Nick Henscheid

The primary goal of a medical imaging system, whether it be an X-Ray CT, a molecular imaging system like SPECT, or an MRI, is to aid doctors in making a useful diagnosis of a patient's condition without damaging the patient in the process. This tradeoff between diagnostic effectiveness and patient safety is very tricky, but it can be understood and tackled using physics, mathematics, statistics and computational simulation. My general philosophy is the following:

• - Let the physics select the relevant mathematical model. The resulting mathematics will always be interesting.
• - Strive to maintain a high level of mathematical rigor, even in the face of complicated models.
• - Use simulation to understand your models, not to avoid understanding them.
• - Evaluate all new fancy models and algorithms using diagnostically relevant figures of merit. Medical imaging systems have an actual purpose, which is to make patients' lives better. The mathematics should reflect this.

One of the fundamental mathematical concepts used in medical imaging is the idea of an inverse problem. Broadly speaking, a lot of classical mathematical physics (mechanics, heat, fluids, gravitation, quantum mechanics) is playing a "predction" game: given a known set of initial conditions, system parameters and system constraints, these theories predict the state of the system at a later time. An inverse problem, again broadly speaking, tries to reverse that process by taking the known or measured states of a system and trying to determine the inital conditions, system parameters or system constraints. This is exactly the situation we face in medical imaging, since we know how different kinds of energy (like electromagnetic waves) propagate through the body, but we don't know what the exact parameters (like density) are inside a given person's body. These parameters are diagnostically relevant, so the whole name of the game in imaging is using indirect system state measurements to make inferences about such parameters.

A fundamental fact of life is that physical systems are random and our measurements of them are noisy and imprecise. This dictates that any rigorous theory of imaging must include a rigorous theory of randomness, which is where stochastic modeling comes in. A stochastic process is simply a mathematical way of making precise the notion of a "random function". In imaging, stochastic processes are used to model patients' anatomical parameters, the imaging process, and the data collected. An exciting new direction in stochastic process theory, due to Michael Unser and his group, is the sparse stochastic modeling framework. This is a way of incorporating random process theory into the very vibrant field of sparse signal processing and compressed sensing.