A practical technique is presented for finding the shape of a smooth, reflecting, convex object from its scattered field. The method assumes the field on the surface of the object satisfies Dirichlet boundary conditions. A surface identifier is computed from the T-matrix of the scatterer, which in turn, is found from data recorded on a surface surrounding the object. We restrict ourselves to two dimensional objects and the scalar wave equation, but the methods will readily extend, for example, to three dimensions or electromagnetic waves.
The Born and Rytov approximations are re-examined, and several necessary conditions for their validity are derived without recourse to extensive numerical computations for simple objects of specific geometries. Some published statements as to the relative merits of the two approximations, as well as the identity of their validity domains, are suggested to be in error.
A new principle to obtain sectional images of a thick specimen by a conventional optical microscope without physically slicing is developed. The optical microscope with an off-axis pupil is used to project a 3-D distribution of the specimen in various directions within the numerical aperture of the objective lens. Since the system is the strictly angularly-limited one, a strong constraint is needed to reconstruct the 3-D structure from the projections. The conjugate gradient method with the object boundary constraint which is a priori known meets this problem. Experimental results with a biological sample verify the capability of the proposed method as optical tomography of microscopic size.
The inverse scattering problem is partitioned into: (i) an inverse source problem, and (ii) a source interpretation problem. The sources are those induced in the scatterer by the incident field. This approach is useful when the scatterer can be characterised by scattering cent(e)r(e)s, so that the sources are discrete. Possible information processing strategies are outlined.
We consider a new computational method, for the solution of the inverse scattering problem of profile reconstruction for an impenetrable obstacle. The technique is iterative, and is based on the well known null field method for direct scattering problems. We direct our attention to the determination of the obstacle boundary, although the method can be extended to determine the spatial variation of refractive index for a certain class of penetrable obstacle.
The use of nonlinear operator equation techniques, and the Newton-Kantorovich method in particular, to solve inverse problems is outlined. The application of the method to two problems - inverse refractive index scattering and an inverse problem of steady-state diffusion - is then considered.
The problem of deducing the temperature profile of a spatially incoherent primary source from measurements of its generalized radiance is addressed. A reconstruction algorithm is presented that allows the spatial intensity profile of the source to be reconstructed from the generalized radiance specified over plane surfaces tangent to a sphere completely surrounding the source. The temperature profile is then determined from the intensity profile via well known formulae relating the temperature profile of an incoherent radiator to its intensity profile'. The entire development is shown to be a generalized approach to ray emission tomography 2 and reduces to the latter in the limit of very short wavelengths.
Renormalization techniques are applied to obtain solutions for inverse scattering problem that are equivalent to the second-order regular perturbation approximations for the exact (Gelfand-Levitan-Marchenko) theory. We have developed an inversion method for reconstructing the permittivity profiles of inhomogeneous dielectric layers from reflection coefficient data. Solutions with increased radii of convergence are obtained. Numerical examples are demonstrated for simulated scattering data from Gaussian, parabolic and triangular dielectric profiles and homogeneous slabs.
Fienup's iterative phase retrieval algorithms are applied to an image (confined to a support having a concave perimeter) whose phase can be chosen either zero, or varying arbitrarily between 0 and 7, or quite arbitrary. Reconstructed images are confined (or, more precisely, we attempt to confine them) to either the support or the rectangle just enclosing it. Both positive reconstructed images are faithful (as expected) when positivity is enforced. Bipolar and complex images reconstructed within the support are recognizable, but are scarcely recognizable when reconstructed within the rectangle. The bipolar reconstructed images are superior to the complex ones. These results reinforce the general experience that a support-constraint tends by itself to be too weak to ensure faithful image reconstruction.
Many phase retrieval problems cannot be readily solved with the Gerchberg-Saxton algorithm. In some cases no solution may exist. In all cases where the implied phase error is large, the algorithm partially converges to a phase afflicted with branch points and then seems unable to eliminate them, although the true phase cannot have branch points.
The analytic properties of two-dimensional band-limited functions are discussed. In practice, only a limited number of intensity samples are available, and so we choose to model the spectrum as a finite degree polynomial. The set of reducible finite degree multi-variate polynomials is of measure zero and unique recovery from noise free Fourier magnitude is expected in almost all cases. We pursue a new algorithm based on finding the complex zeros of 1-D lines of the data set which requires only that the intensity is sampled at twice the Nyquist rate or greater. All solutions compatible with the Fourier magnitude samples are generated, including ambiguities should they exist. An exact solution to the phase retrieval problem, given a polynomial model, may be regarded as factorization. We discuss the relationship of this approach to factorization and iterative procedures and describe problems arising from data truncation and the presence of noise.
In this paper we explore the application of a new iterative algorithm to super resolve one and two-dimensional signals that have been corrupted by zero mean, white, Gaussian noise. This is an example of solving a set of ill-posed linear equations subject to constraints. We demonstrate the superior performance of this algorithm over another proposed algorithm on simulated data. The simulations demonstrate that the algorithm leads to less noise amplification without a concommitant loss in convergence rate. Other applications of the technique are discussed.
The roughness parameters of diamond-turned surfaces are obtained from the Fourier transform of scattered light intensity. Theoretical consideration of the scattered intensity and its Fourier transform is made. It is shown that the Fourier transform of the scattered intensity is equivalent to the autocorrelation of the surface. From this autocorrelation function, we can derive the amplitude and frequency of the periodic tool marks, and the rms roughness and correlation length of the random surface component. Some experimental results for diamond turned aluminum metal surfaces are also given. The comparison of the light scattering data and stylus measurement is made. It shows a qualitative agreement.
A phase retrieval algorithm, developed and implemented by Perkin-Elmer, is used as a technique for in-orbit alignment of the optical system in NASA's Space Telescope. Reasonably accurate estimates of wavefront aberrations are obtained from measurements of the system point spread function (PSF). Further, the accuracy of the estimates increases as the magnitude of the aberrations decreases. Therefore, even if initial aberrations are large and result in crude estimates, a partial correction of the system can be made. A reapplication of the algorithm to the improved PSF will yield more accurate estimates of the uncorrected aberrations. In this fashion, a few iterations of the phase retrieval algorithm will allow perfect system correction from an aberrated state. The algorithm consists of initially defining an error function. The goal is to minimize the error function with respect to a set of parameters. Using Zernike polynomials to describe the phase in the pupil, we are able to write an analytic expression representing the PSF. An error function that can be minimized with respect to the coefficients of the Zernike polynomials is then calculated.