A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operatorμt can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection on a subspace of sufficiently regular functions. Then the eigenfunctions of μt with small eigenvalues have only a very small component in the image of Π." We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lips chitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context. We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.

Automated Assessment of Image Quality and Dose Attributes in Clinical CT Images

Computed tomography (CT) is a valuable technology to the healthcare enterprise as evidenced by the more than 70 million CT exams performed every year. As a result, CT has become the largest contributor to population doses amongst all medical imaging modalities that utilize man-made ionizing radiation. Acknowledging the fact that ionizing radiation poses a health risk, there exists the need to strike a balance between diagnostic benefit and radiation dose. Thus, to ensure that CT scanners are optimally used in the clinic, an understanding and characterization of image quality and radiation dose are essential.

The state-of-the-art in both image quality characterization and radiation dose estimation in CT are dependent on phantom based measurements reflective of systems and protocols. For image quality characterization, measurements are performed on inserts imbedded in static phantoms and the results are ascribed to clinical CT images. However, the key objective for image quality assessment should be its quantification in clinical images; that is the only characterization of image quality that clinically matters as it is most directly related to the actual quality of clinical images. Moreover, for dose estimation, phantom based dose metrics, such as CT dose index (CTDI) and size specific dose estimates (SSDE), are measured by the scanner and referenced as an indicator for radiation exposure. However, CTDI and SSDE are surrogates for dose, rather than dose per-se.

Currently there are several software packages that track the CTDI and SSDE associated with individual CT examinations. This is primarily the result of two causes. The first is due to bureaucracies and governments pressuring clinics and hospitals to monitor the radiation exposure to individuals in our society. The second is due to the personal concerns of patients who are curious about the health risks associated with the ionizing radiation exposure they receive as a result of their diagnostic procedures.

An idea that resonates with clinical imaging physicists is that patients come to the clinic to acquire quality images so they can receive a proper diagnosis, not to be exposed to ionizing radiation. Thus, while it is important to monitor the dose to patients undergoing CT examinations, it is equally, if not more important to monitor the image quality of the clinical images generated by the CT scanners throughout the hospital.

The purposes of the work presented in this thesis are threefold: (1) to develop and validate a fully automated technique to measure spatial resolution in clinical CT images, (2) to develop and validate a fully automated technique to measure image contrast in clinical CT images, and (3) to develop a fully automated technique to estimate radiation dose (not surrogates for dose) from a variety of clinical CT protocols.