Tomo-PIV Reconstruction based on the Alternating Direction Method of Multipliers . 

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From Planar to 3D Measurements

Turbulent flows exhibit complex and highly 3D motions over a wide range of scales and amplitudes. Given the highly tridimensional nature of fluid velocity under certain regimes, planar measurement systems quickly evolved towards systems allowing for the retrieval of the three components of the velocity fields. This section depicts the evolution of planar tracer methods toward 3D probes. We intent, by the present overview, to bring forth certain theoretical and practical tools indispensable for the comprehension of the manuscript. For thorough descriptions of this evolution, the reader should start by addressing the bibliography of PIV by Adrian [3] which covers exhaustive early references on PIV starting with Meynart’s articles released during his Ph.D. thesis [126,127] – marking the introduction of PIV- and scanning up to succeeding developments until 1995. Lateron, the same author flags the 20 years since the term was first coined by pointing out important milestones until the early 3D developments thought stereoscopic-PIV [2].
More recently, book chapter [13] and article [155] review techniques for turning PIV into a 3D velocimetry procedure. Likewise, in [156] a brief survey of the advancement with respect to dimensionality is depicted, together with working principles of the 3D probes. The reader should refer to [153] and [4] for complete theoretical comprehension of the signal processing underpinning particle tracer velocimetry.
A PIV system is historically a quantitative measurement allowing for the retrieval of velocity vectors in a planar domain. Figure 1.3 depicts the standard progression of steps towards these measurements. First, light passive tracers are immersed in the flow with respect to its nature. A camera faces the volume of interest. Then, a light sheet of the latter is illuminated at high frequency by a laser such that the light scattered by the particles is captured on the camera’s plane at each period. These consecutive recordings are then post-processed by means of motion retrieval techniques, provided the particles follow the fluid’s movements, in order to compute the velocity fields within the illuminated plane.

Volume Illumination

As pointed out earlier, the illumination of the volume is enabled by means of laser lighting. The preceding planar PIV techniques need a thin sheet of light aligned to the planes of the captors and parallel to the direction of the flow; there are at most two cameras placed in a stereoscopic configuration (thus very close together); the beam is thinned out by using an intricate system made out of cylindrical lenses and the alignment is realized in order to avoid out-of-plane motion. Concurrently, the volumetric techniques rely on the illumination of a bigger region, ideally uniformly spread on the three dimensions of the space. As a result, there is no need to focus the light on a single sheet; on the contrary, the latter is widened by the use of a beam expander. We give, in this section, the description of the volume illumination profile in the physical scene and a mathematical model which accounts for its behavior.
In a typical optical setup (cf. Figure 2.2 (a)) the light intensity decreases throughout the volume inversely proportional to the square of the distance from the light source. Let the direction of the source light be taken as the −yw = h 0 −1 0 iT axis as depicted by the Figure 2.2. We can model this radial intensity profile along the region of interest as a Gaussian function in the xw = h 1 0 0 iT direction. Let l : R3 ! R be the function depicting the evanescent behavior of the light in the scene at an instant t. The shading intensity function in a point h 2 R3 then writes: l(h) =˜it exp  » − (h1)2 22 f # , (2.3).

Light scattering

The lightning model of the scene described in 2.1.1 can be further refined by taking into account the light scattered model by the seeding particles. In a nutshell, we call the scattered light the part of the incident laser light which is imaged by the particles onto the detectors. Consequently, it is the scattered information which duly enables access to the information on the velocity flow. A sensible choice of the nature of the particles will impact on the quality of the measurements. In fact, the more powerful is the scattered energy, the more contrasted are the images and thus, they will constitute more discriminant assessment. We begin our section by making a small introduction on general notions related to the scattering of the light; next, we present an entry point into the Mie scattering theory adapted for the tomoPIV setting; finally, we formalize related models in order to take into account the scattering of the light by the passive tracers into the experimental scene.

General Notions on Light Scattering

Fundamental notions about light scattering can be found in [6, 33, 179]. The study of the phenomenon is restrained based on several a priori knowledge and assumptions. Let us draw a list based on information compiled from previously cited publications. First of all, the scattering is elastic: the frequency of the scattered light is the same as that of the incident light. Secondly, particles are rarely solely represented in a scene; one particle usually belongs to a collection populating the volume of interest. Implicitly, they are electromagnetically coupled, thus each particle is excited by the resultant field scattered by all the other particles and by the exterior field. The simplification in order assumes single scattering: the number of particles is sufficiently small and their separation sufficiently large that, in the neighborhood of any particle, the total field scattered by all the particles is small compared with the external field. Moreover, we assume incoherent scattering assuming that the separation between the scatterers is random; this implies that there is no consistency in the relation among the phases of the waves scattered by the individual particles. The scattering behavior of the light varies with regard to the obstacles encountered by the latter on its way to the sensors. In [189], a review of elastic light scattering theories is provided based on the specifications of the physical scene. The author states that the theory to compute or approximate the scattered light power should be a function of the obstacle’s shape, its composition and refractive index and its size relative to the wavelength of the incident wave.

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Light Scattering for Small Particles

We address the light scattering problem for our specific optical setting. As outlined in section 2.1.1 when the scene is illuminated with a laser source, the distribution of this energy over the light beam leads to a relatively low energy density. If enhancements to the classical scheme have been brought, the efficiency of the particle scattering is still of utmost importance to the intensification of their projections on the recordings. As discussed in the previous section, we model the particles as spheres with small diameters (e.g., up to 10 microns, for air flows experiments). Given the ratio between the radius of a particle and the wavelength of the illumination source, we embrace the thesis that the scattering of light by these particles occurs in the so-called Mie regime. The latter can be characterized by the normalized diameter dq defined as a function of the particle’s diameter dp and the wavelength of the laser source such that: dq = dp . (2.7).
The Mie scattering model has been fully described by Bohren and Huffman [33]. The problem has been addressed for a linear, homogeneous, isotropic particles in chapter 4 of their book. The authors express the direct problem, which consists in the computation of the field of intensity at any point in the scene, given the precise description of the optical elements of the scene. Based on their breakthroughs, we are interested, in this section, in faithfully modeling the experimental scene, with regard to physical consideration such as dp, and Np, where Np is the refractive index of the particles.
The study of the light scattering of a particle within the tomoPIV context has not received, to our knowledge, high attention when designing the observation model of the scene. The toolbox implemented by Mätzler [121] is based on the model formalized in [33] and computes the scattered intensity with respect to the wavelength of the laser beam , to the refractive index of the particles Np and to the particle’s diameter dp. The toolbox developed by Schäfer during his PhD thesis [159] also relies on the model proposed by [33], but approximates the scattering of the light with respect to the same parameters as above and with respect to the so-called scattering angle; the latter is defined as the angle between the incoming light source and the sensor’s direction in the scattering plane of the particle. This further development allows us to account for a smooth scaling factor in the image formation with regard to the position of the sensor on the scene. In [60], the Mie scattering is accounted for by approximating an angular scattering function for typical PIV parameters basedon the first described toolbox. The authors use however an inexact mode to determine the smoothing scale factor for all the cameras by performing a parametric estimation out of images generated using the Phong reflection model [149]. In [51], the authors perform a parametric study showing the behavior of the scattering function with respect to polydispersion and scattering angles. Further developments including taking into account the scattering behavior into the observation model are stated as future work in the upper-mentioned article and have not been integrated, to our knowledge, at this date.

Table of contents :

1 Experimental Fluid Mechanics 
1.1 Passive Tracers
1.2 Measurements in Flows
1.3 From Planar to tridimensional (3D) Measurements
1.4 Conclusion
2 TomoPIV: Settings and Models 
2.1 Experimental Setup
2.1.1 Volume Illumination
2.1.2 Particles Dynamics Light scattering
2.1.3 Sensors
2.1.4 Small Particles’ Imaging
2.1.5 System Calibration
2.1.6 Summary
2.2 Related Model
2.2.1 General Assumptions
2.2.2 Continuous frame Density Function and Transport Assumption Physical-Based Continuous Model Approximated Continuous Model
2.2.3 Digitized frame Preliminary Notations Blob-Based Density Function Particle-Based Density Function Image Formation Transport Model
2.2.4 Summary
3 Volume Reconstruction
3.1 System Features
3.2 Notations
3.3 Standard Procedures for Tomographic PIV (tomoPIV) Volume Reconstruction
3.4 Inverse Problem
3.4.1 Choices of the Cost Function on the Signal
3.4.2 Choices on the Function Penalizing the Prediction-Observation Discrepancy
3.5 Beyond (M)ART with Proximal Methods
3.5.1 The Gradient Project Method (GPM) and Variants
3.5.2 Proximal Gradient Method (PGM) Applied to Tomo-PIV Problem (3.18)
3.5.3 Nonlinear GPM Applied to Tomo-PIV Problem (3.19)
3.6 Tomo-PIV Reconstruction based on the Alternating Direction Method of Multipliers .
3.6.1 Alternating Direction Method of Multipliers (ADMM)
3.6.2 Application to Tomo-PIV Problem (3.17)
3.7 Other Computational Methods for Sparse Linear Solutions
3.8 Guarantees Related to the `0- and `1-norms
3.9 Pruning
3.10 Assessement
3.10.1 Synthetic Setting
3.10.2 Description of Evaluation Criteria
3.10.3 Nomenclature
3.10.4 Pruning Assessement
3.10.5 Volume Reconstruction Assessement
3.11 Summary
4 Velocity Estimation 
4.1 Features
4.2 Classical Motion Estimation Methods
4.3 Joint Local Method
4.4 Assessement
4.4.1 Synthetic Setting
4.4.2 Description of Evaluation Criteria
4.4.3 Nomenclature
4.4.4 Velocity Reconstruction Assessement
4.5 Summary
5 Conclusion and Perspectives 
A Lagrangian and Eulerian Specification of the Flow Field 
B Mie Scattering Coefficients 
C Synthetic Configuration of the Imaging System 
D Implementation of (2.39) 
E The Proximal Operator 


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