# Module-5 - Module 5 Linear Image Filtering Wraparound and...

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1 Module 5 Linear Image Filtering Wraparound and Linear Convolution Linear Image Filters Linear Image Denoising Linear Image Restoration (Deconvolution) QUICK INDEX

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2 WRAPAROUND CONVOLUTION Modifying the DFT of an image changes its appearance . For example, multiplying a DFT by a zero-one mask predictably modifies image appearance:
3 Multiplying DFTs What if two arbitrary DFTs are (pointwise) multiplied together? or The answer has profound consequences in image processing. Division is a special case which need special handling if contains near-zero or zero values. 1 2 = 1 J I I % % % 1 2 = J I I % % % 2 I %

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4 Multiplying DFTs Consider the product This has inverse DFT 1 2 = 1 J I I % % % N-1 M-1 -ui -vj N M u=0 v=0 N-1 M-1 -ui -vj 1 2 N M u=0 v=0 N-1 M-1 N-1 M-1 N-1 M-1 um vn up vq -ui -vj 1 N M 2 N N M M u=0 v=0 m=0 n=0 p=0 q=0 1 J(i, j)= J(u, v)W W NM 1 = I (u, v)I (u, v)W W NM 1 = I (m, n)W W I (p, q)W W W W NM % % %
5

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6 ( 29 ( 29 ( 29 ( 29 N-1 M-1 N-1 M-1 N-1 M-1 u p+m-i v q+n-j 1 2 N M m=0 n=0 p=0 q=0 u=0 v=0 N-1 M-1 N-1 M-1 1 2 m=0 n=0 p=0 q=0 M-1 1 2 N M n=0 1 = I (m, n) I (p, q) W W NM 1 = I (m, n) I (p, q) NMδ(p+m-i,q+n-j) NM = I (m, n)I i-m , j-n 1 ( 29 ( 29 N-1 m=0 N-1 M-1 1 2 N M p=0 q=0 1 2 = I i-p , j-q I (p, q) =I (i, j) I (i, j) 1 # 1 2 1 2 is the of with wraparound c I I I onvolution I #
7 Wraparound Convolution The summation is also called cyclic convolution and circular convolution . Like linear convolution , it is an inner product between one sequence and a (doubly) reversed, shifted version of the other – except with indices taken modulo-M,N . ( 29 ( 29 1 2 N-1 M-1 1 2 N M m=0 n=0 J(i, j)=I (i, j) I (i, j) = I (m, n)I i-m , j-n #

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8 Depicting Wraparound Convolution Consider hypothetical images and at which we wish to compute the cyclic convolution at (i, j) in the spatial domain (without DFTs). 1 I 2 I (0,0) (0,0) Image I 1 j i Image I 2
9 Without wraparound: Modulo arithmetic defines the product for all 0 < i < N-1, 0 < j < M-1. (0,0) j i Doubly-reversed and shifted I 2 (N-1, M-1)

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10 I 2 Overlay of periodic extension of shifted Summation occurs over 0 < i < N-1, 0 < j < M-1 (0,0) j i (N-1, M-1)
11 Computation of Wraparound Convolution Direct computation of is simple but expensive. For an N x M image: - for each of NM coordinates: NM additions and NM multiplies - or (NM)(NM) multiplies - for N=M=512, this is ( 29 ( 29 1 2 N-1 M-1 1 2 N M m=0 n=0 J(i, j)=I (i, j) I (i, j) = I (m, n)I i-m , j-n # 26 10 2 = 6.9×10

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DFT Computation of Wraparound Convolution Because of FFT , computing 1 in the DFT domain is much faster, provided that N = a power of 2. Simply Computing an (N x M) FFT is 1 [ NM· log (NM) ] , so computation of 1 is as well. We now will discover that
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Module-5 - Module 5 Linear Image Filtering Wraparound and...

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