Discrete Seismic Tomography

Tomographic imaging reveals the interior of an object. Similar to a camera, it images an object by illuminating it with electromagnetic or acoustic waves (or a beam of photons). Due to its non-invasive nature, it has found applications in various fields of sciences and engineering. Recent challenges include fast and accurate reconstructions of high contrasts in the object from a limited number of tomographic measurements. To tackle this, prior information about the object under inspection is beneficial. One particular example is a discrete prior, where an object is made up of only a few homogeneous materials of known grey levels. The tomography that deals with this prior is known as discrete tomography. The discrete assumption holds approximately for many objects. In general, the low-contrast surrounding is not spatially invariant. This leads to a class of objects called partially discrete objects, where high-contrast materials lie in the non-homogeneous low-contrast background. This thesis develops algorithms based on the theory of convex optimization, regularization, and the level-set method to reconstruct discrete and partially discrete objects from limited measurements. In X-ray tomography, the challenge to reconstruct an object from limited measurements stems from the high levels of radiation dose from X-rays. Although the tomographic problem is linear, the discrete prior makes it non-convex. Many heuristic algorithms exist to image discrete objects from a few of their ray projections. These algorithms often require manual tuning of parameters and may suffer in the noisy scenario. We develop a convex program to recover the binary objects (i.e., objects composed of only two materials) that relies on the Lagrangian duality. The resulting problem is a l1-regularized least-squares problem (LASSO) that can be solved quickly. Based on small-scale experiments, we conjecture that if the binary tomography problem admits a unique solution, it can be recovered using the proposed formulation. In the case of multiple solutions, the convex program gives the intersection of the solutions. The proposed algorithm compares favorably to existing state-of-the-art algorithms like total variation and DART. In wavefield imaging, the access to the measurements on one side of the object and the nonlinear interaction of waves with the object leads to challenges in the recovery of high contrast objects. The famous example of high contrast is the salt-bodies in the earth, which are good indicators of hydrocarbon reservoirs. In the first instance, we assume that the grey level of the high contrast material is known and it is homogeneous. We develop a level-set approach that explicitly separates the high-contrast object from the low-contrast surrounding. We make use of a parametrization technique to represent the level-set function and make the problem lower-dimensional and well-behaved. On synthetic phantoms, we show that the high-contrast objects are recoverable under one-sided limited measurements. The parametric level-set approach is extended to partially discrete objects as well. In the second instance, we assume that the data consists of noise, and the noise level is known. We developed a total variation regularized multiscale framework that solves a series of least-squares problems