1.5 Image deblur

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 $B_j\left(k\right)=H\left(k\right)X_j\left(k\right)+N\left(k\right)$ where $B_j\left(k\right)$ and $X_j\left(k\right)$ are the $\text{DFT}$ of the j-th column of the blurred image and the original image, respectively. $N\left(k\right)$ represents the noise. We divide $B_j\left(k\right)$ by $H\left(k\right)$ to obtain $X_j\left(k\right)$ . We then took the inverse $\text{DFT}$ of $X\left(k\right)$ 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 $H\left(k\right)$ in this case is a slightly modified version of the $H\left(k\right)$ 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 $\text{Af}$ whose length matches that of the blurred image $B$ . Then we concatenate $\text{Af}$ with the background contribution vectors to generate the matrix $A$ for the linear system $B=AX$ , 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 $bk=abs(1-Af*ones(n,1\left)\right)$ , where $n$ is the length of the original image. The first m elements in $\text{bk}$ 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 $\text{bk}$ , and the last m elements of the second vectorare the last m elements in $\text{bk}$ . The rest elements of the two vectors are all zeros. These two vectors are then concatenated to $\text{Af}$ 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 $\text{Af}$ .

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.