Invariant polarimetric contrast parameters of coherent light

Many applications use an active coherent illumination and analyze the variation of the polarization state of optical signals. However, as a result of the use of coherent light, these signals are generally strongly perturbed with speckle noise. This is the case, for example, for active polarimetric imaging systems that are useful for enhancing contrast between different elements in a scene. We propose a rigorous definition of the minimal set of parameters that characterize the difference between two coherent and partially polarized states. Indeed, two states of partially polarized light are a priori defined by eight parameters, for example, their two Stokes vectors. We demonstrate that the processing performance for such signal processing tasks as detection, localization, or segmentation of spatial or temporal polarization variations is uniquely determined by two scalar functions of these eight parameters. These two scalar functions are the invariant parameters that define the polarimetric contrast between two polarized states of coherent light. Different polarization configurations with the same invariant contrast parameters will necessarily lead to the same performance for a given task, which is a desirable quality for a rigorous contrast measure. The definition of these polarimetric contrast parameters simplifies the analysis and the specification of processing techniques for coherent polarimetric signals.     

Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations

We propose to analyze Shannon entropy properties of partially coherent and partially polarized light with Gaussian probability distributions. It is shown that the Shannon entropy is a sum of simple functions of the intensity, of the degrees of polarization, and of the intrinsic degrees of coherence that have been recently introduced. This analysis clearly demonstrates the contribution of partial polarization and of partial coherence to the characterization of disorder of the light provided by the Shannon entropy, which is a standard measure of randomness. We illustrate these results on two simple examples.     

Invariant activity detection of a constant magnitude signal with unknown parameters in white Gaussian noise

The authors propose invariant tests for the detection of a complex signal with unknown constant amplitude and unknown phase variation in additive white Gaussian noise (AWGN). The authors show that in this problem, the uniformly most powerful invariant (UMPI) detector does exist only if the number of samples N is two. For more than two samples N ges 3, the authors derive the most powerful invariant (MPI) detector in known signal-to-noise ratio (SNR) and use its performance as the upper bound benchmark for any invariant test. In addition, the authors derive the generalised likelihood ratio (GLR) detector and evaluate its performance against the MPI performance bound. This detector is very simple and represents the ratio of the L ₁-norm to the L ₂-norm of the data. Simulation results illustrate the close performances of the two detectors even at low SNRs, whereas in contrast to the MPI test the SNR is not required in the proposed GLR test. In order to understand why the knowledge of SNR is not so important in this detection problem, the authors also derive the GLR test for the case of known SNR. Interestingly, the resulting GLR detector (derived for the case of known SNR) turns out to be equivalent to the one derived for unknown SNR, i.e. a knowledge of the SNR is not used in any of the GLR tests. This reveals why the knowledge of the SNR is not so useful in this detection problem.     

Observation of Anderson localization of light in three dimensions

Using time-resolved transmission measurements, we have found indications of Anderson localization of light in bulk three-dimensional systems. The observed deviation from classical diffusion is in good accord with theoretical predictions of localization and cannot be explained by absorption or experimental artifacts such as stratification, fluorescence, or background illumination. Moreover, we show that in our samples the control parameter is given by the mean free path times the wavenumber as required by the Ioffe-Regel criterion. This is in contrast to quasi-one-dimensional systems that were studied with microwaves. There, the control parameter is related to the number of modes inside a waveguide, and deviations from classical diffusion are possible due to a small number of modes.     

General state contrast imaging: an optimized polarimetric imaging modality insensitive to spatial intensity fluctuations

In active polarization imaging, one frequently needs to be insensitive to noninformative spatial intensity fluctuations. We investigate a way of solving this issue with general state contrast (GSC) imaging. It consists in acquiring two scalar polarimetric images with optimized illumination and analysis polarization states, then forming a ratio. We propose a method for maximizing the discrimination ability between a target and a background in GSC images by determining the optimal illumination and analysis states. A further advantage of this approach is to provide an objective way of quantifying the performance improvement obtained by increasing the number of degrees of freedom of a GSC imager. The efficiency of this approach is demonstrated on simulated and real-world images.     

Passage of light through a general three dimensional model—equations linking characteristic parameters with system parameters

The determination of the secondary principal stress difference and their orientations for a discrete slice of thickness dz along a particular light path in a three dimensional photoelastic model is discussed in this paper. These quantities are determined in terms of the characteristic parameters¹ and their variations at discrete points distance z and z + dz along the light path. The variations of the characteristic parameters are obtained by deriving the equations linking the characteristic parameters for the light path and the system parameters of the discrete element. The validity of the equations so derived has also been checked by applying these equations to the solution of the problem of a uniformly twisted prismatic bar.     

On the influence of noise statistics on polarimetric contrast optimization

In active scalar polarimetric imaging systems, the illumination and analysis polarization states are degrees of freedom that can be used to maximize the performance. These optimal states depend on the statistics of the noise that perturbs image acquisition. We investigate the problem of optimization of discrimination ability (contrast) of such imagers in the presence of three different types of noise statistics frequently encountered in optical images (Gaussian, Poisson, and Gamma). To compare these different situations within a common theoretical framework, we use the Bhattacharyya distance and the Fisher ratio as measures of contrast. We show that the optimal states depend on a trade-off between the target/background intensity difference and the average intensity in the acquired image, and that this trade-off depends on the noise statistics. On a few examples, we show that the gain in contrast obtained by implementing the states adapted to the noise statistics actually present in the image can be significant.     

Partial polarization in three dimensions

Various different parameters have been introduced to describe the degree of polarization of a partially polarized electromagnetic field in three dimensions. Of these, parameters based on the eigenvalues of the coherency matrix are invariant under a unitary transformation. Here, explicit expressions are presented for the eigenvalues, thus providing a geometrical interpretation of the behavior. These expressions are applied to the Huynen decomposition and allow interrelations between different parameters to be developed.     

Tricubic interpolation in three dimensions

The purpose of this paper is to give a local tricubic interpolation scheme in three dimensions that is both C¹ and isotropic. The algorithm is based on a specific 64 × 64 matrix that gives the relationship between the derivatives at the corners of the elements and the coefficients of the tricubic interpolant for this element. In contrast with global interpolation where the interpolated function usually depends on the whole data set, our tricubic local interpolation only uses data in a neighbourhood of an element. We show that the resulting interpolated function and its three first derivatives are continuous if one uses cubic interpolants. The implementation of the interpolator can be downloaded as a static and dynamic library for most platforms. The major difference between this work and current local interpolation schemes is that we do not separate the problem into three one‐dimensional problems. This allows for a much easier and accurate computation of higher derivatives of the extrapolated field. Applications to the computation of Lagrangian coherent structures in ocean data are briefly discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Stress-intensity factor computation in three dimensions with quarter-point elements

A consistent method for computing stress-intensity factors from three-dimensional quarter-point element nodal displacements is presented. The method is generalized to permit functional evaluation of stress-intensity factors along the crack front. Embedded, surface, and corner crack problems are solved using the proposed technique. Results are compared to previous finite element and boundary element solutions. The comparison shows that use of the functional evaluation technique allows a dramatic decrease in problem size while still maintaining engineering accuracy. Next, a three-dimensional stress-intensity factor calibration of an unusual specimen configuration is presented. By taking advantage of the proposed technique, the calibration was performed with little difference in cost over the more usual two-dimensional approach. Moreover, the three-dimensional solution revealed intersting behaviour that would have been undetected by a two-dimensional solution. Finally, the results of a study on optimum size of the quarter-point element are presented. Surprisingly, Poisson ratio is shown to have marked effect on optimum element size.     

Microstructure and fracture in three dimensions

A high resolution three dimensional (3D) scanning technique called X-ray microtomography was used to measure internal crack growth in small mortar cylinders under compressive loading. Tomographic scans were made at different load increments in the same specimen. 3D image analysis was used to measure internal crack growth during each load increment. Load–deformation curves were used to measure the corresponding work of the external load on the specimen. Fracture energy was calculated using a linear elastic fracture mechanics approach using the measured surface area of the internal cracks created. The measured fracture energy was of the same magnitude that is typically measured in concrete tensile fracture. A nominally bilinear incremental fracture energy curve was measured. Separate components for crack formation energy and secondary toughening mechanisms are proposed. The secondary toughening mechanisms were found to be about three times the initial crack formation energy.     

Finite element Euler computations in three dimensions

An explicit finite element solution procedure for the three dimensional Euler equations is presented. The solution domain is automatically meshed using a tetrahedral mesh generator which is an extension of our previous two dimensional work. Several examples are included to illustrate the performance of the generator and solver. An adaptive mesh regeneration procedure is used for the first time in three dimensions.     

Method for monitoring positions in three dimensions

This article presents a fundamentally new method for controlling the position of an object in three-dimensional space by using a microinterferometer. This method assures high accuracy and efficiency. The method can be used for verifying accuracy of contact gauges in position-measuring machinery. Translated from Izmeritel'naya Tekhnika, No. 4, pp. 17–18, April, 1996.     

Multiple-relaxation-time lattice Boltzmann models in three dimensions

This article provides a concise exposition of the multiple-relaxation-time lattice Boltzmann equation, with examples of 15-velocity and 19-velocity models in three dimensions. Simulation of a diagonally lid-driven cavity flow in three dimensions at Re = 500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of the multiple-relaxation-time lattice Boltzmann equation over the popular lattice Bhatnagar-Gross-Krook equation.     

Adaptive multivlevel methods in three space dimensions

We consider the approximate solution of self-adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. As the performance of hierarchical bases preconditioners deteriorates in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy.     

Universal meshes for smooth surfaces with no boundary in three dimensions

We introduce a method to mesh the boundary Γ of a smooth, open domain in immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh 1 for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. Copyright © 2016 John Wiley & Sons, Ltd.