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Generalized signal averaging

<h2 id="description">Description</h2>
<p>Averaging is a special case of discrete convolution. For a 3 by 3 neighborhood, the convolution mask <i>M</i> is:
</p><p>
  
    
      
        M
        =
        
          
            1
            9
          
        
        
          
            [
            
              
                
                  1
                
                
                  1
                
                
                  1
                
              
              
                
                  1
                
                
                  1
                
                
                  1
                
              
              
                
                  1
                
                
                  1
                
                
                  1
                
              
            
            ]
          
        
      
    
    {\displaystyle M={\frac {1}{9}}{\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\\\end{bmatrix}}}

</p><p>The significance of the central pixel may be increased, as it approximates the properties of noise with a <a href="/facts/Gaussian/PcIz5lFw">Gaussian</a> probability distribution:
</p><p>
  
    
      
        M
        =
        
          
            1
            10
          
        
        
          
            [
            
              
                
                  1
                
                
                  1
                
                
                  1
                
              
              
                
                  1
                
                
                  2
                
                
                  1
                
              
              
                
                  1
                
                
                  1
                
                
                  1
                
              
            
            ]
          
        
      
    
    {\displaystyle M={\frac {1}{10}}{\begin{bmatrix}1&1&1\\1&2&1\\1&1&1\\\end{bmatrix}}}

</p><p>
  
    
      
        M
        =
        
          
            1
            16
          
        
        
          
            [
            
              
                
                  1
                
                
                  2
                
                
                  1
                
              
              
                
                  2
                
                
                  4
                
                
                  2
                
              
              
                
                  1
                
                
                  2
                
                
                  1
                
              
            
            ]
          
        
      
    
    {\displaystyle M={\frac {1}{16}}{\begin{bmatrix}1&2&1\\2&4&2\\1&2&1\\\end{bmatrix}}}

</p><p>A suitable page for beginners about matrices is at:
<a href="https://web.archive.org/web/20060819141930/http://www.gamedev.net/reference/programming/features/imageproc/page2.asp">https://web.archive.org/web/20060819141930/http://www.gamedev.net/reference/programming/features/imageproc/page2.asp</a>
</p><p>The whole article starts on page: <a href="https://web.archive.org/web/20061019072001/http://www.gamedev.net/reference/programming/features/imageproc/">https://web.archive.org/web/20061019072001/http://www.gamedev.net/reference/programming/features/imageproc/</a>
</p>