Goals of Computational Atlases

• Compute the typical or "average" shape of a collection of anatomic structures

• Quantify free-form variations from the mean

Motivation

Question I:

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

• are the differences are limited to size differences?

• or are there differences in shape and geometry?

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 anatomy of a medical structure to a population of similar structures

• correlating abnormal deviations from the average anatomy with diseased states

Overall Approach

• based on matching shapes that are represented by their bounding curves

• compute the average curve in a two step process

• find the optimal correspondence between two curves (see details)

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

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

• Results of averaging profile of the radius bone in the sagittal and coronal directions for 10 male and 10 female subjects

Preliminary Comparison of Profiles

• Use the curve matching/averaging/registration framework to compare medical structures across populations

• Compare the radius bone in male/female subjects

• We compare the average male and female profiles

• Are there any differences between average male/female profiles?

• Are the differences limited to size differences?

• Or are there differences in the underlying shape and geometry?

• Direct comparison of the profiles show  profile of the radius bone is different in males and females

• To see if the differences are limited to size differences

• computed the optimal transformation (rotation/translation/scaling) to align the average profiles

• compared the profiles after optimal alignment

• visually there are no significant differences in the two profiles