Conformal Brain Mapping using Variational Methods and PDEs

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Conformal Brain Mapping using Variational Methods
and PDEs
Tony F. Chan Math Department, UCLA IPAM Brain
Mapping Summer School July 13, 2004
Collaborated with Y. Wang, X. Gu , P. Thompson
and S.T. Yau
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Overview

• Motivation
• Brain Surface Conformal Mapping
• Volumetric Brain Harmonic Map
• Optimize the Conformal Parameterization by
  Landmarks
• Future Work

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Brain Mapping Tasks
Surface conformal mapping
Volumetric harmonic map
Sphere carving algorithm
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Growth patterns in the developing human brain

• Thompson et.al Growth patterns in the developing
  brain detected by using continuum mechanical
  tensor maps, Nature, 2000.

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Overview

• Motivation
• Brain Surface Conformal Mapping
• Volumetric Brain Harmonic Map
• Optimize the Conformal Parameterization by
  Landmarks
• Future Work

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Benefits for Conformal Mapping

• Any surface without holes or self-intersections
  can be mapped conformally onto the sphere
• This mapping, conformal equivalence, is
  one-to-one, onto, and angle preserving
• Locally, distances and areas are only changed by
  a scaling factor (conformal factor)
• A canonical space is useful for subsequent work

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Algorithm Requirement

• Intrinsic, independent of triangulation, tolerant
  to resolution
• Easy to combined with various constraints
• Easy to trace the point correspondence during
  evolvement
• Tolerant to boundaries e.g. easily generalizable
  to surfaces with one boundary.

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Approaches for Conformal and Harmonic Surface
Parameterization
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Approaches for Conformal and Harmonic Surface
Parameterization (Cont.)
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Approaches for Conformal and Harmonic Surface
Parameterization (Cont.)

• Most algorithms work only on genus zero surface
  with one boundary. Tannenbaums method can work
  on genus zero closed surface and genus one
  surface. Circle packing algorithm and conformal
  structure methods can work on surfaces with
  arbitrary topologies.
• Circle packing algorithm only considers the
  connectivity but not surface metric. Conformal
  structure method considers surface metric.

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Previous Work on Brain Conformal Mapping

• Laplacian operator linearization
• Haker et al. 00s
• Joshi Leahy. 02s
• Circle packing
• Hurdal et al. 00s

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Hakers Method

• Linearize Laplacian-Beltrami operator
• Solve the corresponding linear system
• Restriction the target surface must be S2, there
  are big distortions near the north pole.

Haker et al. Conformal Surface Parameterization
for Texture Mapping, IEEE TVCG, Vol. 6, No. 2,
2000
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Joshi and Leahys Method

• Extension of Hakers work.
• Achieving a unique conformal mapping by fixing
  three points. It eliminates six degree freedom
  on translation and rotation.

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Circle Packing Method

• Another way to compute the conformal mapping via
  circle packing.
• It can handle arbitrary topologies.

K. Stephenson, Circle Packing A Mathematical
Tale, Notices of AMS, Vol. 50 No. 11, 2003
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New Method

• Treat surfaces as Riemann surface
• Compute conformal structure based on metric
  tensor
• Dependent on metric continuously

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Definition Conformal Mapping

• Conformal Scaling first fundamental
  form
• Angle preserving
• Geometry similarities in the small

F
M1
M2
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Conformal Mapping Properties

• Intrinsic to geometry
• Independent of triangulation and resolution
• Depends on metric continuously

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Geometric Morphing with Conformal Mapping
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Genus 0 surfaces

• All conformal mappings between two given surfaces
  are equivalent
• Harmonic is equivalent to conformal
• Automorphism group Mobius group

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Mobius Transformation

• Linear rational group on complex plane
• 6 dimensional group

Morphing between two conformal mappings
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Demo of the conformal mapping
Demo of the Mobius Transformation
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Algorithm at a Glance

• Minimize Harmonic Energy
• Use absolute derivative (on tangent plane)
• All computation are on the target surface,
  without projecting to complex plane

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Algorithm Details

• Harmonic energy
• Discrete harmonic energy
• Discrete Laplacian

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Spherical parameterization algorithm for genus
zero surface

• Use Gauss map as the initial degree one map
• Compute the gradient vector of harmonic energy on
  each vertex
• Project the gradient vector to the tangent space
  on S2 at each vertex
• Update the image of each vertex along the
  tangential gradient direction
• Normalize to the surface of the sphere
• Normalize the mapping by shifting the center of
  the mass to the sphere center

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Example
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Brain Conformal Mapping
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Example
Two brain surfaces are of the same subject.
Conformal mapping is robust to the noise.
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Example
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Spherical Harmonic Analysis

• A function f S2?? is called a Spherical Harmonic
  if it is an eigenfunction of Laplace-Beltrami
  operator.
• There is a countable set of spherical harmonics
  which form an orthonormal basis for L2(S2) the
  analytical expressions are known.
• Once brain surface is conformally mapped to S2,
  the surface can be represented as 3 spherical
  functions.
• Useful for geometric compression, matching,
  surface denoising, feature detection shape
  analysis.

original
1/8
1/64
1/256
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More Genus 0 Surfaces
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Application Paint on 3D Brain

• By conformal parameterization results, we can
  directly pick points on a 3D brain surface, e.g.
  landmarks.

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Discussion

• Compared with Hakers and Joshi/Leahys method,
  our method is more geometric no big distortion
  areas more stable good extension ability (e.g.
  it is possible to do brain mapping between two
  brains using our algorithm.)
• Compared with Hurdals method, our algorithm
  considers both connectivity and surface metric.

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Overview

• Motivation
• Brain Surface Conformal Mapping
• Volumetric Brain Harmonic Map
• Optimize the Conformal Parameterization by
  Landmarks
• Future Work

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Volumetric Harmonic Map

• We get a canonical intrinsic volumetric map with
  volumetric harmonic map. It is useful for 3D
  shape registration and its subsequent
  applications.
• 3D registration
• Automatic Segmentation
• MRI / CT image registration

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Motivations for Brain Mapping Research

• Brain surface conformal mapping reseasrch has
  been successful and this motivates our more
  general investigation of 3D volumetric brain
  mapping.
• 3D harmonic mapping of brain volumes to a solid
  sphere can provide a canonical coordinate system
  for feature identification and segmentation, as
  well as anatomical normalization.

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Contributions of the Work

• A sphere carving algorithm which calculates the
  simplicial decomposition of volume adapted to
  surfaces.
• Propose a method which can find harmonic map from
  a 3 manifold to a 3D solid sphere.

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State of Art of Brain Volumetric Model Generation

• Marching cube
• Interval volume Tetrahedralization
• Not too much research on brain volumetric mesh
  construction
• Kikinis group
• Mohamed and Davatzikos

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Sphere Carving Algorithm

• Input (a sequence of volume images and a desired
  surface genus number)
• Output (a tetrahedral mesh whose surface has the
  desired genus number)
• Build a solid handle body tetrahedral mesh
  consisted of tetrahedra, such that the sphere
  totally enclose the 3D data. Let the boundary of
  the solid sphere be S. We cut the model without
  topology changes (using Eulers formula) until we
  get the object 3D tetrahedral model.

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State of Art of Brain Volumetric Analysis

• Thompson et al. (1996) ---- weighted linear
  combination of radial function.
• Gee (1999) ---- generalized elastic matching
  method
• Ferrant et al. (2000) ---- non-rigid registration
• Wang et al. (2004) ---- Volumetric
  parameterization using harmonic foliation

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Harmonic Map

• The map minimizes the stretching energy.
• Geodesics are harmonic maps from a circle to the
  surface.
• Electric magnet fields on surfaces can be
  described as harmonic maps.
• In general, harmonic maps may not exist, or
  unique.
• For genus zero closed surface, harmonic map
  always exists and is equivalent to conformal map.

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Harmonic Map

• Depends on the Riemannian metrics, Independent of
  the embeddings
• Harmonic Energy
• The Euler-Lagrange differential equation is a
  non-linear elliptic partial differential equation

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Harmonic Map

• For 3-manifold, the existence of harmonic maps,
  the uniqueness of harmonic maps, the
  diffeomorphic properties of harmonic maps are
  extremely difficult theoretic problems.
• Between convex 3-disk, harmonic map exists and is
  most likely diffeomorphism.

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Algorithm Details

• Harmonic energy
• Steepest Descent Method

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Volumetric Harmonic Map
Cube surface conformally mapped to a sphere
surface
Cube volume harmonically mapped to a solid sphere
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Volumetric Harmonic Map (Interior View)
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Volumetric Brain Model Construction
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Prostate Volumetric Data Construction
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Volumetric Brain Harmonic Map
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Volumetric Harmonic Map on Prostate Model
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Overview

• Motivation
• Brain Surface Conformal Mapping
• Volumetric Brain Harmonic Map
• Optimize the Conformal Parameterization by
  Landmarks
• Future Work

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Optimize the Conformal Parameterization by
Landmarks

• We define a metric to measure the quality of the
  parameterization.
• Suppose two brain surfaces S1,S2, two conformal
  parameterizations are denoted as f1 S2?S1 and
  f2 S2?S2, the matching metric is defined as

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Optimize the Conformal Parameterization by
Landmarks (Cont.)

• Let ? be the group of Möbius transformations. We
  can compose a Möbius transformation such that
• Landmarks are commonly used in brain mapping.
  They are a set of sulcal curves manually drawn on
  the brain surfaces.
• We can use landmarks to obtain such a Möbius
  transformation.

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With Landmark
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Optimize the Conformal Parameterization by
Landmarks (Cont.)

• Landmarks are represented as discrete point sets.
  We can reduce the brain matching metric by
  reducing the matching metric on landmark sets.
• First we project the sphere onto the complex
  plane. We find a Möbius transformation on the
  complex plane which reduces the matching metric
  on landmark sets. Then we project the results
  back to the sphere.

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Optimize the Conformal Parameterization by
Landmarks (Cont.)

• For a Möbius transformation on the complex plane
  u, since it maps infinity to infinity, it means
  the north poles of the spheres are mapped to each
  other.
• Then u can be represented as a linear form azb.
  Let pi and qi, i1 n, are corresponding landmark
  points. The functional of u can be simplified as
• where zi is the stereo-projection of pi, ?i
  is the stereo-projection of qi, g is the
  conformal factor from the plane to the sphere.

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Experimental Results
The matching metric is reduced after the
landmark matching by Möbius Transformation
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Overview

• Motivation
• Brain Surface Conformal Mapping
• Volumetric Brain Harmonic Map
• Optimize the Conformal Parameterization by
  Landmarks
• Future Work

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Future Work

• Brain Conformal Mapping with Implicit Surface
  Level Set Method
• Automatic brain feature identification
• Brain registration
• Brain structure segmentation
• Brain surface denoising
• Easy surface visualization

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Thank You!