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
What is a Medial Atom?

• 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

3
What is the Medialness of a Medial Atom m?
Medialness M(m) is a scalar function that
measures the fit of a medial atom to image data

• E.g., for slabs
• E.g., for tubes, where V is the set of
  vectors obtained by rotating p about b

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
Medialness Functions

• 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

6
Medial Manifolds

• Medial manifolds of 3D objects are generically
  2D
• If we know were 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 Millers and Kellers 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
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
Branching

• Cores dont branch, so what happens at an
  objects 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

• Fridmans dissertation will try to identify when
  a core is nearing a branch point, and then jump
  across the branch

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