Complex-valued Object Research Articles

Reconstruction of a complex‐valued object under illumination with random light in coherent imaging through a random phase screen

We consider the reconstruction of a complex-valued object that is illuminated with random light (a sequence of speckle distributions) and viewed through a random phase screen. The reconstruction involves a phase retrieval based on the measurement of a series of two shortexposure Fourier intensities: the intensity of the Fourier transform of the image and the intensity of the Fourier transform of the image after transmission through an exponential filter. The method of reconstructing the object consists of the two steps: the reconstruction of the object intensity distribution from the average correlation functions of the short-exposure Fourier intensities of the unfiltered and filtered image fields and the retrieval of the object phase from object intensity distributions reconstructed from the series of the two short-exposure Fourier intensities multiplied by exponential functions. Computer-simulated examples of the reconstruction of 1-D complex objects demonstrate that the reconstruction is robust.

Reconstruction of a complex-valued object in double-passage coherent imaging through a random-phase screen

We consider the reconstruction of a complex-valued object that is coherently illuminated and viewed through the same random-phase screen. The reconstruction is based on two intensity measurements: the intensity of the Fourier transform of the image and the intensity of the Fourier transform of the image when modulated with an exponential filter. The illumination beam has a Gaussian intensity profile of arbitrary width, and the phase screen is assumed to be described by a Gaussian random process of large variance and arbitrary correlation length. Computer-simulated examples of the reconstruction of a two-dimensional complex object demonstrate that the reconstruction is robust.

Noniterative reconstruction of complex-valued objects from two intensity measurements

We present results obtained from the application of a novel phase-retrieval algorithm for recovering a complex-valued object from a set of two intensity measurements. The algorithm requires two intensity measurements at different distances from a weak scatterer, where the total transmitted field is composed of the coherent sum of an incident plane wave and the scattered wave. The algorithm is noniterative and does not have the convergence problems associated with iterative algorithms. The new technique shows great promise for inverse-scattering applications, such as optical diffraction tomography and in-line holography of complex-valued objects, with the aim of eliminating the twin-image problem. Results are presented from a computer simulation of a simple object and from experimental data obtained from a microlens array. Our results obtained using the new algorithm on experimental data compare well with those obtained with a modified form of the Gerchberg-Saxton algorithm, at a significantly reduced computational cost.

Reconstruction of a complex-valued object in coherent imaging through a random-phase screen

We consider the reconstruction of a complex-valued object that is coherently illuminated and then viewed through a random-phase screen. The reconstruction involves a phase retrieval based on two intensity measurements. The first is a measurement of the long-exposure averaged intensity of a Fourier transform of the image; it yields full information on the magnitude of the object Fourier transform but noi nformation on its phase. The second measurement is made with the image field modulated by an exponential function. This modulation has the effect of shifting the Fourier-transform function along the imaginary axis of the complex plane of its argument, thus making its intensity dependent on the unknown object phase. This method is capable of reconstructing the object except for an inherent ambiguity corresponding to a simple displacement. The effects of the noise arising from averaging over finite, instead of infinite, exposure times and the quantum noise were assessed. A computer-simulated example of reconstru ting a two-dimensional object demonstrated that the reconstruction is robust. The reconstruction error increases with an increase of the variance of the random-phase function and with a decrease of its correlation length.

Phase retrieval for a complex-valued object by using a low-resolution image

It is difficult to reconstruct an image of a complex-valued object from the modulus of its Fourier transform (i.e., retrieve the Fourier phase) except in some special cases. By using additionally a low-resolution intensity image from a telescope with a small aperture, a fine-resolution image of a general object can be reconstructed in a two-step approach. First the Fourier phase over the small aperture is retrieved, using the Gerchberg–Saxton algorithm. Then that phase is used, in conjunction with the Fourier modulus data over a large aperture together with a support constraint on the object, to reconstruct a fine-resolution image (retrieve the phase over the large aperture) by the iterative Fourier-transform algorithm. The second step requires a modified algorithm that employs an expanding weighting function on the Fourier modulus.

Uniqueness Features In Sampled Complex-Valued Object Reconstruction From Discrete Intensity Data

A procedure to reconstruct sampled complex-valued objects from intensity data at the output of a general optical system is given. It is shown that both the choice of the sampling points at the output plane (uniform or nonuniform sampling) and the specific values of the intensity at these points determine the uniqueness features of the reconstruction problem.

Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint

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.

Phase retrieval using boundary conditions

It is shown that a priori knowledge of the edges of an object is not sufficient to ensure that it can be uniquely reconstructed from the modulus of its Fourier transform (or from its autocorrelation function). Furthermore, even in those cases for which the ultimate solution is unique, in intermediate steps in the solution by the recursive Hayes–Quatieri algorithm there can be ambiguities. An extension of the recursive algorithm that finds the solution (or solutions) is suggested, and it is shown that the recursive method can be applied to complex-valued objects.

Representation-related errors in binary digital holograms: a unified analysis

Images reconstructed from binary digital holograms are degraded by errors due to the binary representation of the complex-valued object spectrum and by errors due to computational and plotter limitations. In this paper, representation-related errors are analyzed in terms of false images that appear in the desired reconstruction order and in adjacent diffraction orders. It is shown that the false images are strongly dependent on the manner in which the object spectrum is sampled and on the mapping from spectral sample to binary transmittance. Three categories of digital holograms are distinguished: those that sample the object spectrum at the center of each hologram cell; those that sample at the center of each aperture; and those that sample at each resolvable spot. In going from the first to the third category, the reconstruction is successively less degraded by false images. For the third category, there are no false images in the desired reconstruction order and only one false image in each adjacent order. The two methods in this category differ only in the suppression of these false images.