Tomographic Reconstruction of 3D Objects Using Marked Point Process Framework

Abstract

For reconstructing sparse volumes of 3D objects from projection images taken from different viewing directions, several volumetric reconstruction techniques are available. Most popular volume reconstruction methods are algebraic algorithms (e.g. the multiplicative algebraic reconstruction technique, MART). These methods which belong to voxel-oriented class allow volume to be reconstructed by computing each voxel intensity. A new class of tomographic reconstruction methods, called “object-oriented” approach, has recently emerged and was used in the Tomographic Particle Image Velocimetry technique (Tomo-PIV). In this paper, we propose an object-oriented approach, called Iterative Object Detection—Object Volume Reconstruction based on Marked Point Process (IOD-OVRMPP), to reconstruct the volume of 3D objects from projection images of 2D objects. Our approach allows the problem to be solved in a parsimonious way by minimizing an energy function based on a least squares criterion. Each object belonging to 2D or 3D space is identified by its continuous position and a set of features (marks). In order to optimize the population of objects, we use a simulated annealing algorithm which provides a “Maximum A Posteriori” estimation. To test our approach, we apply it to the field of Tomo-PIV where the volume reconstruction process is one of the most important steps in the analysis of volumetric flow. Finally, using synthetic data, we show that the proposed approach is able to reconstruct densely seeded flows.

Appendix A: Data-Driven Energy Calculation Details

If the square of Eq. (4) is developed, the following result is obtained:

$$\begin{aligned} \hbox {MSE}(y \left| o \right. )= \sum _{s \in S} o_{s}^2 + p_{s} ^2 - 2 o_{s} p_{s} \end{aligned}$$

(32)

The first term of the addition in Eq. (32) is constant. The data-driven energy defining the likelihood between the projection of a configuration y and the observed data $o = \left\{ o_s\right\} _{\left\{ s \in S\right\} }$ can be written:

$$\begin{aligned} U_{\text {ext}}(y) = \sum _{s \in S} p_{s}^2 - 2 o_{s} p_{s} \end{aligned}$$

(33)

$U_{\text {ext}}$ defines the quality of a configuration compared to the data: the closer the projection of a population of objects is to the reference image, the lower its energy value. We show below that this energy can be written as a sum of first-order and second-order neighborhood energy terms.

Using Eqs. (5), (33) can be written:

$$\begin{aligned}&\begin{aligned} U_{\text {ext}}(y)&= \sum _{s \in S} \left[ \left( \sum _{y_j , y_j \rightarrow s} p_{y_j \rightarrow s} \right) ^2 \right. \\&\quad \left. -\, 2 \ o_{s} \ \left( \sum _{y_j , y_j \rightarrow s} p_{y_j \rightarrow s} \right) \right] \\ \end{aligned} \end{aligned}$$

(34)

$$\begin{aligned}&\begin{aligned} U_{\text {ext}}(y)&= \sum _{s \in S} \left[ \left( \sum _{y_j , y_j \rightarrow s} p_{y_j \rightarrow s} ^2 \right. \right. \\&\qquad \left. +\, 2 \sum _{ \begin{array}{c} y_j , y_k , j < k \\ y_j \rightarrow s , y_k \rightarrow s \end{array}} p_{y_j \rightarrow s} \ p_{y_k \rightarrow s} \right) \\&\qquad \left. - \,2 \ o_{s} \ \left( \sum _{y_j , y_j \rightarrow s} p_{y_j \rightarrow s} \right) \right] \end{aligned} \end{aligned}$$

(35)

$$\begin{aligned}&\begin{aligned} U_{\text {ext}}(y)&= \sum _{s \in S} \sum _{y_j , y_j \rightarrow s} p_{y_j \rightarrow s} \left( p_{y_j \rightarrow s} - 2 \ o_s \right) \\&\qquad +\, \sum _{s \in S} 2 \sum _{ \begin{array}{c} y_j , y_k , j < k \\ y_j \rightarrow s , y_k \rightarrow s \end{array}} p_{y_j \rightarrow s} \ p_{y_k \rightarrow s} \end{aligned} \end{aligned}$$

(36)

$$ \begin{aligned}&\begin{aligned} U_{\text {ext}}(y)&= \underbrace{ \sum _{y_j \in y} \sum _{\begin{array}{c} s \in S \\ y_j \rightarrow s \end{array}} p_{y_j \rightarrow s} \left( p_{y_j \rightarrow s} - 2 \ o_s \right) }_{first-order\ term}\\&\qquad +\, \underbrace{\sum _{\begin{array}{c} y_j , y_k , j<k \\ y_j \overset{o}{\sim } y_k \end{array}} \sum _{\begin{array}{c} s \in S \\ y_j \rightarrow s \ \& \ y_k \rightarrow s \end{array}} 2 p_{y_j \rightarrow s} \ p_{y_k \rightarrow s}}_{second-order\ term} \end{aligned} \end{aligned}$$

(37)