Sébastien Equis

The empirical mode decomposition: a must-have tool in speckle interferometry?

In a wider and wider range of research and engineering activities, there is a growing interest for full-field techniques, featuring nanometric sensitivities, and able to be addressed to dynamic behaviors characterization. Speckle interferometry (SI) techniques are acknowledged as good candidates to tackle this challenge. To get rid of the intrinsic correlation length limitation and simplify the un-wrapping step, a straightforward approach lies in the pixel history analysis. The need of increasing performances in terms of accuracy and computation speed is permanently demanding new efficient processing techniques. We propose in this paper a fast implementation of the Empirical Mode Decomposition (EMD) to put the SI pixel signal in an appropriate shape for accurate phase computation. As one of the best way to perform it, the analytic method based on the Hilbert transform (HT) of the so-transformed signal will then be reviewed. For short evaluation, a zero-crossing technique which exploits directly the extrema finding step of the EMD will be presented. We propose moreover a technique to discard the under-modulated pixels which yield wrong phase evaluation. This work is actually an attempt to elaborate a phase extraction procedure which exploits all the reliable information in 3D, - two space and one time coordinates -, to endeavor to make the most of SI raw data.

Simulation of speckle complex amplitude: advocating the linear model

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.

Demodulation of spatial carrier images: Performance analysis of several algorithms using a single image

Optical full-field techniques have a great importance in modern experimental mechanics. Even if they are reasonably spread among the university laboratories, their diffusion in industrial companies remains very narrow for several reasons, especially a lack of metrological performance assessment. A full-field measurement can be characterized by its resolution, bias, measuring range, and by a specific quantity, the spatial resolution. The present paper proposes an original procedure to estimate in one single step the resolution, bias and spatial resolution for a given operator (decoding algorithms such as image correlation, low-pass filters, derivation tools …). This procedure is based on the construction of a particular multi-frequential field, and a Bode diagram representation of the results. This analysis is applied to various phase demodulating algorithms suited to estimate in-plane displacements.

Phase Extraction in Dynamic Speckle Interferometry by Empirical Mode Decomposition

In many respects, speckle interferometry techniques are now considered as mature tools in the experimental mechanics circles. These techniques have enlarged considerably the field of optical metrology, featuring nanometric sensitivities in whole-field measurements of profile, shape, and deformation of mechanical rough surfaces. Nonetheless, the phase extraction of speckle interferometry patterns is still computationally intensive, preventing a more widespread use of this technique especially in dynamic experiments. A promising approach lies in the temporal analysis of the pixel signals of photodetector arrays. The basic idea is to extract the instantaneous frequency (IF) of these signals, in order to obtain, in fine, the phase, i.e. the quantity of interest. A pioneering work has been done in this direction in [1] by using the ridge tracking method of Delprat et al [2] applied to the Morlet wavelet transform (MT).

Snap-shot profilometry with the Empirical Mode Decomposition and a 3-layer color sensor.

Remote sensing finds more and more applications, from industrial control, to face recognition, not forgetting terrain surveying. This trend is well exemplified by fringe projection techniques, which enjoyed a considerable development in the recent years. In addition of high requirement in terms of measurement accuracy and spatial resolution, the end-users of full-field techniques show a growing interest for dynamic regimes. We report here what we believe to be the use for the first time of a CMOS 3-layer color sensor (Foveon X3) as the key element of a RGB fringe projection system, together with the processing specifically elaborated for this sensor. The 3-layer architecture allows the simultaneous recording of three phase-shifted fringe patterns and features the precious asset of an unambiguous relationship between the physical sensor pixel and the picture pixel and this for each color layer, on the contrary of common color sensor arrays (Bayer mosaic and tri-CCD). Due to the overlapping of the spectral responses of the layers, color transformation is mandatory to achieve the separation of the three phase-shifted RGB projected fringe patterns. In addition, we propose the use of the Empirical Mode Decomposition to equalize the non-uniform responses of the three layers. Although the conversion of the phase into a height is of primary importance in an actual measurement, it is not treated here, the literature being profuse on the central projection model.

Phase extraction in speckle interferometry by a circle fitting procedure in the complex plane

In speckle interferometry (SI), temporal signals are amplitude- and frequency-modulated signals and exhibit a fluctuating background. Prior to phase computation, this background intensity must be eliminated. Here our approach is to build a complex signal from the raw one and to fit a circle through the points cloud representing its sampled values in the complex plane. The circle fit is computed from a set of points whose length is locally adapted to the signal. This procedure-new to our knowledge in SI-yields the background and the modulation depth and leads to the determination of the instantaneous frequency. The method, applied to simulated and experimental signals, is compared to empirical mode decomposition (EMD). It shows great robustness in the computation of the sought quantities in SI, especially with signals close to the critical sampling or, on the contrary, highly oversampled, situations where the background elimination by EMD is the most prone to errors.

Liquid blending: an investigation using dynamic speckle interferometry

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.

Label free 3D analysis of organelles in living cells by refractive index shows pre-mitotic organelle spinning in mammalian stem cells

Holo-tomographic microscopy (HTM) is a label-free non-phototoxic microscopy method reporting the fine changes of a cell’s refractive indexes (RI) in 3D. By combining HTM with epifluorescence, we demonstrate that cellular organelles such as Lipid droplets and mitochondria show a specific RI signature that distinguishes them with high resolution and contrast. We further show that HTM allows to follow in unprecedented ways the dynamics of mitochondria, lipid droplets as well as that of endocytic structures in live cells over long period of time, which led us to observe to our knowledge for the first time a global organelle spinning occurring before mitosis.