Does GPA require three-dimensional objects?

No. Two-dimensional landmarks and other point configurations are common applications.

Are reflections always allowed?

No. They are a modeling choice and are often excluded in shape analysis.

Does zero Procrustes discrepancy mean identical raw measurements?

No. Configurations can differ in position, orientation, or size while becoming identical after the permitted transformations.

AI Glossary / Statistics and probability / Advanced

Generalized Procrustes analysis

Definition

Generalized Procrustes analysis, or GPA, aligns several configurations of corresponding points to a common consensus by removing specified differences in position, orientation, and, when appropriate, uniform size. The remaining variation describes differences that the permitted transformations cannot explain. It is used in statistical shape analysis and in comparing configurations of the same objects. Reflection, scaling, normalization, and landmark treatment must be specified because they affect what counts as equivalent.

What it means

Two drawings can represent the same shape even if one is moved, rotated, or uniformly enlarged. Comparing their raw coordinates would confuse those changes with actual shape differences. GPA aligns multiple configurations so that comparisons focus on the intended information.

Correspondence is essential: landmark number one must represent the same feature or object in every configuration. Standard GPA does not discover those matches automatically. Nor does it normally permit arbitrary stretching of different axes or free deformation of individual points.

Removing size is a substantive choice. If overall size matters to the scientific or business question, a size-and-shape analysis may be more appropriate. Similarly, allowing reflection treats mirror images as equivalent, which can be unsuitable when handedness matters.

How it works

Represent each configuration as a matrix with corresponding points in rows and spatial dimensions in columns. In a common shape-analysis workflow, center each configuration by subtracting its centroid and normalize its centroid size, the square root of the sum of squared distances from that centroid.

Align configurations to a current reference using permitted rotations, commonly found through matrix factorization. Average the aligned coordinates to update the consensus, normalize as required by the selected formulation, and iterate until convergence.

Examine residual differences, convergence, landmark quality, and sensitivity to influential points. Different GPA formulations use different scale conventions and constraints; the consensus orientation is arbitrary. Subsequent analyses often use an appropriate tangent-space representation rather than assuming aligned coordinates are unconstrained independent variables.

Three practical examples

Computer vision: align corresponding landmarks from multiple objects before studying genuine shape variation or building a statistical shape model.
Product design: compare repeated landmark measurements of components after removing camera-position differences, while retaining size when dimensions are operationally important.
Perceptual research: compare maps of the same products created by different evaluators, distinguishing coordinate orientation from agreement about relative configuration.

Think of it this way

Think of placing transparent drawings on top of one another. You can slide, turn, and perhaps uniformly resize each sheet until corresponding marks align as well as possible. Any remaining mismatch concerns the drawings themselves under those allowed operations, rather than where the sheets initially sat on the table.

Worked example

Take three original triangular configurations with corresponding points in the same order. A contains (0, 0), (2, 0), and (0, 1). B contains (5, −1), (5, 3), and (3, −1). C contains (−2, 4), (−3, 4), and (−2, 3.5).

B is A enlarged by a factor of two, rotated 90 degrees counterclockwise, and translated by (5, −1). C is A reduced by half, rotated 180 degrees, and translated by (−2, 4). Their centroids are respectively (2/3, 1/3), (13/3, 1/3), and (−7/3, 23/6).

A’s squared centroid size is 10/3, so its centroid size is approximately 1.825742. B and C have centroid sizes approximately 3.651484 and 0.912871. After centering, dividing by each centroid size, and undoing the rotations, all three configurations coincide. Independent coordinate calculations confirm residual differences below numerical rounding. Consequently, a GPA allowing those transformations can attain zero alignment error here. It would still not conclude that the physical objects have equal size.

Illustrative case study

Fictional case: FormLab compares component outlines from several production batches. Camera placement varies, and an initial coordinate analysis exaggerates differences between batches.

The team verifies landmark correspondence and performs GPA with translation and rotation. It runs a separate size-normalized analysis to explore shape alone, keeping the original dimensional measurements for engineering tolerances. The report distinguishes pose variation, size differences, and residual shape differences. A small shape discrepancy is not used to certify that a component meets its required physical dimensions.

Common misconceptions and mistakes

Assuming GPA can repair incorrect landmark correspondence.
Removing scale when size is part of the outcome of interest.
Allowing reflection without considering left-right asymmetry or handedness.
Treating alignment as arbitrary nonrigid warping that can erase any shape difference.
Interpreting the consensus as a known true object rather than a summary under chosen transformations.
Treating all aligned coordinate entries as independent measurements in later statistical tests.

Business use

GPA can make comparisons more meaningful when coordinate systems differ for reasons unrelated to the question. Before using it, decide which transformations represent nuisance variation and which remove information the business needs.

Document landmark definitions, missing-point handling, units, transformation permissions, and scale conventions. Inspect aligned configurations and residuals rather than relying only on a small fit statistic. For machine-learning pipelines, avoid allowing test-set configurations to influence a training-only reference when that would leak evaluation information. Alignment quality and downstream predictive usefulness should be assessed separately.

Check your understanding

Answer: 1.

Why must landmark order match across objects?

Answer: The method aligns corresponding rows, so incorrect order compares different features. 2.

What does dividing by centroid size remove?

Answer: Overall uniform size under that normalization. 3.

Would moving one landmark independently always be removable by translation and rotation?

Answer: No; it can change the internal configuration and leave a residual difference.

Related terms

Landmarks
Procrustes analysis
landmark
centroid size
shape analysis
orthogonal transformation
registration
consensus configuration
tangent space

Learn next

Practice aligning two configurations before extending to a multi-configuration consensus. Compare analyses with and without scaling or reflection, then study shape-space geometry and the assumptions behind subsequent principal component or regression analyses.

Sources

Gower, “Generalized Procrustes Analysis” (1975): https://three-mode.leidenuniv.nl/pdf/g/gower1975pm.pdf — original consensus criterion, correspondence, and transformation constraints read.

Dryden, shapes package, procGPA: https://search.r-project.org/CRAN/refmans/shapes/html/procGPA.html — scaling, reflection, and output conventions checked. The example uses explicitly stated centroid-size normalization; its coordinates and calculations are original.

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