Procrustes Analysis

Amy Ross

University of South CarolinaCSCE 790i

What is Shape?

Shape:

“all the geometrical information that remains when location, scale and rotational effects are filtered out from an object.”[1]

Figure 1: The same shape represented in four ways by different Euclidean similarity transforms [1]

How to Maintain Shape?

Types of Euclidean Similarity Transformations:

• Rotation

• Translation

• Isotropic Scaling

Euclidean similarity transforms:

Transforms which maintain shape by allowing the shape to move in a way that filters out the differences while preserving the angles and parallel lines.

How to Describe a Shape?

Landmarks:

A finite set of points on a shapes surface which accurately describes a shape.

Figure 2: Example of how landmarks are used to represent a shape.

Landmarks

Types of Landmarks:

• Anatomical landmarks: expert (i.e. Doctor) assigned points that represent a biological object or objects.

• Mathematical landmarks: points assigned by some mathematical property (i.e. high curvature).

• Pseudo-landmarks: point located between the other two types of landmarks or points around the outline.[2]

Comparing Shapes

Procrustes Analysis:

Minimizes the sum of the squared deviations between matching corresponding points (landmarks) from each of the two data sets (shapes).

Generalized Procrustes Analysis:

The application of Procrustes analysis on more than two data sets (shapes).

Generalized Procrustes Analysis

Steps for GPA:

1. Select one shape to be the approximate mean shape (i.e. the first shape in the set).

2. Align the shapes to the approximate mean shape.

3. Calculate the new approximate mean from the aligned shapes.

4. If the approximate mean from steps 2 and 3 are different the return to step 2, otherwise you have found the true mean shape of the set.

Aligning Shapes

GPA (step 2) alignment process:

1. Calculate the centroid of each shape (or set of landmarks).

2. Align all shapes centroid to the origin.

3. Normalize each shapes centroid size.

4. Rotate each shape to align with the newest approximate mean.

Aligning to the Origin

X: kxm matrix of coordinates of the k landmarks in m dimensions (m = 2 or 3)

Xc: the new coordinates of X centered at the origin

Normalization

Normalize each shape using the already centered coordinates.

X: the new coordinates of X centered at the origin

||X||: the norm of X

Xn: the new normalized and centered coordinates

Rotation

To calculate the correct rotation matrix we must determine the Q that minimizes:

Since, ||A|| = trace(A’A), we have

  ||XQ - || = trace(X’X + ’ ) - 2trace( ’XQ)

||XQ - || → min

X: the coordinates of X centered and normalized

Q: the orthogonal rotation matrix to align X to the average

: the average matrix

Rotation

Since the first part of the rhs doesn’t contain Q, we have

 trace( ’XQ) → max

Using singular value decomposition of ’X=USV’ and the cyclic property of trace we have

 trace( ’XQ) = trace(USV’Q) = trace(SV’QU) = trace(SH)

H = V’QU is an orthogonal (pxp) matrix because it is the product of orthogonal matrices.

Rotation P

trace(SH) = ∑ si hii i =1 

Therefore, since si is non-negative numbers and trace(SH)

is maximum when hii=1 for i=1, 2…p (the maximal value of

an orthogonal matrix), we have

H = I = V’QU

Thus the Q that minimizes ||XQ - || is

Q = VU’

Example Results

Figure 4: Left: 40 unaligned shapes. Right: 40 aligned shapes with the mean given in red. [1]

Procrustes Analysis

Advantages:

• Fairly straightforward approach

• Great for same object alignment

Disadvantages:

• Rigid Evaluation

• Convergence is not guaranteed

• Must have one to one landmark correspondence

References

Mikkel B. Stegmann and David Delgado Gomez. A Brief Introduction to Statistical Shape Analysis, Technical University of Denmark, Lyngby, 2002.

Matthew James Francis Cairns. An Investigation into the use of 3D Computer Graphics for Forensic Facial Reconstruction, Glasgow University, 2000.

John C. Gower and Garmt B. Dijksterhuis. Procrustes Problems, Oxford University Press, 2004.

[1]

[2]

[3]