Medial Methods for Segmentation, Registration, and Shape Measurement  MIDAG - Medical Image Display and Analysis Group
We have learned that the object medial loci that are an important aspect of our research (see Bibliography), have advantages in intuitiveness with their basis on figures (main figures, protrusions, indentations) and ability to achieve locality, in capturing solidity, in their geometric power of carrying local orientation and size information, and in robustness due to their two-sidedness. The disadvantages of medial loci are that if they are not to suffer the traditional weaknesses of sensitivity to boundary detail and image noise, they cannot by themselves represent the object at small scale and the search for a reliable, automatic means of defining a subdivision of an object into figures continues. Also, they cannot handle parts of objects with unpaired boundaries in the image or in the actual geometry. In addition, since boundaries are implied rather than explicit, boundary mechanics such as abutment and intersection are not as easily handled as boundary representations. 

While in the past we have focused on the height ridges of medialness that we have named "cores", we now focus especially on medial methods based on m-reps, which are trees of nets of medial atoms, with each net representing a figure. The m-reps deformation methods that our lab now has in place for both segmentation and shape measurement include coarse-to-fine multiscale approaches based on Markov random fields, both geometric and statistical object difference metrics, and intensity match metrics of both the type reflecting just differences and those reflecting variabilities in the model. We have also begun work on mechanical object difference metrics, aimed at segmentations of a patient based on a model of that very patient. 
wireframe image of the m-rep modelmatching the m-rep model against a slice of the dataset
Fig. 1. An m-rep model for a kidney. Left, in 3D: the m-rep is made from 2 linked figures, each of which is represented by a mesh or chain of medial atoms. Here the two figures represent the whole kidney and the pelvis + ureter. The medial atoms are shown in white with the meshes in green and the medially implied boundaries in blue. Right, in 2D: a cut through the 3D image data showing 2D medial atoms interpolated from the 3D atoms for the figure representing the whole kidney. 

Among the m-reps programs in place are ones for building a 3D multifigure m-reps model (see Fig. 1 for an example) for an object from a training image with the object segmented, as well as a program for segmentation in a target image by deformation, in a coarse-to-fine sequence, via optimization of an objective function summing a structure typicality term and an image match term. The steps of the coarse-to-fine sequence are as follows: similarity transform optimization of the full model, similarity transform of each figure (with the structure typicality term measuring figural deviation from the parent figure), full variation of each medial atom (with the structure typicality term measuring deviation of the atom from the adjacent atoms). In a separate program, soon to be combined with the main program, the final step is boundary offset from the medially implied boundary along its normals. To date in this program both the structure typicality term and the image match term measure "distances" from single training case, with no reflection of variability of the populations. We are applying this method to MRI of the hand (a test case), CT of the kidney, MRI of the cerebral ventricle, CT of the pelvic bones, and CT of the spine. By the time of the site visit in October we anticipate having numerous results with each of these cases and having begun replacing the distance from the training case metrics by statistical metrics based on PCA. 

Related Links
In present dissertation research, 

Martin Styner has begun to develop a method that stabilizes the m-rep description of an object for shape measurement by determining the medial sampling and figural topology through statistics from a medial analysis derived from a spherical harmonics description (Fig. 2). 

Paul Yushkevich has begun the study of coarse-to-fine shape measurement, with the aim of obtaining descriptions at successive levels of locality and thus high intuitiveness to users. 

Jessica Crouch has begun a dissertation on finite-element based mechanical modeling of structural change based on coarse-to-fine m-reps models, intended to achieve two orders of magnitude improvements in speed over previous 3D finite-element methods. 

Andrew Thall is well along in developing methods of computer aided design and rendering of solid objects via m-reps (Fig. 3). 

Robert Katz is largely done with a dissertation focused on medial representations measuring perceptual significance of object sections in 2D. This work has some useful implications for the 3D structural models research, especially the measurement at the medially implied boundary of related effects of adjacent figures and discrimination of continuing figures from subfigures via continuity of figural width and orientation. 
hippocampus/amygdala complex based on spherical                       harmonicsTwo-figure m-rep obtained by deforming m-rep                       fit to average case
Fig. 2 (Styner). Left: Hippocampus/amygdala complex (HAC) based on spherical harmonics boundary representation (SPHARM) from manual segmentation of 3D MRI. Right: Two-figure m-rep of case at right obtained by deforming m-rep fit to average case. Average was computed from population of HACs’ SPHARMs. 
guitar m-rep shown in wireframeguitar m-rep rendered
Fig. 3 (Thall) Computer-aided design of guitar based on m-reps. Left: three meshes of width-proportionally spaced medial atoms. Right: Rendered guitar. 

Figural geometry via statistical analysis of clouds of medial atoms 
In work done here by George Stetten and illustrated in Fig. 4, images are first analyzed in figural terms without the aid of a model. Medial atoms are extracted by thresholding from the image data, and local clouds of the atoms are statistically analyzed to produce information as to the local existence of a piece of figure and if it exists, its width, orientation, and geometric type within the space encompassed by sphere, tube, and slab. 
Fig. 4 (Stetten). Right: Automatic cardiac left ventricle extraction from 3D ultrasound image via axis between mitral valve and left ventricle. Left: As determined from statistical analysis of cloud of medial primitives, mitral valve signalling (by red color) its being a slab and giving its orientation. Middle: Similarly, ventricle body signaling (by green color) its being a tube and giving its orientation and size. 
Cores and Other Medial Height Ridges
Both slabs and tubes can be extracted by locating cores, i.e., height ridges of a graded measure of medial strength that we call "medialness" (Fig. 5). Height ridges are subdimensional maxima and fall into two categories: maximal convexity ridges and optimal parameter ridges, which differ according to the rule used for choosing the subspace in which the medialness maximum is checked for. Subdimensional saddles connect height ridges when they stop, forming "connectors". This work involves singularity theoretic study of height ridges [Damon, Miller, Keller] as well as algorithmic development of height ridge extractors [Fritsch, Aylward, Furst, Fridman]. 
Fig. 5. Left: Tree of blood vessels and aneurysm extracted from magnetic resonance angiogram via height ridges (Aylward, Bullitt). Right: Core (red), with connectors (yellow) for a blood vessel in a (2D) x-ray angiogram (Furst, Fridman).