Image Feature Extraction in MATLAB: Using SIFT and SURF Algorithms

# The Theoretical Foundation of SIFT Algorithm

The Scale-Invariant Feature Transform (SIFT) is an algorithm widely used for image feature extraction, demonstrating robustness against changes in scale, rotation, and affine transformations of images. The theoretical foundation of the SIFT algorithm comprises the following two key steps:

### 2.1 Scale Space Extrema Detection

The SIFT algorithm detects key points in images by constructing a scale space. Scale space is a three-dimensional image representation, where the third dimension denotes the scale of the image. The SIFT algorithm uses the Difference of Gaussian (DoG) operator to detect extremal points in the scale space. The DoG operator is calculated by the difference between Gaussian-filtered images of adjacent scales. Extremal points are the locations of local maxima or minima of the DoG operator within the scale space.

### 2.1.2 Keypoint Orientation Assignment

Once detected, the SIFT algorithm assigns an orientation to each keypoint. This aids the algorithm in maintaining invariance to rotations of the image. The SIFT algorithm determines orientation by calculating a histogram of gradient orientations around the keypoint. The gradient orientation with the highest value in the histogram is assigned to the keypoint.

# The Principle and Practice of SIFT Algorithm

### 2.1 The Theoretical Foundation of SIFT Algorithm

#### 2.1.1 Scale Space Extrema Detection

Scale space extrema detection is one of the key steps of the SIFT algorithm, aimed at finding stable key points across different scales of images. The SIFT algorithm uses Gaussian pyramids and Difference of Gaussian (DoG) pyramids to construct the scale space.

The Gaussian pyramid is obtained by repeatedly applying Gaussian smoothing to the image, with each layer of the Gaussian pyramid corresponding to a different scale. The DoG pyramid is derived by subtracting adjacent layers of the Gaussian pyramid, enhancing edges and corners within the image.

In the DoG pyramid, extremal points refer to points that have the highest or lowest values within their own neighborhood (including both scale space and spatial position). These extremal points correspond to significant features in the image, such as corners, edges, and blobs.

#### 2.1.2 Keypoint Orientation Assignment

After finding the scale space extremal points, an orientation must be assigned to each keypoint for subsequent descriptor computation. The SIFT algorithm uses image gradients to assign orientations.

For each keypoint, the gradient magnitude and orientation are calculated for all pixels within its neighborhood. These gradients are then weighted and summed, with the weights determined by the gradient magnitudes. The result of this weighted sum is the orientation of the keypoint.

### 2.2 The MATLAB Implementation of SIFT Algorithm

#### 2.2.1 Detection of SIFT Feature Points

The `vl_sift` function in MATLAB is used to detect SIFT feature points. This function takes an image as input and returns the coordinates, scale, and orientation of the key points.

% Read image
image = imread('image.jpg');

% Detect SIFT feature points
[frames, descriptors] = vl_sift(image);

% Display key points
figure;
imshow(image);
hold on;
plot(frames(1, :), frames(2, :), 'ro');
hold off;

#### 2.2.2 Computation of SIFT Descriptors