## Mathematics

We have organized the mathematical research described into two Specific Aims. Each Aim tackles a fundamental problem in the computational modeling of biological structures. The proposed solutions and techniques build on our recent advances in image analysis, differential equations, numerical analysis, and differential geometry, as well as our many yearsâ€™ work in mathematics and neuroscience.

Level Sets for Nonlinear Image Registration

We are developing a powerful set of nonlinear image registration approaches that permit the comparison, pooling, and statistical combination of biological image data.

Analyzing Biological Shape
We are creating algorithms to measure, map, and represent variations in biological shape. Modeling of shape is key to detecting group differences in biological images, mapping subtle disease effects on structure, and understanding how structures vary spatially and temporally as they grow or are impacted by disease.

Computational Mathematics
Computational mathematics involves acquiring and processing 3D volumetric images of the brain, extracting relevant anatomical and functional features from these images, reconstructing dynamically varying brain structure from sequences of 3D images, representing the intricate geometry of the anatomical surfaces and volumes within the brain, extracting 3D features from this representation, and mapping these geometries onto standard templates (e.g. spheres, planes, or canonical coordinate systems) for comparison with other brain images. Powerful statistical analysis on these computational surfaces and volumes reveal how the brain varies dynamically with age, in disease, and in response to therapy. Mathematical and computational techniques have been designed to advance each of these steps in the brain mapping process.

Acquisition and Processing
Acquisition of the images involves a variety of mathematical and computational tomographic techniques (e.g. structural and functional MRI, PET, computed tomography). Images are typically mathematically processed to enhance their quality, (e.g. removing noise), and registered to align images from multiple subjects, time-points, or imaging devices. Biomedically significant features, represented as shapes, objects, or curves, often must be extracted from these images to facilitate subsequent processing. This typically involves image segmentation techniques, either automatic or supervised. These processed volumetric images are then combined geometrically to produce 3D structural models of brain surfaces and volumes.

The Computational Mathematics of Geometries
One of the reasons that computational mathematics is so central to brain mapping is that the root problems are geometric by their very nature. A brain map is a geometry, relating a 2D, 3D or 4D coordinate system to some set of anatomical features. Different brain maps yield different geometries, and the problem of their alignment is then a geometric problem of fundamental importance. An important step in developing comprehensive atlases of brain structure and function is to investigate the problem of mapping the geometries of the imaged brain onto standard surfaces and coordinate systems. In order to make comparisons or combinations of different anatomies (either from different subjects or from the same subject over time), we face the problem of mapping the geometries of the imaged brain onto standard surfaces and coordinate systems. Fortunately, many important brain mapping problems can be reduced to mathematical problems on surfaces. In this case, brain maps are treated as surface geometries, and brain mapping becomes the problem of modeling, mapping, and analyzing biological structures that are represented computationally as surfaces.

The Computational Mathematics of Atlases
An atlas is a set of maps (i.e., coordinate systems) and alignments among these (i.e., mappings between coordinate systems). If we view surfaces as maps, and mappings among these as alignments, we get to the computational mathematics just discussed. A computational brain atlas is a framework for analyzing surfaces. Generally, the real value of the brain atlas is its ability to integrate information from multiple sources. Alignment of multiple brain maps is equivalent to the reconciliation of multiple scientific models, and this has great power. In addition to classical operators such as segmentation, there are many useful operators for analyzing surfaces, including for example:

• Finding an “alignment” for a set of surfaces (mappings of each surface to a common canonical space, or mappings among each pair of surfaces)
• Finding a “consensus” for a set of surfaces (an “average” surface, a “best fit” or “maximum likelihood” surface)
• Finding “differences” among a set of surfaces (points of greatest variation, or of “least consensus”)
• Classical “data mining” operators for surfaces (discrimination, classification, clustering)

Collections of operators like these can be combined into an “algebra” or “query language” for surfaces. There is no limit to the number of useful operators. Thus the computational atlas is necessarily open-ended, and the mathematical research pursued in developing it should have a diverse agenda.

Shape Modeling and Analysis
We are constructing new mathematical representations for shapes. This effort is motivated by the CCB driving biological problems. Some of our research is focused on geometry processing such as sampling positions and normals for rendering, querying inside or outside of the geometric object, querying closest point on the object, querying distance to the object, modifying the geometry and/or topology of the object, etc. Efficient geometry processing algorithms always have to be supported by efficient data structures, hence efficient implementation of mathematical representations. There are approaches for shape representation that are only applicable for one particular application — e.g., unique, unambiguous, stable, accurate, concise, local support, invariant under affine transformations, support arbitrary topology, guaranteed continuity, natural parametrization, and efficient display. We study shape representation aiming at a uniform framework that enables modeling and analysis of many different types of objects, where every distinct object has a single distinct representation (i.e., uniqueness), or an object may have different representations, but no two distinct object may have a common representation (i.e., unambiguous representation). An important criterion for a good shape representation is stability, which means small perturbations on the object do not induce large changes on its shape representation.