How Isotropic Scaling Works in Generalized Procrustes Analysis

statistical-shape-modeling

gpa

scaling

A deep dive into isotropic scaling in GPA with formulas, intuition, and potential pitfalls.

Author

Daniel Koska

Published

August 12, 2025

This entry is more of a mental note for my future self than anything else (which I guess most of this blog is tbh).

If you’ve ever worked with Generalized Procrustes Analysis (GPA) in statistical shape modeling, you’ve probably come across the isotropic scaling step. It sounds innocent enough — just scale all coordinates equally — but this step is both powerful and a little sneaky. Let’s unpack what’s really going on.

Where it fits in GPA

The core idea of GPA is to remove differences that are not actual shape variation — things like translation, rotation, and scale — so that PCA or other analyses reflect pure shape.

For each shape (X) compared to a reference (often the current mean shape), GPA does:

Translation: Move the centroid to the origin.
Isotropic scaling: Resize uniformly in all directions to match the reference size.
Rotation: Rotate to minimize the distance to the reference.

Then it updates the mean shape and repeats until everything stops changing much.

What isotropic scaling really is

Isotropic scaling means multiplying all coordinates by the same scalar (s).
No stretching in one direction, no skewing — every axis gets scaled equally.

Mathematically, if (X ^{k m}) is a translated shape (with (k) points in (m) dimensions, usually (m=2) or (3)), the scaled shape is:

\[ X_{\text{scaled}} = s \, X \]

The scalar (s > 0) is chosen so that the scaled shape is as close as possible to the reference (Y).

Deriving the scaling factor

We want to minimize the squared Procrustes distance:

\[ D^2(s) = \|\, sX - Y \,\|_F^2 \]

where (||_F) is the Frobenius norm (sum of squared coordinates, square-rooted).

Expanding and differentiating with respect to (s):

\[ D^2(s) = s^2 \sum_{i,j} X_{ij}^2 - 2s \sum_{i,j} X_{ij} Y_{ij} + \sum_{i,j} Y_{ij}^2 \]

Set derivative to zero:

\[ 2s \sum_{i,j} X_{ij}^2 - 2 \sum_{i,j} X_{ij} Y_{ij} = 0 \]

And solve:

\[ s = \frac{\sum_{i,j} X_{ij} Y_{ij}}{\sum_{i,j} X_{ij}^2} \]

In matrix form:

\[ s = \frac{\mathrm{trace}(X^\mathsf{T} Y)}{\mathrm{trace}(X^\mathsf{T} X)} \]

The centroid size connection

If no specific reference is chosen yet (e.g., the very first iteration), shapes are often scaled to unit centroid size:

\[ C = \sqrt{\sum_{i=1}^k \sum_{j=1}^m X_{ij}^2} \]

and

\[ s = \frac{1}{C} \]

This prevents large shapes from dominating the mean early on.

Why it matters

Without scaling, size differences will dominate PCA.
With isotropic scaling, we remove those size differences, leaving only relative point configurations.
That’s great for pure shape analysis — but it can be risky if size is biologically meaningful.

And here’s the catch: isotropic scaling ties all coordinate axes to a single size factor.
If your sample varies in length or width, this scaling will also change vertical coordinates, even if they were identical before.
That can create artificial deformation in anisotropic structures like weight-bearing feet — a subtle source of bias you might not expect.

--- title: "How Isotropic Scaling Works in Generalized Procrustes Analysis" author: "Daniel Koska" date: 2025-08-12 categories: [statistical-shape-modeling, gpa, scaling] description: "A deep dive into isotropic scaling in GPA with formulas, intuition, and potential pitfalls." format: html: toc: true toc-depth: 2 page-layout: article --- This entry is more of a mental note for my future self than anything else (which I guess most of this blog is tbh). If you’ve ever worked with Generalized Procrustes Analysis (GPA) in statistical shape modeling, you’ve probably come across the *isotropic scaling* step. It sounds innocent enough — just scale all coordinates equally — but this step is both powerful and a little sneaky. Let’s unpack what’s really going on. ### Where it fits in GPA The core idea of GPA is to remove differences that are *not* actual shape variation — things like translation, rotation, and scale — so that PCA or other analyses reflect pure shape. For each shape (X) compared to a reference (often the current mean shape), GPA does: 1. **Translation:** Move the centroid to the origin. 2. **Isotropic scaling:** Resize uniformly in all directions to match the reference size. 3. **Rotation:** Rotate to minimize the distance to the reference. Then it updates the mean shape and repeats until everything stops changing much. ### What isotropic scaling really is Isotropic scaling means multiplying **all** coordinates by the same scalar (s).\ No stretching in one direction, no skewing — every axis gets scaled equally. Mathematically, if (X \in \mathbb{R}\^{k \times m}) is a translated shape (with (k) points in (m) dimensions, usually (m=2) or (3)), the scaled shape is: $$ X_{\text{scaled}} = s \, X $$ The scalar (s \> 0) is chosen so that the scaled shape is as close as possible to the reference (Y). ### Deriving the scaling factor We want to minimize the squared Procrustes distance: $$ D^2(s) = \|\, sX - Y \,\|_F^2 $$ where (\|\cdot\|\_F) is the Frobenius norm (sum of squared coordinates, square-rooted). Expanding and differentiating with respect to (s): $$ D^2(s) = s^2 \sum_{i,j} X_{ij}^2 - 2s \sum_{i,j} X_{ij} Y_{ij} + \sum_{i,j} Y_{ij}^2 $$ Set derivative to zero: $$ 2s \sum_{i,j} X_{ij}^2 - 2 \sum_{i,j} X_{ij} Y_{ij} = 0 $$ And solve: $$ s = \frac{\sum_{i,j} X_{ij} Y_{ij}}{\sum_{i,j} X_{ij}^2} $$ In matrix form: $$ s = \frac{\mathrm{trace}(X^\mathsf{T} Y)}{\mathrm{trace}(X^\mathsf{T} X)} $$ ### The centroid size connection If no specific reference is chosen yet (e.g., the very first iteration), shapes are often scaled to **unit centroid size**: $$ C = \sqrt{\sum_{i=1}^k \sum_{j=1}^m X_{ij}^2} $$ and $$ s = \frac{1}{C} $$ This prevents large shapes from dominating the mean early on. ### Why it matters Without scaling, size differences will dominate PCA.\ With isotropic scaling, we remove those size differences, leaving only relative point configurations.\ That’s great for *pure shape* analysis — but it can be risky if size is biologically meaningful. And here’s the catch: isotropic scaling ties *all* coordinate axes to a single size factor.\ If your sample varies in length or width, this scaling will also change **vertical coordinates**, even if they were identical before.\ That can create *artificial deformation* in anisotropic structures like weight-bearing feet — a subtle source of bias you might not expect.