An Overview of CoresAn Overview of CoresAn Overview of CoresAn Overview of Cores

Yoni FridmanYoni Fridman

The University of North Carolina at Chapel HillThe University of North Carolina at Chapel Hill

Medical Image Display & Analysis GroupMedical Image Display & Analysis Group

Based on work by Fridman, Furst, Damon, Keller, Based on work by Fridman, Furst, Damon, Keller, Miller, Fritsch, PizerMiller, Fritsch, Pizer

Yoni FridmanYoni Fridman

The University of North Carolina at Chapel HillThe University of North Carolina at Chapel Hill

Medical Image Display & Analysis GroupMedical Image Display & Analysis Group

Based on work by Fridman, Furst, Damon, Keller, Based on work by Fridman, Furst, Damon, Keller, Miller, Fritsch, PizerMiller, Fritsch, Pizer

What is a Medial Atom?What is a Medial Atom?What is a Medial Atom?What is a Medial Atom?

A medial atom A medial atom mm = ( = (xx, r, , r, FF, , ) is an oriented ) is an oriented position with two sailsposition with two sails In a 3D image, In a 3D image, mm is eight-dimensional: is eight-dimensional:

A medial atom A medial atom mm = ( = (xx, r, , r, FF, , ) is an oriented ) is an oriented position with two sailsposition with two sails In a 3D image, In a 3D image, mm is eight-dimensional: is eight-dimensional:

xx is the location in 3-space is the location in 3-space r is the radius of two sails, r is the radius of two sails, pp and and ss

FF is a frame that has three is a frame that has three degrees of freedomdegrees of freedom bb is the bisector of the sails is the bisector of the sails

is the object angleis the object angle

xx is the location in 3-space is the location in 3-space r is the radius of two sails, r is the radius of two sails, pp and and ss

FF is a frame that has three is a frame that has three degrees of freedomdegrees of freedom bb is the bisector of the sails is the bisector of the sails

is the object angleis the object angle

xr

p

s

b

E.g.,E.g., for for slabsslabs

E.g., E.g., for for tubes,tubes, where where VV is is the set of vectors the set of vectors obtained by rotating obtained by rotating pp about about bb

E.g.,E.g., for for slabsslabs

E.g., E.g., for for tubes,tubes, where where VV is is the set of vectors the set of vectors obtained by rotating obtained by rotating pp about about bb

What is the Medialness of a What is the Medialness of a Medial Atom Medial Atom mm??

What is the Medialness of a What is the Medialness of a Medial Atom Medial Atom mm??

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sxID

pxID

s

p

Vv

v vxID

),(ˆ

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Medialness M(Medialness M(mm) is a scalar function that ) is a scalar function that measures the fit of a medial atom to image datameasures the fit of a medial atom to image dataMedialness M(Medialness M(mm) is a scalar function that ) is a scalar function that measures the fit of a medial atom to image datameasures the fit of a medial atom to image data

What is a Core?What is a Core?What is a Core?What is a Core?

Cores are critical loci of medialnessCores are critical loci of medialness A core is a description of an image, not a A core is a description of an image, not a

description of the real worlddescription of the real world It is defined based on three choices:It is defined based on three choices:

Dimension of critical loci that are desiredDimension of critical loci that are desired 1D for tubes, 2D for slabs1D for tubes, 2D for slabs Criticality is in co-dimensionCriticality is in co-dimension

Definition of subspace for criticalityDefinition of subspace for criticality Maximum convexityMaximum convexity Optimum parameters: r, Optimum parameters: r, FF, ,

What function is used to compute medialnessWhat function is used to compute medialness

Cores are critical loci of medialnessCores are critical loci of medialness A core is a description of an image, not a A core is a description of an image, not a

description of the real worlddescription of the real world It is defined based on three choices:It is defined based on three choices:

Dimension of critical loci that are desiredDimension of critical loci that are desired 1D for tubes, 2D for slabs1D for tubes, 2D for slabs Criticality is in co-dimensionCriticality is in co-dimension

Definition of subspace for criticalityDefinition of subspace for criticality Maximum convexityMaximum convexity Optimum parameters: r, Optimum parameters: r, FF, ,

What function is used to compute medialnessWhat function is used to compute medialness

Medialness FunctionsMedialness FunctionsMedialness FunctionsMedialness Functions

Originally, medialness was computed by Originally, medialness was computed by integrating over the whole sphere defined by a integrating over the whole sphere defined by a medial atommedial atom

Originally, medialness was computed by Originally, medialness was computed by integrating over the whole sphere defined by a integrating over the whole sphere defined by a medial atommedial atom

Now, we only integrate Now, we only integrate over regions surrounding over regions surrounding the tips of the two sailsthe tips of the two sails

Often use a Gaussian Often use a Gaussian derivative, taken in the derivative, taken in the direction of the sailsdirection of the sails

Now, we only integrate Now, we only integrate over regions surrounding over regions surrounding the tips of the two sailsthe tips of the two sails

Often use a Gaussian Often use a Gaussian derivative, taken in the derivative, taken in the direction of the sailsdirection of the sails

Medial manifolds of 3D objects are generically Medial manifolds of 3D objects are generically 2D:2D:

If we know we’re looking at a tube, we can If we know we’re looking at a tube, we can specify a 1D medial manifold:specify a 1D medial manifold:

Medial manifolds of 3D objects are generically Medial manifolds of 3D objects are generically 2D:2D:

If we know we’re looking at a tube, we can If we know we’re looking at a tube, we can specify a 1D medial manifold:specify a 1D medial manifold:

Medial ManifoldsMedial ManifoldsMedial ManifoldsMedial Manifolds

Maximum Convexity CoresMaximum Convexity CoresMaximum Convexity CoresMaximum Convexity Cores

Two types of cores have been studied: maximum Two types of cores have been studied: maximum convexity cores and optimum parameter coresconvexity cores and optimum parameter cores

For a d-dimensional maximum convexity core For a d-dimensional maximum convexity core located within an n-dimensional space, a height located within an n-dimensional space, a height ridge is found by maximizing medialness over ridge is found by maximizing medialness over the n-d directions of sharpest negative curvaturethe n-d directions of sharpest negative curvature Maximum convexity cores are simpler and their Maximum convexity cores are simpler and their

singularity-theoretic properties have been singularity-theoretic properties have been researched in Miller’s and Keller’s dissertationsresearched in Miller’s and Keller’s dissertations

Two types of cores have been studied: maximum Two types of cores have been studied: maximum convexity cores and optimum parameter coresconvexity cores and optimum parameter cores

For a d-dimensional maximum convexity core For a d-dimensional maximum convexity core located within an n-dimensional space, a height located within an n-dimensional space, a height ridge is found by maximizing medialness over ridge is found by maximizing medialness over the n-d directions of sharpest negative curvaturethe n-d directions of sharpest negative curvature Maximum convexity cores are simpler and their Maximum convexity cores are simpler and their

singularity-theoretic properties have been singularity-theoretic properties have been researched in Miller’s and Keller’s dissertationsresearched in Miller’s and Keller’s dissertations

Optimum Parameter CoresOptimum Parameter CoresOptimum Parameter CoresOptimum Parameter Cores

AlgorithmAlgorithm Medialness is first maximized over the Medialness is first maximized over the

parameter space (r, parameter space (r, FF, , )) The height ridge is then found by further The height ridge is then found by further

maximizing over the spatial directions normal maximizing over the spatial directions normal to the core, as defined by to the core, as defined by FF

Optimum parameter cores seem to represent Optimum parameter cores seem to represent more realistic medial locimore realistic medial loci

AlgorithmAlgorithm Medialness is first maximized over the Medialness is first maximized over the

parameter space (r, parameter space (r, FF, , )) The height ridge is then found by further The height ridge is then found by further

maximizing over the spatial directions normal maximizing over the spatial directions normal to the core, as defined by to the core, as defined by FF

Optimum parameter cores seem to represent Optimum parameter cores seem to represent more realistic medial locimore realistic medial loci

2D cores, calculated by predictor-corrector method of Fritsch2D cores, calculated by predictor-corrector method of Fritsch

3D cores, calculated by marching cubes generalization of Furst3D cores, calculated by marching cubes generalization of Furst

2D cores, calculated by predictor-corrector method of Fritsch2D cores, calculated by predictor-corrector method of Fritsch

3D cores, calculated by marching cubes generalization of Furst3D cores, calculated by marching cubes generalization of Furst

Optimum Parameter CoresOptimum Parameter CoresOptimum Parameter CoresOptimum Parameter Cores

ConnectorsConnectorsConnectorsConnectors

Connectors are height Connectors are height saddles of medialnesssaddles of medialness

Cores can turn into Cores can turn into connectors in one of connectors in one of two situations:two situations: At a branch point of At a branch point of

an objectan object At a location where At a location where

image information is image information is weakweak

Connectors are height Connectors are height saddles of medialnesssaddles of medialness

Cores can turn into Cores can turn into connectors in one of connectors in one of two situations:two situations: At a branch point of At a branch point of

an objectan object At a location where At a location where

image information is image information is weakweak = core

= connector

AlgorithmsAlgorithmsAlgorithmsAlgorithms

Existing algorithms for extracting cores all rely Existing algorithms for extracting cores all rely on core following – determine one medial atom on core following – determine one medial atom and then step to the nextand then step to the next

When does core following stop?When does core following stop? If an object has an explicit end,If an object has an explicit end, the the

end can be signaled by a tri-end can be signaled by a tri- local local endness detectorendness detector

For objects such as blood vessels,For objects such as blood vessels, core core following stops when image information following stops when image information becomes too weakbecomes too weak

Existing algorithms for extracting cores all rely Existing algorithms for extracting cores all rely on core following – determine one medial atom on core following – determine one medial atom and then step to the nextand then step to the next

When does core following stop?When does core following stop? If an object has an explicit end,If an object has an explicit end, the the

end can be signaled by a tri-end can be signaled by a tri- local local endness detectorendness detector

For objects such as blood vessels,For objects such as blood vessels, core core following stops when image information following stops when image information becomes too weakbecomes too weak

BranchingBranchingBranchingBranching

Cores don’t branch, so what happens at an Cores don’t branch, so what happens at an object’s branch point?object’s branch point? In optimum parameter cores, each of the three In optimum parameter cores, each of the three

branches has its own core, and these three cores branches has its own core, and these three cores generically do not cross at a single pointgenerically do not cross at a single point

Cores don’t branch, so what happens at an Cores don’t branch, so what happens at an object’s branch point?object’s branch point? In optimum parameter cores, each of the three In optimum parameter cores, each of the three

branches has its own core, and these three cores branches has its own core, and these three cores generically do not cross at a single pointgenerically do not cross at a single point

Fridman’s dissertation Fridman’s dissertation will try to identify when will try to identify when a core is nearing a a core is nearing a branch point, and then branch point, and then jump across the branchjump across the branch

Fridman’s dissertation Fridman’s dissertation will try to identify when will try to identify when a core is nearing a a core is nearing a branch point, and then branch point, and then jump across the branchjump across the branch

Apply an affine-invariant corner detector to the Apply an affine-invariant corner detector to the image: Limage: LuuuuLLvv, where , where vv is the gradient direction and is the gradient direction and

uu is orthogonal to is orthogonal to vv Medial atoms whose sail tips are at maxima of Medial atoms whose sail tips are at maxima of

“cornerness” are potential branch points“cornerness” are potential branch points

Apply an affine-invariant corner detector to the Apply an affine-invariant corner detector to the image: Limage: LuuuuLLvv, where , where vv is the gradient direction and is the gradient direction and

uu is orthogonal to is orthogonal to vv Medial atoms whose sail tips are at maxima of Medial atoms whose sail tips are at maxima of

“cornerness” are potential branch points“cornerness” are potential branch points

Branch DetectionBranch DetectionBranch DetectionBranch Detection

Jumping to New BranchesJumping to New BranchesJumping to New BranchesJumping to New Branches

MATLAB code exists that MATLAB code exists that uses the techniques uses the techniques presented to follow cores presented to follow cores and detect branch pointsand detect branch points

MATLAB code exists that MATLAB code exists that uses the techniques uses the techniques presented to follow cores presented to follow cores and detect branch pointsand detect branch points

It then uses geometric It then uses geometric information of the information of the extracted core to predict extracted core to predict the two new coresthe two new cores

This is work in progressThis is work in progress

It then uses geometric It then uses geometric information of the information of the extracted core to predict extracted core to predict the two new coresthe two new cores

This is work in progressThis is work in progress