Interferometric signals involving speckle waves invariably exhibit phase indeterminations. These indeterminations
arise at the zero-intensities of the speckle fields, or singularities, and show themselves as a net loss of modulation depth
of the interferometric signals. To bypass the difficulty associated with the processing of low modulated parts of speckle
interferometry signals, we propose a novel approach based on the Delaunay triangulation (DT). The method applies in
both situations of static and dynamic regimes, and is designated respectively by "sine-cosine DT filter" and "3D
piecewise processing" or 3DPP - 3D denoting the temporal and the two spatial coordinates of the recording. The task
consists in discarding purely and simply the under-modulated parts of the signal according to a user-defined binary
criterion, and filling the missing parts by interpolation. This first step provides a grid with nodes randomly occupied by
reliable phase values or empty. At the empty nodes, the computed phase values result from a DT ensuring that the
interpolation relies on the three closest well-behaved neighbors, followed by spline-fitting a smooth surface over them.
In a dynamic regime - where the benefits of the temporal approach are unanimously acknowledged - the empirical mode
decomposition is used to select the valid intervals and the Hilbert transform to compute phase data therein. We give a
detailed description of the DT filtering techniques, show their ability to offer the optimal compromise between spatial
and measurement resolutions depending on the user-chosen binary criterion and highlight some definite advantages over
classical filtering methods in terms of phase error reduction and algorithmic complexity.
The dynamics of liquid-liquid mixing is a difficult problem, encountered in many scientific and engineering branches.
Experiments in this field are mandatory to help building sound mathematical models, finding out the best fit parameters,
evaluating the degree of confidence of these models, or detecting traces of unwanted dangerous substances. The
investigations reported here are driven by water pollution concerns. For analyzing the water-pollutant blending behavior,
dynamic speckle interferometry has been preferred to more standard optical full field methods, like deflectometry, or
classical and holographic interferometry. The choice of this technique is vindicated. The opto-fluidic system is described.
A first series of results is presented, demonstrating the effectiveness of the technique and showing qualitatively how two
liquids blend in controlled conditions. In the last part of the paper, recently appeared processing schemes, including
empirical mode decomposition, Hilbert transform and piecewise treatment, give access to the numerical values of the
phase maps computed for each frame of the recorded sequence. These phase maps represent the refractive index
distributions integrated along the line of sight. They provide a better visualization of the dynamics of the blending
behavior and therefore an improved understanding of the phenomena. These encouraging preliminary results should
open the door to a full characterization of the method and to further flow investigations and diagnostics.
The basic convolution integral, Uf = Uo crossed circle h where Uo is a random object complex amplitude and h the impulse response of the system under consideration, serves to model the observed speckle field Uf. Depending on the choice of h, the simulated field is an objective or a subjective speckle pattern. The computation makes use of two consecutive Fast Fourier Transforms. In the reported examples, the object function represents a pure phase diffuser ruled by a uniform distribution. The probability density functions (PDF) of the simulated intensity and phase patterns fit very well with their analytical counterparts obtained under the classical Gaussian hypotheses. Phase maps exhibit the awaited singularities. Moreover, elements of second order statistics, as the autocorrelation functions, are in very good agreement too. Furthermore, subtle effects, as the dip of contrast in the focused image plane of partially developed speckle patterns, are also suitably disclosed. The linear model thus appears, all together, as conceptually easy, very flexible, computationally simple, very accurate for a wide range of experiments, and endowed with excellent predictive and speculative potentials.
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