Coping with low modulation in speckle interferometry: a novel approach based on the Delaunay triangulation

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.