Download - IPAM Graduate Summer School Mathematics in Brain Mapping UCLA July 17, 2004 Cortical Surface Correction and Conformal Flat Mapping: TopoCV and CirclePack.

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


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• 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


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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


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-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


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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


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• 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]