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Covariance Matrix from Image Patches

Compute the covariance matrix of image patches for PCA-based texture analysis.

Given an image and patch size, extract all overlapping patches, flatten them to vectors, and compute the covariance matrix. This is a fundamental step in texture analysis and eigenface computation.

Algorithm:

Extract all overlapping patches of size $p \times p$ from the image
Flatten each patch into a vector of length $p^2$
Stack vectors as rows of data matrix $X$ (shape: n_patches × p²)
Center the data: $X_c = X - \bar{X}$ (subtract mean vector)
Compute covariance: $\Sigma = \frac{1}{n-1} X_c^T X_c$

The covariance matrix $\Sigma$ captures correlations between pixel positions within patches.

Applications:

PCA for dimensionality reduction
Eigenface/Fisherface recognition
Texture synthesis and analysis

Example:

Input:

image = [[1, 2, 3],
         [4, 5, 6],
         [7, 8, 9]]
patch_size = 2

Output:

[[2.5, 2.5, 2.5, 2.5],
 [2.5, 2.5, 2.5, 2.5],
 [2.5, 2.5, 2.5, 2.5],
 [2.5, 2.5, 2.5, 2.5]]

Reasoning:

Step 1: Extract 2×2 patches From 3×3 image, we get 4 overlapping patches:

Patch 1 (top-left): [[1,2],[4,5]] → flattened: [1,2,4,5]
Patch 2 (top-right): [[2,3],[5,6]] → flattened: [2,3,5,6]
Patch 3 (bottom-left): [[4,5],[7,8]] → flattened: [4,5,7,8]
Patch 4 (bottom-right): [[5,6],[8,9]] → flattened: [5,6,8,9]

Step 2: Create data matrix X (4×4) $X = \begin{pmatrix} 1 & 2 & 4 & 5 \\ 2 & 3 & 5 & 6 \\ 4 & 5 & 7 & 8 \\ 5 & 6 & 8 & 9 \end{pmatrix}$

Step 3: Compute mean and center Mean vector: [3, 4, 6, 7] $X_c = X - \text{mean}$

Step 4: Compute covariance $\Sigma = \frac{1}{3} X_c^T X_c$

All entries are 2.5 due to the regular structure of this simple example.

Constraints:

image: 2D numpy array (grayscale)
patch_size: integer p (patches are p×p)
Return: covariance matrix of shape (p², p²)
Round to 4 decimal places

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.

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