Computational Atlases for Orthopedic Applicaitons

Thomas B. Sebastian, Joseph J. Crisco, Philip N. Klein, Benjamin B. Kimia

Computational Atlases

framework to compare similar anatomic structures

compute the typical or "average" shape of a collection of anatomic structures

quantify free-form variations from the mean

Question I:

Crisco et. al. have shown that there are significant differences in certain carpal kinematic variables between males and females

are the differences limited to size differences, or are there differences in shape and geometry?

to answer these questions we need to first compute an average shape of carpal bones

profile of radius in the sagittal direction for 10 male and 10 female subjects

profile of radius in the coronal direction for 10 male and 10 female subjects

Question II:

Thumb CMC joint in women have a greater predisposition to osteoarthritis than joints in men

Van Mow et. al. [JHS 98] have compared curvature maps to show that the thumb CMC joint in women is less congruent than in men

They hypothesize that this difference in congruence makes women more predisposed to osteoarthritis than men

compare the shape of an anatomic structure to a population of similar structures

correlating abnormal deviations from the average anatomy with diseased states

Previous Approaches

Bookstein [MIA 96/97] uses landmark-based thin-plate spline method

method is sensitive to the landmarks

Rangarajan’s recent work based on matching of landmarks,

Shape as a cloud of point

Christensen et. al. [PNAS 93, IP 96] set the image in an extrinsic coordinate system, which is deformed

deformation is done in extrinsic coordinates and not in intrinsic shape coordinates

deformation is not symmetric

Pizer et. al. [IPMI 97, TMI 99] use a graph of boundary and medialness points for shape representation shape differences are characterized as squared distance of link properties

Model generation needs user assistance

Graph matching does not handle toplogical changes

Taylor et. al. [BMVC 96] represent the shape as a mean (average) shape plus a set of linearly independent variation modes

model is derived using a training set

may not adequately represent abnormal states

Our Approach

based on shape matching

the shape is represented by its bounding curve

compute the average curve in a two step process

find the optimal correspondence between two curves

construct the average curve by averaging the corresponding segments on the curves

Curve Matching

Curve Matching Methods

Duncan et. al. [CVPR 91], Cohen et. al. [ECCV 92], Younes [JAM 96] use an energy minimization method to track anatomical structures

basic premise is to match high curvature points while maintaining a smooth displacement field elsewhere

Given two curves C(s) = (x(s),y(s)), s [0,L] and C'(s')=(x'(s'),y'(s')), s' [0,L'],

for a mapping g : [0,L] --> [0,L'], g(s) = s'

with a cost of alignment , e.g., Cohen's defines

Find optimal alignment

The distance between the curves is

Drawback of Cohen's/Younes' goodness measure

Cohen's and Younes's approach are not invariant to rotation

In addition, Cohen's approach is not invariant to sampling

New goodness measure

we assume that goodness of the optimal match is sum of the goodness of matching subsegments

hence, consider the cost matching two infinitesimal curve segments of C (AB) and C' (A'B')

as intrinsic properties are to be compared, can align start points and their tangents

cost of matching is defined as

the resulting functional is given by

first term penalizes stretching, while second penalizes bending

Drawback of the formulation

inherently asymmetric [Tagare et.al. ICCV95]

a single point on the first curve (C) cannot be mapped to a segment of second curve (C')

does not allow for deletions on the second curve (C')

difficuly is that alignment is represented by a uni-valued function (g)

A Solution to the asymmetry problem

consider the alignment as a pairing of particles one on each of the two curves

then the alignment can be represented in terms of a pair of functions h, h' which relate arclengths of C and C' to a new parameter

when h is invertible,

when h is not invertible, is not defined

Tagare et.al. use this approach with the optimization depending explicitly on two functions

two functions introduce a superfluous degree of freedom

different traversals h,h' give rise to the same alignment, but can represent the same pairing

Our Approach

consider the notion of an alignment curve, defined using coordinates h and h' as

main advantage of using the alignment curve is that it can be represented by a single function, instead of two

alignment curve is specified by the angle between its tangent and x-axis

coordinates of the alignment curve can be obtained by integration

is constrained to lie in

alignment between the curves C, C' is fully represented by the single function

the goodness measure can be expressed in terms of

Properties of this formulation

It is symmetric

This formulation can handle deletions on both curves

A segment of first curve mapped to a single point on the second curve

A segment of second curve mapped to a single point on the first curve

Implementation

functional minimization is done using dynamic programming

let C be sampled by and C' by

let be the cost of matching segments and

discrete version of the distance function is given by

we discretize to nine values

· it takes ~1 sec to match 2 curves sampled with 200 points

Results

matching profiles of the radius bone

matching the outline of metacarpal bones

matching outline of a pair of spine vertebra

Curve Averaging

Goal

Given N curves , find the average curve that minimizes

Difficulties

· Simultaneously computing and the optimal alignments is computationally intractable

· The alignment is not transitive

A Solution

Choose one of the curves (say C1) as the reference curve

Find the optimal alignment between the reference curve and the other curves

Find the average curve by averaging the corresponding segments

To average corresponding segments a linear model is used
As we are interested in averaging intrinsic properties of AB and A'B' we align the start points and their tangents

End points can be expressed

End point of the average curve is computed as

Results

Experimentally verified that the averaging is invariant to choice of reference curve

Results of averaging metacarpal outlines

Results of averaging corpus callosum outlines

We thank Jerome Sanes and James Eliassen, Brown University and Christos Davatzikos, John Hopkins University for providing us with corpus callosum data