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appendix B
Procrustes shape analysis
In a 2D scenario, shape is very commonly used to refer to the appearance or silhouette of an object. Procrustes shape analysis is to analyze the geometrical information that remains when location, scale and rotational effects are filtered out from an object. This appendix briefly reviews the definitions of full Procrustes fit, full Procrustes distance, and full Procrustes mean shape, whose much more complete treatments are in [Dryden and Mardia, 1998].
Assume two shapes or silhouettes in the 2D space are represented by two vectors of complex entries, say and . Without loss of generality, assume these two configurations are centered, i.e., , where means transpose of complex conjugate of and is a length- vector with all components being 1.
Definition B.1 [p40, Dryen and Mardia, 1998]: The full Procrustes fit of onto is
where are chosen to minimize
.
Proposition B.1 [p40, Dryen and Mardia, 1998]: The full Procrustes fit has matching parameters
, , .
So the full Procrustes fit of onto is
.
Proof:
Obviously, the minimizing , are 0. Let us denote , , then
.
So, . Now we have
.
Lastly, by
.
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Definition B.2 [p41, Dryen and Mardia, 1998]: The full Procrustes distance between and is
where the second equation comes from the complex linear regression used in deriving Result 1, and it can be checked that is invariant with respect to translation, rotation, and scaling of configurations of and .
Definition B.3 [p44, Dryen and Mardia, 1998]: The full Procrustes mean shape is obtained by minimizing the sum of squared full Procrustes distances from each configuration to an unkown unit size configuration , i.e.,
.
Note that is not a single configuration, instead, it is a set, whose elements have 0 full Procrustes distance to the optimal unit size configuration .
Proposition B.2 [p44, Dryen and Mardia, 1998]: The full Procrustes mean shape, , can be found as the eigenvector, , corresponding to the largest eigenvalue of the complex sum of squares and products matrix
.
All translation, scaling, and rotation of are also solutions, but they all correspond to the same shape , i.e., have 0 full Procrustes distance to .
Proof:
Under the constraint of , we have
.
The well known linear algebra fact states that is the first eigenvector of matrix .
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Proposition B.3 [p89, Dryen and Mardia, 1998]: The arithmetic mean of the full Procrustes fits, , has the same shape as the full Procrustes mean, i.e.,
.
Proof:
Because is invariant with respect to the scaling of its arguments, so
.
Without restricting the size of , let us denote the Procrustes fit of onto as , then
is a quadratic function with respect to , and is minimized by
.
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