bring together multiple physics components: elasticity of the ventricular wall, electrophysiology, and active contraction of the myocardial fibers.
The full-blown setting of this problem is analogous to a “blind deconvolution” problem, in the sense that neither the model nor the source is fully known. As such, this presents enormous difficulty for the inversion solvers; as in the image registration case, it requires careful formulation and regularization, as well as large-scale computational solvers that are tolerant of ill-conditioning. Recent research18 is following a hybrid approach that interweaves the solution of the image registration and model determination problems.
The story of compressed sensing is an example of the power of the mathematical sciences and of their dynamic relationship with science and engineering. As is often the case, the development of novel mathematics can be inspired by an important scientific or engineering question. Then, mathematical scientists develop abstractions and quantitative models to solve the original problem, but the conversion into a more abstract setting can also supply insight to other applications that share a common mathematical structure. In other words, there is no need to reinvent the wheel for each instantiation of the problem.
Compressed sensing was motivated by a great question in MRI, a medical imaging technique used in radiology to visualize detailed internal structures. MRI is a wonderful tool with several advantages over other medical imaging techniques such as CT or X-rays. However, it is also an inherently slow data-acquisition process. This means that it is not feasible to acquire high-quality scans in a reasonable amount of time, or to acquire dynamic images (videos) at a decent resolution. In pediatrics for instance, the impact of MRI on children’s health is limited because, among other things, children cannot remain still or hold their breath for long periods of time, so that it is impossible to achieve high-resolution scans. This could be overcome by, for example, using anesthesia that is strong enough to stop respiration for several minutes, but clearly such procedures are dangerous.
Faster imaging can be achieved by reducing the number of data points that need to be collected. But common wisdom in the field of biomedical imaging maintained that skipping sample points would result in information loss. A few years ago, however, a group of researchers turned signal processing upside down by showing that high-resolution imaging was possible
18 H. Sundar, C. Davatzikos, and G. Biros, 2009, Biomechanically-constrained 4D estimation of myocardial motion. Medical Image Computing and Computer-Assisted Intervention (MICCAI):257-265.