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This research is motivated by the challenges to be solved in X-ray CT or in emerging tomographic imaging modalities such as X-ray phase CT, Compton camera CT. To this aim, specific solutions dedicated to the various modalities are developed as well as more general algorithms in inverse problems.
Generic methods in Inverse problems (M Robini)
Stochastic optimization: Simulated annealing (SA) is a generic optimization method that is quite popular because of its ease of implementation and its optimal convergence properties. Still, SA is widely reported to converge very slowly, and it is common practice to allow extra freedom in its design at the expense of losing global convergence guarantees. A natural way to increase the flexibility of SA is to allow the objective function and the communication mechanism to vary with temperature, the idea being to gradually reveal the complexity of the optimization problem and to increase the mixing rate of the annealing chain. We call this general class of annealing processes stochastic continuation (SC). We proved that SC inherits the global convergence properties of SA under very weak assumptions [ROBI-10] [ROBI-11a] [ROBI-2012] [ROBI-13b] and we demonstrated the effectiveness of SC in image reconstruction [ROBI-10], graph embedding [ROBI-13b], and inverse treatment planning [ROBI-11b][ROBI-13c].
Edge-preserving restoration with contour-line smoothing: The standard approach to image restoration is to stabilize the inversion process by including an edge-preserving roughness penalty in addition to faithfulness to the data. However, this methodology produces noisy object boundaries and creates a staircase effect. The existing attempts to favor the formation of smooth contour lines take the edge field explicitly into account; they either are computationally expensive or produce disappointing results. We proposed to incorporate the smoothness of the edge field in an implicit way by means of an additional penalty term defined in a multiresolution domain, and we derived an efficient half-quadratic algorithm to solve the resulting optimization problem, including the case when the data fidelity term is non-quadratic and the cost function is non-convex [ROBI-13a].
Here is a presentation.
Phase retrieval in X-Ray Phase CT (M Langer, B Sixou, F Peyrin)
X -Ray Phase contrast imaging is a new modality that increases the sensitivity of the X-ray imaging by exploiting not only the X-ray absorption but the effects of refraction and diffraction. With a coherent synchrotron beam, this technique may be implemented experimentally by propagation, ie by placing the detector at a distance from the object (in-line phase contrast imaging). This technique can be coupled to CT by performing tomographic scans at different detector distances. The reconstruction can be performed in two steps: a “phase retrieval” step where for each projection angle, a phase map is estimated from the 2D radiographs acquired at different distances, and the tomographic reconstruction of the phase maps providing the 3D image.
The phase retrieval problem which consists in estimating a function from the module of its Fresnel transform is a non-linear ill-conditioned inverse problem. Most phase retrieval methods are based on a linearisation of the problem, valid under some physical assumptions. This linearisation can limit the attainable spatial resolution, however, which is especially limiting in high resolution imaging (< 1 μm).
The PhD thesis of Valentina Davidoiu was devoted to studying approaches taking account this non-linearity. A theoretical approach based on the Fréchet derivative of the phase-intensity relationship, leading to an iterative reconstruction algorithm, was proposed [DAVI-11a][DAVI-12d]. The introduction of proper information to regularize the problem was particularly investigated [SIXO-12b]. Various nonlinear iterative approaches were studied and evaluated on simulated and experimental data obtained at the ESRF, Grenoble [DAVI-13a] [DAVI- 13b].
Other works addressed the fundamental problem of low frequency artefact in in-line phase-CT, due to the low contrast transfer function in the low spatial frequency range. We addressed this problem by introducing a priori knowledge on the phase to be reconstructed, so far based on the availability of an attenuation (standard) CT-scan. First, we developed an algorithm based on a homogeneous object assumption which permitted to introduce a priori in the Radon domain [LANG-10]. Many samples of biomedical interest are heterogeneous, however. Therefore, we developed algorithms that permit multimaterial objects [LANG-12b] and heterogeneous objects under some limitations [LANG-10d], where priors are introduced in the object domain [LANG-14].
Here is a presentation.
Tomographic reconstruction for the Compton gamma-camera (V Maxim, with R Prost team 2)
During the acquisition process with the Compton gamma-camera, integrals of the intensity distribution of the source on conical surfaces are measured. Image reconstruction from cone-surface projections is a topic currently addressed by teams in Europe (M. Rafecas, IFIC, Spain), US (J. Fessler, A. Zoglauer, J. Polf), Asia (Korea, Japan). For the analytical inversion of the Compton transform we first proposed a Fourier-Slice theorem [MAXI-09]. We have shown how different projections are related together and how they may be combined in a filtered back-projection reconstruction algorithm [MAXI-14a]. Iterative List Mode Maximum Likelihood Expectation Maximization algorithms are more widely employed and use to give better results especially for small extent cameras. Different methods for the calculation of the system matrix were proposed in the PhD thesis of Xavier Lojacono [LOJA-13c], funded by the European project ENVISION. Analytical and iterative methods were applied to data simulated for the prototype under construction at IPNL-CNRS, designed for hadrontherapy online monitoring. A median value of 5 mm was found for the PSF for a central, large energy spectrum source with as few as 500 events [LOJA-13b]. A comparison on the basis of data simulated by the Dresden partner from the ENVISION project is in progress. The evaluation of the algorithms and of the performances of the Compton camera for hadrontherapy monitoring is currently done on data that we simulate with MEGAlib (collaboration with A. Zoglauer, Berkeley, US), GATE or a combination of both [FRAN- 10a].
Here is a presentation.
Regularized CT and sparse methods (B Sixou, S Rit, F Peyrin)
Compressive sensing developed in signal processing is revolutionizing classical sampling theory based on the Shannon theorem. They rely on a sophisticated mathematical theory but a basic idea is that the signal (or image) can be represented by a small number of coefficients (ie admits a sparse representation) in a well suited basis. This has led the developments of new algorithms to reconstruct a signal under sparsity constraints.
Tomographic reconstruction methods based on compressive sensing have been developed in the PhD work of C Mory (CIFRE Philips) in the context of cardiac rotational angiography. 3D Cardiac C-arm imaging is an application where only a few x-ray projections are available per time phase for the reconstruction of dynamic CT images. Cyril Mory’s proposed during his PhD a new algorithm to tackle this highly undersampled problem, the 4D RecOnstructiOn using Spatial and TEmporal Regularization (4D ROOSTER) [MORY-14, MORY-14b, MORY- 14c]. The sparsity constraints in space and in time produced clinically relevant images from real cardiac acquisitions.
Preliminary work exploiting Total Variation (TV) regularisation methods are also considered in the PHD of A Toma for the restoration of blurred undersampled images (cf perspectives)[TOMA-14].
Here is a presentation.
Motion-compensated X-ray CT (S Rit, D Sarrut)
The team has a long experience in motion-compensated CT with applications in the compensation of cardiac and respiratory motion. Innovative methods for breathing [RIT-09, RIT-09b] have been transferred to a clinical solution in the Netherlands following an evaluation on a large cohort of patients [RIT-11a]. This clinical solution has been improved during the PhD of Vivien Delmon [DELM-13a].
Here is a presentation.
Fluorescence Diffuse Optical Tomography (N Ducros, F Peyrin).
Fluorescence Diffuse Optical Tomography (FDOT) is an emerging technique that aims at visualizing molecular processes in vivo and represents a fundamental tool for medical research by addressing the molecular bases of diseases. Novel approaches for FDOT reconstruction using time-resolved data have been proposed [DUCR-09] [DUCR-09b] [DUCR-10].
Here is a presentation.