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# ds2 - ECE 178 Digital Image Processing Discussion Session#2...

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ECE 178 Digital Image Processing Discussion Session #2 Mehmet Emre Sargin [email protected] January 19, 2007 Linearity and Time Invariance: Consider a system L { . } that relates the input x [ m, n ] to y [ m, n ], y 1 [ m, n ] = L { x 1 [ m, n ] } y 2 [ m, n ] = L { x 2 [ m, n ] } The system L { . } is called linear if Ay 1 [ m, n ] + By 2 [ m, n ] = L { Ax 1 [ m, n ] + Bx 2 [ m, n ] } The system L { . } is called time invariant if y [ m - m 0 , n - n 0 ] = L { x [ m - m 0 , n - n 0 ] } 1

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DS#2 ECE 178 Mehmet Emre Sargin [email protected] Importance of Linearity and Time Invariance: Consider a system L { . } that relates the input x [ m, n ] to y [ m, n ]. y [ m, n ] = L { x [ m, n ] } (1) We can write down x [ m, n ] as x [ m, n ] = X k,l x [ k, l ] δ [ m - k, n - l ] Then the Equation 1 becomes: y [ m, n ] = L { X k,l x [ k, l ] δ [ m - k, n - l ] } If L { . } is linear: y [ m, n ] = X k,l x [ k, l ] L { δ [ m - k, n - l ] } If L { . } is time invariant and the response of L { . } to the δ [ m, n ] is h [ m, n ], i.e. h [ m, n ] = L { δ [ m, n ] } The Equation 1 boils down to y [ m, n ] = X k,l x [ k, l ] h [ m - k, n - l ] Which is the convolution of x [ m, n ] and h [ m, n ]. y [ m, n ] = x [ m, n ] * h [ m, n ] = X k,l x [ k, l ] h [ m - k, n - l ] 2
DS#2 ECE 178 Mehmet Emre Sargin [email protected] Exercise: 1

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ds2 - ECE 178 Digital Image Processing Discussion Session#2...

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