# 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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 views: 47 posted: 4/16/2012 language: English pages: 19