[Foundation] Image Pyramids in MATLAB: Constructing Image Pyramids and Multi-scale Analysis

# 2.1 Principles and Types of Image Pyramids

An image pyramid is a hierarchical data structure that represents an image as a sequence of sub-images with different resolutions. It is based on the hierarchical decomposition principle in image processing, decomposing the image layer by layer, forming a pyramid structure.

The types of image pyramids are mainly divided into two categories:

***Gaussian Pyramid:** Constructed through Gaussian filtering and downsampling to smooth the image, where each layer's image size is halved, and the image smoothness is increased, used for image feature extraction.
***Laplacian Pyramid:** Constructed by the differences between adjacent layers of the Gaussian pyramid, preserving the image's detailed information, used for image reconstruction and fusion.

# 2. Constructing Image Pyramids

### 2.1 Principles and Types of Image Pyramids

An **image pyramid** is a hierarchical data structure that decomposes the original image into a series of images with gradually decreasing resolutions. Each level of the pyramid corresponds to a smaller version of the original image, and the more levels there are, the lower the image resolution.

The construction principle of the pyramid is achieved through **image downsampling**. ***mon downsampling methods include:

- **Average Pooling:** Averaging neighboring pixels in the image to generate a new pixel.
- **Max Pooling:** Taking the maximum value among neighboring pixels in the image as the new pixel.
- **Bilinear Interpolation:** Using the weighted average of surrounding pixels to generate a new pixel.

Depending on the downsampling method, image pyramids can be divided into two types:

- **Gaussian Pyramid:** Constructed using average pooling or bilinear interpolation, resulting in a continuously smooth pyramid.
- **Laplacian Pyramid:** Constructed using the differences between adjacent layers of the Gaussian pyramid, resulting in a pyramid with more prominent edge and texture information.

### 2.2 Pyramid Construction Algorithms: Gaussian Pyramid and Laplacian Pyramid

#### 2.2.1 Gaussian Pyramid

The construction algorithm for a Gaussian pyramid is as follows:

1. Start with the original image, and use average pooling or bilinear interpolation to reduce the image size by half.
2. Use the reduced image as the input for the next layer, and repeat step 1 until the desired number of layers is reached.

**Code Block:**

% Constructing a Gaussian Pyramid
original_image = imread('image.jpg');
gaussian_pyramid = impyramid(original_image, 'reduce');

% Displaying the pyramid
figure;
for i = 1:length(gaussian_pyramid)
    subplot(1, length(gaussian_pyramid), i);
    imshow(gaussian_pyramid{i});
end

**Logical Analysis:**

- The `impyramid` function is used to construct the image pyramid, with the `'reduce'` option specifying the use of average pooling for downsampling.
- Loop through each level of the pyramid and use the `imshow` function to display the images.

#### 2.2.2 Laplacian Pyramid

The construction algorithm for a Laplacian pyramid is as follows:

1. Construct a Gaussian pyramid.
2. Subtract the current layer from the adjacent layers of the Gaussian pyramid to generate each layer of the Laplacian pyramid.

**Code Block:**

% Constructing a Laplacian Pyramid
laplacian_pyramid = impyramid(gaussian_pyramid, 'reduce');

% Displaying the pyramid
figure;
for i = 1:length(laplacian_pyramid)
    subplot(1, length(laplacian_pyramid), i);
    imshow(laplacian_pyramid{i});
end

**Logical Analysis:**

- The `impyramid` function is used again to construct the image pyramid, but this time with the `'reduce'` option specifying the use of Laplacian dow