IPAM Graduate Summer School
Mathematics in Brain Mapping
UCLA July 17, 2004
Cortical Surface Correction and
Conformal Flat Mapping:
TopoCV and CirclePack
Monica K. Hurdal
Department of Mathematics,
Florida State University, Tallahassee
mhurdal@math.fsu.edu
http://www.math.fsu.edu/~mhurdal
Mapping the Brain
• Functional processing mainly on cortical surface
• Surface-based (2D) analysis methods desired
rather than volume-based (3D) methods =>
Cortical Flat Maps
• Impossible to flatten a surface with intrinsic
curvature (such as the brain) without
introducing metric or areal distortions: “Map
Maker’s Problem”
• Metric-based approaches (i.e. area or length
preserving maps) will always have distortion
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
2
Flattening Surfaces with
Intrinsic Curvature
• What about preserving angles?
– a conformal map preserves angles and angle direction
• Mathematical theory (Riemann Mapping Theorem,
1850’s) says there exists a unique conformal mapping
from a simply-connected surface to the plane,
hyperbolic plane (unit disk) or sphere
• Conformal maps offer a number of useful properties
including:
– mathematically unique
– different geometries available
– canonical coordinate system
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
3
Conformal Mapping
• Conformal maps preserve angles and
q
angle sense (angle direction) where
an angle between 2 curves is defined to be the angle
between the tangent lines of the 2 intersecting curves.
f : z  exp( z )
•
In the discrete setting, mapping a surface embedded in R3
conformally to the plane corresponds to preserving angle
proportion: the angle at a vertex in original surface maps
so that it is the Euclidean measure rescaled so the total
angle sum measure is 2p
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
4
Cortical Surface “Flattening”
• Extract region of interest from MRI volume
• Cortical regions defined by various lobes and
fissures color coded for identification purposes
• Create triangulated surface of neural tissue from
MRI volume
• ALL flattening methods required a surface
topologically equivalent to a sphere or disc:
surface topology corrected
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
5
Cortical Surfaces
• Few algorithms (but more emerging) are available
for creating topologically correct cortical surfaces;
widely used algorithms, such as the marching
cubes/tetraheda algorithms, generate surfaces with
topological errors
• Cortical surface topologically corrected
– a topological sphere has no boundary components
– a topological disc has a single closed boundary
component formed by the boundary edges
• Cortical flattening: a single closed boundary cut
may be introduced into a topological sphere to act
as a map boundary under flattening
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
6
TopoCV: Software for Correcting
Cortical Surface Topology
• Topological problems are detected and repaired
using topological invariants and examining the
connectivity of the surface
• If S is a surface then:
– the Euler characteristic of S is defined to be
 (S )  v  e  f
where v, e, f are the numbers of vertices, edges and faces
of S respectively
1
– the genus of S is defined to be g ( S )  2   ( S )  m( S ) 
2
where m(S) is the number of boundary components of S
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
7
Topological Invariants
• If a surface is topologically correct, then it is a
topological sphere if and only if (S) = 2 and it is
a topological disc if and only if (S) = 1
• NOTE: a surface with (S) = 2 is NOT
necessarily topologically correct
• The number of handles in S = g(S)
• Typical topological problems include multipleconnected components, non-manifold edges,
holes and handles
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
8
Types of Topological Problems
• Unused vertices
– vertices which do not belong to any triangle face
– checked/corrected by examining the connectivity of the
surface
• Duplicate triangles
– triangle faces which are listed twice: fm = fn
– checked/corrected by examining the vertex components of fk
• Non-manifold edges
– each edge of S is required to be an interior edge or a
boundary edge
– edges which occur 3 or more times are non-manifold edges
i.e. walls, ridges, bubbles
– checked/corrected using a region growing algorithm
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
9
Surface Problems Continued
• Multiple-connected components
– every triangle belongs to the main part of the surface
– checked/corrected using a region growing algorithm
• Triangle orientation
– all triangles are required to be oriented in the same
direction; by convention, orientation assumed to be
counter clockwise
– checked/corrected by examining each interior edge
3
1
4
2
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
Triangle 1-2-3 gives
edges 1-2, 2-3, 3-1
Triangle 2-4-3 gives
edges 2-4, 4-3, 3-2
IPAM - Mathematics in Brain Imaging, July17, 2004
10
Surface Problems Continued
• Vertex problems: singularity (pinched surface)
– the surface touches itself at a single point
– checked/corrected using vertex neighbors which form a
vn-1
vertex flower:
v
v
vn
v3
v1 v2
closed flower
v: v1, v2, …, vn, v1
v3
vn
v1 v2
open flower
v: v1, v2, …, vn
– each vertex should be the center of at most 1 flower
v8 v7
– if vertex vi belongs to flower of vj, then
vertex vj must belong to the flower of vi
vi: v1, v2, v3, vj, v7, v8
vj: v3, v4, v5, v6, v7, vi
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
v1
v5
vj
vi
v2
v6
v3
v4
IPAM - Mathematics in Brain Imaging, July17, 2004
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• Holes
– a surface should contain at most one closed boundary component;
additional boundary components are holes
– checked/corrected by examining the boundary components and
“filling” in missing regions with additional triangles
• Handles (or Tunnels)
– each handle contributes -2 to the Euler characteristic
– the number of handles can only be determined after all other
topological problems have been corrected
– a handle can be corrected in 2 possible ways:
• cut handle and then “cap” off ends or
• fill in handle “tunnel”
– unless a priori information about handles are known or assumed,
then handle correction should be guided by volumetric data used to
create the surface
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
12
-1
-3
-5
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
13
Fixing Handles
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
14
CirclePack: Software for
Discrete Conformal Flat Mapping
•
•
A circle packing is a configuration of circles with a specified
pattern of tangencies
Discrete Uniformization Theorem, Circle Packing Theorem
and Ring Lemma guarantee that this circle packing is unique
and quasi-conformal
–
•
If triangulated surface consists of equilateral triangles, then the
discrete packing converges to the conformal map in the limit.
Given a simply-connected triangulated surface:
–
–
–
represent each vertex by a circle such that each vertex is located at the
center of its circle
if two vertices form an edge in the triangulation, then their
corresponding circles must be tangent
assign a positive number to each boundary vertex
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
15
A Circle Packing Example
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
16
The Algorithm
• Iterative algorithm has been proven to converge
• Surface curvature is concentrated at the vertices
• A set of circles can be “flattened” in the plane if
the angle sum around a vertex is 2p
• To “flatten” a surface at the interior vertices:
Positive curvature
or cone point
(angle sum < 2p
Zero curvature
(angle sum = 2p
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
Negative curvature
or saddle point
(angle sum > 2p
IPAM - Mathematics in Brain Imaging, July17, 2004
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For all faces v, u, w containing vertex v:
rw w
rv
v
rw
rv
ru
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
u
ru
IPAM - Mathematics in Brain Imaging, July17, 2004
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The Algorithm (Continued)
• This collection of tangent circles is a circle packing and
gives us a new surface in R2 which is our quasiconformal flat mapping.
• Easy to compute the location of the circle centers in R2
once the first 2 tangent circles are laid out.
• Each circle in the flat map corresponds to a vertex in the
original 3D surface.
• Similar algorithm exists for hyperbolic geometry.
• No known spherical algorithm: use stereographic
projection to generate spherical map.
• NOTE: A packing only exists once all the radii have been
computed!
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
19
The Cerebellum
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
20
Euclidean & Spherical Maps
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
21
Hyperbolic Maps
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
22
Flat Maps
of Different
Subjects
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
23
Left
Hemisphere
and
Occipital
Lobe
Maps
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
24
The Visible
Man Cerebrum
Occipital Lobe
Temporal Lobe
Parietal Lobe
Frontal Lobe
Limbic Lobe
Boundary
Parcellated data courtesy of David Van Essen, Washington U. School of Medicine, St. Louis
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
25
Animation
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
26
Twin Study
Collaboration with Center for Imaging Science (Biomed. Eng.), Johns
Hopkins U. & Psychiatry and Radiology Departments, Washington U.
School of Medicine
• As with non-twin brains, identical twin brains have
individual variability i.e. brains are NOT identical in the
location, size and extent of folds
• Aim: determine if twin brains more similar than nontwin brains
– if so, this can be used to help identify where a disease
manifests itself if one twin has a disease/condition that the
other does not
• Flat maps can help identify similarities and differences
in the curvature and folding patterns
• Examining medial prefrontal cortex
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
27
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
28
VMPFC Coordinate System
Mean
Curvature
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
29
Circle Packing Flexibility:
Rectangular Discrete Conformal Maps
• An advantage of conformal mapping via circle packing is the
flexibility to map a region to a desired shape.
• Boundary angles, rather than boundary radii are preserved.
• For a rectangle: 4 boundary vertices are nominated to act as the
corners of the rectangle.
• Aspect ratio (width/height) is a conformal invariant of the
surface (relative to the 4 corners) and is called the extremal
length.
• Conformal modulus represents one measure of shape.
• Two surfaces are conformally equivalent if and only if their
conformal modulus is the same.
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
30
Euclidean Maps:
Specify Boundary Radius or Angle
Left
Twin A
Right
Left
Twin B
Right
3D
Surface
Euclidean Map:
Boundary Radius
Euclidean Map:
Boundary Angle
0.750
0.802
Extremal Length
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
0.731
0.806
IPAM - Mathematics in Brain Imaging, July17, 2004
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Cortical Region Containing VMPFC
CG = cingulate gyrus
CS = cingulate sulcus
GR = gyrus rectus
IRS = inferior rostral
sulcus
LOS = lateral orbital
sulcus
MOG = medial orbital
gyrus
OFC = orbital frontal
cortex
OFS = olfactory sulcus
ORS = orbital sulci
PS = pericallosal sulcus
SRS = superior rostral
sulcus
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
32
Future Mathematical Issues
• comparing maps between subjects: metrics
• alignment of different regions
– align one volume or surface in 3-space to another and then
quasi-conformally flat map
– quasi-conformally flat map 2 different surfaces and then
align/morph one to the other (in 2D)
• analysis of similarities, differences between different map
regions
• projecting functional data onto cortical surface
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
33
Summary
• A neuroscientific research goal: understand functional
processing of human brain & make conclusions regarding
individual differences in functional organization -- may lead
to better diagnostic tools
• Functional processing mainly on brain surface
• 2D analysis methods rather than volumetric analysis methods
needed -- brain flat maps
• Conformal flat maps are mathematically unique with a variety
of features and may lead to localization of functional regions
in normal & diseased subjects
• Conformal extremal length measures need to be investigated
• 150 year-old mathematics being used in medical research
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
34
Software
The software TopoCV (Topology Checker and
Viewer) is available for download from
http://www.math.fsu.edu/~mhurdal and runs on
Linux. SGI, SUN and AIX platforms.
• Links to the conformal flattening software CirclePack
Software (written by Ken Stephenson) can also be found
there. The TopoCV software will also reformat
surfaces into the format required by CirclePack.
• TopoCV can read in and output surfaces in a variety of
file formats (including byu, obj, off, vtk, CARET,
CirclePack and FreeSurfer).
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
35
Acknowledgements
De Witt Sumners, Phil Bowers (Math, FSU)
Ken Stephenson, Chuck Collins (Math, UTK)
Kelly Rehm, Kirt Schaper, Josh Stern, David Rottenberg (Radiology,
Neurology, UMinnesota)
Michael Miller, Center for Imaging Science (Biomed, JHU) & Kelly
Botteron (Radiology, Psychiatry, WashU), Agatha Lee
NIH Human Brain Project Grant EB02013
NSF Focused Research Group Grant DMS-0101329
More information on Brain Mapping:
URL: http://www.math.fsu.edu/~mhurdal
Email: mhurdal@math.fsu.edu
© 2004 Monica K. Hurdal
URL: www.math.fsu.edu/~mhurdal
Department of Mathematics, Florida State University
IPAM - Mathematics in Brain Imaging, July17, 2004
36