Hybrid boundarymedial shape description for biologically variable shapes Martin Styner, Guido Gerig

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Hybrid boundary-medial shape description for
biologically variable shapes Martin Styner,
Guido GerigUNC Chapel Hill MMBIA -2000
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Motivation

• Detecting 3D shape changes in clinical studies to
  determine
• normal biological shape variability
• pathological shape changes
• quantitative similarity of two related shapes

Focus application Statistical 3D shape analysis
of anatomical structures in the human brain with
respect to neurological diseases, e.g.
schizophrenia, autism and others.
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Goal

• Development of a shape description scheme
  suitable for doing statistical analysis of shape
  changes in neuro-morphological studies.

Shape description requirements

• Allow comparisons between objects, i.e. a
  correspondence has to be given.
• Express morphological changes intuitively.
• Low number of features for more stable
  statistical analysis.
• Capture both fine and coarse scale properties

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Related, previous work

• Davatzikos et al. Shape analysis by comparison
  of elastic 2D transformations.
• Joshi / Miller/ Christensen Shape analysis by
  comparison of elastic 3D transformations.
• Kelemen / Székely / Gerig Shape analysis by
  comparison of spherical harmonic coefficients
  describing boundary.
• Golland / Grimson Shape analysis by comparison
  of sampled 2D skeletons.
• Pizer et al. Shape analysis by comparison of
  medial figures in 2D and 3D.
• Ayache / Subsol et al. Shape analysis by
  comparison of crest lines in 3D.

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My choice Hybrid boundary-medial shape
description

• A) Parametric boundary
• description at fine scale
• SPHARM (Brechbühler, Székely), set of spherical
  harmonic coefficients

B) Sampled medial description at coarse
scale M-reps (Pizer), set of figures, each
sampled by medial atoms mi (x,r, F, ?)
(location, thickness, orientation)
Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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Hybrid boundary-medial description

• 3D correspondence by alignment of first ellipsoid
• Captures small shape changes
• Analytical geometric properties
• No intuitive interpretation of coefficients and
  its changes

• 3D correspondence given if medial graph (figures
  atoms) is fixed
• Captures coarse scale changes
• Features are intuitive, local changes in
  location, thickness and orientation.

Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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Hybrid boundary-medial description

• Individual object is described by
• A) the SPHARM parameterized boundary
• B) the warped M-rep model (with fixed medial
  graph)

• Statistical analysis of a population
• SPHARM combined set of coefficients
• M-reps analysis of location, thickness and
  orientation information

Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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Main Problem

• How to generate a stable M-rep model in the
  presence of biological shape variability ?
• Given Population of anatomical objects
• Goal Determine a stable M-rep model
  automatically
• So far Human experts select one representative
  sample object and manually extract the M-rep
  model

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M-rep model generation scheme
Segmented object
Step 1 Define shape space
Step 2 Extract common topology
Step 3 Compute minimal sampling
Step 4 Determine M-rep statistics
Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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A medial model in 4 steps Step 1 - Shape space

• 1. Define a shape space by the average object and
  the major modes of deformation (covering more
  than 95 variability). Determined via PCA on
  SPHARM.

Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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A medial model in 4 steps Step 2 - Common
topology

• 2. Determine an appropriately stable
  figural/sheet topology in the shape space (via
  pruned Voronoi skeletons)
• ? set of medial sheets

Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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A medial model in 4 steps Step 3 - Minimal
sampling

• 3. Determine grid sampling of medial sheets
  (given maximal error in shape space, find minimal
  grid).
• ? The common topology and a set of grid
  parameters (ni,mi) determine the common medial
  model.
• The common medial model is a collection of
  sampled sheets from shapes in the shape space

Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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A medial model in 4 steps Step 4 - M-rep
statistics

• 4. Determine distribution of medial features in
  shape space via fitting the common medial model
  to the shape space

Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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Step 2 - Common topology - refinement

• Surface sampled via uniform spherical
  icosahedron-sampling of parameterization
• The full 3D Voronoi skeleton is generated from
  the sampled boundary
• Preprocessing of the skeleton
• Grouping/Merging/Pruning of skeleton
• Individual Sheet extraction
• Common Sheet Model

Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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2-A. Grouping/Merging/Pruning scheme

• Group the Voronoi faces into a set of 2D medial
  manifolds ? medial sheets (algorithm influenced
  by Naefs grouping algorithm, 96).
• Merge similar sheets based on a combined
  radial/geometric continuity criterion
• Prune sheets initially using a sheet-area-based
  criterion
• Prune sheets using a volumetric contribution
  criterion
• Every pruning changes the topology of the medial
  manifold ? recalculate grouping ? recalculate
  pruning... until no more changes
• Computationally expensive

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2-A1. Pruning via sheet criterions

• Number of Voronoi faces per sheet area
  contribution of the Voronoi sheet to the whole
  skeleton (Naef). It is used only as fast, very
  conservative ( 0.1), initial pruning
• Size of a medial manifold has no direct link to
  the importance of the manifold to the shape.

1211 sheets
61 sheets
Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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2-A2. Pruning via sheet criterions

• Volumetric contribution of sheet to overall
  volume 1. conservative threshold (0.1). 2.
  Non-conservative threshold (1)

61 sheets
6 sheets
6 sheets
2 sheets
Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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2-A. Pruning via sheet scheme

• Human
• Hippocampus
• From 1200 to
• 2 sheets

Human left lateral ventricle From 1600 to 3
sheets
volumetric overlap between reconstruction and
original shape gt 98
Human Hippocampus data from M. Shenton, R.
Kikinis, Boston Lateral ventricle data from D.
Weinberger, Washington
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2-B. Determine common topology
For all shape samples in the shape space

• Spatial match of sheets (distance criterion)
• NO structural (graph) topology match
• Use correspondence on boundary to transform
  medial sheets of different shapes into a common
  frame (thin plate spline warp).

Warp medial sheets using SPHARM correspondence on
boundary
Initial topology (average case)
Match medial sheets ? Extract common topology
Human Hippocampus data from M. Shenton, R.
Kikinis, Boston
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Conclusion / Discussion

• New approach to 3D shape representation in the
  presence of biological variability
• 3D correspondence enables statistical analysis
• Automatic scheme to determine a M-rep model
  taking into account the variability in a set of
  training shapes.
• Drawbacks
• Sphere topology ? simply connected objects
• Pathological cases outside of shape space not
  appropriately represented

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Outlook

• Ongoing research ? full implementation of
  automatic model building
• Statistical analysis Application to clinical
  studies (schizophrenia, monocygotic twins) in
  progress