...on segmentation

Scalar Level-Set Methods

The goal of this sub-project is to extract manageable quantities of meaningful information from large-scale N-dimensional datasets. In order to make the visualization of such large datasets practical, more effective, more automated tools for higher-dimensional data segmentation are needed. One effective way to locate geometric features in higher-dimensional data sets is to use deformable models. Deformable models can seek out geometric features while maintaining shapes that adhere to certain geometric conditions, such as smoothness and continuity, thereby offsetting noise and other shortcomings of the data. The difficulty with higher-dimensional deformable models is achieving a surface representation that is sufficiently expressive to allow for a wide range of possible object shapes. An alternative to a parametric deformable model is an implicit model, i.e., specifying a model as a level set of a scalar function, phi, which is represented as a discrete sampling on a rectilinear grid. The level-set formulation provides a set of methods that describe how to manipulate the greyscale values on the grid, so that the level surfaces of phi move in a prescribed manner.

Our goal in the proposed work is threefold. First is to extend the multiresolution method to the proposed applications by the use of intervals. That is, to use the known properties of the data at coarse resolution to determine bounds on the values of phi, and thereby restrict the calculation of finer-resolutions to areas where phi pertains to the level set of interest. Our second goal is to achieve real-time user interaction of level-set models. There are several aspects to this. One is to implement the current, efficient, numerical techniques in a parallel computing environment. The second is to extend the notions of volume sculpting to level-set models, i.e., define modes of user interaction with level-set models. The third aspect is the real-time volume rendering of the implicit function phi. Our third goal is to develop the specific mathematical methods needed to segment N-dimensional datasets. We assume that the measured data has some degree of noise, and therefore some regularization, i.e. smoothing, of the surface models is appropriate. Furthermore, we intend to combine this use of vector-valued features with the user interaction to generate nonparametric tools, that can infer properties of interesting features from user-defined segments.

Vector-Valued Level-Set Methods

We will develop mathematical and computational methods for analyzing vector- and tensor-valued datasets. MRI can produce a vector of image density ``gray'' values that correlate with the chemical properties of the specimen on a voxel-by-voxel basis. Tensor values are particularly interesting because they can give us information about the orientation of the tissue structure, derived from the transformation rules for Cartesian tensors. We will find tissue boundaries by finding analogous level sets within the transitions from one vector value to another.

In addition, we will use the new level-set methods to develop multi-dimensional volume-morphing algorithms. In this part of the work, instead of focusing on interpolating between surfaces without internal structure, we will develop methods to interpolate the whole volumetric model, voxel by voxel. Applications include the smooth interpolation of a series of volume datasets, separated by changes in configuration or development. We can create a 4D dataset (as an evolving 3D dataset that varies in time) that we can then ``cut'' at particular instances of time. For instance, we will use these methods to visualize the development of mouse and bird embryos using an in-house MRI microscope. Biologists would like to track the development of the homologous internal structures of an embryo as it grows and deforms during its development. The homologous tissue structures will have the same vector ``fingerprint'' even if a structure splits and turns into two structures of the same tissue type.

In part of the work, we will also use Bayesian probability theory to estimate the posterior probability based on conditional and prior probabilities derived from our assumptions about what we are measuring and how the measurement process works. With this information we identify multiple materials contained within each voxel based on the sample values for the voxel and its neighbors. The sampling theorem allows us to reconstruct a continuous vector function from the samples. We then represent all of the vector function values within a small region containing the pixel by creating a continuous vector histogram. Our methods will handle measurements containing three or four materials that share surface or line boundaries. We will also investigate methods to compute the position and orientation of most-probable subvoxel fibers and diffusion tensor orientations.