7/12/2004 jhu/iacl jerry l. prince image analysis and communications laboratory dept. of electrical...
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JHU/IACL 7/12/2004
Jerry L. Prince
Image Analysis and Communications Laboratory
Dept. of Electrical and Computer Engineering
Johns Hopkins University
Cortical Surface Segmentation and Topology
7/12/2004JHU/IACL
Acknowledgments
• Chenyang Xu• Dzung Pham• Xiao Han• Duygu Tosun• Bai Ying• Daphne Yu• Kirsten Behnke• Xiaodong Tao
• Susan Resnick• Mike Kraut• Maryam Rettmann• Christos Davatzikos• Nick Bryan• Aaron Carass• Ulisses Braga-Neto
Funding sources: NSF, NIH/NINDS, NIH/NIA
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Outline
• Introduction
• Fuzzy Classification
• Nested Surface Segmentation
• Spherical Mapping and Partial Inflation
• Sulcal Segmentation
• Applications
7/12/2004JHU/IACL
Outline
• Introduction
• Fuzzy Classification
• Nested Surface Segmentation
• Spherical Mapping and Partial Inflation
• Sulcal Segmentation
• Applications
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Brain Cortex Reconstruction
Magnetic Resonance Images (MRI)
Cortical Surface
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• Study geometry of cortex– relation to function
– changes in aging and disease
• Use in function mapping– EEG/MEG/PET signals
– localization on surface instead of volume
• Surgical planning– Automatic labels
– geometric plan
Why Cortex Reconstruction?
Extracranial Tissue
Cerebrospinal Fluid (CSF)Gray Matter (GM)
Outer Pial Surface
Central SurfaceInner WM/GM Surface
White Matter (WM)
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Nested SurfacesInner
Central Outer
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Some Difficulties
• Highly convoluted cortical folds Highly convoluted cortical folds • Image noiseImage noise • Image intensity inhomogeneity Image intensity inhomogeneity • Partial volume effect Partial volume effect
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Some Requirements• Topology correctness • Valid 2D manifold
X
X
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Four Steps
1. Fuzzy classification
2. Nested surface segmentation
3. Spherical mapping and partial inflation
4. Sulcal segmentation
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Outline
• Introduction
• Fuzzy Classification
• Nested Surface Segmentation
• Spherical Mapping and Partial Inflation
• Sulcal Segmentation
• Applications
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Preprocessing
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Fuzzy Segmentation[Pham & Prince TMI 1999]
Gray matter
GM
White matter
WM
Cerebrospinal fluid
CSF
• Yields continuous-valued fuzzy membership functions, with values in the range of [0, 1]
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Published Algorithms
• AFCM: Adaptive fuzzy c-means– smooth gain field; fuzzy clusters; yields pseudo
partial volume segmentation
• AGEM: Adaptive generalized Expectation Maximization– smooth gain field; MRF label smoothness;
posterior density is “fuzzy segmentation
• FANTASM– Fuzzy segmentation with smooth membership
functions and gain field
Pham and Prince
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Membership Improvements
• White Matter– Modifications to fill interior, remove
extraneous surfaces, remove connectivity errors, and correct topology
• Gray Matter– Modification to provide evidence of CSF in
tight sulci
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WM Isosurface
• Approximates WM/GM boundary
• Problems:– undesired surfaces– connectivity errors– handles
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Autofill• WM isosurface should represent the
GM/WM interface of the cortex only
isosurface of WM segmentationbefore filling
isosurface of WM segmentationafter filling
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Autofill WM Volume
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WM Isosurface Principle
• 0.5 of WM membership approximates WM/GM interface
• 0.5 of WM+GM membership approximates GM/CSF interface
0.5WM GM CSF
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Marching Cubes Isosurface
• Consider values on corners of voxel
• Label as– above isovalue– below isovalue
• Determine position of triangular mesh surface passing through voxel
• Linear interpolation
> 0.5< 0.5
Voxel values
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Connectivity Errors
• Multiple meshes – select the largest mesh
• Touching vertices, edges, and faces– isovalue choice, or– adjust pixel values by epsilon
• Ambiguous faces and cubes– use saddle point methods, or– use connectivity consistent MC algorithm
Most isosurface algorithms use rules that lead to connectivity errors
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Ambiguous Faces
Two possible tilings:
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Ambiguous Cubes
Two possible tilings:
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Digital Connectivity
• Consistent pairs: (foreground,background) → (6,18), (6,26), (18,6), (26,6)
6-connectivity
18-connectivity 26-connectivity
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Connectivity Consistent MC Algorithm
• (black,white)• (18,6) choose b, f• (26,6) choose b, e
(a) (b) (c)
(d) (e) (f)
AmbiguousFace
AmbiguousCube
• (6,18) choose c, f• (6,26) choose c, f
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Remaining Problem: Handles
multiple surfacesshared verticesshared edgesshared facesconnectivity errors
• handles
Taken from actual white matter
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Removes Handles by Editing WM
Fill the backgroundCut the foreground OR
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Euler Number
– Euler number of a triangular mesh:
– A simple closed surface is topologically equivalent to a sphere iff
– genus is handle
tunnel
A surface handle
Illustration
• Handles: easy to detect by computing the Euler number of the surface mesh
• Euler number provides no information about the location of the handles
2/1 g
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GTCA Flow DiagramBODY
RESIDUE
SE
Opening
CTE
4
56
7
1
23
Component Labeling andConnection Analysis
Graph
Construction1
4
23
7
56Cycle
Breaking1
4
23
7
6
New Object
Original Object
Illustration of the basic ideas
(A) (B)
(C)(D)
Recycling
Illustration of our topology correction filter
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1
23
4
56
78
Morphological Opening
structuring element
“body” “residue”
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After Opening
• Divides object into two components:– “body”– “residue”
• Build graph? Throw out residue pieces? NO!– residue are often very large, but thin sheets– opening may create holes that did not exist
before
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Conditional Topological Expansion• Grow body by adding “nice” points from
residue: prohibits creation of handles; allows filling of holes
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Build a Graph
1
23
4
56
7
1
23
4
5
6
7
connected components
connectivity
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Detect and Remove Cycles
• Find a cycle using depth-first search
• Find the smallest residue connected component in the cycle and remove it
• Repeat until no more cycles remain
1
23
4
5
6
7X
X
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Restore Residue
• Add remaining residue connected components back to body
• Run conditional topological expansion again.– restores some points
that were discarded prior to graph construction.
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Success?
• Compute isosurface of binary volume
• Compute Euler number– If less than 2; repeat on background
• Compute Euler number again– If less than 2; repeat with larger structuring
element, and so on…
• Is isosurface algorithm consistent with digital topology?– wrong algorithm connectivity paradoxes
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Topology Correction: Result
Before Topology Correction After Topology Correction
¹WM ¹WM
^
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Results: Quantitative
Ratio of voxels changed to original genus is around 2
Genus of resulting volume.
Brain S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15
Init. Genus 724 955 1376 744 1031 776 562 886 688 825 986 597 1944 1280 801
b1 46 31 31 39 31 24 16 33 26 23 20 17 57 36 20f1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0b2 : : : : 1 : : 0 : : : : : : :f2 : : : : 0 : : : : : : : : : :
Changes 1371 1915 2526 1434 1984 1352 1049 1576 1257 1493 1717 1051 3812 2477 1498
ANVCPH 1.89 2 1.84 1.93 1.92 1.74 1.87 1.78 1.83 1.81 1.74 1.76 1.96 1.93 1.87
Number of voxels changed in volume.
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GM/WM Interface• Topologically correct• No self intersections• Sub-voxel resolution• Close to
– WM/GM surface– GM central surface– pial surface
• Represented by – triangle mesh, or– level set function
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Gray Matter Isosurface
• Misses tight sulci
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Partial Volume Effect
Imaging
GMCSF
partial volumeaveraging
WM
GM CSF
WM
Gyri
Sulci
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Weighted Distance Skeleton
Distance functionfrom the GM/WM
interface in
Compute its Laplacian and normalize to [ , ]0 1
L( )in
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Anatomically Consistent Enhancement (ACE)
GM GMold
in ( ( ))1 L
if in 0 CSF CSFold
GMold
in L( )
if in 0
Outside
^
^
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ACE Result
Original GM ACE GM
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Outline
• Introduction
• Fuzzy Classification
• Nested Surface Segmentation
• Spherical Mapping and Partial Inflation
• Sulcal Segmentation
• Applications
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Deformable Surface Model
• Want to move the initial WM/GM mesh
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Nested Deformable Surfaces
Pial Surface
Inner Surface
Central Surface
TGDM-3
Initial WM Isosurface
TGDM-2TGDM-1
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• Parametric deformable models (PDMs)
─ Represent curves or surfaces through explicit parameterization
─ e.g. curves tessellated with nodes,
surfaces tessellated with triangles
• Geometric deformable models (GDMs)
– Implicit implementation – uses level set numerical
method
Deformable Models
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Parametric Deformable Models
p = location on contour
[Kass, Witkin, & Terzopolous, 1987]
• Curves/surfaces that deform with a speed law derived Curves/surfaces that deform with a speed law derived from image information and prior knowledge about object from image information and prior knowledge about object shape (e.g. boundary smoothness and continuity)shape (e.g. boundary smoothness and continuity)
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x
y
One Extra Dimension
C p t( , )
z 0
z
xy
z x y t( , , )
Level Set Method
C p t x x t x R R( , ) { | ( , ) }, ) 0 2 3(or
[Osher and Sethian 1988]
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Advantages of GDMs
• Produce closed, non-self-intersecting contours
• Independent of contour parameterization
• Easy to implement: numerical solution of PDEs on regular computational grid
• Stable computation
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Parametric to Geometric[Osher & Sethian 1988]
0||||
Ft
Level Set PDE:
Contour Deformation:
0
t
C
t
0)),,(( ttpC
||||
FF
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Topology Behavior of Deformable Contour Models
• Parametric self intersection problem
• Geometric cannot control topology
• TGDM (ours) preserves topology
Parametric Geometric TGDM
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Digital Embedding of Contour Topology
White Points:
0)( x
Black Points:
• Contour topology is determined by signs of the level set function at pixel locations
• Topology of the implicit contour is the same as the topology of the digital object
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Connectivity Rule of Contour
• Topology of digital contour determined by connectivity rule
n n 4 8, n n 8 4,
Same digital object, different topologies
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Topology Preservation Principle
• Preserving contour topology is equivalent to maintaining the topology of the digital object
• The digital object can only change topology when the level set function changes sign at a grid point
• Which sign changes can be allowed, and which cannot?
• To prevent the digital object from changing topology, the level set function should only be allowed to change sign at simple points
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Simple Point• Definition: a point is simple if adding or removing the point
from a binary object will not change the object topology • Determination: can be characterized locally by the
configuration of its neighborhood (8- in 2D, 26- in 3D) [Bertrand & Malandain 1994]
SimpleNon-
Simple
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x is a Simple Point
0)( x
x
0)( x
xx
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x is Not a Simple Point
n n 4 8,
0)( x 0)( xX
X
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Topology Preserving Geometric Deformable Model (TGDM)
• Evolve level set function according to GDM• If level set function is going to change sign,
check whether the point is a simple point– If simple, permit the sign-change– If not simple, prohibit the sign-change
(replace the grid value by epsilon with same sign)– (Roughly, this step adds 7% computation time.)
• Extract the final contour using a connectivity consistent isocontour algorithm
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SGDM TGDM
A 2D Demonstration
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PDM Result TGDM Result
No Self-intersections
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A 3D TGDM DemonstrationOriginalObject
SGDMInit #1
#1
#2
SGDMInit #2
TDGMInit #1
TDGMInit #2
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TGDM for Inner Surface
Initial WM Isosurface Final GM/WM Interface
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TGDM for Inner Surface
• Evolution Equationt R x x ( ( ) ( )) 1 2
( ) ( )x
Mean Curvature:
1 2and are weighting factors
R x x( ) ( ) 2 1WMRegion Force:
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TGDM for Central Surface
Initialize with GM/WM surface Final Central Surface
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TGDM for Central Surface
• Gradient Vector Flow [Xu & Prince TIP98]
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TGDM for Central Surface
( ) ( )x
Mean Curvature:
Gradient Vector Flow Force:
F xGVF GMGVF( ) ( )
1 3, and are weighting factors2
Region Force:
R xx
x x( )
,
( ) ( ),
if ( ) 0.5
otherwiseGM
WM CSF
0
• Evolution Equation
t R x x F x ( ( ) ( )) ( ) 1 2 3 GVF
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Nesting Constraint
• Nested surfaces:– Central is outside GM/WM– Pial is outside central
• If level set function wants to go negative to positive – allow if inner level set function is positive – otherwise set to small positive epsilon
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TGDM for Outer Surface
Final Pial SurfaceStart from Central Surface
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TGDM for Outer Surface
• Evolution Equationt R x x F x ( ( ) ( )) ( ) 1 2 3 GVF
R x x x( ) ( ) ( ) GM CSFRegion Force:
( ) ( )x
Mean Curvature:
Gradient Vector Flow Force:
F xGVF GM WMGVF( ) ( ( ) ) 1 3, and are weighting factors2
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Coronal
Results Visual Inspection
Sagittal
• Slice views of three surfaces overlaid on cross-sections of the original image
Axial
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Repeatability Analysis
• 3 subjects, each scanned twice
• Surface pairs rigidly registered
• Average errors:– signed distance– absolute distance
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Repeatability Results (mm)
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Landmark Validation Study
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Landmark Validation Analysis
• Raters: 12• Brains: 2 • Landmarks: 10 per
region• Sulci: 33 / brain• Geometry: 11 fundi, 11
gyri, 11 banks• Surface: Inner & Pial• Statistical software: “R”
version 1.8.1
• CRUISE surfaces are reference surfaces: yield “landmark offset”– signed and absolute
• Membership values– white matter– gray matter
• Statistical factors:– Brain– Geometry– Sulci
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Landmark Validation: Results
• MANOVA revealed significant factors: – geometry & sulci, but
not brain
• Landmark offset– mean = - 0.35 mm– std = 0.65 mm– 16% farther than 1
mm from reference
• ACE regions show smaller offsets
• Signed distance consistently negative
• outward bias of CRUISE– differs for geometry
(largest for fundi)– differs for surface
• Note: we are optimizing parameters
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Nested Surface Segmentation• Nearly fully automated
– skull-stripping is semi-automated (10 minutes)– AC & PC need to be picked manually (5 minutes) – The rest is fully automated
• Less than 25 minutes for each brain – (Previous PDM version takes 2-3 hours)
• More than 200 brain datasets processed so far – average error is about 1/3 voxel– highly repeatable scanner errors dominate
Han et al, 2004
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Outline
• Introduction
• Fuzzy Classification
• Nested Surface Segmentation
• Spherical Mapping and Partial Inflation
• Sulcal Segmentation
• Applications
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Spherical and Partial Flattening[Tosun et al, 2003]
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Surface Inflation
• Coarsen shape• More regular mesh
structure• Use relaxation
operator:
• Check norm of mean curvature:
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Atlas Registration
• Simpler surface registered using modified ICP
• Atlas labels transfer easily
Atlas Subject
(a)
(b)
(c)
(d)
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Spherical Mapping
• Single conformal map from atlas
• Inverse stereographic projection
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Automatic Labelling
• Brains mapped to sphere• Segmented sulci compared to labelled atlas• Simple voting scheme leads to >90%
accuracy
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Outline
• Introduction
• Fuzzy Classification
• Nested Surface Segmentation
• Spherical Mapping and Partial Inflation
• Sulcal Segmentation
• Applications
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Sulcal SegmentationGoals: • Automatically segment sulci • carry out cortical parcellation
Applications:• Localizing activation sites in functional images• Brain registration• Understanding morphological changes in normal aging and disease
Principle:• Based on depth from “outer” surface
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Sulcal RegionsDefined as buried cortical regions that
surround sulcal spaces
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Classifying Gyral and Sulcal Regions
• Generate a shrink-wrap surface• Sulcal regions distinguished
from gyral regions based on distance to shrink-wrap surface
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Sulcal/Gyral Classification
sulcal regions (red)andgyral regions (blue)
Euclidean distance to outer surface
sulci > 2 mmfrom outer surface
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Watershed Segmentation
• Classification does not separate sulci
• Further segmentation is required
• Watershed by immersion is intuitive idea:
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Geodesic Distance Computation• use Fast Marching (Kimmel and Sethian, ’98)
• initial contour at time zero is gyral/sulcal boundary
• Propagation at unit speed in normal direction on mesh
• geodesic distance is arrival time of evolving contour
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Watershed Computation
Each local minimumproduces acatchment basin (CB).
Critique:• true sulci are separated • single sulci are over-segmented.
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Merging Algorithm
• Addresses over-segmentation problem
• Small ridges in sulcal regions result in formation of separate CBs
• Criterion for merging CBs:
1) height of ridge
2) size of CB
• Provides different “levels” of merging
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Sulcal Segmentation Results
Height threshold = 1 cmSize threshold = 3 cm2
Rettmann et al. MMBIA 2000
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Sulcal Segmentation Results
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Cross-Sections
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Outline
• Introduction
• Fuzzy Classification
• Nested Surface Segmentation
• Spherical Mapping and Partial Inflation
• Sulcal Segmentation
• Applications
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Repeat Scan Validation
Superiorfrontal sulcus
scan 1 scan 2 scan 3
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Shape Analysis
Left
Right
Cingulate
Subject 1 Subject 2
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Geometric Features
mean curvature
geodesic depth
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Cortical Thickness[Yezzi et al, 2003]
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Baltimore Longitudinal Study of Aging
• PI: Susan Resnick (NIA)
• 1994-2003
• Ages 55-85, 158 participants
• >1000 separate scans, 1 per year per subject
• volumetric SPGR brain scans
• 0.9375x0.9375x1.5mm voxel size
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Thickness Map from CRUISETypical Thickness Map
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Cross-sectional Study of Cortical Thickness
• Preliminary study on 35 subjects
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The END
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