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2D convolution

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					       Let x(m,n) be the 2-D signal (Image) and h(m,n) be the
       impulse response of the system



            x(m,n)                        h(m,n)                            y(m,n)




                      y(m, n)  h(m, n)  x(m, n)


Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
     Let us take an example
        n                                               n


             1 4 1                                           1 1
             2 5 3                                           1 -1
                                   m                                           m


            x(m,n)                                          h(m,n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA        Digital Image
   n                                  n                                n


        1 1                           1 1
        1 -1                         -1 1                             -1 1
                              m                                m       1 1                  m


       h(m,n)                             h(-m,n)                           h(-m,-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
                                         n



                                              1 1
                                              1 -1
                                                                     m


                                             h(-m,n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA        Digital Image
                                         n


                                        1 1
                                       -1 1
                                                                     m


                                             h(-m,-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA        Digital Image
                    y(m, n)  h(m, n)  x(m, n)
                              M 1 N 1
                 y (m, n)     h(m  m, n  n) x(m, n)
                              m  0 n   0


   Length of m=M+ number of columns in h(m,n) =4

   Length of n=M+ number of columns in h(m,n)= 3
                                          0≤m≤3

                                          0≤n≤2

Jagadish Nayak    Department of Electronics and Communication Engineering,
                      Manipal Institute of Technology, Manipal, INDIA        Digital Image
      y(0,0)   h(m,n) x(m, n) = 2x1=2
        n                                   n


             1 4 1
             2 5 3                        -1 1
                                   m
                                   m       1 1                       m


            x(m,n)                              h(-m,-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA        Digital Image
     y(0,1)   h(m,1  n) x(m, n) = 2x1+1x1=3
        n                                      n


             1 4 1                            -1 1
             2 5 3                             1 1
                                   m
                                                                        m


            x(m,n)                                 h(-m,1-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA        Digital Image
    y(0,2)   h(m,2  n) x(m, n)= 1x1=1
        n                                       n

                                               -1 1
             1 4 1                              1 1
             2 5 3
                                   m
                                                                            m


            x(m,n)                                  h(-m,2-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
     y(1,0)   h(1  m,n) x(m, n) = 2x-1+5x1=3
        n                                        n


             1 4 1
             2 5 3                                   -1 1
                                   m
                                                      1 1                   m


            x(m,n)                                   h(1-m,-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
    y(1,1)   h(1  m,1  n) x(m, n)= 2x1+5x1+1x-1+4x1=10
        n                                         n


             1 4 1                                    -1 1
             2 5 3                                     1 1
                                   m
                                                                            m


            x(m,n)                                    h(1-m,1-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
   y(1,2)   h(1  m,2  n) x(m, n)= 1x1+4x1=5
        n                                         n

                                                      -1 1
             1 4 1                                     1 1
             2 5 3
                                   m
                                                                            m


            x(m,n)                                    h(1-m,2-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
    y(2,0)   h(2  m,n) x(m, n)= 5x-1+3x1=-2
        n                                         n


             1 4 1
             2 5 3                                       -1 1
                                   m                      1 1               m


            x(m,n)                                    h(2-m,-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
   y(2,1)   h(2  m,1  n) x(m, n) =5x1+3x1+4x-1+1x1=5
        n                                         n


             1 4 1                                       -1 1
             2 5 3                                        1 1
                                   m
                                                                            m


            x(m,n)                                    h(2-m,1-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
   y(2,2)   h(2  m,2  n) x(m, n)=4x1+1x1=5
        n                                         n

                                                         -1 1
             1 4 1                                        1 1
             2 5 3
                                   m
                                                                            m


            x(m,n)                                    h(2-m,2-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
    y(3,0)   h(3  m,n) x(m, n) =3x-1=-3
        n                                         n


             1 4 1
             2 5 3                                            -1 1
                                   m
                                                              1 1           m


            x(m,n)                                    h(3-m,-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
    y(3,1)   h(3  m,1  n) x(m, n) =3x1+1x-1=2
        n                                         n


             1 4 1                                           -1 1
             2 5 3                                    1       1 1
                                   m
                                                                            m


            x(m,n)                                    h(3-m,1-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
   y(3,2)   h(3  m,2  n) x(m, n) =1x1=1
        n                                         n

                                                             -1 1
             1 4 1                                            1 1
             2 5 3                                    1
                                   m
                                                                            m


            x(m,n)                                    h(3-m,2-n)



Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA            Digital Image
                                      n

                                           1 5 5 1
                                           3 10 5 2
                                           2 3 -2 -3
                                                                  m



                                          y(m,n)




Jagadish Nayak   Department of Electronics and Communication Engineering,
                     Manipal Institute of Technology, Manipal, INDIA        Digital Image

				
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posted:4/16/2012
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