Template-Based Image Reconstruction from Sparse Tomographic Data

Abstract

We propose a variational regularisation approach for the problem of template-based image reconstruction from indirect, noisy measurements as given, for instance, in X-ray computed tomography. An image is reconstructed from such measurements by deforming a given template image. The image registration is directly incorporated into the variational regularisation approach in the form of a partial differential equation that models the registration as either mass-or intensity-preserving transport from the template to the unknown reconstruction. We provide theoretical results for the proposed variational regularisation for both cases. In particular, we prove existence of a minimiser, stability with respect to the data, and convergence for vanishing noise when either of the abovementioned equations is imposed and more general distance functions are used. Numerically, we solve the problem by extending existing Lagrangian methods and propose a multilevel approach that is applicable whenever a suitable downsampling procedure for the operator and the measured data can be provided. Finally, we demonstrate the performance of our method for template-based image reconstruction from highly undersampled and noisy Radon transform data. We compare results for massand intensity-preserving image registration, various regularisation functionals, and different distance functions. Our results show that very reasonable reconstructions can be obtained when only few measurements are available and demonstrate that the use of a normalised cross correlation-based distance is advantageous when the image intensities between the template and the unknown image differ substantially.

Template based image reconstruction facing different topologies

“…Contributions In this article, we study the variational and regularizing properties of the proposed model (8) and develop efficient numerical schemes for solving it. Furthermore, we demonstrate experimentally that the modification indeed allows to obtain reconstructions of similar high quality as in [22], but with the advantage that new structures can appear due to the additional source term z. Clearly, the amount of details that new structures will have depends on the available amount of data, i.e., we are not able to reconstruct detailed new objects out of nothing.…”

“…Our discretization of the ODE constraint in ( 6) is based on the Lagrangian methods developed in [22,23] and builds up on FAIR [29]. As our problem is non-smooth due to our regularizer choice for z, we can not employ the proposed Gauss-Newton-Krylov solver and we use the iPALM algorithm [35] instead.…”

“…If g are tomographic measurements, solving (1) is often referred to as image reconstruction. Several possibilities to solve such problems under the presence of an additional template image T ∈ L 2 (Ω) are described in [10,18,20,22,32,33]. For all of these models, the transformation between the template T and the unknown reconstruction R is modeled in terms of a PDE constraint.…”

“…where D : Y × Y → R ≥0 is a data fidelity term quantifying the misfit of the reconstruction against the measurements, and the regularizer E : V → R ≥0 enforces the required smoothness of v, see Chen and Öktem [10] and Lang et al [22]. Note that the model can be simplified by using linearized deformations ϕ and a Taylor expansion around some initial flow field, leading to optical flow based models [1,6,21,33].…”

Regularization with metric double integrals for vector tomography

“…In contrast to [8], where simple image analysis tasks, such as denoising and inpainting have been investigated, the focus of this paper is on solving vector tomography problems which involve the Radon R or the ray transform D, respectively. Nonlinear imaging tasks, such as registration (see for instance [3,11,13,[15][16][17]23], to name but a few) and tomographic displacement estimations [20], fit in the framework of this paper, but are not considered here.…”