A variational reconstruction method for undersampled dynamic x-ray tomography based on physical motion models

Abstract

Abstract. In this paper we study the reconstruction of moving object densities from undersampled dynamic X-ray tomography in two dimensions. A particular motivation of this study is to use realistic measurement protocols for practical applications, i.e. we do not assume to have a full Radon transform in each time step, but only projections in few angular directions. This restriction enforces a space-time reconstruction, which we perform by incorporating physical motion models and regularization of motion vectors in a variational framework. The methodology of optical flow, which is one of the most common methods to estimate motion between two images, is utilized to formulate a joint variational model for reconstruction and motion estimation.We provide a basic mathematical analysis of the forward model and the variational model for the image reconstruction. Moreover, we discuss the efficient numerical minimization based on alternating minimizations between images and motion vectors. A variety of results are presented for simulated and real measurement data with different sampling strategy. A key observation is that random sampling combined with our model allows reconstructions of similar amount of measurements and quality as a single static reconstruction.

Undersampled Dynamic X-Ray Tomography With Dimension Reduction Kalman Filter

“…Another option is based on optical flow methods that try to calculate the motion between two consecutive image frames. For example this option was considered in a recent study (5) with variational methods. In filtering context, one could for example calculate the displacement field from images x est k−1 and x est k , and then use this field to predict x p k+1 = M flow x est k .…”

Enhancing Compressed Sensing 4D Photoacoustic Tomography by Simultaneous Motion Estimation

“…(3.4). While this is computationally advantageous, studies generalizing this to L p -norms, e.g., for p = 1, have shown promising results and directions for future research [20,10,9]. The main drawback of the concrete TVTVL2 model we used here (3.5) is that it leads to a challenging, large-scale bi-convex optimization problem.…”

“…analysis of this approach. While it was used for 2D dynamic computed tomography reconstruction in [9], we present the first application to a challenging, large-scale 3D dynamic problem with experimental data, which also requires the development of tailored numerical optimization schemes.…”

A Variational Model Dedicated to Joint Segmentation, Registration, and Atlas Generation for Shape Analysis

“…In recent years, joint image processing models have experienced increasing attention, including combined segmentation/registration models [30,34] (joint phase field approximation and registration), [45] (model based on metric structure comparison), [26,61] (level set formulation that merges the piecewise constant Mumford-Shah model with registration principles), [33] (grounded in the expectation maximisation algorithm), [25] (based on a nonlocal characterisation of weighted-total variation and nonlocal shape descriptors), or [1,43,52,55,63,68]; joint image reconstruction and motion estimation [9,14,19,51,57,62,13,46,6]; joint reconstruction and registration for post-acquisition motion correction [22] with the goal to reconstruct a single motion-free corrected image and retrieve the physiological dynamics through the deformation maps, joint optical flow estimation with phase field segmentation of the flow field [12], or joint segmentation/optimal transport models [10] (to determine the velocity of blood flow in vascular structures). This can be attributed to several factors: (i) the will to limit error propagation.…”