In optical metrology, the phase-shifting technique is used to retrieve the phase information from interferograms. Such displacement can be performed by mirrors attached to electromechanical devices (such as piezoelectric or moving mounts), gratings, or polarizing components, which need to be calibrated to associate the displacement of the device with respect to the induced phase shift. For this purpose, we present a closed-form formula to calculate of the original phase step between two randomly shifted fringe patterns by extending the Gram–Schmidt orthonormalization algorithm. To demonstrate its feasibility, we perform an evaluation that consists of three cases that represent different fringe pattern conditions. First, we evaluate the accuracy of the method in the orthonormalization process by estimating the test step using synthetic normalized fringe patterns with no background, a constant amplitude, and different noise levels. Second, we evaluate the formula with a variable amplitude function on the fringe patterns and a constant background. Third, we evaluate non-normalized noisy fringe patterns in which we include the comparison of prefiltering processes such as the Gabor filters bank, Hilbert–Huang transform, and isotropic normalization process and a high-pass filter to emphasize how they affect the calculation of the phase step.
Phase unwrapping is an important problem in the areas of optical metrology, synthetic aperture radar (SAR) image analysis, and magnetic resonance imaging (MRI) analysis. These images are becoming larger in size and, particularly, the availability and need for processing of SAR and MRI data have increased significantly with the acquisition of remote sensing data and the popularization of magnetic resonators in clinical diagnosis. Therefore, it is important to develop faster and accurate phase unwrapping algorithms. We propose a parallel multigrid algorithm of a phase unwrapping method named accumulation of residual maps, which builds on a serial algorithm that consists of the minimization of a cost function; minimization achieved by means of a serial Gauss–Seidel kind algorithm. Our algorithm also optimizes the original cost function, but unlike the original work, our algorithm is a parallel Jacobi class with alternated minimizations. This strategy is known as the chessboard type, where red pixels can be updated in parallel at same iteration since they are independent. Similarly, black pixels can be updated in parallel in an alternating iteration. We present parallel implementations of our algorithm for different parallel multicore architecture such as CPU-multicore, Xeon Phi coprocessor, and Nvidia graphics processing unit. In all the cases, we obtain a superior performance of our parallel algorithm when compared with the original serial version. In addition, we present a detailed comparative performance of the developed parallel versions.
We present recent applications of robust half-quadratic regularization for solving common problems in interferogram analysis. Specifically, we review half-quadratic cost functions for: phase unwrapping noisy and discontinuous wrapped phase maps, phase refinement process for computed coarse phases, as those obtained from partial-field patterns with a full-field method for open fringes analysis, for phase retrieval from closed-fringe interferograms and phase denoising. The cases of convex and the non-convex cost functions are analyzed: The convex formulation produces unique and noise-free solutions, but such solution are deteriorated by real discontinuities in phase maps; therefore, we also present a non-convex formulation which, with an extra computational cost, shows a superior performance.
One of the powerful approaches to demodulate a single fringe-pattern is the regularized phase tracking (RPT) technique. We present here some modifications in the algorithm, which consist in the addition of the modulation estimation and the modification of the minimization algorithm. With these changes, the RPT technique can be used in the demodulation of non-normalized fringe patterns with significative improvement in the processing time.
Analysis of fringe patterns with partial-field and/or closed fringes is still a challenging problem that requires the development of robust methods. This paper presents a method for fringe pattern analysis with those characteristics. The method is initially introduced as a phase refinement process for computed coarse phases, as those obtained from partial-field patterns with a full-field method for open fringes analysis. Based on the phase refinement method, it is proposed a propagative scheme for phase retrieval from closed-fringe interferograms.
KEYWORDS: Diffusion, Signal to noise ratio, Image registration, Surface conduction electron emitter displays, Anisotropy, Magnetism, Tissues, Associative arrays, Radiology, In vivo imaging
The aim of this study was to examine the registration of diffusion tensor magnetic resonance images. A method for estimating a smooth, continuous mapping between two tensor images is presented. This method includes a tensor-to-tensor measure of similarity as well as a neighborhood similarity measure intended to preserve the relative position of adjacent structures. Additionally, tensor reorientation is integrated into the algorithm in order to insure that the structural information provided by the diffusion tensor is retained. This method was tested on a variety of synthetic data sets. Experiments indicate that the orientation similarity term plays an important role in both accuracy and speed. Additionally, an investigation of the effect of signal to noise ratio (SNR) was conducted to insure the usefulness of the methods at clinically obtainable values. Qualitative examination of the results obtained with this method suggest its potential usefulness in the examination of in vivo human data, but some extension of the method as well as further testing will be necessary to fully understand its limitations for use on clinical data.
This paper proposes a robust method for computing discontinuous phase maps. The proposed algorithm is based on the minimization of a edge-preserving regularized cost functional. We use a robust regularized potential based on the paradigm of the Plate with Adaptive Rest Condition (PARC). Our algorithm computes the phase from fringe patterns with discontinuities on the fringe pattern due to steps on the phase and changes on the illumination component. The method is presented in the context of calibrating Gauge Blocks by interferometric method. The performance of the method is demonstrated by numerical experiments on both synthetic and real data.
A well-founded and computationally fast method is presented for filtering and interpolating noisy and discontinuous wrapped phase fields that preserves both the 2(pi) discontinuities that come from the wrapping effect and the true discontinuities that may be present. It also permits the incorporation of an associated quality map, if it is available, in a natural way. Examples of its application to the computation recovery of discontinuities phase fields from speckle interferometry fracture measuring are presented.
The task of image segmentation implies estimation of the number and associated parameters of the classes within an image, and the class label for each image voxel. In this work, an over-segmentation of the data is first obtained using a Bayesian restoration algorithm. The method incorporates a novel spatial interaction prior, in which neighboring voxels can be classified differently so long as the distance between the centroids of their intensity distributions are within a certain extent. The corresponding functional is iteratively minimized using a series of local optimizations for the label field and a half-quadratic algorithm for the restoration. Redundant classes are then grouped in a second step by making use of information obtained in the initial restoration about the degree of affinity or interaction between the classes. The method is demonstrated on MRI images of the head.
The wavefronts in some optical systems like radial Gradient Index (GRIN) rods and aspherical lenses have large radial slopes at the circular edge of the pupil. Then some special mathematical tools are needed to represent these wavefronts. An idea of the mechanisms employed to fabricate the GRIn rods is helpful when designing these tools. In this article we propose the use of a gaussian function to represent these wavefront deformations at the edge of the pupil. This is a compact representation and it is also very convenient for radial GRIN rods. A data fitting is performed with a regularization process introducing the a priori known wavefront characteristics.
We present a new class of models, derived form classical Markov Random Fields, that may be used for the solution of ill-posed problems in image processing and computational vision. They lead to reconstruction algorithms that are flexible, computationally efficient and biological plausible. To illustrate their use, we present their application to the reconstruction of the dominant orientation field and to the adaptive quantization and filtering of images in a variety of situations.
A very powerful technique for solving the kind of inverse problems that often arise in the processing of fringe pattern images is based on Bayesian Estimation with prior Markov Random Field models. In this approach, the solution of a processing problem is characterized as the minimizer of a cost function which has two types of terms: terms that specify that the solution should be compatible with the available observations and terms that impose certain constraints on the solution. In this paper we show that by the appropriate choice of these terms, one can use this approach in almost every processing step for accurate interferogram demodulation. Specifically, one can construct: robust smoothing filters that are almost insensitive to edge effects; operators that automatically determine a mask that indicates the shape of the region where valid fringes are available; adaptive quadrature filters for phase recovery from single and multi-phase stepping interferograms and robust phase unwrapping algorithms.
It is proposed a quadratic cost functional for calculating a consistent gradient field from an inconsistent one. This
inconsistent gradient in obtained by wrapping the first difference ofthe wrapped phase. The derivation ofthe respective Euler
equations of the proposed functional is presented. A discrete Fourier transfonn based technique for solving the Euler
equations is shown. The calculated unwrapped phase may then be calculated by integrating the consistent gradient with any
integration algorithm. This fast (transform) method has the following advantages over existing Fourier technique: their
transfer functions are close to one and, the boundary errors are estimated and minimized. So that the global estimation error
is reduced.
Keywords: Phase miwrapping, Finite differences, Regularization.
A Fizeau interferometer has been built to test aspherical and concave telescope mirrors. Results of one method used to evaluate the maximum number of fringes when the interferogram image is projected on a CCD device are mentioned. Some complementary techniques have been utilized to analyze interferograms using this kind of interferometer. A deterministic approach to the regularization term added by Marroquin and Rivera (1995) to the least-squares unwrapping technique is shown.
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