IPAM Graduate Summer School Mathematics in Brain Mapping UCLA July 17, 2004 Cortical Surface...

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 [email protected]

http://www.math.fsu.edu/~mhurdal

IPAM - Mathematics in Brain Imaging, July17, 2004© 2004 Monica K. Hurdal URL: www.math.fsu.edu/~mhurdal Department of Mathematics, Florida State University 2

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


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


Conformal Mapping• Conformal maps preserve angles and

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.

)exp(: zzf

• 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 2


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


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


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

where v, e, f are the numbers of vertices, edges and faces of S respectively

– the genus of S is defined to be

where m(S) is the number of boundary components of S

fevS )(

)()(22

1)( SmSSg


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 multiple-connected components, non-manifold edges, holes and handles


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


0

• 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

Surface Problems Continued

1 2

3 4 Triangle 1-2-3 givesedges 1-2, 2-3, 3-1

Triangle 2-4-3 givesedges 2-4, 4-3, 3-2


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• Vertex problems: singularity (pinched surface)– the surface touches itself at a single point

– checked/corrected using vertex neighbors which form a vertex flower:

– each vertex should be the center of at most 1 flower

– if vertex vi belongs to flower of vj, then vertex vj must belong to the flower of vi

v1

v2 v3 v4

v5

v6v7v8

vjvi

Surface Problems Continued

vi: v1, v2, v3, vj, v7, v8

vj: v3, v4, v5, v6, v7, vi

v

v1 v2

v3vn

closed flowerv: v1, v2, …, vn, v1

vvn

v1 v2

v3

vn-1

open flowerv: v1, v2, …, vn


2

• 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


3

-1

-3

-5


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Fixing Handles


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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


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A Circle Packing Example


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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 2

• To “flatten” a surface at the interior vertices:Positive curvature or cone point (angle sum < 2

Zero curvature(angle sum = 2

Negative curvature or saddle point (angle sum > 2


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vrv

rv wrw

rw

ururu

For all faces containing vertex v:wuv ,,


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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 quasi-conformal 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!


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The Cerebellum


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Euclidean & Spherical Maps


2

Hyperbolic Maps


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Flat Maps of Different Subjects


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Left Hemisphere

and Occipital

Lobe Maps


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Occipital Lobe

Temporal Lobe

Parietal Lobe

Frontal Lobe

Limbic Lobe

BoundaryParcellated data courtesy of David Van Essen, Washington U. School of Medicine, St. Louis

The Visible Man Cerebrum


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Animation


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Twin Study

• 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 non-twin 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

Collaboration with Center for Imaging Science (Biomed. Eng.), Johns Hopkins U. & Psychiatry and Radiology Departments, Washington U.

School of Medicine


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VMPFC Coordinate System

MeanCurvature


0

• 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.

Circle Packing Flexibility: Rectangular Discrete Conformal Maps


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0.731

Euclidean Maps: Specify Boundary Radius or Angle

Left RightRight LeftTwin A Twin B

0.750 0.802 0.806Extremal Length

3DSurface

Euclidean Map: Boundary Radius

Euclidean Map: Boundary Angle


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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


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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


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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


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SoftwareThe 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).

http://www.math.fsu.edu/~mhurdal


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AcknowledgementsDe 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 LeeNIH Human Brain Project Grant EB02013 NSF Focused Research Group Grant DMS-0101329More information on Brain Mapping:

URL: http://www.math.fsu.edu/~mhurdalEmail: [email protected]

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