# Analysis of Blur Measure Operators for Single Image Blur Segmentation
## Appendix A Blur Measure Operators
### Statistical-Based Operators  
\([STA01]\) _Singular Value Decomposition_  
This blur measure is based on the eigen-values which are computed for each pixel in the image by placing that pixel in the center of a small image neighborhood window \(\Omega(x,y)\). [PERSON] et al. [16] utilized singular value decomposition of an image neighborhood window as a blur measure by exploiting the fact that larger singular values correspond to the the overall look of the neighborhood window while smaller singular values represent fine details. The proposed blur measure is  
\[M_{SVD}(x,y)=1-\frac{\sum_{i=1}^{k}\lambda_{i}}{\sum_{i=1}^{n}\lambda_{i}}, \tag{12}\]  
where \(k\) largest eigen-values have been considered. \([STA02]\) _Sparsity of Dark Channel_  
[PERSON] et al. [64] enforces the sparsity of dark channel to deblur an image \(I\). The dark channel for a pixel \((x,y)\) in a neighborhood window \(\Omega(x,y)\) is defined as  
\[M_{SDC}(x,y)=\min_{(i,j)\in\Omega(x,y)}\ \bigg{(}\min_{c\in\{r,g,b\}}\ I^{c}(i,j) \bigg{)}, \tag{13}\]  
where \(I^{c}\) is the \(c\)-th color channel. \([STA03]\) _Total Variation_  
By measuring the diversity in intensity values of image pixels in a small neighborhood, the blur can be estimated for that patch. Ref. [65] suggested such a measure:  
\[M_{TV}(x,y)=\frac{1}{4}\ \max_{\xi\in\Omega(x,y)}\ (TV(\xi)), \tag{14}\]where \(TV(\xi)=\frac{1}{255}\sum_{\xi_{i},\xi_{j}\xi\in\xi}|I(\xi_{i})-I(\xi_{j})|\) is the total variation of smaller blocks (let \(\xi\) is of \(2\times 2\)) in the image neighborhood window \(\Omega\). \([STA04]\) _Local Binary Pattern_  
[PERSON] et al. [14] exploits the observation that local neighborhood windows in blurry regions, in comparison with the sharp regions, have significantly lesser amount of higher LBPs (6 to 9 in case of 8-bit LBP), and proposed a blur measure given by  
\[M_{LBP}(x,y)=\frac{1}{N}\sum_{i=6}^{9}n(LBP_{8,1}^{riu2}i), \tag{15}\]  
where \(n(LBP_{8,1}^{riu2}i)\) is the number of rotation-invariant uniform 8-bit LBP pattern of type \(i\), and \(N\) is the total number of pixels in the image neighborhood window. \([STA05]\) _Gray-Level Variance_  
The variance of the image gray-levels as used in autofocus [66] can be taken as the blur measure for the central pixel \((x,y)\) in a local window \(\Omega(x,y)\)  
\[M_{GLV}(x,y)=\sum_{(i,j)\in\Omega(x,y)}\Big{(}I(i,j)-\mu\Big{)}^{2}, \tag{16}\]  
where \(\mu\) is the mean gray-level of pixels in the neighborhood window \(\Omega(x,y)\). \([STA06]\) _Gray-Level Local Variance_  
The local variance of gray-levels has been proposed as a focus measure [61] and it can be reformulated as a blur measure  
\[M_{GLL}(x,y)=\sum_{(i,j)\in\Omega(x,y)}\Big{(}L_{\
u}(i,j)-\overline{L_{\
u}} \Big{)}^{2}, \tag{17}\]  
where \(L_{\
u}(i,j)\) and \(\overline{L_{\
u}}\) are the variance and mean value within \(\Omega(x,y)\)  
\([STA07]\) _Normalized Gray-Level Variance_  
The image sharpness and blurriness can be differentiate by the normalized gray-level variance [67]  
\[M_{NGLV}(x,y)=\frac{\sum_{(i,j)\in\Omega(x,y)}\Big{(}I(i,j)-\mu\Big{)}^{2}}{ \mu(x,y)}, \tag{18}\]  
where \(\mu(x,y)\) is the mean value for computed over the neighborhood window \(\Omega(x,y)\). \([STA08]\) _Histogram Entropy_  
Entropy and range of the histogram [66] of the image indicate the diversity of information and can be utilized as the sharpness measure  
\[M_{Hist}(x,y)=-\sum_{k=1}^{L}P_{k}\log(P_{k}), \tag{19}\]where \(P_{k}\) is the frequency of the \(k\)-th gray-level within \(\Omega(x,y)\). \([STA09]\)_DCT Energy Ratio_  
[PERSON] and [PERSON] [68] used the DC/AC ratio of discrete cosine transform (DCT) for focus measure and it can be used as the blur measure  
\[M_{DCTE}(x,y)=\sum_{(i,j)\in\Omega(x,y)}f(i,j)^{2}, \tag{20}\]  
where \(f(i,j)=\frac{\sum_{u=0}^{M-1}\frac{\sum_{u=0}^{N-1}F(u,\
u)^{2}}{F_{0,0}^{2}}}{ F_{0,0}^{2}}\) is the DCT in an \(M\times N\) sub-block. \([STA10]\)_DCT Reduced Energy Ratio_  
[PERSON] et al.