An Overview of Cores

1. An Overview of Cores Yoni Fridman The University of North Carolina at Chapel Hill Medical Image Display & Analysis Group Based on work by Fridman, Furst, Damon, Keller, Miller, Fritsch, Pizer

2. A medial atom m = (x, r, F, q) is an oriented position with two sails In a 3D image, m is eight-dimensional: x is the location in 3-space r is the radius of two sails, p and s F is a frame that has three degrees of freedom b is the bisector of the sails q is the object angle b q r s x p What is a Medial Atom?

3. x n p s -q q b n x b What is the Medialness of a Medial Atom m? • E.g., for slabs • E.g., for tubes, where V is the set of vectors obtained by rotating p about b Medialness M(m) is a scalar function that measures the fit of a medial atom to image data

4. What is a Core? • Cores are critical loci of medialness • A core is a description of an image, not a description of the real world • It is defined based on three choices: • Dimension of critical loci that are desired • 1D for tubes, 2D for slabs • Criticality is in co-dimension • Definition of subspace for criticality • Maximum convexity • Optimum parameters: r, F, q • What function is used to compute medialness

5. Originally, medialness was computed by integrating over the whole sphere defined by a medial atom Now, we only integrate over regions surrounding the tips of the two sails Often use a Gaussian derivative, taken in the direction of the sails Medialness Functions

6. Medial Manifolds • Medial manifolds of 3D objects are generically 2D: • If we know we’re looking at a tube, we can specify a 1D medial manifold:

7. Maximum Convexity Cores • Two types of cores have been studied: maximum convexity cores and optimum parameter cores • For a d-dimensional maximum convexity core located within an n-dimensional space, a height ridge is found by maximizing medialness over the n-d directions of sharpest negative curvature • Maximum convexity cores are simpler and their singularity-theoretic properties have been researched in Miller’s and Keller’s dissertations

8. Optimum Parameter Cores • Algorithm • Medialness is first maximized over the parameter space (r, F, q) • The height ridge is then found by further maximizing over the spatial directions normal to the core, as defined by F • Optimum parameter cores seem to represent more realistic medial loci

9. Optimum Parameter Cores • 2D cores, calculated by predictor-corrector method of Fritsch • 3D cores, calculated by marching cubes generalization of Furst

10. = core = connector Connectors • Connectors are height saddles of medialness • Cores can turn into connectors in one of two situations: • At a branch point of an object • At a location where image information is weak

11. Algorithms • Existing algorithms for extracting cores all rely on core following – determine one medial atom and then step to the next • When does core following stop? • If an object has an explicit end, the end can be signaled by a tri- local endness detector • For objects such as blood vessels, core following stops when image information becomes too weak

12. Cores don’t branch, so what happens at an object’s branch point? In optimum parameter cores, each of the three branches has its own core, and these three cores generically do not cross at a single point Fridman’s dissertation will try to identify when a core is nearing a branch point, and then jump across the branch Branching

13. Branch Detection • Apply an affine-invariant corner detector to the image: LuuLv, where v is the gradient direction and u is orthogonal to v • Medial atoms whose sail tips are at maxima of “cornerness” are potential branch points

14. Jumping to New Branches • MATLAB code exists that uses the techniques presented to follow cores and detect branch points • It then uses geometric information of the extracted core to predict the two new cores • This is work in progress