空间滤波基础

图像滤波是一种常见的图像处理技术，用于平滑图像、去除噪音和边缘检测等任务。图像滤波的基本原理是在进行卷积操作时，通过把每个像素的值替换为该像素及其邻域的设定的函数值来修改图像。

预备知识：可分离滤波核、边缘填充。

一、线性滤波器

1、盒式滤波器（方框滤波器）
盒式核是最简单的低通滤波器核。盒式核中各像素点的系数相同（通常为1）。盒式滤波器因为也满足秩为1，所以也是可分离核，计算也可使用分离核进行加速。
$$K=\alpha \left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1\end{array}\right]当\alpha =\{\begin{array}{ll}\frac{1}{ksize.width\ast ksize.height}& \text{if }normalize=true\\ 1& \text{if }其他\end{array}$$

OpenCV函数：

void cv::boxFilter(InputArray src, OutputArray dst, int ddepth, Size ksize, Point anchor = Point(-1,-1), bool normalize = true, int borderType = BORDER_DEFAULT)Parameters
src				input image.
dst				output image of the same size and type as src.
ddepth			the output image depth (-1 to use src.depth()).
ksize			blurring kernel size.
anchor			anchor point; default value Point(-1,-1) means that the anchor is at the kernel center.
normalize		flag, specifying whether the kernel is normalized by its area or not.
borderType		border mode used to extrapolate pixels outside of the image, see BorderTypes. BORDER_WRAP is not supported.

2、均值滤波器
均值滤波器是特殊的盒式滤波器，目标图像中的每个值都是源图像中相应位置一个窗口（核）中像素的平均值。
$$K=\frac{1}{ksize.width\ast ksize.height}\left[\begin{array}{cccccc}1& 1& 1& ...& 1& 1\\ 1& 1& 1& ...& 1& 1\\ ...& & & & & \\ 1& 1& 1& ...& 1& 1\end{array}\right]$$
OpenCV函数：

void cv::blur(InputArray src, OutputArray dst, Size ksize, Point anchor = Point(-1,-1), int borderType = BORDER_DEFAULT)	Parameters
src				input image; it can have any number of channels, which are processed independently, but the depth should be CV_8U, CV_16U, CV_16S, CV_32F or CV_64F.
dst				output image of the same size and type as src.
ksize			blurring kernel size.
anchor			anchor point; default value Point(-1,-1) means that the anchor is at the kernel center.
borderType		border mode used to extrapolate pixels outside of the image, see BorderTypes. BORDER_WRAP is not supported.

3、高斯滤波器
高斯滤波器是通过根据高斯函数来选择权值的线性平滑滤波器的方式，对随机分布和服从正态分布的噪声有很好地滤除效果。高斯滤波器比盒式滤波器产生的边缘更加平滑，因为高斯滤波器的权重服从二维高斯分布，越靠近窗口中心点权重越大。
高斯核公式：
$$k(s,t)=K{e}^{-\frac{s^2+t^2}{2{\sigma }^2}}$$

OpenCV函数：

void cv::GaussianBlur(InputArray src, OutputArray dst, Size ksize, double sigmaX, double sigmaY = 0, int borderType = BORDER_DEFAULT)	Parameters
src				input image; the image can have any number of channels, which are processed independently, but the depth should be CV_8U, CV_16U, CV_16S, CV_32F or CV_64F.
dst				output image of the same size and type as src.
ksize			Gaussian kernel size. ksize.width and ksize.height can differ but they both must be positive and odd. Or, they can be zero's and then they are computed from sigma.
sigmaX			Gaussian kernel standard deviation in X direction.
sigmaY			Gaussian kernel standard deviation in Y direction; if sigmaY is zero, it is set to be equal to sigmaX, if both sigmas are zeros, they are computed from ksize.width and ksize.height, respectively (see getGaussianKernel for details); to fully control the result regardless of possible future modifications of all this semantics, it is recommended to specify all of ksize, sigmaX, and sigmaY.
borderType		pixel extrapolation method, see BorderTypes. BORDER_WRAP is not supported.

二、非线性滤波器

1、中值滤波器
中值滤波器用中心像素的邻域内的灰度值的中值替换中心像素的值。中值滤波器对冲激噪声（椒盐噪声）特别有效，并且对图像的模糊程度比线性平滑滤波器要小得多。

OpenCV函数：

void cv::medianBlur(InputArray src, OutputArray dst, int ksize)	Parameters
src				input 1-, 3-, or 4-channel image; when ksize is 3 or 5, the image depth should be CV_8U, CV_16U, or CV_32F, for larger aperture sizes, it can only be CV_8U.
dst				destination array of the same size and type as src.
ksize			aperture linear size; it must be odd and greater than 1, for example: 3, 5, 7 ...

2、双边滤波器
双边滤波器可以很好地减少不必要的噪声，同时保持边缘相当锐利。然而，与大多数过滤器相比，它非常慢。

OpenCV函数：

void cv::bilateralFilter(InputArray src, OutputArray dst, int d, double sigmaColor, double sigmaSpace, int borderType = BORDER_DEFAULT)	parameters
src				Source 8-bit or floating-point, 1-channel or 3-channel image.
dst				Destination image of the same size and type as src .
d				Diameter of each pixel neighborhood that is used during filtering. If it is non-positive, it is computed from sigmaSpace.
sigmaColor		Filter sigma in the color space. A larger value of the parameter means that farther colors within the pixel neighborhood (see sigmaSpace) will be mixed together, resulting in larger areas of semi-equal color.
sigmaSpace		Filter sigma in the coordinate space. A larger value of the parameter means that farther pixels will influence each other as long as their colors are close enough (see sigmaColor ). When d>0, it specifies the neighborhood size regardless of sigmaSpace. Otherwise, d is proportional to sigmaSpace.
borderType		border mode used to extrapolate pixels outside of the image, see BorderTypes