cortical cartography using the discrete conformal approach ... · introduction the human brain can...

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NeuroImage 23 (2004) S119–S128

Cortical cartography using the discrete conformal approach

of circle packings

Monica K. Hurdala,* and Ken Stephensonb

aDepartment of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USAbDepartment of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA

Available online 23 September 2004

Cortical flattening algorithms are becoming more widely used to assist

in visualizing the convoluted cortical gray matter sheet of the brain.

Metric-based approaches are the most common but suffer from high

distortions. Conformal, or angle-based algorithms, are supported by a

comprehensive mathematical theory. The conformal approach that

uses circle packings is versatile in the manipulation and display of

results. In addition, it offers some new and interesting metrics that may

be useful in neuroscientific analysis and are not available through

numerical partial differential equation conformal methods.

In this paper, we begin with a brief description of cortical bflatQmapping, from data acquisition to map displays, including a brief

review of past flat mapping approaches. We then describe the

mathematics of conformal geometry and key elements of conformal

mapping. We introduce the mechanics of circle packing and discuss its

connections with conformal geometry. Using a triangulated surface

representing a cortical hemisphere, we illustrate several manipulations

available using circle packing methods and describe the associated

bensemble conformal featuresQ (ECFs). We conclude by discussing

current and potential uses of conformal methods in neuroscience and

computational anatomy.

D 2004 Elsevier Inc. All rights reserved.

Keywords: Circle packing; Computational anatomy; Conformal map;

Cortical flat map; Mapping; Riemann Mapping Theorem; Surface

flattening

Introduction

The human brain can be divided into various regions based on

function and anatomy. The surface is a highly convoluted sheet,

with the folds (gyri) and fissures (sulci) varying in size, position,

and extent from person to person. It is estimated that 60–70% of the

cortical surface is buried in folds and hidden from view. The gray

matter, or cerebrum, is the thin layer of cortical surface where most

of the functional processing of the brain occurs and is about 3–5 mm

1053-8119/$ - see front matter D 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.neuroimage.2004.07.018

* Corresponding author. Fax: +1 850 644 4053.

E-mail address: [email protected] (M.K. Hurdal).

Available online on ScienceDirect (www.sciencedirect.com.)

thick. Due to the two-dimensional sheet topology of the cortex,

some methods have been proposed that bunfoldQ and bflattenQ thecerebrum. These bflat mapsQ of the brain are useful not only as

visualization methods, but may facilitate registration, comparison,

and recognition of structural and functional cortical relationships.

In this paper, we discuss a discrete mapping approach that uses

circle packings to produce bflattenedQ images of cortical surfaces in

the sphere, the euclidean plane, and the hyperbolic plane. This

approach is founded upon a rich body of classical mathematics that

is now becoming available in discrete, computable form via circle

packing. Our maps exploit the conformal geometry of cortical

surfaces and are in fact quasi-conformal approximations of classical

conformal maps (as are bconformalQ maps produced via other

approaches); we refer to them as discrete conformal (flat) maps.

Cortical flattening overview

The main processing sequence for cortical flattening is (1)

acquisition of 3D data; (2) extraction, discretization, and triangu-

lation of the relevant surface(s); (3) flat mapping computations;

and (4) analysis, manipulation, or display of results.

1. Acquisition

For brain flattening applications, the data processing sequence

typically begins with the acquisition of a 3D magnetic resonance

imaging (MRI) volume of a region of interest. There are many data

processing steps involved, including correcting for inhomogene-

ities, stripping extraneous tissues such as the skull and scalp, the

identification and perhaps color-coding of structures of interest,

and possibly alignment of functional data with anatomical data.

Although these steps represent major tasks, they are considered

preprocessing for purposes of this paper; the reader is referred to

the literature for more details on these methods.

2. Surface extraction

A topologically correct, triangulated surface mesh is required

before flattening can begin. The surface of interest may cover the

M.K. Hurdal, K. Stephenson / NeuroImage 23 (2004) S119–S128S120

entire cortex, a hemisphere, or a region within a hemisphere, such

as a lobe. A surface representation is created from the preprocessed

3D volumetric data, typically using an isosurface or threshold-

based algorithm that approximates the white matter–gray matter

interface, the gray matter–cerebrospinal fluid (CSF) interface, or a

surface midway between. We assume here that the surface of

interest is either a full cerebral hemisphere (or cerebellum) and

hence a topological sphere, a surface region that is a topological

disc (i.e., without holes), or a multiply connected surface region (a

topological disc with one or more holes) such as a topological

annulus. The most common algorithms for surface extraction are

marching cubes (Lorensen and Cline, 1987) and its derivatives

(e.g., marching tetrahedra) due to their speed and ease of

implementation. Inadequate spatial resolution of MRI scans, as

well as certain ambiguities that can arise during isosurface creation,

means that the initial surface extraction will generally require

correction. In addition to routine artifacts, there are several

common topology errors, including holes in the surface (more

than one boundary component), multiple connected components,

nonmanifold edges (edges contained in three or more triangles),

and handles. Many defects can be detected via the Euler

characteristic formula (Massey, 1967) and automatically corrected

(Hurdal, 2004b); these issues are actively being pursued by Hurdal.

There is also progress on extraction methods which directly

generate topologically correct surfaces (Han et al., 2002; Mac-

Donald et al., 2000; Mangin et al., 1995; Stern et al., 2001).

3. Flattening

We repeat that all flattening approaches require the final

surface that is embedded in R3 to be a topologically correct

triangulated mesh. (Meshes can be generated with nontriangular

faces, but our method requires that those faces be further broken

into triangles.) Flattening refers to the creation of a one-to-one

simplicial map (i.e., triangles go to triangles) between the

triangulated 3D surface and a triangulation in bflatQ space.

Flat space is interpreted to mean a space of constant Gaussian

curvature: the sphere, the euclidean plane, or the hyperbolic plane.

Surfaces with boundary edges can be mapped to any one of these;

surfaces without boundary must be mapped to the sphere unless a

boundary is introduced by removing one or more topological

discs. (Note: bflat mapQ refers equally to the mapping itself as a

function and to its image.)

Flattening methods fall into three broad categories: graph-

based, metric-based, and conformal-based methods. Combinato-

rially speaking, triangulated meshes are planar graphs, and every

planar graph has a straight-line embedding (i.e., flattening) (Fary,

1948; Wagner, 1936). There is a substantial body of established

embedding techniques with various target criteria, such as

convexity or resolution, often having extremely efficient imple-

mentations. These graph-based methods disregard geometric

information in the surface and hence have not been used in

cortical flattening.

Metric-based algorithms were the first implemented for the

cortical flattening problem, in part, because metric features—

euclidean lengths and areas—are the most familiar of surface

features. As a result, metric-based approaches are the most popular

and well-known. These algorithms typically involve an iterative

strategy that attempts to minimize some combination of areal,

linear, and angular distortion between the original 3D mesh and its

flattened counterpart. Van Essen et al. (Drury et al., 1996; Van Essen

and Drury, 1997; Van Essen et al., 1998) have developed methods

based on longitudinal and torsional forces; Fischl et al. (1999a,b)

use gradient methods incorporating geodesic distance and area.

Other methods using similar premises have also been developed for

cortical flattening (Goebel, 2000; Wandell et al., 2000) and in

visualization and computational geometry. We make two points vis-

a-vis our later discussion: First, flat maps that preserve metric

information do not exist; algorithms attempt to minimize distortion;

some even introduce ad hoc cuts into the surfaces for this purpose.

Second, performance of these algorithms in terms of distortion (be it

linear, areal, or angular) is judged based on noncanonical and

blocalQ features.Conformal-based methods include our circle packing approach

and alternative numerical partial differential equation (PDE)

methods. bConformalQ notions are not widely known outside of

mathematics, and a principal goal of this paper is to introduce and

illustrate the central concepts. Among conformal methods, PDE

approaches rely on standard numerical techniques; they tend to be

very fast, but they apply in only limited settings. The circle packing

approach is known for its versatility, as will be demonstrated in our

illustrations. It is more soundly grounded in theory and more

comprehensive in the features of conformal geometry that it can

bring to bear in applications.

4. Results

After flat maps have been computed, they can be used for

manipulation, analysis, display, and even storage. We show many

images in this paper, but wish to emphasize that flat mapping is

much more than (perhaps not even primarily) a visualization

method. The desire to encode binformationQ about the target

surfaces, which will be useful in applications such as neuroscience,

is driving interest in conformal mapping methods. In the next

section, we discuss the nature of bconformalQ information; later we

discuss its potential uses.

Conformal surfaces

The mathematical study of surfaces recognizes properties of

two types: those intrinsic to the surface itself and those having to

do with how the surface sits in 3-space. Surface-based techniques

naturally target the intrinsic properties. The most familiar are

metric properties: lengths of surface curves, areas of surface

patches, Gaussian curvature. (Note: mean curvature is not intrinsic,

though useful for coloring of flat maps.)

Less well-known intrinsic properties are those termed

bconformalQ, which are grounded in the ability to measure angles

between curves on the surface at points where they intersect. If

there is a consistent way to measure such angles, the surface is

termed a Riemann surface and is said to have a conformal

structure. All surfaces encountered in cortical studies may be

assumed to be Riemann surfaces.

Flattening can be used purely for visualization, but the main

scientific value will lie in the ability of a flattening method to

preserve binformationQ about the surface, so that properties

observed in the flat image reflect properties inherent in the surface

rather than artifacts of the flattening process. It has been known

since Gauss that a surface with variable Gaussian curvature (such

as a cortical surface) cannot be flattened in a way that preserves

metric structure, lengths, and areas (Polya, 1968). On the other

M.K. Hurdal, K. Stephenson / NeuroImage 23 (2004) S119–S128 S121

hand, it has been known since Riemann (Gauss’s student) that a

surface can be flattened in a way that preserves conformal

structure. Mappings between surfaces that preserve angles between

intersecting curves (in magnitude and orientation) are known as

conformal maps.

The three classical geometric spaces (those of constant

Gaussian curvature) are the sphere P, the plane C, and the

hyperbolic plane D. In computations, P is the unit sphere at the

origin in R3, C is R2 identified as the complex plane, and

D = f x; yð Þ: r bffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pb 1g is the unit disc which is treated as

the Poincare model for the hyperbolic plane. It is well-known

how to measure angles in these spaces; moreover, these are

archetypes for the geometries of all Riemann surfaces, as

indicated in the famous Riemann Mapping Theorem (RMT) of

1851 (as extended to surfaces by Koebe) (Riemann, 1876).

Theorem 3.1 (RMT). If S is a simply connected Riemann surface,

then there exists a one-to-one conformal mapping of S onto one of

the standard spaces P; C or D.

There are several observations regarding conformal geometry

and the mathematics of conformal mapping(s) that is pertinent to

scientific uses:

(a) Conformal structure has been a core concept of mathematics

for nearly two centuries, from number theory and the famous

Riemann Hypothesis to analytic functions to the latest

conformal string theory. Teichmqller theory for Riemann

surfaces is an example of the exquisitely nuanced information

resident in conformal structures.

(b) For maps between plane regions, being conformal is

equivalent to being analytic. Such maps are characterized

as those having derivatives (in complex arithmetic) or

equivalently those satisfying the Cauchy–Riemann PDEs.

The maps are also intimately related to the Laplace operator

and harmonic functions.

(c) Conformal maps between plane regions have been central to

physical and engineering studies for decades—fluid flow,

electrostatics, airfoil design, residue theory. There are literally

thousands of papers on numerical approximation of such

planar conformal maps (many predating computers).

(d) Only in the last decade have methods become available for

approximating conformal mappings for general (i.e., non-

planar) surfaces, motivated by mathematical applications,

visualization p imaging, and computational geometry. Circle

packing was among the first and remains the most compre-

hensive and versatile approach. Others involve numerical PDE

methods for solving the Cauchy–Riemann equations, har-

monic energy minimization for solving the Laplace–Beltrami

equation, and differential geometric methods based on

approximation of holomorphic differentials (Angenent et al.,

1999; Gu and Yau, 2002; Gu et al., 2003; Ju et al., 2004; Levy

et al., 2002). These methods do not require equilateral surface

triangulations and are still able to obtain conformal maps.

(e) In applications, all flat mapping methods involve (1)

discretization of the surface (as with triangulations) followed

by (2) numerical flattening of the result. However, computed

maps are never conformal in the strict sense—angle is a local

infinitesimal feature that can survive neither (1) nor (2). In

technical terms, all computed maps are quasi-conformal

rather than conformal. One measure of the extent to which a

mapping distorts conformal information is reflected in a

parameter j z 1 known as dilatation. A j-quasi-conformal

map is one whose worst local dilatation is bounded above by

j. One can get a sense of this by picturing a map of a circle to

an ellipse having ratio (major axis–minor axis) = j; the closerj is to one, the less distortion. The objective goal is the

construction of flat maps with j as close to 1 as possible. A 1-

quasi-conformal map is in fact conformal. However, that goal

is more nuanced in practice. Regarding a given conformal

method: Can one estimate j? Can large j be avoided? Can jbe brought close to 1? Close to 1 outside some small set? j is

in a sense a worst-case measure of how bconformalQ a

mapping is, and it is likely that experience will lead to more

useful constructs.

Exploiting bconformalQ notions in applications implies exploit-

ing nonlocal and global features, which are fellow travelers with

angle. These reside somehow in the bconformal structuresQthemselves. We will call these ensemble conformal features

(ECFs) and will discuss and illustrate several with examples in the

Illustration section. The goal in numerical conformal flat mapping

is to bring the ensemble conformal features through the surface

discretization and flattening stages of the approximation process to

the maximum extent possible. The scientific goal is then to learn

the uses, and perhaps meaning, of this conformal information.

Circle packing and its mechanics

Conformal flat mapping was initiated by our group in the

context of cortical flattening. However, our circle packing

methods lie within a much broader theory, tracing back to a

conjecture of Field’s Medalist William Thurston, circa 1985.

Circle packing has rapidly developed into a comprehensive

discrete analytic function theory that both parallels and approx-

imates the classical continuous theory. We provide some basics

on circle packing mechanics along with some images, and hope

that this will convey some of the key intuition; see the surveys

(Stephenson, 1999; Stephenson, 2002) and the forthcoming book

(Stephenson, in preparation) for details.

A circle packing is a configuration of circles with a prescribed

pattern of tangencies. Our packings lie in the geometric spaces P,

C and D and each will be univalent, meaning that its circles have

mutually disjoint interiors. The pattern of tangencies will be

prescribed by the surface triangulation T. Abstractly, T is a

simplicial 2-complex, that is, an encoding of the connectivity

involving vertices, edges, and triangular faces. A circle packing P

associated with T in a geometry G (one of P, C, or D) must have a

circle cv for each vertex v of T, so that if hv, ui is an edge of T, then
cv and cu are tangent, and if hv, u, wi is a positively oriented face
of T, then hcv, cu, cwi is a positively oriented triple of mutually

tangent circles. In other words, the triangulation in space tells us

the pattern of tangencies required of the circles in P. The bcarrierQ,carr(P), is the collection of triangles in G obtained by connecting

the centers of tangent circles of P, so these triangles necessarily

have the same pattern as T. Later, we also discuss inversive

distance packings in which tangency is replaced by a conformally

invariant notion of separation between pairs of circles. An

example of a circle packing is shown in Fig. 1.

Definition 4.1. Suppose P is a circle packing for T in one of P, C

or D. The associated discrete conformal flat map refers to the


continuous and one-to-one simplicial map identifying each triangle

of the surface triangulation with the corresponding triangle in

carr(P).

A typical triangulation of a cerebral hemisphere involves

150,000 vertices, each of which is associated with a circle when

computing P. One may be forgiven for being skeptical that this

many circles can so tightly choreograph their sizes as to meet the

combinatorics dictates of T. However, that is the content of the

Discrete Riemann Mapping Theorem (Rodin and Sullivan, 1987;

Beardon and Stephenson, 1990).

Theorem 4.2 (DRMT). Given any triangulation T of a simply

connected surface S , there exists an associated univalent circle

packing P in G (one of P, C, or D) with the combinatorics of T.

Hence, there exists a discrete conformal flat map f:S Y G from Sto G.

Overview of packing mechanics

The process begins with the abstract combinatorics of the

given triangulation T, metric data from the 3D surface (if using

inversive distance packings), and choice of a geometry. If T is a

topological disc, then boundary conditions are set, meaning that

the radii of boundary circles or the sum of the angles in the faces

meeting at a boundary vertex are specified. The main computa-

tional task is in finding radii of the circles. This is an iterative

process driven by a packing condition that must be achieved for

all interior circles; it is nonlinear, accounting for the sometimes

length computations. Nonetheless, a typical cortical lobe may

involve 40,000 circles and takes only seconds on a PC for the

tangency packing computation. Once radii are known, a packing

P can be laid out. Spherical packings are stereographically

projected from packings in C. The software CirclePack is

available for computing circle packings of topologically correct

surfaces (Stephenson, 1992–2004), and algorithm details can be

found in Collins and Stephenson (2003). The software TopoCV

is also available (Hurdal, 2004a) for checking and correcting

surfaces with topological errors before using CirclePack.

Typical surface interaction features, such as color coding,

geodesic tracing, area computations, interactive data exchange

between the 3D and flat images, can be applied between the

original cortical surface in R3 and its flattened image. Our aim in

Fig. 1. Circle packing mechanics illustrated. Left: 3D surface. Middl

this paper is to show the mapping options and ensemble

conformal features (ECFs).

The extent to which our circle packing maps are bconformalQ innature and j-quasi-conformal in fact involves considerations that

we can only briefly describe here, and not all are yet fully

resolved.

(1) If the triangles forming the surface S in 3D were equilateral,

then one could define an in situ circle packing Q on S . In the

fuller discrete theory, the map f:Q Y P is a discrete

conformal map: there are known bounds on its dilatation j,and with well-established refinement procedures, one can

obtain improved maps that actually converge to a classical

(i.e., truly) conformal map of S to G. (As the first general

realization of the 150-year-old Riemann Mapping Theorem,

this is a singular accomplishment.)

(2) In point of fact, cortical triangulations are determined by

numerous factors, from image resolution to data correction

and sampling, and are never equilateral. When the data

reaches our group, the principal impacts on triangle shape

are due to surface extraction and smoothing. If the triangles

are moderately equilateral, then circle packing computations

can be adjusted by replacing tangency packings with

inversive distance packings, as described shortly. (There

are several inversive distance packings among our illustra-

tions.) The inversive distance theory is not as complete as

that for tangency packings, but in practice, inversive

distance packings are becoming routine for lobe-size data

sets and they considerably reduce the quasi-conformal

distortion in the associated conformal flat maps. More

details of inversive distances can be found in Bowers and

Hurdal (2003).

(3) With wire-frame or marching cube methods, and especially

with ad hoc smoothing, surface triangulations tend to be less

equilateral—often spectacularly so. Discrete conformal meth-

ods compensate only in part and the results have greater (but

quantifiable) distortion. A growing body of experience

indicates that the combinatorics alone—irrespective of

triangle shapes—capture many ECFs. This is in part due to

elements of randomness: noncoherent distortions tend to

cancel, leaving an ECF largely in tact. This effect is not yet

quantified and a key goal in ongoing work concerns stability

e: tangency packing, P1. Right: inverse diastance packing, P2.


of ECFs concerning triangle shapes, retriangulations, refine-

ments, and so forth.

Illustrations

Fig. 1 displays discrete conformal flat map images f:S Y C on

a small surface fragment S . The packing P1 is a tangency packing

(Fig. 1, middle); one sees the circles, the edges connecting tangent

circles, and the triangles forming the carrier, the flat image. The

angle sum for a circle cv in P is the sum cv of angles in the faces

meeting at its center; hv must be 2p for interior circles—this is the

packing condition we alluded to above. In this example, angle sum

hw = p was prescribed for all boundary circles cw, save hw = p/2for the four corner circles. The next packing, P2, is an inversive

distance packing (Fig. 1, right); each edge of the carrier runs

between two circle centers, and the separation between those

circles was prescribed in advance based on 3D metric data in S .The color coding in P1 and P2 reflects quasi-conformal dilatation jfrom 1 to roughly 2 (light to dark red). Loosely speaking, j = 1.50

represents 50% distortion in the local bworst-caseQ sense. Noticethat the quasi-conformal distortion is reduced considerably in the

inversive distance packing.

The least familiar of the standard geometries is that of the

hyperbolic plane, represented as the unit disc D, famous for being

bnon-euclideanQ. The packings on the left in Fig. 2 are maximal

packings because they fill the plane—their outer circles are

horocycles, internally tangent to the unit circle and of infinite

hyperbolic radius. Several features are attractive in the hyperbolic

setting, among them, a common canonical domain and manipu-

lation with Mfbius transformations, conformal mappings whose

actions are illustrated on the right.

Figs. 3 and 4 are based on a single data set. The gray matter–

CSF surface representing the left cerebral hemisphere was

Fig. 2. Hyperbolic packings and manipulations.

extracted from a 1 � 1 � 1 mm MRI scan of a normal human

adult and parcellated into lobes. The resulting triangulation T

(146,922 vertices) is a topological sphere.

We first discuss Fig. 3. Top right are the lateral and medial

views of the full triangulated surface S in 3D; the surface

extraction method gives triangles only moderately nonequilateral.

To the left are the associated discrete conformal flat maps in P,

C, and D; the latter two are based on the triangulated topological

disc L obtained from the triangulation of S by removing the

11,600 vertices of the corpus callosum and ventricle (i.e., nongray

matter). The euclidean packing of L preserves the edge lengths of

boundary edges from S . The packing in the unit disc is treated in

hyperbolic geometry; this is the essentially unique maximal

packing for L.

In the lower part of the figure, we have isolated the occipital

lobe from the euclidean flat packing. This is in fact an inversive

distance packing, as one sees in the blowup. Four physical

landmarks have been indicated on the boundary of the occipital

lobe and we have created a new flat mapping to a rectangle that

maps those points to the corners. This is displayed using the faces

of the carrier rather than its circles. A grid imposed on the

rectangle can be pulled back to this lobe on any of the flat maps

and to the surface as well.

In Fig. 4, we have added three orange patterns on the occipital

lobe in various of its manifestations. This could represent, for

example, an overlay of functional MRI (fMRI) data obtained on

this subject. These are pure simulations, however, chosen with

large, identifiable shapes for convenience in this demonstration.

The identical patches are shown in blowups as they appear on (1)

the occipital lobe of the 3D cortical surface; (2) the spherical map;

(3) the hyperbolic maximal packing; (4) the euclidean inversive

distance occipital map; (5) the rectangular occipital map. Our point

in this is to let the reader compare the similarity of the bshapesQ andrelative positions of the patches in the various settings. At the

bottom, we have manipulated the occipital lobe based on these

patterns: the rectangular packing from earlier, nowwith the patterns;

cutting out the largest of the islands and mapping the resulting

annulus to a round annulus, A = f ðx; yÞ: r bffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pb Rg;

cutting out all three islands; and mapping to a rectangle with three

round holes.

Ensemble conformal features

Underlying all our work are two fundamental facts regarding

conformal maps (both classical and discrete): They exist and they

are canonical (vis-a-vis the given surface), meaning that they are

uniquely determined (based on boundary conditions and up to

standard Mfbius normalization). Conformal invariants are

surface quantities that remain unchanged under conformal maps,

such as curve intersection angles. However, as mentioned earlier,

angles are not preserved in numerical flattening (since angles are

a local infinitesimal feature); this is the reason why we are

concentrating on bensembleQ features. Due to space limitations,

we emphasize these two principal ECFs.

Extremal length of annuli

The extremal length of the round annulus A = f ðx; yÞ: r bffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pb Rg is EL(A) = log(R/r)/2p. Annuli A and B are

conformally equivalent if and only if EL(A) = EL(B). Every


annular subregion A of a (Riemann) surface can be mapped

conformally onto a unique (up to similarity) round annulus A;

hence, one defines EL(A) = EL(A). Extremal length, large

or small, means roughly that the annulus is fat or thin,

respectively.

Fig. 3. Circle packings of a left cerebral hemisphere. Top right: lateral and med

euclidean maps. Bottom left: inversive distance (euclidean) packing of occipital lo

Extremal length of quadrilaterals

The extremal length of the rectangle Q = [0,L] � [0,W] of

length L and width W is EL(Q) = L/W. Rectangles Q and R are

conformally equivalent (with bendsQ identified) if and only if

ial views of cerebral hemisphere. Top left: spherical, hyperbolic disc, and

be along with an enlargement. Bottom right: rectangle map of occipital lobe.

Fig. 4. Manipulations of simulated activation sites. Top: lateral view of cortical hemisphere and an enlargement of the occipital lobe containing simulated

activations. Counterclockwise from left: spherical and hyperbolic hemisphere maps with activations along with enlargements of the occipital region;

euclidean hemisphere map; euclidean and rectangle maps of occipital lobe with activations; rectangle map with activations cut out and mapped to holes;

annulus map of occipital lobe with one activation cut out.


Fig. 5. Cortical surfaces and rectangle maps for left ventral medial

prefrontal cortical region.


EL(Q) = EL(R). A quadrilateral subregion Q of a surface with

designated boundary arcs as its ends can be mapped conformally

onto a unique (up to similarity) rectangle Q; hence, one defines

EL(Q) = EL(Q). Extremal length, large or small, means roughly

that the quadrilateral is long or wide, respectively, relative to the

designated ends.

These numerical extremal lengths may be treated as a form of

conformal bsizeQ due to bmonotonicityQ and bconvergenceQproperties and bquasi-invarianceQ. For instance, if B and A are

annuli and B is contained in A and separates its two boundaries,

then EL(B) b EL(A). If f:A Y B is j-quasi-conformal and A and

B are both annuli (or both quadrilaterals), thenEL Að ÞEL Bð Þ a 1=j; j½ .

The annular region A (see Fig. 4, lower right) is an example; this

bsizeQ of EL(A) has nothing to do with area, diameter, or how

contorted A is within the parent surface. It represents intrinsic

information that is not otherwise accessible, for example, visually

or intuitively. The same is true for the size of quadrilateral

regions in our examples.

Note that extremal lengths are numerical quantities readily

approximated using circle packing—independently of whether the

packings themselves are to be displayed or not. In processing

cerebral hemispheres, for example, one could routinely compute

(no visualization involved) EL(A), where A is the annular region,

say, between the occipital lobe and the corpus callosum; this is an

intrinsic measure of the bdistanceQ between these two regions.

Concerning visualization, Fig. 4 is intended to show the quasi-

stability of bshapeQ under conformal flat mappings. The appearance

of features that are deep in the interior of flattened images is largely

independent of the parent region or of the mode of flattening. There

are as yet only rudimentary methods for quantifying this, but it is

clear that the effect is at least visually helpful. The mathematics of

shape is an intense area of investigation in mathematics, though

largely in studies of visualization and recognition problems.

There are many other types of ECFs accessible via circle

packing. (These are amenable to PDE methods only in limited

special cases.) Extremal lengths associated with four-connected

regions (for example, discs with three holes as shown in Fig. 4,

lower right) are sets of numbers; likewise extremal lengths for point

distributions (discs with, say, n distinguished points), extremal

lengths of regions with slits, etc. Undoubtedly, experience, now that

we have computable flat maps, will suggest other intrinsic

quantities. Related conformal concepts such as capacities, harmonic

measures, and more general extremal lengths are candidates.

Conformal mapping and its applications in neuroscience

Our circle packing methods have been applied to a number of

different cortical regions. With Dr. Michael Miller’s laboratory (see

http://www.cis.jhu.edu), we are using conformal flat mapping to

look at small cortical regions including the plenum temporal

(Ratnanather et al., 2003), the medial prefrontal cortex (Hurdal et

al., 2003), as well as the superior temporal gyrus and the occipital

lobe. Focusing on smaller regions of cortex enables us to use

cortical flat mapping to identify and demarcate specific regions of

interest, and also impose local coordinate systems, while reducing

the larger distortions that result from flattening a cortical hemi-

sphere. In some of these studies, we are exploiting the ECF

extremal length metrics to quantify anatomical differences across

subjects. Fig. 5 illustrates sample results from the ventral medial

prefrontal cortex. The 3D surface representing the ventral medial

prefrontal cortex from young adult female twins is shown along

with rectangular euclidean flat maps. The surfaces are colored

according to mean curvature.

In collaboration with Dr. David Rottenberg’s group (see http://

www.neurovia.umn.edu), we are using conformal flattening to

localize functional activity on the cerebellum, as well as the

cortical hemispheres (Hurdal et al., 1999; Ju et al., 2004).

There is, in fact, one well-known example of conformal

mapping in neuroscience in which the angle-preserving property

is the target. This example is the retinotopic mapping between the

visual cortex and the retina p visual field. A number of papers in

the literature approximate the retinotopic mapping in monkeys and

humans by a conformal map and suggest mapping functions to

model experimental observations (see for example Drasdo, 1991;

Ermentrout, 1984; Horton and Hoyt, 1991; Hurdal, 1998;

Schwartz, 1977; Sereno et al., 1995).

Summary

Conformal mapping as applied to cortical data of the human

brain is in its infancy. However, the extensive mathematical history

and theory of conformal mapping offers many potential features

that can be exploited in a neuroscientific context. A unique

advantage of conformal methods over metric-based approaches is

that conformal mappings are canonical and hence mathematically

unique. In addition, by the RMT, conformal mappings exist and

they can be computed in the euclidean, hyperbolic, and spherical

geometries. Perhaps what is most exciting, and potentially useful,

are the ensemble conformal features. Conformal invariants, such as

the extremal length of an annulus or a rectangle, can provide new

bsizeQ and bshapeQ metrics for comparing cortical features across

subjects. Currently, only the circle packing approach allows these

kinds of conformal metrics to be computed and may provide new

ways of analyzing functional and structural data of the human

brain.

http:\\www.cis.jhu.edu

http:\\www.neurovia.umn.edu


Software

The software TopoCV is available from http://www.math.

fsu.edu/~mhurdal (Hurdal, 2004a) for checking and correcting

surfaces with topological errors. It can be read in and output

surfaces in a variety of file formats (including byu, obj, vtk,

CARET, CirclePack and FreeSurfer formats) and can be used

to convert existing data formats into the format used by

CirclePack. The software CirclePack is available from

http://www.math.fsu.edu/~kens (Stephenson, 1992–2004) for

computing discrete conformal maps via circle packings. TopoCV

was used to process the data used in this paper and

CirclePack was used to generate the flat map images in

this paper.

Acknowledgments

This work has been supported in part by an NSF Focused

Research Group grant DMS-0101329 and NIH grant P20

EB02013. The authors would like to thank David Rottenberg,

Kelly Rehm, and Lili Ju (U. Minnesota) for providing the MRI data

used in Figs. 2 and 3, and to Michael Miller (Johns Hopkins U.)

and Kelly Botteron (Washington U. School of Medicine) for

providing the MRI data in Fig. 5 (under NIH grants R01

MH62626-01 and P41-RR15241). In addition, they would like to

thank Charles Collins (U. Tennessee, Knoxville) for assistance

with Fig. 3.

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