Image Feature Dimensionality Reduction in MATLAB: Applying Principal Component Analysis (PCA)

# 1. Overview of Image Feature Dimensionality Reduction

Image feature dimensionality reduction is a technique aimed at decreasing the dimensionality of image features while preserving their primary information. In image processing and computer vision, images often possess high-dimensional features, posing challenges in terms of computation and storage. Dimensionality reduction addresses these issues by projecting high-dimensional features onto a low-dimensional subspace, thus simplifying data analysis and processing.

Dimensionality reduction techniques are widely applied in fields such as image classification, retrieval, compression, and recognition. By reducing the feature dimensions, algorithm efficiency is improved, computational costs are lowered, and the robustness of image representations is enhanced.

# 2.1 Mathematical Principles of PCA

### 2.1.1 Covariance Matrix and Eigendecomposition

A covariance matrix measures the correlation between random variables. For a given dataset, its covariance matrix is defined as:

Cov(X) = E[(X - μ)(X - μ)ᵀ]

Where:

- X is the dataset
- μ is the mean of the dataset
- E is the expectation operator

The covariance matrix is symmetric, with diagonal elements representing the variance of each feature and off-diagonal elements representing the covariance between features.

Eigendecomposition is the process of decomposing the covariance matrix into eigenvalues and eigenvectors. Eigenvalues are the roots of the characteristic polynomial of the covariance matrix, and eigenvectors are the corresponding unit orthogonal vectors.

### 2.1.2 Calculation of Principal Components

Principal components are the eigenvectors of the covariance matrix. Their directions represent the directions of maximum variance in the data. The number of principal components equals the number of eigenvectors, which is the same as the dimension of the dataset.

The calculation of principal components can be completed through the following steps:

1. Calculate the covariance matrix.
2. Perform eigendecomposition on the covariance matrix.
3. Take the eigenvectors corresponding to the largest eigenvalues as the principal components.

The order of the principal components indicates their importance. The first principal component contains the maximum data variance, followed by others.

[V, D] = eig(Cov(X));

Where:

- V is the matrix of eigenvectors
- D is the matrix of eigenvalues

# 3. Implementation of PCA in MATLAB

### 3.1 Using the PCA Function

#### 3.1.1 Syntax and Parameters of the pca() Function

MATLAB provides the `pca()` function to implement the PCA algorithm. Its syntax is:

[coeff,score,latent,tsquared,explained,mu] = pca(X, 'NumComponents', n)

Where:

- `X`: Input data matrix, with each row representing a sample and each column representing a feature.
- `'NumComponents'`: Specifies the number of principal components to retain.
- `coeff`: Matrix of principal component coefficients, with each column representing a principal component.
- `score`: Matrix of dimension-reduced data, with each row representing a sample and each column representing a principal component.
- `latent`: Vector of eigenvalues, arranged in descending order.
- `tsquared`: Hotelling's T² statistic, used to evaluate the similarity between dimension-reduced data and the original data.
- `explained`: Vector of retained variance percentages.
- `mu`: Mean vector of the input data.

#### 3.1.2 Obtaining Dimension-Reduced Data

Dimension-reduced data can be obtained through