Catalog

1 Definition of convolution operation and transpose convolution

1.1 Convolution operation

1.2  Transposition convolution

1.3  The steps of transpose convolution

2  The understanding of transpose convolution

2.1 stride=1 The understanding of transpose convolution

2.1.1 ​​​​​​​ One dimensional form

2.1.2  Two dimensional form

2.1.3 Formula calculation  

2.2 stride>1 The understanding of transpose convolution  

2.2.1 One dimensional form ​​​​​​​

2.2.2  Two dimensional form

2.2.3  Formula calculation

3 summary

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        Transposition convolution （Transpose Convolution）, Some places are also called “ deconvolution ”, In deep learning, it is expressed as an inverse process of convolution , According to the size of convolution kernel and the size of output , Restore the image size before convolution , Instead of restoring the original value .

1 Definition of convolution operation and transpose convolution

1.1 Convolution operation

        For an input, the size is Image , The convolution kernel size is ：

 

        Calculate the output    The specific implementation method is matrix multiplication .

        The convolution kernel is expressed as a sparse matrix

        

          Each line vector represents the convolution operation at one position ,0 Filling indicates the area not covered by the convolution kernel .  

        Enter    Expand to column vector ：

 

        Then the convolution operation can be expressed as ： 

        

        Output vector    The size is    The column vector , Rewritten as a matrix is   . 

1.2  Transposition convolution

        Now? , Think about how we can change the direction of calculation . in other words , We want to build on a matrix 1 Values and... In another matrix 9 Relationship between two values . This is it. image In the reverse operation of convolution , This is the core idea of transpose convolution .

       ( translator's note ： From the perspective of information theory , Convolution is irreversible . So it's not from output Matrix and kernel The matrix calculates the original input matrix . Instead, it calculates a matrix that maintains the positional relationship .) 

          Transpose convolution is to    The inputs and outputs in the are interchangeable ：

          

• Represents matrix transpose , Now the size is   .
• by    The input of   , After transpose convolution , The output size is    The column vector   , here Of   The value is different from the original , Just consistent in shape .

1.3  The steps of transpose convolution

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Example ：

How to calculate convolution ： 

Calculation steps of transpose matrix ：

 

 

 

 

2  The understanding of transpose convolution

2.1 stride=1 The understanding of transpose convolution

2.1.1 ​​​​​​​ One dimensional form

        In this way, we can get the same result as transpose convolution . It's actually the input for transpose convolution    Add extra padding, The convolution kernel is mirror flipped . It's still Compare the correspondence of element position relationship , Regardless of the numerical value . 

What's the use of such an understanding ？

        Because the convolution kernel of transposed convolution is also learning Of , Given the size of the input and convolution kernel , We can think of transpose convolution as having padding The general convolution of . in other words , So called transpose convolution , Equivalent to ordinary convolution , No more than is padding large , It can achieve the effect of up sampling .

2.1.2  Two dimensional form

2.1.3 Formula calculation  

        about The general convolution of , The output size of the equivalent transpose convolution is ：. At the same time, the input of transpose convolution is required to add . That is, the output of transpose convolution becomes larger is the influence of convolution kernel .

 

2.2 stride>1 The understanding of transpose convolution  

2.2.1 One dimensional form

2.2.2  Two dimensional form

        It is also equivalent to adding holes to the input , additional padding, Ordinary convolution with horizontal and vertical flip of convolution kernel ： 

 

2.2.3  Formula calculation

The following figure shows the different in transpose convolution s and p The situation of ：

s=1, p=0, k=3	s=2, p=0, k=3	s=2, p=1, k=3

 

 

 

 

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3 summary

        Transposed convolution and ordinary convolution have the same essence ： The relationship between some values is established . It's just , The relationship established by transpose convolution is the same as that established by ordinary convolution , In the opposite direction .

        We can Upsampling using transpose convolution . also , The weights in transpose convolution can be learned . therefore , We don't need any interpolation method to do up sampling .

        Although it is called transpose convolution , However, this does not mean that we take the transpose of an existing convolution matrix as the weight matrix to carry out the transpose convolution operation . Compared with ordinary convolution ,intput and output The relationship is reversed ( Transpose convolution is 1 For more than , Instead of ordinary many pairs 1), Is the essence of transpose convolution .

        Because of this , Strictly speaking, transpose convolution is not convolution . But we can put input Fill in some positions in the matrix with 0 And perform ordinary convolution to obtain the same result as transpose convolution output matrix . You can find a lot of articles on how to use this method to realize transpose convolution . However , Because before the convolution operation , Go to input Fill in a lot of 0, This method actually has some efficiency problems .

        Warning ： Transpose convolution will cause checkerboard effect in the generated image (checkerboard artifacts).. In this paper, it is recommended to use transpose convolution for up sampling operation, and then use a common convolution to reduce this kind of problem . If you are mainly concerned with how to generate images without checkerboard effect , You need to read paper.

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