The open curve representation and brain image analysis

There are about a hundred sulci in the human brain and over a hundred billion white matter fibers. These two anatomical structures differ in many respects but they share a common geometric description: they are both open continuous curves.
To study how the physical attributes of open curves can be used to advantage in the many varied quantitative applications of white matter fibers and sulci, we begin by considering the open curve. The brain imaging landscape from this window has its own particular challenges, open problems and possibilities. The view offers the following:

Related to the representation
The choice of representation restricts our analysis to sulci, white matter fibers and the veins and arteries of the vascular system in the brain. These are the three anatomical structures that can be modeled by open curves in 3D.

An open curve is one of the simplest ways to mathematically represent an individual sulcus or white mater fiber tract. An (open) surface, which may also be used, is more difficult to model and implement. The closed curve is a third option, but it is also a mathematically more complex representation. Anatomical structures are not easy abstractions of geometrical primitives. Were this the case, a closed curve representation of a square or circle would be the right choice.

The details of the mathematical modeling aside, for the purposes of analysis (of sulci or white matter fibers), an open curve is the more useful representation. In the case of sulci, the fact that they are highly variable in shape and structure also means that there is no additional information in a surface representation. The bottom curve (the section of the sulcus closest to the inner cortex) is the most stable part of a sulcus and this is what we use in our analysis. For white matter fiber analysis too, a curve model enables us to analyze individual tracts and there are benefits to this analysis over a volume-based one.

An open curve may be represented in a Euclidean or Riemannian space. In a Euclidean space, a spline (a parameterized curve defined piecewise by polynomials) can easily approximate shapes of complex curves. The L2 distance we compute between two such curves can be used for clustering or other classification tasks. The main advantage with this approach is the ease of implementation and the fact that existing methods from machine learning and statistics can be readily incorporated into the analysis.

In a Riemannian analysis, the open curve can be represented in different ways. In the model I use, the open curve is represented as a point in the space of continuous functions.This space of functions is infinite dimensional and nonlinear and advanced methods such as those from functional analysis, differential geometry and group theory are needed to represent and analyze curves. The advantage of a Riemannian representation is that we can use the intrinsic (Riemannian) geometry of the space to compute distances between two points (curves). Open curves are essentially nonlinear structures. The space of open curves is also nonlinear. Linear
measurements in such spaces are not well-defined and the results may not be consistent.

Related to the geometrical analysis

The open curve is a geometric structure and we use it in analysis in this context. Geometrical analysis in medical imaging is most commonly a shape study although the term itself covers other possibilities. We use a mathematical model for open curves that allows us to incorporate shape, orientation, scale and position, the physical features associated with the curve. We expect to be able to apply this mathematical model to new kinds of problems or to reinterpret old ones. (I have done some work on a new study design which I will be publishing shortly. It is essentially a study of morphological changes, but my tools enable me to study the problem in a new way.)

White matter DTI fibers bundles have a well-defined shape and structure which is determined by the regions they connect and the constraints of the surrounding anatomy. Sulci, on the other hand, are not easily interpreted in a geometrical study. This is due to the variability which we can also interpret as a problem of selecting a feature that is unique to a class of sulci. The possibility of shape analysis has been explored before. It is my view that sulcal variability, as it relates to identication and labeling, is better addressed by a graph pattern matching paradigm.