IPAM Workshop on Image Processing for Random Shapes: Applications to Brain Mapping, Geophysics and Astrophysics UCLA May 21-25, 2007 Geometric Shape Descriptors.

IPAM Workshop on Image Processing for Random Shapes: Applications to Brain Mapping, Geophysics and Astrophysics

UCLA May 21-25, 2007

Geometric Shape Descriptors for Shape Analysis and Sulcal Classification of MRI Data

Monica K. HurdalDepartment of Mathematics,

Florida State University, [email protected]

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

Motivation• Increasingly, diseases that affect brain function are

being correlated to structural brain changes.• Classification and identification of differences in brain

anatomy (i.e. differences in shape) due to disease is difficult due to individual variability in folding patterns.

• Parcellation and labeling of cortical features are critical for brain cartography and conveying information to compare subjects or populations.

• Parcellation and labeling are manually intensive processes confounded by variability in folding patterns.

• Methods are needed to quantify properties of the shape of the cortical surface on a global as well as local level.

Brains are Heterogeneous

Slide from Duke course

Important Questions• How much variation is there in the size, shape

or location of an individual healthy brain structure?

• To what extend does function and disease depend on the anatomy of the brain?

• Potential of using structural abnormality as a measure of development, normal aging, and disease

Anatomical LocalizationSulci and Gyri

gray matter (dendrites & synapses)

white matter (axons)

FUNDUS

BA

NK

SULCUS

GY

RU

SS

ULC

US

gray

/whi

te b

orde

r

pial

sur

face

FISSURE

Source: Ludwig & Klingler, 1956 in Tamraz & Comair, 2000

Shape Analysis, Curve Shape Descriptors & Features

• Properties of shape curves and surfaces in 3D can be described by invariant features.

• Sulcal or gyral curves that fall in the same class exhibit geometric resemblance, but significant intra-class variations have been observed as well.

• Thus, starting with a collection of shape descriptors, it is essential to develop algorithmic procedures to extract (learn) the features that can be used effectively for parcellation, labeling and disease quantification.

Motivation for the Measures• We notice that the brain sulci

are a set of space curves. • Interested in geometric

invariant features of polygonal curves such as writhe, mathematical moments, extremal length.

• Methods are inspired by generalized Gauss Integrals.

• Methodology proposed is concerned with crossings seen in planar projections of curves.

• Thickness and length are also considered.

Centralsulcus

Calcarinesulcus

Superior Temporalsulcus

Superior Frontalsulcus

Centralsulcus

Calcarinesulcus

Superior Temporalsulcus

Superior Frontalsulcus

Precentralsulcus

Geometric Shape Descriptors

• Generalized Gauss Integrals

• Thickness and Ropelength

Crossing Signs

At a crossing point c of an oriented regular diagram of a knot (or curve), we have two possible configurations. Either sign(c)=+1 or sign(c)=-1.

Projected Writhe -1

-1

-1

Wr(γ, ) = -3where γ =closed space curve projected into the plane in the direction of the vector

Writhing NumberIn general, almost all projections of a curve into the plane in the direction of the vector will be regular, where at most two projected edges intersect, and that intersection is transverse.

For any regular projection, each crossing c can be assigned a value of according to the right hand rule.

The Writhe (or projected writhing number), Wr(), is defined as the average over all projections of the projected writhing number

The projected writhing number of a closed space curve γ, denoted by Wr(γ), can be calculated using generalized Gauss integrals.

).,( Wr

Calculating Writhe using a Gauss Integral

Given a simple closed curve , the writhe is defined by the double integral:

where D is the diagonal of .

D

dtdttt

ttttWr

\

21321

2211 ,|)()(|

)(')),()((),('

4

1)(

• What about polygonal curves?

dWrS ),(

4

12

Polygonal Open Curves

Source: Knotplot by R Scharein

Consider an Oriented Polygonal Curve

• For any pair of edges ei, ej, consider the set of points u on the unit sphere such that ei and ej are crossing when looking in the direction of the vector u.

WritheThe writhe can be computed as:

where w(i, j) is the area spanned by ei and ej .

Since w(i, j) = w(j, i),

,),(4

1)(

,0

ji

nji

jiwWr

nji

jiwWr0

),(2

1)(

I(1,2) & I|1,2|

We set

Here w(i1,i2) is the contribution to the writhing number coming from the i1th and the i2th line segments. w(i1,i2) is equal to the probability from an arbitrary direction to see the i1th and the i2th line segment cross, multiplied by the sign of this crossing. Thus, geometrically this notion of writhe number is still the projected writhing number averaged over all projections.

Taking the absolute value inside the sum one gets the average crossing number:

.),(210

21)2,1(

Nii

iiwWrI

.|),(|210

21|2,1|

Nii

iiwI

Higher Order Writhing Numbers: A Family of Measures

Nii

ii

iiwiiwI

43

2104321)4,3)(2,1( ),(),()(

Niiii

iiwiiwI

43210

4231)4,2(|3,1| ),(|),(|)(

where N is the number of vertices for the polygonal curve .

Reference: P. Rogen, B. Fain;2003 PNAS, Vol 100 No. 1, 119-124

By combining these numbers, we can make a whole set of structural higher order writhing measures, e.g.

Importance

• Plane curves A and B possess the same writhe and average crossing number.

• The higher-order invariants, however, distinguish the two curves.

Reference: P. Rogen, B. Fain;2003 PNAS, Vol 100 No. 1, 119-124

A set of measures given by:

divided by the length L of the curve were computed.

)6,5)(3,2)(4,1(

)5,4)(6,2)(3,1()6,4)(5,2)(3,1()6,5)(4,2)(3,1(

)5,4)(6,3)(2,1()6,4)(5,3)(2,1()6,5)(4,3)(2,1(

|3,2||4,1||3,2|)4,1()3,2(|4,1|)3,2)(4,1(

|4,2||3,1||4,2|)3,1()4,2(|3,1|)4,2)(3,1(

|4,3||2,1||4,3|)2,1()4,3(|2,1|)4,3)(2,1(

|2,1|)2,1(

,,,

,,,

,,,,

,,,,

,,,,

,,

I

III

III

IIII

IIII

IIII

II

Geometric Shape Descriptors

• Generalized Gauss Integrals

• Thickness and Ropelength

Thickness and Ropelength

• The thickness of a knot is the radius of the largest embedded normal tube around the knot.

• The ropelength is the quotient of its length and its thickness.

Source: Knotplot by R Scharein

Methods: Cortical Surface Extraction &

Sulcal Curve Generation• Extract region of interest from MRI volume (e.g. cortical

hemisphere)• Create topologically correct triangulated surface of

neural tissue from MRI volume • Compute principal curvatures• Consider curves which are oriented polygonal curves in

Euclidean 3-space: trace lines of principal curvature using dynamic programming

• Compute shape features: writhe and higher order writhing numbers, thickness, ropelength

Data: Cortical Hemispheres• 15 normal right-handed subjects from a static force

experiment (7 females, 8 males; ages 19-45)• Topologically correct cortical left and right

hemispheres reconstructed from MRI data

• Five sulcal curves tracked on each pial surface hemisphere: calcarine, central, precentral, superior frontal, superior temporal sulci

Methodology

(0.3,

0.5,

…

…

…

0.7)

Brain surface Sulcus paths Vector of Geometric Measures

Centralsulcus

Calcarinesulcus

Superior Temporalsulcus

Superior Frontalsulcus

Subject 2

Subject 1 Writhe Invariants

Notice that the orders of magnitude of most invariants are similar, yet the values of the invariants are different.

This suggests two facts:1. It might possible to identify a specific sulcus for labeling

2. It might be possible to classify populations.

Classification Methods & Results• A vector of 24 features composed of scaled Gauss

integrals, ropelength, thickness, length were used as the basis for classification.

• Experiment 1: Sulcal curves were separated into two classes: i) left and ii) right hemisphere (5 sulcal paths per hemisphere; 15 curve sets from left hemisphere, 15 curve sets from right; 75 curves per class)

• Experiment 2: Sulcal curves separated into two classes: i) male and ii) female subjects (sulci from one hemisphere only; 8 males, 7 females)

• Discrimination via Multiple Discriminant Analysis (MDA): – The projection maximizes the distance between the means

of the classes while minimizing the variance within each class.

Left-Right Hemisphere Distinction

• Data divided into two classes: Left and right• 5 sulcus paths per hemisphere• 15 curve sets from left hemisphere, 15 from

right

• Then we compute the vectors of geometric measures, and project the values in the plane using MDA

Left-Right Hemispheres

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2

MDA 1

MD

A 2

.

Left

Right

Male-Female Distinction

• We divided the data into two classes: Male and Female.

• We consider only sulci from one hemisphere (5 sulci per brain).

• Compute the vectors of geometric measures, and project by using MDA.

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

-0.25 -0.24 -0.23 -0.22 -0.21 -0.2 -0.19 -0.18 -0.17

MDA 1

MDA 2

.

Female

Male

Similar results obtained for data from the left hemisphere

Male-Female, Right Hemisphere

Classification Using Machine Learning Methods

• MDA results suggest classification scheme possible

• WEKA: open source machine learning software• Left-Right Hemisphere: 50% considered for

training; a number of methods classified 100% correctly (including Ridor, Lazy LWL)

• Male-Female: 75% considered for training; 100% accuracy using Voted Perceptron; 75% accuracy using Ridor; as amount of training decreases, percentage of accuracy drops

• Other scenarios, including sulcal curve classification have been successful (see poster)

Conclusions

• We used a novel set of geometric measures that seem to capture the anatomy of the brain.

• Originally for closed curves, these measures have been extended to open and broken curves.

• We perform a number of experiments to test the performance of the Gauss integral measures, length and thickness.

Conclusions

• We have a method that could allow a criteria for separating their functional information on the basis of the gross neuroanatomy parameters.

• Our method get us one step closer to identify classes of brain anatomy and categorizing function and behavior.

Future Directions• Future direction of this research is to refine the features

in order to make discrimination more sensitive, e.g. labeling of sulci on the same hemisphere.

• Develop methods that determine which features and algorithms are optimal for classification. These methods need to be tailored to specific applications.

• Evaluate robustness of features due to alternate surface reconstructions, other sulci or gyri.

• With these invariant features, it might be possible to identify specific sulci for labeling.

• Use these measures to differentiate between specific populations.

AcknowledgementsJuan Gutierrez, Aaron Kline, Christian Laing, Debbie Smith (Math,

FSU)David Rottenberg (Radiology, Neurology, U. Minnesota) for 15

subject MRI data

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

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