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Image Deblur

Image Resizing

Before implement the deblurring, we resize the blurred image B by mk to compensate for the difference between blur size k and code length m. Inthe resized image, the blur size is m instead of k.

Frequency Domain Deblur

  • Coded: We obtain the frequency response of the filter H(k) and the frequency content of the blurred image B(k) through DFT. Thesystem can be represented in frequency domain as B j ( k ) = H ( k ) X j ( k ) + N ( k ) where B j ( k ) and X j ( k ) are the DFT of the j-th column of the blurred image and the original image, respectively. N ( k ) represents the noise. We divide B j ( k ) by H ( k ) to obtain X j ( k ) . We then took the inverse DFT of X ( k ) to obtain the recovered image.
  • Boxcar: We use almost the same frequency domain deblur approach as used for the coded filter. The primary difference betweenthe two approaches is the following: the frequency response of a boxcar filter is a sinc function, which contains zeros at certainpoints. We added a small number to the zeros to avoid division-by-zero problems. So H ( k ) in this case is a slightly modified version of the H ( k ) used for the coded filter.

Time Domain Deblur

  • Coded: We first generate a circulant matrix whose characteristic vector contains the code sequence. Then we kick out thewrap-around part of the circulant matrix and take only the left n columns, which construct the foreground smear matrix Af whose length matches that of the blurred image B . Then we concatenate Af with the background contribution vectors to generate the matrix A for the linear system B = A X , where B is the blurred image and X is the original image. Then we use least square estimation to find the solution for this system,which gives the deblurred image X , the estimate of X . Background contribution vectors are b k = a b s ( 1 - A f * o n e s ( n , 1 ) ) , where n is the length of the original image. The first m elements in bk account for the background contribution at the top of the blurred image, and the last m elements account for that at the bottom.Thus we generate two vectors of length n + m - 1 . The first m elements of the first vector are the first m elements in bk , and the last m elements of the second vectorare the last m elements in bk . The rest elements of the two vectors are all zeros. These two vectors are then concatenated to Af to generate the smear matrix A .
  • Boxcar: We use basically the same approach as used in time domain deblurring, except that we use the 52-length boxcar as the characteristic vector ofthe matrix Af .

Over-exposure Compensation:

The deblurred image might have some element values above 1 or below 0. We set all the values below 0 to 0 and all above 1 to 1.

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Source:  OpenStax, Elec 301 projects fall 2015. OpenStax CNX. Jan 04, 2016 Download for free at https://legacy.cnx.org/content/col11950/1.1
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