Overcomplete tomography: A novel approach to imaging

Regularised least-squares tomography offers a straightforward and efficient imaging method and has seen extensive application across various fields. However, it has a few drawbacks, such as (i) the regularisation imposed during the inversion tends to give a smooth solution, which will fail to reconstruct a multi-scale model well or detect sharp discontinuities, (ii) it requires finding optimum control parameters, (iii) it does not produce a sparse solution. This paper introduces ‘overcomplete tomography’, a novel imaging framework that allows high-resolution recovery with relatively few data points. We express our image in terms of an overcomplete basis, allowing the representation of a wide range of features and characteristics. Following the insight of ‘compressive sensing’, we regularise our inversion by imposing a penalty on the L1 norm of the recovered model, obtaining an image that is sparse relative to the overcomplete basis. We demonstrate our method with a synthetic and a real X-ray tomography example. Our experiments indicate that we can reconstruct a multi-scale model from only a few observations. The approach may also assist interpretation, allowing images to be decomposed into (for example) ‘global’ and ‘local’ structures. The framework presented here can find application across a wide range of fields, including engineering, medical and geophysical tomography.