Computational anatomy (CA) is a discipline within medical imaging focusing on the study of anatomical shape and form at the visible or gross anatomical scale of morphology. The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, including medical imaging, neuroscience, physics, probability, and statistics. It focuses on the anatomical structures being imaged, rather than the medical imaging devices. The central focus of the sub-field of computational anatomy within medical imaging is mapping information across anatomical coordinate systems most often dense information measured within a magnetic resonance image (MRI). The introduction of flows into CA, which are akin to the equations of motion used in fluid dynamics, exploit the notion that dense coordinates in image analysis follow the Lagrangian and Eulerian equations of motion. In models based on Lagrangian and Eulerian flows of diffeomorphisms, the constraint is associated to topological properties, such as open sets being preserved, coordinates not crossing implying uniqueness and existence of the inverse mapping, and connected sets remaining connected. The use of diffeomorphic methods grew quickly to dominate the field of mapping methods post Christensen's original paper, with fast and symmetric methods becoming available.
The main statistical model
The central statistical model of Computational Anatomy in the context of medical imaging has been the source-channel model of Shannon theory; the source is the deformable template of images , the channel outputs are the imaging sensors with observables (see Figure). The importance of the source-channel model is that the variation in the anatomical configuration are modelled separated from the sensor variations of the Medical imagery. The Bayes theory dictates that the model is characterized by the prior on the source, on , and the conditional density on the observable
conditioned on .
In deformable template theory, the images are linked to the templates, with the deformations a group which acts on the template; see group action in computational anatomy For image action , then the prior on the group induces the prior on images , written as densities the log-posterior takes the form
The random orbit model which follows specifies how to generate the group elements and therefore the random spray of objects which form the prior distribution.
The random orbit model of computational anatomy
The random orbit model of Computational Anatomy first appeared in modelling the change in coordinates associated to the randomness of the group acting on the templates, which induces the randomness on the source of images in the anatomical orbit of shapes and forms and resulting observations through the medical imaging devices. Such a random orbit model in which randomness on the group induces randomness on the images was examined for the Special Euclidean Group for object recognition in which the group element was the special Euclidean group in.
For the study of deformable shape in CA, the high-dimensional diffeomorphism groups used in computational anatomy are generated via smooth flows which satisfy the Lagrangian and Eulerian specification of the flow fields satisfying the ordinary differential equation:
with the vector fields on termed the Eulerian velocity of the particles at position of the flow. The vector fields are functions in a function space, modelled as a smooth Hilbert space with the vector fields having 1-continuous derivative . For , the inverse of the flow is given by
and the Jacobian matrix for flows in given as
To ensure smooth flows of diffeomorphisms with inverse, the vector fields must be at least 1-time continuously differentiable in space which are modelled as elements of the Hilbert space using the Sobolev embedding theorems so that each element has 3-square-integrable derivatives. Thus embed smoothly in 1-time continuously differentiable functions. The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:
where with a linear operator defining the norm of the RKHS. The integral is calculated by integration by parts when is a generalized function in the dual space .