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 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

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

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

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

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

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

Maximum Convexity Cores ä Two types of cores have been studied: maximum convexity cores and optimum parameter cores ä 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 ä Maximum convexity cores are simpler and their singularity-theoretic properties have been researched in Miller’s and Keller’s dissertations ä Two types of cores have been studied: maximum convexity cores and optimum parameter cores ä 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 ä Maximum convexity cores are simpler and their singularity-theoretic properties have been researched in Miller’s and Keller’s dissertations

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

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

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

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

BranchingBranching ä Cores don’t branch, so what happens at an object’s branch point? ä 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 ä Cores don’t branch, so what happens at an object’s branch point? ä 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 ä Fridman’s dissertation will try to identify when a core is nearing a branch point, and then jump across the branch

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

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