# Cortical Surface Reconstruction via Unified Reeb Analysis of Geometric and Topological Outliers in Magnetic Resonance Images.

**ABSTRACT** In this paper we present a novel system for the automated reconstruction of cortical surfaces from T1-weighted magnetic resonance images. At the core of our system is a unified Reeb analysis framework for the detection and removal of geometric and topological outliers on tissue boundaries. Using intrinsic Reeb analysis, our system can pinpoint the location of spurious branches and topological outliers, and correct them with localized filtering using information from both image intensity distributions and geometric regularity. In this system, we have also developed enhanced tissue classification with Hessian features for improved robustness to image inhomogeneity, and adaptive interpolation to achieve sub-voxel accuracy in reconstructed surfaces. By integrating these novel developments, we have a system that can automatically reconstruct cortical surfaces with improved quality and dramatically reduced computational cost as compared with the popular FreeSurfer software. In our experiments, we demonstrate on 40 simulated MR images and the MR images of 200 subjects from two databases: the Alzheimers Disease Neuroimaging Initiative (ADNI) and International Consortium of Brain Mapping (ICBM), the robustness of our method in large scale studies. In comparisons with FreeSurfer, we show that our system is able to generate surfaces that better represent cortical anatomy and produce thickness features with higher statistical power in population studies.

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**ABSTRACT:**In this paper we develop a novel approach for computing conformal maps between anatomical surfaces with the ability of aligning anatomical features and achieving greatly reduced metric distortion. In contrast to conventional approaches that focused on conformal maps to the sphere or plane, our method computes the conformal map between surfaces in the embedding space formed the intrinsically defined Laplace-Beltrami (LB) eigenfunctions. Utilizing the power of LB eigenfunctions as informative descriptors of global geometry, the conformal maps computed by our method can effectively align anatomical features on cortical surfaces. By computing such feature-aware conformal maps to a group-wisely optimal atlas surface, which is also computed with metric optimization in the LB embedding space, we develop a fully automated system for cortical labeling with the fusion of labels on a large number of atlas surfaces. In our experiments, we build our system with 40 labeled surfaces and demonstrate its excellent performance with leave-one-out cross validation. We also applied the automated labeling system to cortical surfaces reconstructed from MR scans of 50 patients with Alzheimer's disease (AD) and 50 normal controls (NC) to illustrate its robustness and effectiveness in clinical data analysis.Information processing in medical imaging: proceedings of the ... conference 01/2013; 23:244-55. - SourceAvailable from: Rongjie LaiYonggang Shi, Rongjie Lai, Danny J. J. Wang, Daniel Pelletier, David Mohr, Nancy Sicotte, Arthur W. Toga[Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research. Using the Laplace-Beltrami eigensystem, we represent each surface with an isometry invariant embedding in a high dimensional space. The key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding. By minimizing a distance measure in the embedding space with metric optimization, our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features. Besides pairwise surface maps, we also extend the metric optimization approach for group-wise atlas construction and multi-atlas cortical label fusion. In experimental results, we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies. For cortical labeling, our method achieves excellent performance in a crossvalidation experiment with 40 manually labeled surfaces, and successfully models localized brain development in a pediatric study of 80 subjects. For hippocampal mapping, our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects.IEEE transactions on medical imaging. 03/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we propose a novel algorithm for the efficient search of the most similar brains from a large collection of MR imaging data. The key idea is to compactly represent and quantify the differences of cortical surfaces in terms of their intrinsic geometry by comparing the Reeb graphs constructed from their Laplace-Beltrami eigenfunctions. To overcome the topological noise in the Reeb graphs, we develop a progressive pruning and matching algorithm based on the persistence of critical points. Given the Reeb graphs of two cortical surfaces, our method can calculate their distance in less than 10 milliseconds on a PC. In experimental results, we apply our method on a large collection of 1326 brains for searching, clustering, and automated labeling to demonstrate its value for the "Big Data" science in human neuroimaging.Machine learning in medical imaging. MLMI (Workshop), author. 8679:306-313.

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 3, MARCH 2013 511

Cortical Surface Reconstruction via Unified Reeb

Analysis of Geometric and Topological Outliers

in Magnetic Resonance Images

Yonggang Shi, Rongjie Lai, Arthur W. Toga*, and Alzheimer’s Disease Neuroimaging Initiative

Abstract—In this paper we present a novel system for the

automated reconstruction of cortical surfaces from T1-weighted

magnetic resonance images. At the core of our system is a uni-

fied Reeb analysis framework for the detection and removal of

geometric and topological outliers on tissue boundaries. Using

intrinsic Reeb analysis, our system can pinpoint the location of

spurious branches and topological outliers, and correct them with

localized filtering using information from both image intensity

distributions and geometric regularity. In this system, we have

also developed enhanced tissue classification with Hessian features

for improved robustness to image inhomogeneity, and adaptive

interpolation to achieve sub-voxel accuracy in reconstructed sur-

faces. By integrating these novel developments, we have a system

thatcan automatically reconstruct corticalsurfaceswith improved

quality and dramatically reduced computational cost as compared

with the popular FreeSurfer software. In our experiments, we

demonstrate on 40 simulated MR images and the MR images

of 200 subjects from two databases: the Alzheimer’s Disease

Neuroimaging Initiative (ADNI) and International Consortium of

Brain Mapping (ICBM), the robustness of our method in large

scale studies. In comparisons with FreeSurfer, we show that our

Manuscript received September 11, 2012; accepted October 09, 2012. Date

of publication October 15, 2012; date of current version February 27, 2013.

This work was in part supported by the National Institute of Health (NIH) under

Grant K01EB013633 and Grant 5P41RR013642. Data collection and sharing

from ADNI was funded by National Institutes of Health Grant U01 AG024904.

ADNI is funded by the National Institute on Aging, the National Institute

of Biomedical Imaging and Bioengineering, and through generous contri-

butions from the following: Abbott; Alzheimer’s Association; Alzheimer’s

Drug Discovery Foundation; Amorfix Life Sciences Ltd.; AstraZeneca;

Bayer HealthCare; BioClinica, Inc.; Biogen Idec Inc.; Bristol-Myers Squibb

Company; Eisai Inc.; Elan Pharmaceuticals Inc.; Eli Lilly and Company;

F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; GE

Healthcare; Innogenetics, N.V.; Janssen Alzheimer Immunotherapy Research

& Development, LLC.; Johnson & Johnson Pharmaceutical Research & De-

velopment LLC.; Medpace, Inc.; Merck & Co., Inc.; Meso Scale Diagnostics,

LLC.; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Servier; Synarc

Inc.; and Takeda Pharmaceutical Company. The Canadian Institutes of Health

Research is providing funds to support ADNI clinical sites in Canada. Private

sector contributions are facilitated by the Foundation for the National Institutes

of Health. The grantee organization is the Northern California Institute for

Research and Education, and the study is coordinated by the Alzheimer’s

Disease Cooperative Study at the University of California, San Diego. ADNI

data are disseminated by the Laboratory for Neuro Imaging at the University

of California, Los Angeles. Data collection and sharing from ADNI was also

supported by NIH Grant P30 AG010129 and Grant K01 AG030514.

Y. Shi is with the Laboratory of Neuro Imaging, Department of Neurology,

UCLA School of Medicine, Los Angeles, CA 90095 USA (e-mail: yshi@loni.

ucla.edu).

R. Lai is with the Department of Mathematics, University of Southern Cali-

fornia, Los Angeles, CA 90089 USA (e-mail: rongjiel@usc.edu).

*A. W. Toga is with the Laboratory of Neuro Imaging, Department of

Neurology, UCLA School of Medicine, Los Angeles, CA 90095 USA (e-mail:

toga@loni.ucla.edu).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMI.2012.2224879

system is able to generate surfaces that better represent cortical

anatomy and produce thickness features with higher statistical

power in population studies.

Index Terms—Cortical surface reconstruction, Laplace–Bel-

trami eigenfunctions, Reeb graph, tissue classification, topology

correction.

I. INTRODUCTION

C

which provides the geometric foundation for measuring cortical

morphometry and tissue integrity. While many sophisticated

algorithms were developed for its solution [1]–[11], significant

challenges remain in improving the accuracy, robustness, and

speed of cortical reconstruction. In this work, we develop

a novel system for the automated reconstruction of cortical

surfaces from T1-weighted MR images based on intrinsic

analysis of geometry and topology using the Reeb graph of

Laplace–Beltrami (LB) eigenfunctions. We demonstrate that

our system can robustly reconstruct high quality cortical sur-

faces on large scale data sets.

The successful reconstruction of cortical surfaces requires

the development of a complicated image analysis system that

includes preprocessing steps such as inhomogeneity correction

[12], nonlinear registration [13]–[16], skull stripping [17]–[19],

and tissue classification [20]–[23]. Even though different

combinations of preprocessing methods were made in various

systems, a popular choice for surface formation is that a smooth

white matter (WM) surface, which represents the boundary

between WM and gray matter (GM), with correct topology is

first reconstructed, and then deformed to find the GM surface

that represents the boundary of the GM and cerebrospinal

fluid (CSF). Due to limited image resolution, partial failure

of preprocessing steps, or anatomical variability across pop-

ulation, however, geometric outliers, such as spurious spikes,

and topological outliers, such as handles and tunnels, occur

frequently during the surface reconstruction process. To build

high quality surface representations of tissue boundaries, the

challenge is the removal of both types of outliers without

sacrificing accuracy.

Geometric and topological outliers were handled separately

in previous works. To avoid geometric outliers in the recon-

structed surface, smoothness regularization was applied glob-

ally together with data terms in surface evolution [4], [5], [7],

[8], [11]. This regularization-based method, however, has two

ORTICAL surface reconstruction from magnetic reso-

nance (MR) images isa criticalproblem in brain mapping

0278-0062/$31.00 © 2012 IEEE

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512IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 3, MARCH 2013

problems. First, it only helps but does not guarantee the occur-

rence of geometric outliers can be prevented, which could be

duetoleakageintononcorticalareasnotremovedbyskull-strip-

ping. Second, shrinkage can arise, especially in deep sulcal re-

gions, because the regularization is enforced everywhere on the

surface,evenatplacesitisunnecessary.Thiscouldaffecttheac-

curacyincorticalthicknessmeasurementsandleadtodecreased

power in statistical analysis.

To ensure genus-zero topology in reconstructed surfaces,

topology-preserving deformations can be used [1], [3], [4],

[24], [25] after topological outliers are removed. There are

mainly two different approaches for topological outlier de-

tection and correction. The first approach works in the voxel

space and uses graph analysis and morphological operations

for topology analysis [22], [26], [27], however cut decisions

are made according to the geometric size of outlier branches

only. The second approach uses the triangular mesh repre-

sentation of the surface and detects topological outliers from

overlapping triangles after mapping the surface to the sphere

[28], [29]. Cut/fill decisions are then made according to image

intensity and curvature distributions. The drawback of this ap-

proach is that it relies on the spherical mapping for outlier

detection, which is computationally expensive and not suitable

for complicated surfaces with large number of handles and

tunnels.

In this work, we present a novel system for automated

reconstruction of cortical surfaces from MR images. At the

core of our system is a unified approach for the correction of

geometric and topological outliers based on intrinsic geometry.

Withwell-composedboundarydeformationdrivenbyevolution

speeds derived from the MR image [30], [31],our system builds

manifold representations of tissue boundaries and computes

the LB eigenfunctions [32]–[43] as Morse functions for Reeb

analysis [44]–[49], where we develop a novel algorithm for

the efficient construction of Reeb graphs to capture manifold

geometry and topology. By analyzing the loops and branches

of the intrinsically defined Reeb graph, we develop a unified

approach for the detection and correction of geometric and

topological outliers. For geometric outliers, intrinsic Reeb

analysis naturally leads us to perform localized outlier filtering

without causing unintended shrinkage to other parts of the

cortex. By computing paired boundary estimates with different

topology constraints, our system carefully takes into account

information from tissue analysis and geometric regularity for

the cut or fill decisions in the correction of topological outliers.

Besides the novel method for geometric and topological

outlier correction, there are two other key components in our

system. The first is the enhancement of tissue classification

with geometric features to help overcome ambiguities in tissue

classification. By using features derived from the Hessian of

images [50], [51], this improvement enables us to rely on infor-

mation in the data instead of artificial assumptions to separate

touching gyri and alleviate the impact of partial volume effects

on misclassification. The second key element is an adaptive

interpolation algorithm that achieves sub-voxel accuracy for

the final surface. With locally estimated tissue property, we

position the surface at sub-voxel locations for the accurate

representation of cortical anatomy.

Fig. 1. Overview of our cortical reconstruction system.

By integrating these novel developments, we build a robust

and efficient system for automated cortical reconstruction.

An overview of our system is shown in Fig. 1. Given a

skull-stripped MR image, we perform enhanced tissue classifi-

cation and nonlinear registration as described in Section II to

design initial evolution speeds for the WM and GM surface.

Using methods developed in Sections III and IV, geometric and

topological outliers in the WM evolution speed are then cor-

rected with our unified approach, which produces an evolution

speed for the estimation of a clean WM boundary. To obtain the

GM boundary, we deform the WM boundary outward with the

GM evolution speed. Geometric outliers in the GM boundary

are removed with intrinsic Reeb analysis. Finally adaptive

interpolation developed in Section V is applied to both the

WM and GM boundary to generate the WM and GM cortical

surfaces with sub-voxel accuracy.

Therestof the paper isorganized asfollows.InSection II,we

present the tissue enhancement algorithm and derive the evo-

lution speeds for WM and GM boundary estimation. The con-

structionofthemanifoldrepresentationoftissueboundariesand

their Reeb graphs are developed in Section III. Using paired

boundary estimates, we develop in Section IV the unified ap-

proach for geometric and topological outlier correction with in-

trinsic Reeb analysis and localized filtering. In Section V, the

adaptive interpolation algorithm is developed for sub-voxel ac-

curacy. Experimental results are presented in Section VI, where

we present detailed comparisons with FreeSurfer [5], [52] and

demonstrate that our system can achieve better performance on

both simulated and real MR images from two databases. Finally

conclusions are made in Section VII.

II. ENHANCING TISSUE CLASSIFICATION

Tissue classification typically uses statistical models to map

intensities to three tissue types in the brain: gray matter (GM),

white matter (WM), and cerebrospinal fluid (CSF) [20]–[23],

but inhomogeneity and partial volume effects usually compro-

mise the effectiveness of such models in correctly identifying

thin structures. To enhance fractions of CSF between touching

gyri, skeletons of weighted distance transforms from WM were

proposed to open up CSF regions [7], but the challenge is that

the WM map by itself has inherent uncertainty. In this section,

we propose to use features directly derived from the data, which

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SHI et al.: CORTICAL SURFACE RECONSTRUCTION VIA UNIFIED REEB ANALYSIS OF GEOMETRIC AND TOPOLOGICAL OUTLIERS

513

is theeigen-structureof the Hessian,to enhance tissue mapsand

capture thin structures. Instead of just enhancing the CSF maps,

ourmethodappliestothinstructuresinbothWMandCSFmaps,

which leads to more integral reconstructions of both WM and

GM surfaces. Based on the enhanced tissue maps and atlas la-

bels, we then define evolution speeds for boundary estimation.

Let

denote the skull-stripped MR image, where

is the lattice of voxels that we represent as

. With partial

volume models, a tissue classifier typically uses the intensity at

each voxel to assign it as either one of three brain tissue types:

GM, WM, CSF, or a weighted combination of WM and GM

at the WM boundary, and GM and CSF at the GM boundary.

Let

denote the individual tissue maps of GM,

WM, and CSF, which we compute in this work with the FAST

tool in the FSL software [21]. Tissue classification is a critical

step in our system as it provides the basis for our enhancement

algorithm. We designed our system with the flexibility of taking

inputs from any software that generates tissue maps with partial

volume models. Here we choose the FAST tool because of its

robustness across various datasets. It is also relatively efficient

and can compute results in less than 10 min. At each voxel,

the individual tissue map represents the fraction of the voxel

belonging to the corresponding tissue. Voxels where all tissue

maps are zero are labeled as background (BG). Using the tissue

maps, we can define a composite tissue map as

if

if

if

if

if.

(1)

for all

way of representing the partial volume tissue models. As an

example, the composite tissue map of an MR image in Fig. 2(a),

which is from a patient with Alzheimer’s disease, is plotted in

Fig. 2(e).

Because tissue classification relies on intensity information

to assign tissue types, it does not necessarily follow the geo-

metric assumptions of partial volume models. For example,

voxels with small factions of WM tissue should be either on the

boundary of definite WM voxels or ridges of the image. When

neighboring gyri touch each other, voxels in between with tiny

fractions of CSF may exhibit as valleys in image intensities.

Even though such ridges and valleys have distinctive geometric

characteristics, partial volume modeling with intensity distribu-

tions may not be sensitive enough to detect them. To overcome

this difficulty, we will compute an enhanced tissue map based

on Hessian features of the image, which is well known for ridge

detection in image analysis [50], [51].

At a voxel

in the MR image

gradient as

, and Hessian as

. This composite map provides a simple

, we denote its

(2)

Fig. 2. Tissue enhancement with Hessian features and atlas labels. Using en-

hanced tissue maps and atlas labels, evolutions speeds are designed to recon-

struct the WM (red) and GM (blue) boundaries of the left hemisphere (LH) and

right hemisphere as plotted in (g) and (h).

where the entries are the second order gradients of

voxel. The eigenvalue of

tude is denoted as

is denoted as

. At each voxel, we define the tissue en-

hancement feature as

at this

with the maximal magni-

, and the corresponding eigenvector

(3)

where the inner product between

gradient of

along

of ridges or bottom of valleys, we have

. Using the sign of

ridge or valley. With this tissue enhancement feature, we can

help resolve ambiguities in intensity-based tissue classification.

As an example, we show in Fig. 2(b) the tissue enhancement

feature. Compared with the image in Fig. 2(a) and the original

andis the

at the voxel. For voxels at the top

, we can determine whether it is a

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514IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 3, MARCH 2013

TABLE I

RIDGE/VALLEY DETECTION ALGORITHM

tissue map in Fig. 2(e), we can see that this feature successfully

detects ridges and valleys, but there are also false positives that

need to be cleaned.

To filter out these false positives, we use a nonlinear reg-

istration software ANTS [16] to register the image to the

LPBA40 atlas [53] and warp the probabilistic GM atlas, which

assigns to each voxel a probability of belonging to the GM,

and anatomical labels to the image space. With the help of

nonlinear registration, we can take advantage of atlas infor-

mation to improve the robustness of our tissue enhancement

algorithm and guide the design of evolution speeds for surface

reconstruction. Here we choose the ANTS tool for nonlinear

warping because of its efficiency in obtaining high quality

MR image registration [54]. Typically we can obtain very

accurate warped labels and GM atlas in around 30 min with

the ANTS software. Let

in the image space. As an illustration, the GM atlas

is shown in Fig. 2(c), and the warped anatomical labels are

shown in Fig. 2(d). Using the anatomical labels, we denote

denote the warped GM atlas

as the

set of voxels belonging to the sub-cortical structures putamen,

caudate, and ventricles. Using the tissue enhancement feature

and atlas labels, we canfilter out outliers and detect CSF valleys

and WM ridges.

Let

be the set of candidate voxels on the CSF valley, where

a threshold we typically choose as 2. Because the CSF between

touching gyri will occupy tiny fractions of voxels, we apply a

thinning algorithm as listed in Table I to

the ridge/valley detection feature. Starting from the boundary

voxel of

with the least feature value

algorithm iteratively removes voxels if its feature

thethreshold

andisasimplepoint[24],[55],i.e.,peelingit

off will not result in a topological change. As a result, we obtain

a thin set of voxels

with the maximal tissue enhancing

feature.

For partial WM voxels, we detect voxels on the top of ridges

to follow the geometry of partial volume models. Let

is

usingas

, this thinning

is above

be the

set of candidate voxels on the ridge, where

for ridge detection that we typically choose as

is a threshold

. Applying

the thinning algorithm to the set

valley detection feature, we obtain a thin set of voxels

with partial WM.

Using these results, we can define an enhanced tissue map

as

withas the ridge/

if

if

else.

(4)

For the MR image shown in Fig. 2(a), the enhanced tissue map

is shown in Fig. 2(f). Compared with the original tissue map in

Fig. 2(e), we can see that it successfully opens up deep sulci in

buried gyral regions, and enhances thin WM regions such as the

top of the superior frontal gyrus.

Using the enhanced tissue map, we can design evolution

speeds for surface generation based on the fast evolution

algorithm [31], [56], [57]. By dividing the image into left

hemisphere (LH) and right hemisphere (RH) using the warped

anatomical labels as shown in Fig. 2(d), we reconstruct the

surfaces of each hemisphere separately. In each hemisphere,the

evolution speed defined over the grid

is

for the WM boundary

if

otherwise

or

(5)

and the GM boundary is

if

otherwise.

or

(6)

The thresholds in (5) and (6) are chosen for the robustness

to remaining inhomogeneities and minimization of mesh dis-

tortion in reconstructed surfaces. While the correction of in-

homogeneity is applied before tissue classification, some re-

maining inhomogeneities can still exist in various regions of

the brain. For example, in regions with hyper-intensities, large

blocks of CSF voxels could be included to the interior of the

GM boundary if the threshold is selected toward one in (6). On

the other hand, in regions with relatively low intensities, large

areas of GM voxels could be left out if the threshold is chosen

toward two in (6). To achieve a balance in both situations, we

choose the thresholds in (5) and (6) such that only voxels con-

taining at least 50% of the WM or GM tissue are included in

the WM or GM speed. As illustrated in Fig. 2(g) and (h), the

final reconstructed boundaries are optimally determined with

the adaptive interpolation method in Section V, which shifts the

position of voxel boundaries with locally estimated tissue prop-

erties. By putting voxels with more than 50% of the WM or GM

tissue to the interior of the boundary and voxels with less than

50%oftheWM orGMtissuetotheexterioroftheboundary,we

restrict the movement of all boundary points to be around half

of the voxel resolution. This avoids large distortion of triangles

and leads to improved mesh quality in reconstructed WM and

GM surfaces.

Inthenexttwosections,wewilldeveloptheunifiedapproach

for the removal of both geometric and topological outliers in

the evolution speeds. After that, sub-voxel accuracy will be

achieved with the adaptive interpolation scheme in Section V

to generate the final cortical surfaces.

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SHI et al.: CORTICAL SURFACE RECONSTRUCTION VIA UNIFIED REEB ANALYSIS OF GEOMETRIC AND TOPOLOGICAL OUTLIERS

515

III. INTRINSIC REEB GRAPHS ON TISSUE BOUNDARIES

In this section, we develop numerical algorithms for the con-

struction of intrinsic Reeb graphs on tissue boundaries that cor-

respond to the interface of different tissue types. We first build

a mesh representation of tissue boundaries using the evolution

speedscomputedinSectionII.WiththeLBeigenfunctionasthe

Morse function, a novel numerical algorithm is then developed

to construct Reeb graphs that intrinsically capture the geometric

and topological structure of the tissue boundary.

A. Mesh Representation

Starting from an initial mask, we run a fast boundary evo-

lution algorithm, which we developed in our previous work

[31], [56], [57], following the evolution speed in (5) or (6). The

boundaryevolutionprocessseparatesthelatticeofvoxels

two regions: the object region

. To extract the continuous representation of the boundary

between regions, we consider each grid point

as the center point of a rectangular cuboid

into

and the background region

where

the

representation, each voxel has six rectangular faces.

Because the well-composedness condition [30] is satisfied

in our reconstruction [31], the boundary between

is a manifold and composed of a set of rectangular faces

. Each face

of the cuboids of two voxels:

establishes a map from boundary faces to voxels in the ob-

ject region

background region

dividing each rectangular face

and , we have a triangular mesh representation of the

boundary, where

are the set of vertices and triangles.

For each face

, we know it is on the boundary face

, where denotes the greatest integer less

than or equal to . We can define the maps from each triangular

face to voxels in

andas:

and

freely between the surface representation and voxel representa-

tion of the tissue boundary under consideration.

,, and

direction of the MR image. Under this cuboid

are the spatial sampling resolutions in

and

and

is the intersection

and. This

withand the

. Bywith

into two triangular faces

and

with

. These maps allow us to move

B. Laplace–Beltrami Eigen-System

With the triangular mesh representation of the boundary

, we can compute its LB eigen-system by solving

a matrix eigenvalue problem [31]–[33]

(7)

where

and the two matrices

is the eigenvalue,is the eigenfunction,

are formed using the finite and

element method [58]. More specifically, the matrices are

defined as

if

if

otherwise

if

if

otherwise

where

of

is the set of vertices in the one-ring neighborhood

isthesetoftrianglessharingtheedge

is the angle in the triangle

is the area of the th triangle

The LB eigen-system is discrete and the eigenvalues can be

ordered as

eigen-functions are

trinsically defined on manifolds and has the nice property of

being isometric invariant. It has been applied successfully for

various shape analysis tasks in computer vision and medical

imaging [32]–[43]. In particular, we will use the first noncon-

stant LB eigenfunction

here as the Morse function for Reeb

graph construction [59], which is the solution of the minimiza-

tion problem

,,

,opposite to the edge

and

.

. Correspondingly, the

. The LB eigen-system is in-

(8)

and can be viewed as the smoothest map from the manifold

totherealline .Asanillustration,theLBeigenfunction

WM boundary surface is plotted in Fig. 5(a). Similar to our pre-

vious work in hippocampal modeling with LB eigenfunctions

[39], we can see the LB eigenfunction intrinsically models the

front-to-posterior trend of the cortical surface.

ona

C. Reeb Graph

Given a Morse function

defined as follows [44].

Definition 1: Let

is the quotient space with its topology defined through the

equivalent relation

if

Following this abstract definition, the Reeb graph is intu-

itively a graph of level contours of

tion of Reeb graphs on surfaces with general topology, pre-

vious methods typically need to scan through the whole mesh

to detect topological changes of level contours [46]–[48]. Be-

cause level contours change topology only at critical points of

, Reeb graph is essentially a graph of critical points. By using

levelcontoursatsaddlepoints,anefficientReebgraphconstruc-

tion algorithm was proposed in [49], but it uses triangle-based

region growing and cannot handle densely distributed saddle

points. The partition generated by the method in [49] produces

nonmanifold regions which are not suitable for further anal-

ysis with intrinsic geometry. To overcome these drawbacks, we

develop a novel method for Reeb graph construction on high

genus meshes. The key idea here is that we augment the mesh

on the mesh, its Reeb graph is

. The Reeb graphof

for.

on. For the construc-

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516IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 3, MARCH 2013

with level contours crossing the upper and lower neighborhood

of saddle points to process close saddle points. In our method,

the surface partition generated by the Reeb graph ensures each

surface patch is a manifold, so that further analysis of geom-

etry and topology, such as the computation of geodesics, can be

performed.

The critical points of

can be classified into maximum, min-

imum, and saddle points. For a vertex

neighborhood

. Let

and

the lower and upper neighbors in the 1-ring neighborhood of a

vertex

. Letdenote the number of connected components

in a set. Using the number of connected components in

and, we can classify the vertices as follows:

, and its one-ring

denote

otherwise.

(9)

Let

on the surface

and sort the critical points according to the critical values such

that

withoutperfectsymmetry,wefindthisassumptionalwaysholds

in our experience. For synthetic shapes with perfect symmetry,

we can perturb the metric [37], [43] and make sure this assump-

tion is valid. To accurately represent the partition of the surface

by neighboring saddle points on the Reeb graph, which could

have very subtle differences in the function values, we will aug-

ment the original mesh by splitting its triangles along the level

contoursduringtheReebgraphconstructionprocess.Wedenote

this augmented mesh as

augmented set of trianglesand verticesof

represented as

, where

isthesetofcriticalpointsandactasnodesofthegraph,and

is the set of arcs. Each arc in

as

are the start and end node,

longing to this arc in

,

on the boundary of the arc. The set of triangles

the subset of

enclosed by

fold with boundaries.

To construct the Reeb graph, we need to find the arcs con-

necting these critical points. Let

that is initialized as

. The augmented set of triangles

vertices

areinitializedastheoriginaltriangleset

set

, respectively. We also assign a label function

and initialize it to befor all vertices in

these critical points sequentially as follows to build the Reeb

graph.

If

is a minimum, we increase the current arc label

one and create a new arc

. We set the start node of

, and its start level contour as

set

. Note that this arc is incomplete and we

need to find the end node

, the end level contour

the triangles

.

be the set of critical points of

. We assume all critical values are different

. For anatomical shapes

, whereandare the

.TheReeb graphis

is represented

, where

is the set of triangles be-

are the level contours and

represent

and and form a mani-

denote the current arc label

and

andvertex

. We scan through

by

as

. We also

, and

Fig. 3. Mesh augmentation by splitting triangles. Original edge: black. Level

contour: red. Inserted edge: green. (a) Keep the triangle. (b) Split into two tri-

angles. (c) Split into three triangles.

If is a saddle point, we first define the isovalues of level

contours for arcs entering and leaving this node. For the in-

coming arcs, the isovalue is defined as

(10)

For outgoing arcs, the isovalue is defined as

(11)

The definition ensures that there is no interference from other

critical points and that every arc in the Reeb graph would have

a manifold structure. This enables us to use existing algorithms

for surface analysis such as finding geodesics [60].

For each component in the lower neighborhood

thesaddlepoint

,wetracealevelcontouratthevalue

on the mesh. From the definition of

this level contour crosses all edges between

this component. This level contour is represented as a polyline

composed of points intersecting edges

of

at the level

this set of new vertices by adding edges. Let

two consecutive points that intersect the two edges of a triangle

in. As shown in Fig. 3, we either

keep this triangle intact or split it into two or three triangles and

addthemto

.Repeatingthisprocessforalllinesegmentinthis

levelcontour,weaddallpointsto

Starting with these new vertices in

using the algorithm in Table II and complete an arc

Reeb graph

. We set

which are the set of triangles in

that belong to this arc of the Reeb graph. We repeat this process

until all arcs connecting

and

contour.

For each component in the upper neighborhood

of the saddle point, we trace a level contour at the value

that cross all edges connecting

this component. Similarly, this level contour is represented

as

and all intersecting points in this

level contour are added to the augmented vertex set

mesh

. A new, but incomplete, arc in the Reeb graph is

created and we increase the current arc label

,, and set

added vertices

. This process is repeated until all edges

connecting

is crossed.

of

, we know that

and vertices in

. We augment the mesh with

andbe

asnew verticesinthemesh.

, we then grow backward

in the

,, and,

in the augmented mesh

is crossed by a level

and vertices in

of the

by one. We set

for all newly

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517

Fig. 4. Reeb graph construction of a double torus using a LB eigenfunction. (a) An illustration of the level contours in the neighborhood of a saddle point. (b) An

illustration of the arc construction process in building the Reeb graph. (c) The final partition of the surface with the Reeb graph.

TABLE II

REEB GRAPH ARC GROWING

If is a maximum, we grow backward with it as the

starting point using the algorithm in Table II to complete an

arc

. We set,

.

Byrepeatingtheaboveprocessforallcriticalpoints,wecom-

plete the construction of the Reeb graph

graphisdefinedontheaugmentedmesh

form triangles generated from the triangle-splitting process, we

can build a map

from

to the original mesh

For each triangle

, the map

in

such that is a subset. This is a many-to-one map since

the triangles in

is obtained by splitting triangles in

tablishing a map between the triangles of

a bridge between the Reeb graph on the augmented surface

and the evolution speed defined in the voxel space, which we

want to modify with intrinsic Reeb analysis. Once the outliers

detected by Reeb analysis are corrected, we can generate the

final WM and GM surfaces with the regular mesh structure of

to produce high quality mesh representations.

Asanexample,weshowinFig.4theconstructionoftheReeb

graph on a double torus. The Morse function, which is the first

, and the triangle set

. While this Reeb

,whichhasnon-uni-

to relate the properties derived

that has regular triangles.

denotes the triangle

. By es-

, we build and

nonconstant LB eigen-function of the double torus, is plotted in

Fig. 4(a), where the level contours at a saddle point

trated. One level contour is generated that crosses edges con-

necting the lower neighborhood

Two level contours are generated crossing the edges connecting

the upper neighborhood

process is illustrated in Fig. 4(b), where all arcs are plotted in

differentcolors.For eacharc,wealsoplotitstwonodesasblack

dots. In the first step, an arc between a saddle point and a min-

imum is constructed. In the second, third, and fourth step, arcs

between saddle points are added sequentially to the Reeb graph.

In the final step, an arc between a maximum and a saddle point

is constructed. The final Reeb graph is visualized in Fig. 4(c) by

plottingtrianglesoneacharcof

used in Fig. 4(b), where the graph structure is evident from the

neighboring relation of the arcs.

is illus-

and this saddle point.

and . The arc construction

withcorrespondingcolors

IV. UNIFIED ANALYSIS OF GEOMETRIC AND

TOPOLOGICAL OUTLIERS

Similar to previous works [5]–[7], our system first recon-

structs a clean WM surface with the correct topology and then

deform it to obtain the GM boundary. In this section, we de-

velop the unified approach for the removal of geometric and

topological outliers on the WM boundary. With topology-pre-

serving evolution, the cleaned WM boundary with genus-zero

topology is deformed to generate the GM boundary. While we

develop this unified approach in the context of WM surface re-

construction, the method is general and applicable to the anal-

ysis of surfaces of arbitrary topology. In particular, we apply

it to the GM boundary with spherical topology to remove geo-

metric outliers.

To remove outliers on the WM boundary, we iteratively

modify the WM evolution speed with intrinsic Reeb anal-

ysis. Using the WM evolution speed, we first compute paired

estimates of the WM boundary with and without topology

constraints. By comparing the paired boundary estimates, we

derive filling voxels for topological artifacts, which provide

the basis for cut or fill decisions in topological analysis if the

operation is consistent with underlying tissue properties. By

analyzing the intrinsically defined Reeb graph on the WM

boundary, we detect geometric and topological outliers and

design modifications to the WM evolution speed for their

removal. With the modified evolution speed, the above steps

are repeated until no change is necessary. Next we describe the

details of each step.

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Fig. 5. Geometric and topological outlier detection with the Reeb graph of the LB eigenfunction. Zoomed views of geometric and topological outliers are plotted

in (c) and (d), respectively. Yellow: Geometric outliers. Red: Topological outliers.

A. Paired Boundary Estimates

Given the WM evolution speed

tion algorithm described in [31] to compute a pair of estimates

of the WM boundary for topological analysis. Starting from

a genus-zero mask, we evolve the mask toward the boundary

under the topology-preserving constraint, which is ensured by

only updating points satisfying the simple and well-composed-

ness condition. This generates a genus-zero estimate of the ob-

ject boundary. The region enclosed by the boundary is denoted

as

and the background region is denoted as

gular meshrepresentation oftheboundaryis

For topological analysis, we turn off the topology-preserving

constraint and continue the boundary evolution process under

the same speed

to obtain the second boundary estimate,

which can have arbitrary topology. The object and background

region determined by this boundary are denoted as

and the boundary mesh is denoted as

By comparing the paired boundary estimates

wecanlocatepairedpatcheson

tunnel in

. For all faces on

the background region of

, i.e.,

map

was defined in Section III-A, we perform a connected

componentlabelinganddenotethesetofconnectedcomponents

as

nent

, we define a connected component (CC) label

. To find paired patches on

tunnel or handle, we want to find the set of filling voxels that

connect different components in

dles or tunnels.For every voxel

of all interior voxels of faces in

the paired labels of connected components it bridges. For each

, we use the fast evolu-

. The trian-

.

and

.

and

,

,

thattogetherfillahandleor

with their interior voxel in

, where the

. For faces in theth compo-

that jointly fill a

and fill topological han-

,which is the set

, we define a set

in

as

face

Note that there can be at most two components sharing an in-

terior voxel, so there cannot be more than two labels in the

paired label set

for any voxel

the voxels were included in

ness constraint, they do not directly touch two opposing patches

in

. For each

its six-neighborhood

. If there is a voxel

that satisfies

gether form a bridge to connect two patches, we set

as the union of these two sets. For each face

, we assign it the paired label

rior voxel if

group triangles with the same paired labels into paired compo-

nents

and denote

of filling voxels for topological artifacts. As an illustration, we

show in Fig. 7(b) and (d), the filling voxels for a handle and

tunnel, respectively. By analyzing the underlying tissue proper-

ties on filling voxels, cut or fill decisions could be made in topo-

logical analysis with the inclusion of information from tissue

classification.

For the unified analysis of geometric and topological outliers

on the WM boundary

, we compute its LB eigenfunctions

and construct the Reeb graph. Let

function of

, as illustrated for a WM boundary in Fig. 5(a),

and

the Reeb graph of

denotes the set of critical points, and

augmented mesh from the Reeb graph construction is denoted

as

. The map from the triangles on

is denoted as

, we first assign its CC label to.

. Because a subset of

to satisfy the well-composed-

with, we search

, which means they to-

and

of its inte-

. Using paired labels, we

as the set

be the first LB eigen-

on, where

the set of arcs. The

to

. Each arc of the Reeb graph

, where

is the set of triangles

are

the start and end node, and

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519

on this arc in

start and end node.Thedegree of a nodeinthe Reeb graph isthe

number of arcs that it is either the start or the end node. A node

is called a leaf node if its degree is one, and an internal node if

its degree is greater than one. The arc connected to a leaf node

is called a leaf arc.

, and are the level contours at the

B. Geometric Outlier Correction

We detect geometric outliers from the set of leaf arcs in the

Reeb graph. Left

be a leaf arc with

and end level contour. Using the level contour

we define an arc feature

andas the start

and,

as follows:

(12)

where

is the outward normal on the surface

are the mean coordinates of points on

, it meansis an outward leaf pointing toward the

exterior of the cortex; otherwise, it is an inward leaf that points

toward the interior of the cortex.

We consider this leaf arc as an outlier if it satisfies two con-

ditions

,and

. If and

(13)

where

parameters

and small outliers. To further localize geometric outliers, we

project the original mesh

functions

and calculate the area distortion

of triangles. For each triangle in

(ADF) is defined as

is the sum of area of all triangles in

are thresholds selected to identify sharp

, the

and

onto a subset of its LB eigen-

, its area distortion factor

(14)

where

andare the area of the triangle

and the projected meshin the original mesh

For an outlier leaf arc, we define the set of outlier triangles

as

is the set of triangles obtained by mapping the trian-

gles in to the original mesh

threshold used for further localization of geometric outliers to

triangles that exhibit large area distortions during the projection

ontothesubspaceofLBeigenfunctions[31].Tolocalizeoutliers

on sub-cortical surfaces, the method in [31] relies exclusively

on eigen-projection and needs to use hundreds of eigen-func-

tions to form the subspace. For the much more complicated cor-

tical surfaces, we observe in our experiments that a much larger

number of eigen-functions are needed to achieve similar local-

ization of outliers. This is computationally very expensive and

clearly infeasible for efficient processing because the number of

vertices on a cortical surface is typically two orders of magni-

tudelarger thanmanysub-corticalstructures.If weusethesame

.

, where

, and the parameteris a

subspace

method in [31] would produce a large number of false positives,

especially on the frontal lobe, as shown in Fig. 5(b). In contrast,

we show in Fig. 5(c) and (d) that we can achieve much better

performance with the Reeb analysis approach proposed here

using the same number of eigen-functions. A zoomed view of

detected geometric outliers is plotted in Fig. 5(c), which shows

that the proposed method achieves accurate outlier localization

with very few eigen-functions.

Toremovetheoutlier,wemodifytheevolutionspeed

follows. If is an outward leaf, we set

for all so that its interior will be removed.

Ifis an inward leaf, we set

such that its exterior will be filled and

the outlier can be removed. In contrast to previous methods

thatrelyonglobalsmoothnessregularizationtoavoidgeometric

outliers, our method only modifies the evolution speed locally

whileleavingother partsoftheboundaryintact.Thisgreatlyde-

creases shrinkage effect and help improve the accuracy in sur-

face reconstruction.

ofthefirstfoureigen-functions,theeigen-projection

as

for all

C. Topological Outlier Correction

For topological analysis, we first remove duplicated arcs in

. For any two arcs with the same start and end node, we re-

move the one with smaller size from

logical outlier to a set we denote as

resent the Reeb graph

as a matrix

graph searchalgorithmstolocatetheremaining topological out-

liers. For any arc

, we set

where

andare the indexes of the start and end critical

points, respectively, and

We detect topological outliers from redundant paths between

saddle points in the Reeb graph. Using the matrix represen-

tation of the Reeb graph, we analyze paths starting at each

node to locate topological outliers. For a node

rected graph

, if there are more than one out-going arcs, we

use breadth-first-search (BFS) to test if an outlier exists. Let

denote the set of neighboring nodes which satisfies

and add it as a topo-

. After that, we can rep-

and apply standard

,

is the area of this arc.

in the di-

. For every outgoing arc, we

and perform BFS with

node. This generates a spanning tree of nodes reachable from

after the removal of the arc

recorded as an array

. For each node

the parent node in this spanning tree if it is reached by

. By repeating this step for all outgoing arcs at

we generate a set of BFS trees

section node

of these trees is defined as the node with the

smallest index such that

Starting from the intersection node

to the current node

on each BFS tree

path as

this

is the total area of all arcs on this path. Among all

paths we pick the one with the smallest cost and add all arcs

on this path to the outlier set

illustration, the topological outliers detected with Reeb analysis

on the surface in Fig. 5(a) are plotted in red in Fig. 5(c) and (d).

A zoomed view of a topological outlier is plotted in Fig. 5(d).

set

as the starting

. The spanning tree is

,is

from

,

. The inter-

for all

, we trace backward

and record the

. The cost of

.

as topological outliers. As an

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520IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 3, MARCH 2013

Fig. 6. Reeb graph can capture a topological outlier with an arc in the form of

a handle (a) or tunnel (b).

To provide further localization of topological operations, we

compute cutting paths on the handles and tunnels detected by

Reeb analysis. For each arc in the outlier set

cide if this is a handle or tunnel on the boundary and then apply

differentanalysisstrategiestocalculateacuttingpath.Forbetter

illustration,weshow inFig.6thetwo possiblecasesthatatopo-

logicaloutliercanberepresentedintheReebgraph.InFig.6(a),

the arc of the Reeb graph, which is composed of the triangles

plotted in red, captures this outlier attached to a box as a handle.

On the other hand, this outlier could also be captured by an arc

that forms a tunnel as plotted in Fig. 6(b). To differentiate these

two cases, we compute the arc feature

foreacharc

.If

of the start and end level contours are enclosed by the surface

as in the case plotted in Fig. 6(a), we classify

If

, which means the centroids of the start and end

level contours fall outside the surface as shown in Fig. 6(b), we

classify

as a tunnel. For a handle or tunnel, we use a different

method to find a cutting path with minimal length if a cut op-

eration is needed. If

is a handle, we uniformly sample a set

of level contours of the eigenfunction

tour

and end contour

least length as the cutting path. For example, a handle on a WM

boundary is plotted in red in Fig. 7(a) and the cutting path as

the shortest level contour is plotted in Fig. 7(b). For a tunnel,

we combine two geodesic paths to form a cutting path. The first

geodesic goes from the start to the end node within the triangles

. The second geodesic goes from the start to the end node

without passing through the triangles in

a WM boundary as shown in Fig. 7(c), the cutting path that is

composed of two geodesics is plotted in Fig. 7(d).

We use information from tissue classification and geometric

regularity to make cut/fill decisions about topological outliers.

Based on tissue maps, we enforce the anatomical knowledge

that gyri on WM surfaces should not enclose CSF. Using

the area distortion during eigen-projection, we apply the as-

sumption that the WM surfaces should have similar geometric

regularity as a folded sheet. From the analysis of the paired

boundary estimates

and

on the genus-zero boundary estimate

These patches fill the handles and tunnels on

paired patches, its interior voxels are

, we first de-

as defined in (12)

,whichmeansthecentroids

as a handle.

between the start con-

, and pick the contour with the

. For a tunnel on

, we have paired patches

.

. For each

and they are

Fig. 7. Topological analysis of handles and tunnels detected with Reeb anal-

ysis. (a) A handle detected with Reeb analysis. (b) Red: arc of the Reeb graph.

Green: filling voxels. (c) A tunnel detected by Reeb analysis. (d) Red: arc of the

Reeb graph. Green: filling voxels.

filled inside

illustrated in Fig. 7 for a handle and tunnel. At a paired patch

, a cut decision should be made if either of the following

two conditions is met.

• If the number of voxels

greater than

.

• If the number of triangles

is greater than.

Both parameters

and

choose empirically to identify topological outliers that are not

consistent with the underlying image intensity distributions and

geometric regularity of cortical surfaces. The first condition

checks if filling a handle or tunnel needs more than

voxels classified as CSF in the enhanced tissue map. The

second condition measures the geometric regularity of the

filling patches by calculating if there are more than

triangles with its ADF, which is defined in (14), greater than

the threshold

used for geometric outlier detection.

To implement the cut/fill decision from the analysis of paired

patches, we use the handles and tunnels detected by the Reeb

analysis process. To cut open a paired component in

pick a cutting path on a handle or tunnel that is connected to

this paired component. For the cutting path

in, we denote

on

. We consider a paired component

cutting path

if the set

empty, which means the exterior voxels of faces on the cutting

path intersects with the interior voxels of the paired patches.

Among all cutting paths intersecting the paired component, we

choose the one with the shortest length and denote it as

To cut it open, we modify the evolution speed as

to satisfy the genus-zero constraint, which are

with is

with

are thresholds we

, we

of an arc

as the set of triangles it passes

connected to a

is not

.

(15)

Similar to the modification for the removal of geometric out-

liers, this change to the evolution speed is local and peels off a

layer of voxels along the cutting path. To completely cut open

a large outlier, this process might be applied to the same outlier

during each iteration of our unified outlier correction algorithm.

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521

Fig. 8. WM and GM surface before and after sub-voxel calculations. (a) WM

boundary after removal of outliers. (b) WM surface with sub-voxel accuracy.

(c) GM boundary after removal of outliers. (d) GM surface with sub-voxel

accuracy.

The number of iterations it takes for the topology correction al-

gorithm to converge depends on the complexity of the topolog-

ical outliers in the mask boundary. In our experience from ex-

periments on various MR datasets, this process typically takes

less than five iterations to converge.

D. WM and GM Boundary Estimate

With the modified speed obtained from the geometric and

topological analysis process in Sections IV-B and IV-C, we can

repeat the above steps in Sections IV-A, IV-B and IV-C until no

changes are made to the WM speed. The genus zero estimate

then gives a clean estimate of the WM boundary. For the

example in Fig. 2(a), the cleaned WM boundary after the re-

moval of outliers shown in Fig. 5 is plotted in Fig. 8(a).

In order to reconstruct the GM boundary, we evolve the

cleaned WM boundary outward using the GM speed defined

in (6) under the topology-preserving constraint. To remove

geometric outliers on the GM boundary estimate, the Reeb

analysis is applied to this genus zero surface and the geometric

outlier correction process described in Section IV-B is applied

iteratively. By removing the spurious outliers, we obtain a clean

estimate of the GM boundary. As an illustration, the cleaned

GM boundary corresponding to the example in Fig. 2 is plotted

in Fig. 8(c).

V. SUB-VOXEL ACCURACY WITH ADAPTIVE INTERPOLATION

After removing geometric and topological outliers, we obtain

acleanreconstructionoftheboundarybetweentwotissuetypes.

For the WM surface, it is the boundary between WM and GM.

FortheGMsurface,itistheboundarybetweenGMandCSF.To

achieve sub-voxel accuracy in the final surface reconstruction,

we develop an adaptive interpolation method in this section.

For a tissue boundary between a high intensity tissue

low intensity tissue

, we denote

as the set of rectangular faces and

gular mesh representation of the boundary surface as defined

in Section III-A. The pair

and

the trian-

could be (3, 2) for the WM

boundary or (2, 1) for the GM boundary. For a face

terior voxel neighbor is

is

. We denote

and exterior neighboring voxels of all rectangular faces on the

boundary, respectively.

For each voxel

faces in each of the , , and

of faces for partial volume modeling, we count the number of

faces that the intersection of

in the , ,

direction and denote them as

, respectively. We also estimate locally the image inten-

sity

for the high intensity tissue

sity of voxels classified as

in the

centered at

. Similarly, the image intensity

intensity tissue

is estimated locally as the mean intensity of

voxels classified as

in the same neighborhood. The fraction

of high intensity tissue

contained in this voxel

computed with linear interpolation as

, its in-

and exterior voxel neighbor

as the set of interior and

, its cuboid has two

direction. To calculate the shift

and, i.e.,, have

, and

as the mean inten-

neighborhood

of the low

can then be

if

if

otherwise.

(16)

To account for this partial volume effect, we will shift faces on

the boundary. Let

be the total number of directions that we

will shift faces. The shrink factor in each direction is then

.

For a face

, if the tissue map of its interior voxel

, it needs to move inward according

to the shift factor

with

. If

move outward according to the partial volume shift factor

computed from

denote the coordinate of the point

also denote

as the projection of

face

, which is the nearest point on the face

this face is

computed

, it needs to

. Let

. We

onto the

. The shift for

if

if

if

if

if

if

(17)

where the shift in each direction is multiplied by the corre-

sponding spatial resolution, and divided by the number of faces

that

has in that direction.

Given the shift of all faces in

vertex

as the average shift of its neighboring faces

, we calculate the shift for a

(18)

where

borhood of the vertex

angular faces in this set.

denote the set of rectangular faces in the neigh-

, andis the number of rect-

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522IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 3, MARCH 2013

Letdenote the vector of coordinates of all vertices, and

the shift of all vertices. We compute the final vertex coordinates

with sub-voxel accuracy by minimizing the following energy

function:

(19)

where

Laplacian matrix of the mesh, and

eter. Note that we have chosen the Laplacian instead of the gra-

dientoperatorintheregularizationtermtoavoidlargeshrinkage

effectsfrom thegradient operator.Thesolution of thisquadratic

problem gives us the coordinates of vertices on the smoothed

surface

is the optimized vertex coordinate,is the discrete

is a regularization param-

(20)

By applying the sub-voxel calculation to both the WM and

GM surfaces, we obtain our final solution for cortical surface

reconstruction. As an illustration, we have plotted the surfaces

before and after sub-voxel interpolation in Fig. 8. Clearly we

can see the improved quality in the reconstructed surfaces after

the application of the adaptive interpolation scheme.

VI. EXPERIMENTAL RESULTS

In this section, we present experimental results on 40 sim-

ulated MR images and T1-weighted real MR images of 200

subjects to demonstrate that our method can reconstruct high

quality cortical surfaces on large scale data sets. We also com-

pare our results with surfaces reconstructed with FreeSurfer [5],

[52], a widely used software for cortical surface reconstruction

and analysis in brain imaging research, to demonstrate that our

method can obtain better results with significantly less compu-

tational cost.

The simulated dataset includes a set of 20 simulated MR

images from a publicly available dataset [61], and a set of 20

images we simulated with known gray matter atrophy. The

real T1-weighted MR images used in our experiments are

from two publicly available databases. The first dataset has

100 baseline MR images, from 50 normal controls (NC) and

50 Alzheimer’s disease (AD) patients, from the Alzheimer’s

Disease Neuroimaging Initiative (ADNI) database [62]1. The

second dataset includes MR images of 100 subjects from the

International Consortium of Brain Mapping (ICBM) [63]. The

100 subjects have a wide age range from 19 to 80.

All MR images were first automatically skull-stripped with a

meta algorithm [64] in the LONI pipeline [65]. Using the skull-

1Part of the data used in the preparation of this article were obtained from the

Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.ucla.

edu). ADNI was launched in 2003 by the National Institute on Aging (NIA),

the National Institute of Biomedical Imaging and Bioengineering (NIBIB), the

Food and Drug Administration (FDA), private pharmaceutical companies and

nonprofit organizations, as a $60 million, five year public-private partnership.

The primary goal of ADNI has been to test whether serial magnetic resonance

imaging(MRI),positronemissiontomography(PET),otherbiologicalmarkers,

andclinicalandneuropsychologicalassessmentcanbecombinedtomeasurethe

progression of mild cognitive impairment (MCI) and early Alzheimer’s disease

(AD). Determination of sensitive and specific markers of very early AD pro-

gression is intended to aid researchers and clinicians to develop new treatments

and monitor their effectiveness, as well as lessen the time and cost of clinical

trials.

Fig. 9. Iterative process of outlier removal with Reeb analysis.

stripped images, our system and FreeSurfer automatically re-

constructedtheWMandGMsurfacesontheleftandrighthemi-

spheres in all subjects. For our method, the same set of param-

eters are used:

,

, . For FreeSurfer, the default settings

were used. All computations are carried out on the grid of the

LONI pipeline. In general, our system takes around 2–4 h to re-

constructsurfacesfromeachsubject.ForFreeSurfer,thesurface

reconstruction process typically takes around 10–20 h, and the

whole workflow, including surface labeling, can take 20–30 h.

,,,

A. Qualitative Comparisons

As an illustration, we first present in this experiment a de-

tailed comparisonbetween our method and FreeSurferusingthe

same MR scan plotted in Fig. 2. This is an MR image of an AD

patient from the ADNI dataset. Representative examples from

both the ADNI and ICBM datasets will then be presented to

demonstrate that our method can reconstruct surfaces that better

preserve the integrity of cortical anatomy.

For the MR image of an AD patient, we applied our method

and FreeSurfer to reconstruct the WM and GM surfaces on both

hemispheres. Computationally our method took around 3 h and

is much more efficient than FreeSurfer, which took over 10 h.

BothourmethodandFreeSurfersuccessfullyreconstructedcor-

tical surfaces with genus-zero topology. The unified outlier re-

moval processof our methodtookaround20minforeachhemi-

sphere. The topological correction process in FreeSurfer took

more than one and half hour for each hemisphere. As a more

detailed illustration of the Reeb analysis process for outlier re-

moval, we plotted in Fig. 9 the number of nodes in the Reeb

graph, the number of geometric outliers, and the number of

total handles and tunnels at each iteration of our algorithm.

With the increase of iteration, we can see the complexity of

the Reeb graph decreases with the number of outliers in the

boundary. For this image, it took three iterations for the uni-

fied outlier correction algorithm to convergewhen all geometric

outliers were removed and no more cuts to be made. After that,

a topology-preserving evolution was applied to reconstruct a

genus-zero boundary. The adaptive interpolation scheme devel-

oped in Section V was finally applied to reconstruct a smooth

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Fig. 10. Comparison of WM surfaces reconstructed by our method and FreeSurfer. Surfaces are colored by their mean curvature and plotted in both lateral and

medial views. (a), (d): Our result. (b), (e): FreeSurfer result. (c), (f): Intersection of an image slice with circled regions in (a), (b) and (d), (e), respectively. Red

contour: Our result. Blue contour: FreeSurfer result. Green arrows in (c) and (f) highlight the differences.

and accurate representation of the cortical surface. As an illus-

tration, we plotted the intersection of the image slice shown in

Fig. 2(a) with the WM and GM surfaces reconstructed by our

method in Fig. 2(g) and (h). From the results we can see that our

method is able to robustly reconstruct the WM and GM bound-

aries even in the presence of tissue degeneration such as the

white matter degeneration on the right hemisphere of this AD

patient.

To compare the results from our method and FreeSurfer,

the left hemispherical WM surfaces reconstructed by both

methods are plotted in Fig. 10. The two WM surfaces have

similar number of vertices because they are both derived from

the WM mask boundary. The FreeSurfer WM surface has

137309 vertices, and our WM surface has 138130 vertices.

In regions highlighted in the dashed circles in Fig. 10, we can

see that our method produces more complete reconstruction of

the boundary, which is better illustrated with the intersections

of surfaces and two axial slices shown in Fig. 10(c) and (f).

The left hemispherical GM surfaces reconstructed with our

method and FreeSurfer are plotted in Fig. 11. Because the

FreeSurfer GM surface is obtained by deforming the WM mesh

to the GM boundary, it has the same number of vertices as

its WM surface and typically has irregular triangles. On the

contrary, our method follows the boundary between GM and

CSF to generate a more uniform mesh representation of the

GM surface. The GM surface reconstructed with our method

here has 145136 vertices, which has slightly more vertices

than the WM surface. This is natural in order for the GM

surface to have the same level of vertex density and accuracy

as the WM surface because it has a larger area than the WM

surface. From the highlighted mesh structures of four ROIs

in Fig. 11, we can see that FreeSurfer produced broken gyri

in these regions while our method is able to generate more

Fig. 11. Comparison of the GM surface reconstructed by our method and

FreeSurfer. (a), (b): Lateral and medial views of our result. (c), (d): Lateral

and medial views of FreeSurfer result. The mesh structures of four highlighted

ROIs are plotted to demonstrate differences between these two results.

complete reconstruction at these gyri. The intersections of the

GM surfaces reconstructed by our method and FreeSurfer with

three sagittal slices are shown in Fig. 12, which illustrate that

our surface can better capture deep sulcal regions. This shows

that the localized outlier detection and filtering approach in our

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524IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 3, MARCH 2013

Fig. 12. Intersection of image slices with GM surfaces reconstructed with our method (red contours) and FreeSurfer (blue contours). Green arrows point to

locations where our surface can better capture deep sulcal regions.

Fig. 13. Comparison of cortical reconstruction results on four ADNI and four ICBM cases. For each case, the result from our method is plotted on the top, and

the result from FreeSurfer is plotted on the bottom. Different views are selected for different subjects to better illustrate differences in regions highlighted with

dashed circles.

method is able to avoid shrinkage and produce a more accurate

surface representation in these regions.

To further demonstrate that surfacesgenerated by our method

are able to better represent the integral anatomy of the cortex,

we have plotted in Fig. 13 more representative reconstruction

results from ADNI and ICBM data. For each subject, the left

hemispherical GM surfaces reconstructed by our method and

FreeSurfer are plotted with differences highlighted in dashed

circles. Compared with results from our method, we can see

that FreeSurfer results failed to reconstruct a complete gyrus

in the regions enclosed in the dashed circles. This shows that

our method is able to produce a more complete representation

of cortical anatomy.

B. Atrophy Detection on Simulated MR Images

In this experiment, we will compare our method and

FreeSurfer on the detection of simulated atrophy in MR images

where the ground truth of WM, GM, and CSF tissue maps are

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Fig. 14. Comparison of our method and FreeSurfer on a simulated image with

known WM tissue map. (a) The WM surface reconstructed by our method. (b),

(c) A zoomed view of the circled region in (a) before topology correction and

after final reconstruction. (d) The WM surface reconstructed by FreeSurfer. (e),

(f): A zoomed view of the circled region in (d) before topology correction and

after final reconstruction. (g), (h): Intersection of reconstructed surfaces with

the true WM tissue map. Red contour: Our method. Blue contour: FreeSurfer.

known. By matching reconstructed cortical surfaces with the

underlying tissue boundaries, we can analyze the performance

of different topology correction strategies in our method and

FreeSurfer. Using the reconstructed WM and GM surfaces,

we can calculate cortical thickness and compare quantitatively

the ability of these two methods in detecting sub-voxel tissue

atrophy.

To simulate longitudinal brain atrophy, we first downloaded

from BrainWeb [66] a set of 20 simulated MR images of normal

subjects with known WM, GM, and CSF tissue maps [61] and

used them as the baseline images. All images have an isotropic

spatial resolution of 1 mm. Using these tissue maps, we simu-

lated a 0.1 mm GM atrophy everywhere on the cortex by mod-

ifying intensities in the MR images. For each voxel with more

than10%GMtissueandontheboundaryofGMandCSF,wein-

troducedasub-voxelatrophybysubtractingitsintensityby10%

of the difference between the average GM intensity and CSF

intensity in the image because we assumed the lost GM tissue

in this voxel will be replaced by CSF. By applying this proce-

dure to all baseline images, we obtained a set of 20 images with

simulated GM atrophy, which we denoted as follow-up scans of

these 20 subjects.Both our method and FreeSurfer were applied

to the 40 simulated images to reconstruct the WM and GM cor-

tical surfaces.

From the examples shown in Figs. 10 and 13, we see various

cases that our method is able to generate a more integral recon-

struction of the superior temporal cortex than FreeSurfer. With

simulated images, we have the opportunity to demonstrate this

on images with known tissue boundaries. In Fig. 14(a) and (d),

Fig. 15. Projection of FreeSurfer gyral labels onto the GM surface recon-

structed by our method.

we plotted the left WM surfaces reconstructed from one of the

baseline images with our method and FreeSurfer, where we can

clearly see that our result has a more integral representation of

the superior temporal cortex. For both methods, we extracted

the WM boundary before topology correction to investigate the

causeofthisdifference.InFig.14(b)and(d),weplottedthecor-

responding WM boundary without topology correction in the

region highlighted by the dashed ellipses in Fig. 14(a) and (d),

respectively. In both cases, we can see a hole is clearly present

on the superior temporal gyrus. Different correction strategies

were adopted by these two methods: our method chose to fill

the hole, while FreeSurfer decided to cut it open. As a result,

different surface reconstruction results were obtained as shown

in Fig. 14(c) and (f). The intersection of the reconstructed sur-

faces with two sagittal slices of the underlying true WM tissue

map were plotted in Fig. 14(g) and (h), where our result was

plotted in red and the FreeSurfer result was plotted in blue. As

highlighted by the green arrows in these two pictures, we can

see that our result provides a more faithful representation of the

underlying tissue boundary at the superior temporal cortex.

To compare the performance of our method and FreeSurfer

in the detection of simulated longitudinal atrophy, we parcel-

lated all cortical surfaces into gyral regions and tested for GM

thickness changes in each region. In this work, we used the

gyral labels on GM surfaces generated by FreeSurfer for a

localized comparison. In order to project the FreeSurfer label

onto our surface, we aligned both surfaces in the image space.

For each vertex on our surface, we found the nearest vertex on

the FreeSurfer surface such that the normal directions of both

vertices have positive inner products and pulled back the label

on this vertex. As an illustration, we plotted in Fig. 15 the labels

on the GM surfaces generated by FreeSurfer and our method as

shown in Fig. 11. We can see that we have successfully mapped

the gyral labels onto the GM surface generated by our method.

At each point of a GM surface, we used its distance to the

closest point on the WM surface as the measure of thickness

[7], [67]. Numerically, we first computed the signed distance

function of the WM surface with the fast marching method

[68], and then calculated the thickness at each vertex of the GM