Louis A. Romero

Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods

Two-dimensional (2D) phase unwrapping continues to find applications in a wide variety of scientific and engineering areas including optical and microwave interferometry, adaptive optics, compensated imaging, and synthetic-aperture-radar phase correction, and image processing. We have developed a robust method (not based on any path-following scheme) for unwrapping 2D phase principal values (in a least-squares sense) by using fast cosine transforms. If the 2D phase values are associated with a 2D weighting, the fast transforms can still be used in iterative methods for solving the weighted unwrapping problem. Weighted unwrapping can be used to isolate inconsistent regions (i.e., phase shear) in an elegant fashion.

Phase encoding of shot records in prestack migration

Frequency‐domain shot‐record migration can produce higher quality images than Kirchhoff migration but typically at a greater cost. The computing cost of shot‐record migration is the product of the number of shots in the survey and the expense of each individual migration. Many attempts to reduce this cost have focused on the speed of the individual migrations, trying to achieve a better trade‐off between accuracy and speed. Another approach is to reduce the number of migrations. We investigate the simultaneous migration of shot records using frequency‐domain shot‐record migration algorithms. The difficulty with this approach is the production of so‐called crossterms between unrelated shot and receiver wavefields, which generate unwanted artifacts or noise in the final image. To reduce these artifacts and obtain an image comparable in quality to the single‐shot‐per‐migration result, we have introduced a process called phase encoding, which shifts or disperses these crossterms. The process of phase encoding...

Cellular-automata method for phase unwrapping

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.

Minimum Lp-norm two-dimensional phase unwrapping

We develop an algorithm for the minimum Lp-norm solution to the two-dimensional phase unwrapping problem. Rather than its being a mathematically intractable problem, we show that the governing equations are equivalent to those that describe weighted least-squares phase unwrapping. The only exception is that the weights are data dependent. In addition, we show that the minimum Lp-norm solution is obtained by embedding the transform-based methods for unweighted and weighted least squares within a simple iterative structure. The data-dependent weights are generated within the algorithm and need not be supplied explicitly by the user. Interesting and useful solutions to many phase unwrapping problems can be obtained when p< 2. Specifically, the minimum L0-norm solution requires the solution phase gradients to equal the input data phase gradients in as many places as possible. This concept provides an interesting link to branch-cut unwrapping methods, where none existed previously.

Lossless laser beam shaping

The Fresnel approximation is used in the design of a lens that turns a beam with one intensity distribution into a beam with a different distribution at the focal plane of a Fourier transform lens. In general, this cannot be done exactly, so one must be satisfied with approximate solutions to this problem. It is shown that the difficulty of this problem depends on a parameter β that is a dimensionless measure of how well the geometrical optics approximation holds. An analytical method is given for determining the lens when β is large. The sensitivity of this solution to various imperfections in the system alignment is also analyzed.

Direct phase estimation from phase differences using fast elliptic partial differential equation solvers

Obtaining robust phase estimates from phase differences is a problem common to several areas of importance to the optics and signal-processing communities. Specific areas of application include speckle imaging and interferometry, adaptive optics, compensated imaging, and coherent imaging such as synthetic-aperture radar. We derive in a concise form the equations describing the phase-estimation problem, relate these equations to the general form of elliptic partial differential equations, and illustrate results of reconstructions on large M by N grids, using existing, published, and readily available FORTRAN subroutines.

Normalized correlation for pattern recognition

The normalization of the correlation filter response effects intensity invariance. We discuss the implications of a normalization based on the Cauchy-Schwarz inequality for the discrimination problem. It is shown that normalized phase-only and synthetic discriminant functions do not provide the discrimination/recognition obtained with the classical matched filter.

Tippe Top Inversion as a Dissipation-Induced Instability

By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell--Bloch equations. We revisit previous work done on this problem and follow Ors mathematical model [SIAM J. Appl. Math., 54 (1994), pp. 597--609]. A linear analysis of the equations of motion reveals that the only equilibrium points correspond to the inverted and noninverted states of the tippe top and that the modified Maxwell--Bloch equations describe the linear/spectral stability of these equilibria. We supply explicit criteria for the spectral stability of these states. A nonlinear global analysis based on energetics yields explicit criteria for the existence of a heteroclinic connection between the noninverted and inverted states of the tippe top. This criteria for the existence of a heteroclinic connection turns out to agree with the criteria for spectral stability of the inverted and noninverted states. Throughout the work we support the analysis with numerical evidence and include simulations to illustrate the nonlinear dynamics of the tippe top.

BIFURCATION TRACKING ALGORITHMS AND SOFTWARE FOR LARGE SCALE APPLICATIONS

We present the set of bifurcation tracking algorithms which have been developed in the LOCA software library to work with large scale application codes that use fully coupled Newtons method with iterative linear solvers. Turning point (fold), pitchfork, and Hopf bifurcation tracking algorithms based on Newtons method have been implemented, with particular attention to the scalability to large problem sizes on parallel computers and to the ease of implementation with new application codes. The ease of implementation is accomplished by using block elimination algorithms to solve the Newton iterations of the augmented bifurcation tracking systems. The applicability of such algorithms for large applications is in doubt since the main computational kernel of these routines is the iterative linear solve of the same matrix that is being driven singular by the algorithm. To test the robustness and scalability of these algorithms, the LOCA library has been interfaced with the MPSalsa massively parallel finite element reacting flows code. A bifurcation analysis of an 1.6 Million unknown model of 3D Rayleigh–Benard convection in a 5 × 5 × 1 box is successfully undertaken, showing that the algorithms can indeed scale to problems of this size while producing solutions of reasonable accuracy.

Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings

We give an analytical basis for the theory of optimal beam splitting by one-dimensional gratings. In particular, we use methods from the calculus of variations to derive analytical expressions for the optimal phase function.