【Basic】Image Contour Detection in MATLAB: Using Edge Detection and Contour Extraction

# 2.1 Sobel Operator

### 2.1.1 Principle and Formula

The Sobel operator is a first-order differential operator used for detecting edges in images. It works by calculating the gradient vector for each pixel in the image. The direction of the gradient vector points towards the direction of fastest brightness change in the image, while its magnitude represents the rate of that brightness change.

The Sobel operator uses two 3x3 convolution kernels, one for horizontal and one for vertical gradient computation. The horizontal kernel is:

[-1 0 1]
[-2 0 2]
[-1 0 1]

And the vertical kernel is:

[-1 -2 -1]
[0 0 0]
[1 2 1]

For each pixel in the image, these two kernels are convolved with the local 3x3 region of the image to produce the horizontal and vertical gradient components. Together, these components make up the gradient vector.

# 2. Edge Detection Techniques

Edge detection is a fundamental technique in image processing, aiming to detect discontinuities between pixels in an image, thereby extracting edge information from the image. There are many edge detection algorithms, among which the Sobel operator, Canny operator, and Laplace operator are three commonly used edge detection operators.

### 2.1 Sobel Operator

The Sobel operator is a first-order derivative edge detection operator that detects edges by calculating the amplitude of the pixel gradients in the image. It uses two convolution kernels to calculate the gradients in the horizontal and vertical directions:

% Horizontal Sobel Operator
Gx = [-1 0 1; -2 0 2; -1 0 1];

% Vertical Sobel Operator
Gy = [-1 -2 -1; 0 0 0; 1 2 1];

The formula for the gradient magnitude of the Sobel operator is:

G = sqrt(Gx.^2 + Gy.^2)

### 2.1.1 MATLAB Implementation

In MATLAB, the `imgradientxy` function can be used to perform Sobel operator edge detection:

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

% Sobel Operator Edge Detection
[Gx, Gy] = imgradientxy(I);
G = sqrt(Gx.^2 + Gy.^2);

% Display edge detection results
imshow(G, []);

### 2.2 Canny Operator

The Canny operator is a multi-stage edge detection operator that detects edges through steps such as Gaussian smoothing, gradient computation, non-maximum suppression, and hysteresis thresholding. The Canny operator has good noise immunity and accurate edge localization.

### 2.2.1 Principle and Formula

The principle and formula of the Canny operator are as follows:

1. **Gaussian Smoothing:** Use a Gaussian filter to smooth the image, removing noise.
2. **Gradient Computation:** Use the Sobel operator to calculate the gradient magnitude and direction of the image.
3. **Non-Maximum Suppression:** Suppress non-maximum gradient magnitudes along the gradient direction, keeping only local maxima.
4. **Hysteresis Thresholding:** Use two thresholds (high and low) to threshold the gradient magnitudes after non-maximum suppression. Pixels with gradient magnitudes higher than the high threshold are considered edge pixels, while those lower than the low threshold are considered non-edge pixels. Pixels between the high and low thresholds are judged based on the edge state of their neighboring pixels.

### 2.2.2 MATLAB Implementation

In MATLAB, the `edge` function can be used to perform Canny operator edge detection:

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

% Canny Operator Edge Detection
edges = edge(I, 'canny');

% Display edge detection results
imshow(edges, []);

### 2.3 Laplace Operator

The Laplace operator is a second-order derivative edge detection operator that detects edges by calculating the second derivative of image pixels. The Laplace operator has good edge localization accuracy but poor noise immunity.

### 2.3.1 Principle and Formula

The principle and formula of the Laplace operator are as follows:

L = ∇^2 I = ∂^2 I / ∂x^2 + ∂^2 I / ∂y^2

Where:

* L: Laplace operator
* I: Image
* ∇^2: Laplacian operator
* ∂^2 I / ∂x^2: Second derivative of the image in the x direction
* ∂^2 I / ∂y^2: Second derivative of the image in the y direction

### 2.3.2 MATLAB Implementation

In MATLAB, the `laplacian` function can be used to perform Laplace operator edge detection:

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

% Laplace Operator Edge Detection
L = laplacian(I);

% Display edge detection results
imshow(L, []);

# 3. Contour Extraction Algorithms

### 3.1 Boundary Tracking Algorithm

#### 3.1.1 Principle and Steps

The boundary tracking algorithm is a contour extraction algorithm based on pixel connectivity. Its basic principle is to trace along the edges of the image pixel by pixel until returning to the starting point. The specific steps are as follows:

1. **Initialization:** Select an edge pixel in the image as the starting point.
2. **Direction Determination:** Determine the direction with the largest gradient value among the eight neighboring pixels around the starting point.
3. **Movement:** Move to the next pixel in the direction with the largest gradient value.
4. **Judgment:** Determine if the current pixel is an edge pixel. If it is, continue moving; otherwise, return to the starting point.
5. **Repeat:** Repeat steps 2-4 until returning to the starting point.

#### 3.1.2 MATLAB Implementation

function boundary = boundaryTracking(image)
     % Get the image gradient
     [Gx, Gy] = gra