Image Classification in MATLAB: Applying Support Vector Machines for Image Classification

# 2.1 Basic Principles and Mathematical Model of SVM

Support Vector Machine (SVM) is a binary classification algorithm whose fundamental concept is to map data points into a high-dimensional feature space and to seek a hyperplane that segregates the two classes of data points. The mathematical model of SVM can be represented as:

f(x) = w^T x + b

Where:

* x is the input data point
* w is the weight vector of the hyperplane
* b is the bias term of the hyperplane

The goal of SVM is to find a hyperplane that maximizes the margin between the two classes of data points. The margin can be represented as:

margin = 2 / ||w||

Where:

* ||w|| is the norm of the weight vector

To find the hyperplane with the largest margin, SVM uses the following optimization problem:

min 1/2 ||w||^2
subject to y_i (w^T x_i + b) >= 1, for all i

Where:

* y_i is the label of the ith data point (+1 or -1)
* x_i is the ith data point

# 2. Theoretical Basis of Support Vector Machine (SVM)

### 2.1 Basic Principles and Mathematical Model of SVM

Support Vector Machine (SVM) is a powerful machine learning algorithm, particularly suited for solving classification problems in high-dimensional, small-sample scenarios. The basic principle of SVM is to map input data into a high-dimensional feature space and to find a hyperplane that separates samples of different categories.

**Mathematical Model:**

Given a training dataset:

{(x₁, y₁), (x₂, y₂), ..., (xₙ, yₙ)}

Where:

* xᵢ ∈ R^d represents the d-dimensional feature vector of the ith sample
* yᵢ ∈ {-1, 1} represents the category label of the ith sample (-1 denotes the negative class, 1 denotes the positive class)

The goal of SVM is to find a hyperplane:

wᵀx + b = 0

That separates the two classes of samples, where w is the weight vector of the hyperplane and b is the bias term.

To find the optimal hyperplane, SVM employs a strategy that maximizes the distance of samples from the hyperplane on both sides (called the margin). The margin can be represented as:

γ = min{||xᵢ - wᵀxᵢ - b|| : yᵢ(wᵀxᵢ + b) < 0}

Where ||·|| denotes the Euclidean norm.

### 2.2 Kernel Functions and Classification Methods of SVM

**Kernel Function:**

In some cases, samples that are linearly inseparable in the original feature space may be linearly separable in a high-dimensional feature space. To achieve this, SVM uses kernel functions to map the samples to a high-dimensional feature space:

Φ(x)

Where Φ(·) ***monly used kernel functions include the linear kernel, polynomial kernel, and Gaussian kernel.

**Classification Method:**

Once mapped into a high-dimensional feature space, SVM finds the optimal hyperplane by solving the following optimization problem:

min 0.5||w||² + C∑ᵢξᵢ

Where:

* C is the regularization parameter, controlling the complexity of the model
* ξᵢ is the slack variable, allowing some samples to violate the margin constraints

After solving this optimization problem, the weight vector w and bias term b of the hyperplane can be determined. Then, a new sample x can be classified according to the following rule:

y = sign(wᵀΦ(x) + b)

# 3.1 Preprocessing and Feature Extraction of Image Dataset

Before performing image classification, preprocessing and feature extraction of the image dataset are necessary. The purpose of preprocessing is to remove noise and interference from images, enhancing image quality. Feature extraction is about extracting features that can distinguish between different categories from the images.

#### Image Preprocessing

Image preprocessing typically includes the following steps:

- **Image Size Normalization:** Adjust the image to a uniform size for easier subsequent processing.
- **Image Grayscale Conversion:** Convert color images to grayscale images to reduce color information interference.
- **Image Denoising:** Use filters to remove noise from images, such as Gaussian filters or median filters.
- **Image Enhancement:** Enhance the contrast and clarity of images using techniques like contrast enhancement or histogram equalization.

#### ***

***mon feature extraction methods include:

- **Histogram Features:** Calculate the grayscale distribution histogram of pixels in the image as features.
- **Texture Features:** Extract texture information from the image, such as Local Binary Patterns (LBP) or Gray-Level Co-occurrence Matrices (GLCM).
- **Shape Features:** Extract shape features from the image, such as contours, area, and perimeter.

In MATLAB, the following functions can be used for image preprocessing and feature extraction:

% Image Size Normalization
image = imresize(image, [224, 224]);

% Image Grayscale Co