Detection of non-affine shape outliers for matched-pair shape data

Abstract

Cleft lip/palate (CLP) is a relatively common birth defect so disfiguring
that nowadays it is almost always corrected surgically as early as possible.
The postnatal surgical correction does not, however, result in a normally growing
upper jaw, but instead, owing to scar tissue, one that grows abnormally. It is
important to decide if a clinical treatment group is homogeneous. The example
involves data from digitally processed lateral X-ray films of 48 boys who have complete
unilateral CLP but no other malformation. 22 landmarks were represented
by their Procrustes shape coordinates, principal components of matched-pair differences
were examined, and the distribution of the 48 shape changes was studied
for outliers in the affine and non-affine subspaces of the full Procrustes shape
and form space. To separate outliers from inliers we use bagplots. There are no
outliers apparent in the affine subspace. In the non-affine subspaces, we found
no outliers in the subspace of bending patterns at large scale but four outliers
in the subspace of local changes at small scale. Almost the same outliers were
found by form-space PCA. These latter are associated with possible creases of
the corresponding thin-plate splines. In those cases we can use the same spline
formalism to relax the outlying form to an inlier by optimal relaxation along
the curve d´ecolletage that weighs bending energy against Procrustes distance
and stop relaxation on the fence. These maneuvers suggest a possibly novel and
interesting fusion of the Procrustes-spline toolkit with outlier detection. They
also have practical implications for craniofacial management of CLP follow-up as
well as suggestive implications for outlier detection in applied craniometrics and
anthropometrics more generally.