Some Methods for Modeling Cortical Surface

Yuai Hua

2010.10.29

What is a cortical surface like?

Gyral region, sulcal region, gyral crest, sulcal fundi

Sulcal Region

Gyrus Crest Line

Gyral Region

Sulcal Fundi

Sulcal basin & Gyral basin

Sulcal Basin

Gyral Basin

(cross-section)

What will I present ?

• Cortical sulcal parcellation

• Cortical fundi extraction

• Cortical gyral parcellation

• Cortical sulcal bank segmentation

• Gyral folding pattern analysis

Part A. Cortical sulcal parcellation

Key Technique:

Principal direction flow field tracking

method

(proposed by Gang Li et al.)

Goal: Finding sulcal region

sulcal basin&

Some Basic Concepts

----- the maximum and minimum values of

curvatures at a point p on a surface.

----- the vectors along which the curvatures

are principal.

Principal curvatures

Principal directions

principal directions

Triangulated cortical surface

sulcal regions & sulcal basins?

How to find

Stages:

(1) Estimate principal curvatures and

principal directions at each point;

(2) Finding sulcal regions;

(3) Finding sulcal basins.

Step1. Calculate the normal vectors of each triangle face.

X

X1

X2

X5

X4 X3

X6

(1) Estimate principal curvatures and principal directions at each point.

• Weingarten Matrix is a symmetric matrix.

• Its eigenvalues are the principal curvatures.

• Its eigenvectors are the principal directions.

Weingarten Matrix

Step2. Calculate Weingarten Matrix in each triangle face.

Step4. Calculate the eigenvalues and eigenvectors of each Weingarten Matrix. Those are the principal curvatures and principal directions.

Step3. Calculate Weingarten Matrix at each vertex by weighted averaging its adjacent faces.

X

X1

X2

X5

X4 X3

X6

Only the

maximum principal curvatures

and

its

corresponding principal directions

are adopted.

Calculating the directional derivative of

maximum principal curvature along the

corresponding principal direction and ensuring

it decreases by choosing appropriate principal

direction.

Keeping in mind……

fundi

Principaldirection

• The principal direction points towards the s

ulcal fundus from the gyral crest;

• The principal curvatures are large positive

and negative values at gyral crown and sulc

al fundi.

Thus

(2) Finding sulcal regions

),,,( 21 NxxxX

Letn = the number of the total vertices on the cortical surface

.regiongyralatobelongsif,1

region;sulcalatobelongsif,0

i

ixi

i

yi

vertexat value

curvature principal maximum the

),,,( 21 NyyyY

In order to segment cortical surface into sulc

al regions and gyral regions,

we should solve

)(maxarg^

YXPXX

Problem : )( YXP is unknown

ii xyKnowing: is a normal random variable

)(XP

&

can be estimated.

Hence, according to the Bayes theory and a

special method (proposed by Zhang et al., 2

010), we can estimate X by solving

)()(maxarg^

XPXYPXX

During this process, hidden Markov random

field model, expectation maximization method

and iterated algorithms are used.

So far, a cortical surface is segmented into a

series of sulcal regions and gyral regions.

(3) Finding sulcal basins

Idea: following the maximum principal directions from a gyral crown region until to the sulcal fundus. The vertices that converge to the same fundus are grouped together, and these vertices form a sulcal basin. Thus the cortical surface are segmented into different sulcal basins.

fundi

Principaldirection

Idea: at a sulcal fundus and gyral crown, the pro

duced flow field should be close to the original dir

ection field, and at flat areas, it should vary

smoothly.

Step1 Estimating flow field

Problem: at a flat cortical region, the two

principal curvatures might be very small, so

we may not find the exactly maximum

principal direction.

Method: Principal direction flow field diffusion

SXnXV

dXXPXVXCXVXV22

0)()()()()(maxarg)(

n : normal vector

Estimating flow field V(X)

:)(XC maximum principal curvature

:)(XP maximum principal direction

: weighting parameter

: gradient operator

where

Using calculus of variation to solve the equation

Step2 Sulcal basin segmentation

Method: Principal direction flow field tracking

—Searching for flow trajectories

X

),(XV

Given a vertex on a flow trajectory with

is calculated as

the principal direction

i

i

X XX

XXXVX

i

),(arccosminarg

:iX Xthe one-ring adjacent vertex of

.

the next vertex

X6

V(X)

X5

X4

X1

X3

X2

X

=x′

stops,0

;continoues,0)(),( XVXV

V(X)

X

V(X′)

V(X)

X

V(X′)

x′

The region at which the flow field tracking pr

ocedure stops should be a fundus.

Thus every vertex flows to a fundus.

Those vertices flow to the same fundus are

grouped together as a sulcul basin.

Cortical sulcal parcellation is over

Part B. A pipeline for cortical fundi extraction

(Proposed by Gang Li et al)

Step1 Estimating curvatures and curvature derivatives

Step2 Detecting sulcal fundi segments

:maxc The maximum principal curvature

:minc)( minmax cc

The minimum principal curvature

,pmax :pmin

,

The principal directions.

:pcd

max

maxmax

Directional derivative

Criterion for fundus point :

0p,0,0max

maxmaxmax

ddc

Fundus point

Procedure for fundi segmenting

(1) Procedure for finding fundi points:

in the cortical surfaceFor each triangle

321 ,, vvv(three vertices are )

(2) Connect the adjacent fundi points to form fundi segments:

Candidate fundi segment: if there is any

candidate fundi point in it;

Strict fundi segment: if there is no candidate

fundi point in it;

Two types of

Provisional fundi segments

b) Expanding again

For every vertex with negative maximum curvature

in a fundi curve, connect itself with its adjacent

vertex which is in another fundi curve, obtain a new

fundi curve.

(3) Linking sulcal fundi segments

a) Starting from a strict fundi segment, adding the adjacent segments to it, and go on, a fundi curve will be obtained.There may be more than one fundi curve.

c) Pruning the fundi curves less than three

segments.

The remain fundi curves may include some very

short ones.

How to tell the different kinds of the short fundi curve?

Two types of short fundi curves: interrupted and inherent.

Next, it is necessary to connect those interrupted short fundi curves to the long ones.

d) From each endpoint of the extracted fundi curve,

searching the geodesic region to find whether there

exists another fundi curve in the region. If any, conn

ect the endpoints to the newly found sulcal fundi cu

rve.

e) Smoothing the extracted sulcal fundi curves.Due to the numerical error, the extracted sulcal fundi curves may exist sharp bumps, it should be smoothed. Method : minimizing the geodesic distance between the endpoints or junction points of each piece of the extracted sulcal fundi.

Part C. Cortical Gyral Parcellation

Technique:

Using probabilistic atlas and graph cuts

(proposed by Gang Li et al.)

• Each gyral patch is a part of gyral basin

• Each gyral patch is bounded by adjacent sulcal f

undi and interrupted at junctions of gyral basins

• Each gyral patch belongs to only one gyral basin

• Each gyral basin is composed of one or more gy

ral patches.

Characteristics of a gyrus:

Every kinds of gyral patchs have been labeled b

y experts in the form of Probabilistic Atlas.

What we know:

Probabilistic atlas is a series of maps of

human brain anatomic regions. These maps

were produced from a set of whole-head MRI.

Each MRI was manually delineated to identify a

set of 56 structures in the brain, most of which

are within the cortex.

• Each type of region has a label.

n: number of gyral patches

:p

:k

:)( pR

:)(kR

:)( kpR kp

:)(

)()(

pR

kpRkhp

p k

p-th gyral patch in sulcal surface

k-th gyral structure in the Probabilitic Atlas

Area of k-th gyral structure in the P-Atlas

Area of the intersection

Likelihood of belonging to

.

Area of p-th gyral patch in sulcal surface

:pl The label which the p-th gyral patch in sulcal surface be assigned corresponding to the gyral basin in the P-Atlas

:)( ipc ip

0,0

,0,1)(

x

xx

:

)12

))()((

exp(),(

,

,

qp

qp

qcpc

qpw

),( qpw

Maximum principal curvature at vertex

in gyral patch

.

Set of all neighboring vertices between two neighboring gyral patches.

represents the weight between two neighboring gyral patches.

)1(),(),( qpqpqp llqpwllV

qpqpqp llllV if0),(

Nqp

qpqpp

pp llVlhE,

, ),()(ln

)),()(ln:(

,,

Nqp

qpqpp k

p llVkhEOriginal

:

Elp

p minarg

.

N: the set of neighboring gyral patch pairsan adjust parameter.

by using graph cuts method.

The cortex is segmented into different gyral basins

Part D. Cortical sulcal bank segmentation

Sulcal bank: Each side of a sulcal basin.

• A sulcal basin has two opposite sulcal banks.

Goal: Segment a sulcal basin into two sulcal banks.

Technique: Graph partition (proposed by Gang Li et al.)

Step1: Segment a cortical surface into sulcal basins.

:),( 11 EVG The triangular mesh of a sulcal basin

Procedure

Step2: Rough sulcal bank segmentation

ji,

)nn1(2

1),( ji jiwn :n,n ji

),( jiwn

:, jiAngular similarity between two vertices

:

unit normal direction)

will be large if

otherwise, small.

(

are in the same

sulcal bank;

:, ji

),(),(),( jidjidjiw ged

:),( jide

:),( jid g

),( jiwdji,

Distance similarity between two vertices

: Euclidean distance;

geodesic distance (the shortest path

will be large if

sulcal bank; otherwise, small.

connecting two vertices along the

triangular mesh of a sulcal basin)

are in the same

:, ji

),()1(),(),( jiwjiwjiw nd

10

),( jiw ji vv ,

Similarity between two vertices

will be large if

otherwise, small.

( is a weight parameter)

are in the same sulcal

bank;

BvAu

vuwBAcut,

),(),(

VvAu

vuwVAassoc,

),(),(

VvBu

vuwVBassoc,

),(),(

),(

),(

),(

),(),(

VBassoc

BAcut

VAassoc

BAcutBANcut

),(minarg, BANcutBAA

,

Rough graph partition:

Then set

Firstly, using Normal cuts method to divide the sulcal baisn into two opposing sulcal banks

A and B

Technique: construct a energy function and minimize it.

Step3: Fine sulcal bank segmentation

Goal: making the boundary clearer.

Cortical sulcal bank segmentation is over

(proposed by Kaiming Li, et al)

Part E. Gyral folding pattern analysis

Gyral folding patterns:

Gyral crown

Gyrus

Sulcus

Hinge Line

Hinge

Different parts of cortex

Five different parts of cortex

Blue: sulcus basin

Red: gyrus crown

Yellow: sub gyrus crown

Green： Central area

Light blue: sub sulcus basin

Goal: 1. segment cortex into five different classes

2. detect the hinge.

Firstly, build a coordinate system as follows:

O: Any vertex on a sulcal surface

N: The normal direction at O

Ro: A randomly selected vector on tangent plane

Then build a polar coordinate system in tangent Plane, then build a Cartesian-Polar Coordinate System.

Cartesian-Polar Coordinate System

N

P

Rα

RO

C(α,x, y)

αy

x

o

A profile

N

iif

Nf

1

1

gyrus crown: 1; sub gyrus crown: 2central area: 3; sub sulcus basin: 4sulcus basin: 5

fi represents the value of i-th point, then calculate

the value of the profile:Set some threshold, the gyral surface can b

e divided into five regions and the hinge point

s can be detected.

Assign each class a value:

For each profile, suppose there are N points on it,

Procedure

ReferencesGang Li et al. ,Automatic cortical sulcal parcellation based on surface principal direction flow field tracking. (http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WNP-4VXDTX7-1&_user=130907&_coverDate=07%2F15%2F2009&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1522441145&_rerunOrigin=scholar.google&_acct=C000004198&_version=1&_urlVersion=0&_userid=130907&md5=6abccc308c9b83e504db8c4f51819d29&searchtype=a)

Gang Li et al. ,An automated pipeline for cortical fundi extraction.

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W6Y-4Y9XM1S-1&_user=130907&_coverDate=06%2F30%2F2010&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1522445802&_rerunOrigin=google&_acct=C000004198&_version=1&_urlVersio

n=0&_userid=130907&md5=74a63ac5ec24e920f55e874ba48d64ea&searchtype=a)

Gang Li et al. , Automatic cortical gyral parcellation using probabilistic atlas and graph cuts.

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WNP-4VXDTX7-1&_user=130907&_coverDate=07%2F15%2F2009&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1522455825&_rerunOrigin=scholar.google&_acct=C000004198&_version=1&_urlVersion=0&_userid=130907&md5=1d5aa287fe1024dddbf3c85e02287e05&searchtype=a)

http://en.wikipedia.org/wiki/Principal_curvature

http://planetmath.org/encyclopedia/WeingartenMatrix.html

http://www.loni.ucla.edu/Atlases/LPBA40

http://en.wikipedia.org/wiki/Cut_(graph_theory)

Gang Li et al. , Cortical sulcal bank segmentation via geometric similarity based graph partition.(http://www.springerlink.com/content/l5213t641r587l83/)

Kaiming Li et al. , Gyral folding pattern analysis via surface profiling.(http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2915584/)

Thank you