Image filter from the perspective of convolution

         Image filter can be understood as image interpretation of convolution operation , translation 、 rotate 、 Scale transformation 、 wave filtering 、 Enhancement and so on , Can be achieved by convolution .

1、 Mean filtering

         Mean filtering , As the name suggests, it is to take the average value in the window instead of the original value . The convolution template is used to express the following ：

         Every point in the image , Take the average of the eight neighborhood centered on itself as the filtered value . Expressed mathematically ,f by m*n Size image ,g by k * l The convolution kernel of magnitude , as follows ：

2、 translation

         Image with a center point of 1, Others are 0 Convolution of , The result is itself （ The boundary treatment is inconsistent with the original drawing ）.

         translation , Such as right translation , On the right of convolution kernel is 1, Others are 0, The effect is as follows ：

3、 edge 、 sharpening

3.1、 Edge extraction

         What is the edge ？ Original picture - The graph after average filtering and smoothing = Marginal graph .

         Turn to convolution operation expression ：

3.2、 sharpening

         What is sharpening ？ Original picture + Marginal graph = sharpening .

         Turn to convolution operation expression ：

3.3、 Laplace Gauss sharpening

Mathematical expression ：

4、 Convolution operation properties

4.1、 Image and convolution operations have the following properties

1） linear ：

2） Translation invariance ：

4.2、 Convolution kernel properties

1） Commutative law ：

2） Associative law ：

3） Distributive law ：

4） scale ：

5） The convolution center is 1 The nuclear result of is itself ：

5、 Gaussian convolution kernel

         The mean filtering of square template is easy to make the image ring , as follows ：

         therefore , In practice, we prefer the template to be smooth , The weight high point close to the current point , The weight low point far away . Gaussian function meets such a demand , Two dimensional Gauss is expressed as follows ：

         notes ： Here, it is required that the sum of all template weights is 1, To ensure that the overall brightness of the image remains unchanged . therefore , Generally, it will be normalized after the Gaussian template .

         Three elements ： Window width 、 variance 、 Template normalization .

         When the window width remains unchanged , The greater the variance , The flatter the shape , The more concentrated the intermediate value . Smooth is not so powerful .

         With the variance unchanged , The size of the window width is only affected by the template normalization in the last step . General situation 3o The confidence level is 98%,2o Degree of confidence 90%. Generally, half the window width is equal to 3o.

6、 Properties of Gaussian convolution kernel

         nature 1： The image is convoluted twice by Gaussian convolution kernel , Equivalent to using a root 2 Convolution kernel of times variance once . That is, a large Gaussian convolution kernel can be obtained by continuous convolution of two small convolution kernels .

         nature 2： The two-dimensional Gaussian kernel can be decomposed into two one-dimensional Gaussian kernels .