| An image processing application involving prenatal medical ultrasound is considered, and the following situation arises: A 3-dimensional object is known except for location, scale, and rotation. Based upon noisy, 2-dimensional images taken in predefined planes within a 3-dimensional space, one wishes to estimate the location, scale and rotation of the object. The space of rotations SO(3), being a manifold, has traditionally made such object recognition problems difficult.;In this work we present, and mathematically justify, algorithms--some of which are of the simulated annealing type--constructed to search through the space of locations, scales, and rotations in a semi-intelligent manner. These algorithms are discretizations of time homogeneous and time inhomogeneous diffusions on a manifold, M. Specifically, those considered are the diffusions on M generated by the second order differential operator ;Justification for the use of these algorithms in the estimation of the true location, scale, and rotation of an object is provided by two main theorems which establish the weak convergence of the diffusion processes mentioned above. In the time homogeneous case, c(t) is taken to be constant and the function h on M is taken to be the negative log of the density of a probability measure ;In the time inhomogeneous case, c(t) is taken to be a function which decreases to zero sufficiently slowly. The function h can be any smooth function on M with certain restrictions, but in relation to the time inhomogeneous (simulated annealing) algorithm, h is considered to be the negative log of a likelihood function. It is shown that under certain circumstances, the transition probability describing the time inhomogeneous diffusion process converges weakly to a probability measure concentrated on the set of global minima of h. |
