The acquisition of high resolution multi-dimensional image data is becoming increasingly important in several bio-medical applications (e.g., spectroscopic imaging and cardiac imaging). Very often, imaging devices are pushed to their limits in the quest for high spatio-temporal/spatial-spectral resolution, resulting in several artifacts and SNR loss. Recently, the recovery of the image data from sub-Nyquist sampled measurements using constrained image models has emerged as a promising alternative. A challenge in using pre-determined models is the misfit between the representation and dataset; many coefficients are often required to represent the signal at hand. The main focus of this work is to develop a novel theoretical framework and efficient algorithms to adapt image representations to under-sampled measurements.
We specifically focus on blind or adaptive representations, which are a significant departure from classical approaches based on pre-determined dictionaries. By adapting the signal model to measurements, we expect to obtain unbiased reconstructions from far smaller numbers of measurements. We formulate the joint estimation of the representation and the signal from the entire under-sampled data as a single optimization problem, where the criterion is only dependent on the recovered signal. This enables the development of efficient optimization algorithms, performance optimization using tailored cost functions, and determination of the conditions for perfect recovery. This approach is expected to considerably improve the resolution in several multi-dimensional imaging schemes, which will facilitate several basic science and clinical applications of very high significance; the proposed research is truly transformative. The impact of this work is strengthened by the sharing of software and data-sharing, integration of research and teaching, and well-designed out-reach program.