Digital image processing---Basics of spatial filtering

Spatial filtering concept

Filtering: By modifying or suppressing specific frequency components of a given image, low-pass filters and high-pass filters are common.
Spatial filtering: Replace the pixel value with the value of the pixel value and its neighborhood. There are two replacement methods: linear filter and nonlinear filter.

linear filter

The general method is to use a matrix (also called a kernel) to perform a similar convolution operation with the image to do the product.
The values inside the matrix determine the function of the filter.

(The picture below is from Digital Image Processing 4th Edition=DIGITAL IMAGE PROCESSING,FOURTH EDITION)

separable filter core

Decompose a 2D matrix into the product of two 1D vectors.
$G(X,Y)=G_1(X)G_2(Y)$
如
$w=\begin{equation} \left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ & 1 & 1 \ end { array } \ right ] \ end { equation }$
Defaults
$w_1=\begin{equation} \left[\begin{array}{ccc} 1 \\ 1 \\ 1 \end{array}\ right ] \ end { equation } \hspace { 1.5cm } w_2= \ begin { equation } \ left [ \ begin { array } {ccc } 1 & 1 & 1 \ \ end { array } \ right ] \ end { equation } }$
The product of the two.

PS:判断一个核是否是可分离卷积核，即判断该卷积核是否是秩为1.若为1，即可分离，否则不可。

The benefit of convolution being separable:
Assume the image size is $M * N$ , the convolution kernel size is $m * n$ , the calculation amount of direct convolution includes $MN mn$ times multiplication and $MN mn$ additions. If separable convolution is used, $w$ minute $w_1*w_2$ , and $w_1$ is $m * 1$ ， $w_2$ is $1 * n$ , combine the image matrix with $w_1$ sum $w_2$ Do convolution separately to get the amount of operation: $MN m + MN n$ ，即 $MN (m + n)$ . Compared to the previous $MN mn$ , greatly reducing the amount of calculation.

Determination of separable vectors:
According to the properties of the matrix, for a matrix with rank 1, the difference between each row is only a constant multiple, and the difference between each column is only a constant multiple. Then we can find any row and column to solve. Of course It is necessary to ensure that both the row and column are non-zero vectors.

filter type

linear filter

Low pass spatial filter
$\hspace{0.5cm}$ It mainly acts on areas that change slowly, such as smooth floors; table edges, etc. are high-frequency areas.
1.1 Box filter kernel
$\hspace{0.5cm}$ The value of this filter kernel is all 1, which can produce a smooth effect on the picture.
1.2 Low-pass Gaussian filter kernel
$\hspace{0.5cm}$ 高斯核 $G(s,t)=Ke^{-\frac{s^2+t^2}{2\sigma^2}}$
$\hspace{0.5cm}$ Among them $s$ and $t$ is the coordinate (with the core as the origin), $(s^2+t^2)^{\frac{1}{2}}$ is the distance to the original center.

In particular, the Gaussian kernel is also separable.
High pass filter
$\hspace{0.5cm}$ Act on images through edge enhancement and other methods.

nonlinear filter

Median filter
$\hspace{0.3cm}$ Sort a certain pixel and the pixels in its neighborhood. The value of the central pixel is determined by the intermediate value of the sorted pixels in the neighborhood. It is especially effective in eliminating salt and pepper noise.
Maximum filter and minimum filter
$\hspace{0.3cm}$ Maximum filter: sorts a pixel and the pixels in its neighborhood. The value of the center pixel is determined by the maximum sorted pixel value in the neighborhood.
$\hspace{0.3cm}$ Minimum filter: Sorts a pixel and the pixels in its neighborhood. The value of the center pixel is determined by the maximum sorted pixel value in the neighborhood.
Midpoint filter
$\hspace{0.3cm}$ Sort a pixel and the pixels in its neighborhood. The central pixel value is determined by the average of the maximum value and the minimum value. It is suitable for processing randomly distributed noise, such as Gaussian noise or uniform noise.
Otherwise $A l pha mean$ filter
$\hspace{0.3cm}$ An improvement on the median filter, which deletes part of the lowest gray value and the highest gray value from a certain pixel and its neighborhood, and calculates the arithmetic mean of the remaining pixel values as the central pixel value, which is suitable for processing mixed noise , such as a mixture of salt-and-pepper noise and Gaussian noise.

adaptive filter

Adaptation focuses on considering changes in different feature points of the image.

Adaptive local noise reduction filter
$\hspace{0.3cm}$ The parameters that need to be considered for this filter involve the image pixel value and noise variance with noise; the local pixel mean and local variance in the filter.
The pixel value of the image with noise is $f (x, y)$ , the noise variance is $\sigma_f$ , the local mean is $\hat{f_{xy}}$ , the local variance is $\sigma_{xy}$
The calculation method of the four parameters can be obtained by multiplying the frequency and correlation value of the histogram.
The adaptive expression is $\bar{f}(x,y)=f (x,y)-\frac{\sigma_f^2}{\sigma_{xy}^2}[f(x,y)-\hat{f_{xy}}]$
The characteristic of this formula is
$\hspace{0.6cm}$ If the variance of the noise is 0, the filtering result is $f (x, y)$ ；
$\hspace{0.6cm}$ If the variance of the noise is equal to the local variance, the filtering result is the mean of the local pixel values of the image;
Adaptive median filter
$\hspace{0.3cm}$ Optimization of median filtering
Smoothes non-impulsive noise (non-dramatic changes) while preserving image details
Similar to the adaptive local noise reduction filter, calculations are performed in local areas.
In the local area, $z_{min}$ Represents the minimum gray value, $z_{max}$ Represents the maximum gray value, $z_{med}$ Represents the median gray value, $z_{xy}$ is the coordinate $(x,$ The gray value at $y$ $)$ $S_{max}$ Indicates the size of the local area.
Algorithm steps:
1. 若 $z_{min}$ < $z_{med}$ < $z_{max}$ , then go to the second step; otherwise, increase the local area size, if the local area size is less than $S_{max}$ , repeat step 1, otherwise output $z_{med}$
2. 若 $z_{min}$ < $z_{xy}$ < $z_{max}$ , then output $z_{xy}$ , otherwise output $z_{med}$
  The main idea of the algorithm is to eliminate impulse noise through two layers of judgment. If you switch from the first step to the second step, $z_{med} can be guaranteed$ is non-impulsive noise; in the second step, regardless of the output $z_{med}$ Or $z_{xy}$ All achieved the purpose of eliminating impulse noise. But if the output ends up in the first step, there is no guarantee $z_{med}$ Whether it is impulse noise.
  PS:所以该算法旨在尽力消除，不保证全部消除。