In tomographic problems, a preferred method to construct parametrized models of the indirectly constrained physical property is the regularised least-squares inversion. However, it has a few drawbacks, such as: (i) the regularization 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. It was recently proved with the approach ‘compressive sensing’ that if the model is sparse, i.e., has only a few non-zero coefficients, it can be recovered with high-resolution with fewer but randomly chosen samples by minimizing the L1 norm of the recovered model.
We introduce ‘overcomplete tomography,’ a novel method that adapts the concept of compressive sensing to inverse problems and finds a sparse representation of the model using an ‘overcomplete basis’. We demonstrate our method with a synthetic and a real X-ray tomography example. The results show that we can reconstruct a multi-scale model and detect sharp discontinuities with a very small number of randomly sampled data. It can also successfully separate the local and global parts of the model in a respective basis. We compare our results with the least-squares inversion and show that our method enables excellent recovery. We also explore several intriguing geophysical applications, such as low-artifact imaging of systems containing features at multiple scale lengths.