimensional representation of faces in higher dimensions of the face space

Abstract: Faces can be represented efficiently as a weighted linear combination of the eigenvectors of a covariance matrix of face images. It has also been shown [ J. Opt. Soc. Am.4, 519– 524 ( 1987)] that identifiable faces can be made by using only a subset of the eigenvectors, i.e., those with the largest eigenvalues. This low-dimensional representation is optimal in that it minimizes the squared error between the representation of the face image and the original face image. The present study demonstrates that, whereas this low-dimensional representation is optimal for identifying the physical categories of face, like sex, it is not optimal for recognizing the faces (i.e., discriminating known from unknown faces). Various low-dimensional representations of the faces in the higher dimensions of the face space (i.e., the eigenvectors with smaller eigenvalues) provide better information for face recognition.

Parametric representation of Stiles–Crawford functions: normal variation of peak location and directionality

Abstract: Evidence suggests that the psychophysically determined Stiles-Crawford effect of the first kind (SCE) reflects waveguide properties of human photoreceptors. The peak of the SCE data set is assumed to reflect the principal alignment tendencies, and the spread (e.g., rho value, the curvature or width at half-height) is assumed to reflect the directionality (i.e., interreceptor differences in alignment) of the population of photoreceptors being tested. As such, disruption of the normal SCE can be used and/or has been used (1) to document early natural history of retinal pathology involving the photoreceptors, (2) to provide a firm rationale for therapeutic intervention, and (3) to provide a method for monitoring therapies designed to alter the natural course of retinal-disease processes. We report large-sample norms for foveal SCE peak location and spread (horizontal peak location, nasal 0.51 +/- 0.72, horizontal rho value 0.047 +/- 0.013, vertical peak location, superior 0.20 +/- 0.64, vertical rho value 0.053 +/- 0.012), compare these norms with values determined in other laboratories, and discuss the various mathematical forms used for the empirical description of SCE data sets.

Effects of aging in retinal image quality

Abstract: The retinal image quality characterized by the modulation-transfer function of the eye was measured for two groups of subjects aged in the late twenties and mid sixties, respectively. In both groups, we obtained modulation transfer functions by using a double-pass method under the same experimental conditions: 4-mm artificial pupil, paralyzed accommodation, and objective control of the refractive state and centering. Results showed lower values of modulation in the retinal image for older subjects compared with the younger subjects. The modulation transfer function ratio is similar to that previously found for contrast-sensitivity measurements with subjects in the same age groups. These results suggest that a significant fraction of the loss in spatial vision with age has an optical origin. Apart from the well-known increase in intraocular scattering, there also appears to be an increment in ocular aberration that causes an additional reduction in the contrast of retinal images.

Detection mechanisms in L-, M-, and S-cone contrast space

Abstract: Detection thresholds were obtained for a 2° Gaussian-blurred spot flashed for 200 ms on an 8.9° white adapting field of 1070 trolands. The spot’s contrast was represented in an L-, M-, and S-cone contrast space. Detection thresholds were obtained for many vectors close to specific but theoretically important planes within this space. A three-dimensional surface was fitted to the data generated by the probability summation of three mechanisms, each a weighted sum of cone contrasts. The fit revealed a red–green chromatic mechanism driven by ΔL/L − ΔM/M with no S-cone input that was 1 order of magnitude more sensitive than the two other mechanisms. The latter consisted of a luminance mechanism with little S-cone input and a blue–yellow chromatic mechanism with the S cone opposed to L and M cones.

Modulation transfer of the human eye as a function of retinal eccentricity

Abstract: We measured the monochromatic image quality of the eye across a wide visual field (120°), with natural pupil (4 mm) and accommodation (3 diopters). The method is based on the acquisition and the posterior processing of double-pass aerial images of a point source imaged on the retina, which was kept at a fixed distance from the eye at all retinal eccentricities. The two-dimensional modulation transfer functions (MTF’s) computed from the aerial images show that astigmatism is the dominant monochromatic aberration in both the fovea and the periphery and is also the major cause of variability among individuals. We found a slower decline in optical quality with eccentricity than had been found by previous measurements. Our foveal results are in close agreement with those of Campbell and Gubisch [ J. Physiol. (London)186, 558– 578 ( 1966)], but off-axis optical quality is much better than found previously by Jennings and Charman [ Am. J. Optom. Physiol. Opt.55, 582– 590 ( 1978);Vision Res.21, 445– 454 ( 1981)]. The optical system of the eye seems to follow a wide-angle lens design: the optical quality in the center (fovea) is not particularly good (it is far from the diffraction limit at this pupil size), but the modulation transfer function remains roughly constant for a wide visual field.

Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes

Abstract: A new method has been developed for testing the optical quality of ground-based telescopes Aberrations are estimated from wideband long-exposure defocused stellar images recorded with current astronomical CCD cameras An iterative algorithm is used that simulates closed-loop wave-front compensation in adaptive optics Compared with the conventional Hartmann test, the new method is easier to implement, has similar accuracy, and provides a higher spatial resolution on the reconstructed wave front It has been applied to several astronomical telescopes and has been found to be a powerful diagnostic tool for improving image quality

Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media

Abstract: Light reflectance by semi-infinite turbid media is modeled by a hybrid of Monte Carlo simulation and diffusion theory, which combines the accuracy of Monte Carlo simulation near the source and the speed of diffusion theory distant from the source. For example, when the turbid medium has the following optical properties—absorption coefficient 1 cm^(-1), scattering coefficient 100 cm^(-1), anisotropy 0.9, and refractive-index-matched boundary—the hybrid simulation is 7 times faster than the pure Monte Carlo simulation (100,000 photon packets were traced), and the difference between the two simulations is within 2 standard deviations of the Monte Carlo simulation.

Appearance of colored patterns: pattern-color separability.

Abstract: We have measured how color appearance of square-wave bars varies with stimulus strength and spatial frequency. Observers adjusted the color of a uniform patch to match the color appearance of the bars in square-wave patterns. We used low-to-moderate square-wave patterns, from 1 to 8 cycles per degree (c/deg). The matches are not photoreceptor matches but rather are established at more central neural sites. The signals at the putative central sites obey several simple regularities. The cone contrast of the uniform patch is proportional to square-wave stimulus strength (color homogeneity) and additive with respect to the superposition of equal-frequency square waves containing different colors (color superposition). We use the asymmetric matches to derive, from first principles, three pattern–color-separable appearance pathways. The matches are explained by two spectrally opponent, spatially low-pass mechanisms and one spectrally positive, spatially bandpass mechanism. The spectral mechanisms that we derive are similar to luminance and opponent mechanisms that are derived with entirely different experimental methods.

Numerical solution of diffraction problems: a method of variation of boundaries

Abstract: Previously ( Proc. R. Soc. Edinburgh122A, 317– 340, 1992) we established that solutions to problems of diffraction of light in a periodic structure behave analytically with respect to variations of the interface. We present an algorithm based on this observation for the numerical solution of the problem. The principal component of the algorithm is a simple recursive formula for the derivatives of the efficiencies with respect to the height of the grating. A conformal mapping mechanism is introduced to enhance the convergence of the series. This allows us to deal with the types of profile and wavelength usually considered in practice. To illustrate our method, we give numerical results for sinusoidal and echelette gratings.

Color constancy. I. Basic theory of two-stage linear recovery of spectral descriptions for lights and surfaces

Abstract: Changing a scene’s illuminant causes the chromatic properties of reflected lights to change. This change in the lights from surfaces provides spectral information about surface reflectances and illuminants. We examine conditions under which these properties may be recovered by using bilinear models. Necessary conditions that follow from comparing the number of equations and the number of unknowns in the recovery procedure are not sufficient for unique recovery. Necessary and sufficient conditions follow from demanding a one-to-one relationship between quantum catch data and sets of lit surfaces. We present an algorithm for determining whether spectral descriptions of lights and surfaces can be recovered uniquely from reflected lights.

Eigenvector method for maximum-likelihood estimation of phase errors in synthetic-aperture-radar imagery

Abstract: We develop a maximum-likelihood (ML) algorithm for estimation and correction (autofocus) of phase errors induced in synthetic-aperture-radar (SAR) imagery. Here, M pulse vectors in the range-compressed domain are used as input for simultaneously estimating M − 1 phase values across the aperture. The solution involves an eigenvector of the sample covariance matrix of the range-compressed data. The estimator is then used within the basic structure of the phase gradient autofocus (PGA) algorithm, replacing the original phase-estimation kernel. We show that, in practice, the new algorithm provides excellent restorations to defocused SAR imagery, typically in only one or two iterations. The performance of the new phase estimator is demonstrated essentially to achieve the Cramer–Rao lower bound on estimation-error variance for all but small values of target-toclutter ratio. We also show that for the case in which M is equal to 2, the ML estimator is similar to that of the original PGA method but achieves better results in practice, owing to a bias inherent in the original PGA phase estimation kernel. Finally, we discuss the relationship of these algorithms to the shear-averaging and spatial correlation methods, two other phase-correction techniques that utilize the same phase-estimation kernel but that produce substantially poorer performance because they do not employ several fundamental signal-processing steps that are critical to the algorithms of the PGA class.

Aperture realizations of exact solutions to homogeneous-wave equations

Abstract: Several new classes of localized solutions to the homogeneous scalar wave and Maxwell’s equations have been reported recently. Theoretical and experimental results have now clearly demonstrated that remarkably good approximations to these acoustic and electromagnetic localized-wave solutions can be achieved over extended near-field regions with finite-sized, independently addressable, pulse-driven arrays. We demonstrate that only the forward-propagating (causal) components of any homogeneous solution of the scalar-wave equation are actually recovered from either an infinite- or a finite-sized aperture in an open region. The backward-propagating (acausal) components result in an evanescent-wave superposition that plays no significant role in the radiation process. The exact, complete solution can be achieved only from specifying its values and its derivatives on the boundary of any closed region. By using those localized-wave solutions whose forward-propagating components have been optimized over the associated backward-propagating terms, one can recover the desirable properties of the localized-wave solutions over the extended near-field regions of a finite-sized, independently addressable, pulse-driven array. These results are illustrated with an extreme exampl—one dealing with the original solution, which is superluminal, and its finite aperture approximation, a slingshot pulse.

Measure of goodness of a set of color-scanning filters

Abstract: Accurate scanning of a color image, which is absolutely essential for good color reproduction, can ensure that all relevant information about the color stimulus of a signal is obtained. The set of scanning filters is hence an important component of a color reproduction system. In this paper we introduce a measure of the goodness of a set of color-scanning filters. This measure relates the space spanned by the scanning filters to the human visual subspace. The q factor of a single color-scanning filter is shown to be a particular case of the measure. Experimental results are presented to justify the appropriateness of the measure.

Optical reversibility theorems for polarization : application to remote control of polarization

Abstract: Using Jones’s formalism, we prove three optical reversibility theorems that relate the polarization ellipticity at the output of an optical system to the polarization of the retroreflected light at the input. We describe how these theorems can be used to measure the ellipticity of a polarization remotely and thus to control it remotely. As an example, we use this method to create a linear or a circular polarization after a total internal reflection inside a prism, and the impurity of polarization is found to be better than 10−3. Finally we describe the use of this remote control to create polarization configurations that are useful for laser cooling of atoms.

Extended Jones matrix method. II

Abstract: We derive an extended Jones matrix method to treat the transmission of light through birefringent networks, where the incident angle of light and the optical axis of the birefringent media are arbitrary. As an example, we employ the method to analyze the leakage problem of a twisted nematic liquid-crystal display and to suggest its possible solutions. A generalization of the method covers all dielectric media, including uniaxial and biaxial crystals and also gyrotropic materials that exhibit optical rotation and Faraday rotation.

Probes for scanning tunneling optical microscopy: a theoretical comparison

Abstract: In near-field optical microscopy two kinds of probe are used: a dielectric tip and a small-aperture tip. The purpose of this paper is to compare theoretical images of the same sample obtained with these two probes. We describe the use of a scanning tunneling optical microscope when the sample is a transparent dielectric rough surface illuminated by total internal reflection. The dielectric tip is modeled as a small scattering dipolar center. The intensity detected by the small-aperture probe is calculated with use of the diffraction theory of Bethe [ Phys. Rev.66, 163 ( 1944)] and Bouwkamp [ Rep. Phys.27, 35 ( 1954)]. It is shown that the two probes do not detect the same information: The dielectric tip picks up the square modulus of the electric near field. The small-aperture probe is sensitive to both the electric and the magnetic fields. The models are used for calculating and comparing images of a periodic grating and of a two-dimensional object (a letter) that are smaller than the wavelength. The images are quite different, and polarization of the incident light is an important parameter for scanning tunneling optical microscope images, with different behavior for the two tips.

Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography

Abstract: We compare two methods of reconstructing the complex index-of-refraction distribution of a scattering object from optical scattering data obtained in a set of scattering experiments employing incident monochromatic plane waves. The first method generates an approximate reconstruction directly from the far-field intensity, which is measured as a function of scattering angle for each incident plane wave. The second method uses an iterative phase-retrieval algorithm to extract the phase of the scattered field over any given plane from the measurement of the intensity of the total field over that plane and from a priori object-support information. The reconstruction is then performed from the scattered field that is so determined by using the filtered backpropagation algorithm of diffraction tomography. We compare the performance of the two procedures on computer-simulated and experimental scattering data.

Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements

Abstract: For stratified samples where material is being uniformly deposited or removed at a known rate, I show that the dielectric function ∊o of the outermost region is determinable exactly and analytically at any given wavelength from the value and the thickness derivative of the complex reflectances for p-polarized, s-polarized, or normally incident light without any knowledge of the underlying structure. This minimal-data approach greatly simplifies analysis compared with the standard procedure, in which dielectric functions are determined sequentially from a combination of new data and previously established sample parameters. It is also robust, eliminating cumulative error and error propagation that can cause conventional analysis to fail. An interferometric method for acquiring complex-reflectance data is proposed, although to achieve the necessary level of accuracy with present technology would be a formidable challenge. For ellipsometric measurements these technical obstacles do not exist, but an equivalent exact solution is not possible. However, I develop a common-pseudosubstrate approximation (CPA) that in applications to semiconductor crystal growth is accurate to better than 0.1%. The minimal-data approach also provides new insights about how sample parameters are determined from measured optical functions. For example, to determine deposition rates one needs to establish the second derivative (curvature) as well, which places additional constraints on measurement accuracy and/or the amount of data required. Also, the small-term expansion of the ellipsometrically determined pseudodielectric function 〈∊〉, originally derived as a thin-film limit of the three-phase model, is shown to be more generally valid. This result provides a theoretical basis for the direct analysis of several phenomena, including interface mixing, from 〈∊〉 data obtained during epitaxial growth. Using the CPA, I derive expressions that allow one to assess whether the performance of a given ellipsometer is adequate for growth control. Finally, the influence of selvage layers on determined values of ∊0 is briefly discussed.