According to Compressive Sensing (CS) theory, sparse signals can be accurately recovered from a small set of linear measurements using efficient L1-norm minimization techniques. The success of such techniques depends upon the “quality” of the sensing matrix as determined by some metric. The Restricted Isometry Property (RIP) establishes some of the tightest reconstruction guarantees that are currently known. These guarantees come at a cost: it is NP-hard to evaluate the RIP. To overcome this issue, researchers typically utilize random matrix theory to create matrices that satisfy the RIP with a high probability. Unfortunately, practical constraints prevent this method from being applied to many multi-physics imaging applications.
In this dissertation, we propose the use of several optimization algorithms for practical CS applications. First, we present a novel sensing matrix design method based upon minimizing the mutual coherence. Although it provides looser reconstruction guarantees than the RIP, the mutual coherence is a much more practical metric, which can be evaluated in polynomial time. Although we investigate this design method in the context of wave-based imaging, it can be applied to any CS application where the sensing matrix is differentiable with respect to the design variables over the feasible set.
Second, we extend the coherence minimization technique to problems that exhibit block sparsity. By maximizing the minimum capacity of sub-matrices obtained by selecting groups of columns from the full sensing matrix, this design method indirectly minimizes the block restricted isometry constants, which significantly improves the reconstruction accuracy of joint L2/L1 minimization techniques. The improved reconstruction accuracy is demonstrated using several numerical examples.
Third, we introduce a general optimization method for designing so-called “compressive antennas” utilized in electromagnetic imaging applications. In this method, the design variables are optimized so that a combination of the channel capacity and efficiency of the transmitting antennas, receiving antennas, and sensing matrix is maximized. The method is investigated using a practical forward model known as the modified equivalent currents approximation (MECA), which is well-suited for large imaging applications.
Fourth, we present a novel reconstruction paradigm that combines CS with unmixing algorithms. Traditional methods reconstruct the constitutive parameters of the underlying sensing system. This approach is sub-optimal when one has additional a-priori knowledge. The unmixing paradigm considers the object at each voxel to be mixture of several material types. If one knows models for computing the constitutive parameters of each material type, the unmixing algorithm can be used to directly reconstruct the mixture proportions. This approach allows one to jointly reconstruct the unknowns using multiple sensing modalities.
Finally, we conclude the dissertation with an extension of the minimum capacity-based design method to problems that contain time-varying dynamics. By exploiting the known temporal dynamics, one can enhance the reconstruction capabilities of the system. We primarily focus on two design frameworks, one that is adaptive and one that is non-adaptive, although only the former is useful in practice due to the large computational complexity of the latter.
- Professor Jose Martinez Lorenzo (Advisor)
- Professor Dana Brooks
- Professor Mario Sznaier