Filter/ Convolution Kernel and Its Operation

Filter / Convolution Kernel and Its Operation | 卷积核及其操作

1. What is a Convolution Kernel? | 什么是卷积核？

A convolution kernel, also known as a filter, is a small matrix used to apply effects like blurring, sharpening, edge detection, or other transformations to an image. It moves (slides) over the input image and performs element-wise multiplication with the image pixels, followed by summing up the results to produce a new pixel value. 卷积核，也称为滤波器，是一个用于对图像进行模糊、锐化、边缘检测等效果的小矩阵。卷积核在输入图像上滑动，并对图像像素进行逐元素相乘，随后将结果相加，生成一个新的像素值。

2. How Does Convolution Work? | 卷积操作是如何进行的？

When applying a convolution, the filter is slid over the input image. For each position of the filter:

• Multiply each element in the filter (kernel) by the corresponding pixel in the image.

• Sum all the products.

• The sum becomes the new pixel value in the output image at that location.

This process is repeated across the entire image, and the filter will shift pixel by pixel, applying the same operation to every part of the image. 当进行卷积操作时，卷积核会在输入图像上滑动。对于卷积核的每一个位置：

• 将卷积核中的每个元素与图像中对应的像素相乘。

• 将所有乘积相加。

• 相加的结果成为输出图像在该位置的新像素值。

这个过程会在整个图像上重复，卷积核会逐像素地移动，对图像的每个部分应用相同的操作。

3. Example of Convolution | 卷积操作的例子

Consider a simple 3x3 kernel applied to a grayscale image:

$$\text{Kernel} = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 0 & -1 \\ 1 & 0 & -1 \end{bmatrix}$$

This kernel can be used for detecting vertical edges. It slides over the image and computes new pixel values by multiplying the kernel values with the corresponding image pixels. 考虑一个简单的 3x3 卷积核应用于灰度图像：

$$\text{卷积核} = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 0 & -1 \\ 1 & 0 & -1 \end{bmatrix}$$

这个卷积核可以用于检测垂直边缘。它在图像上滑动，通过将卷积核的值与对应的图像像素相乘来计算新的像素值。

4. Cross-Correlation vs. Convolution | 互相关与卷积的区别

In mathematics, convolution requires flipping the kernel both horizontally and vertically before applying it to the image, while cross-correlation does not involve flipping. However, in deep learning, the term "convolution" is often used, even though mathematically it is closer to cross-correlation because no flipping is applied. 在数学上，卷积需要先将卷积核在水平和垂直方向上翻转后再应用于图像，而互相关则不需要翻转。然而，在深度学习中，即使没有进行翻转，"卷积" 这个术语仍然常常使用，尽管数学上这更接近互相关。

Common Convolution Kernels | 常见的卷积核

1. Identity Kernel | 单位映射卷积核

• Purpose | 目的: This kernel does nothing to the image. It simply outputs the original image as it is.

• Kernel | 卷积核:

  $$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$
• Explanation | 解释: The identity kernel leaves the pixel values unchanged. It multiplies the center pixel by 1 and all others by 0.

2. Mean Smoothing Kernel | 平滑均值卷积核

• Purpose | 目的: This kernel is used for blurring or smoothing an image by averaging neighboring pixel values.

• Kernel | 卷积核:

  $$\frac{1}{9} \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$
• Explanation | 解释: This kernel takes the average of the surrounding pixels (including the center pixel), which results in a smooth or blurred version of the image.

3. Sharpening Kernel | 锐化卷积核

• Purpose | 目的: This kernel is used to enhance edges and details, making the image appear sharper.

• Kernel | 卷积核:

  $$\begin{bmatrix} 0 & -1 & 0 \\ -1 & 5 & -1 \\ 0 & -1 & 0 \end{bmatrix}$$
• Explanation | 解释: The center pixel is enhanced by reducing the surrounding pixel values, making edges more distinct.

4. Edge Detection (Sobel) Kernel | 边缘检测（Sobel）卷积核

• Purpose | 目的: Sobel kernels are commonly used to detect horizontal or vertical edges in images.

• Vertical Sobel Kernel | 垂直边缘检测卷积核:

  $$\begin{bmatrix} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 \end{bmatrix}$$
• Horizontal Sobel Kernel | 水平边缘检测卷积核:

  $$\begin{bmatrix} -1 & -2 & -1 \\ 0 & 0 & 0 \\ 1 & 2 & 1 \end{bmatrix}$$
• Explanation | 解释: The Sobel kernels highlight changes in pixel intensity along the vertical or horizontal axis, which helps detect edges.

5. Gaussian Blur Kernel | 高斯模糊卷积核

• Purpose | 目的: This kernel is used for applying a Gaussian blur, which smooths the image by giving more weight to the center pixel and progressively less to surrounding pixels.

• Kernel (3x3 example) | 卷积核（3x3 示例）:

  $$\frac{1}{16} \begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{bmatrix}$$
• Explanation | 解释: This kernel applies a Gaussian blur by taking a weighted average of the surrounding pixels, with the center pixel contributing the most to the final result.

6. Laplacian Kernel | 拉普拉斯卷积核

• Purpose | 目的: The Laplacian kernel is used for edge detection by highlighting regions of rapid intensity change.

• Kernel | 卷积核:

  $$\begin{bmatrix} 0 & -1 & 0 \\ -1 & 4 & -1 \\ 0 & -1 & 0 \end{bmatrix}$$
• Explanation | 解释: The Laplacian kernel detects edges by looking for regions where pixel values change significantly from their neighbors.

7. Prewitt Kernel | Prewitt 卷积核

• Purpose | 目的: Similar to the Sobel operator, Prewitt kernels are used for edge detection, typically less sensitive to noise.

• Vertical Prewitt Kernel | 垂直 Prewitt 卷积核:

  $$\begin{bmatrix} -1 & 0 & 1 \\ -1 & 0 & 1 \\ -1 & 0 & 1 \end{bmatrix}$$
• Horizontal Prewitt Kernel | 水平 Prewitt 卷积核:

  $$\begin{bmatrix} -1 & -1 & -1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$
• Explanation | 解释: Prewitt kernels are simpler than Sobel kernels and are used to detect vertical or horizontal edges.