Template-Based Image Reconstruction from Sparse Tomographic Data
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Template-Based Image Reconstruction from Sparse Tomographic Data Lukas F. Lang1 · Sebastian Neumayer2 · Ozan Öktem3 · Carola-Bibiane Schönlieb1
© The Author(s) 2019
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 mass- and 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. Keywords Inverse problems · Optimal control · Tomography · LDDMM · Image registration
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Lukas F. Lang [email protected]
Extended author information available on the last page of the article
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Applied Mathematics & Optimization
1 Introduction In medical imaging, an image can typically not be observed directly but only through indirect and potentially noisy measurements, as it is the case, for example, in computed tomography (CT) [41]. Due to the severe ill-posedness of the problem, reconstructing an image from measurements is rendered particularly challenging when only few or partial measurements are available. This is, for instance, the case in limited-angle CT [22,41], where limited-angle data is acquired in order to minimise exposure time of organisms to X-radiation. Therefore, it can be beneficial to impose a priori information on the reconstruction, for instance, in the form of a template image. However, typically neither its exact position nor its exact shape is known. In image registration, the goal is to find a reasonable deformation of a given template image so that it matches a given target image as closely as possible according to a predefined similarity measure, see [39,40] for
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