# Analysis of Blur Measure Operators for Single Image Blur Segmentation
## 3 Material and Method
### Methodology
#### 3.2.1 Blur Measures  
Let \(I_{b}(x,y)\) be the blurred input image for which the amount of blurriness is to be computed at each pixel location. This can be achieved by the application of a blur measure operator \(B\) in the local neighborhood window-wise environment around a pixel \(\Omega(x,y)\). For this, \(B\) is applied on a local image neighborhood window \(\Omega(x,y)\) centered at pixel \(\Omega(x,y)\) to provide the blur measurement for this pixel. By sliding this neighborhood window for all the image pixels to be at the center of the neighborhood window one-by-one, the whole image can be traversed to result into the generation of the initial blur map \(M(x,y)\).  
\[M(x,y)=B(I_{b}(x,y)). \tag{1}\]  
Then, this initial blur map is normalized linearly so that \(M(x,y)\in[0,1]\). The resultant normalized blur map can be expressed as  
\[\hat{M}(x,y)=\frac{M(x,y)-min(M)}{max(M)-min(M)}, \tag{2}\]  
where \(min(M)\) and \(max(M)\) are the minimum and maximum measurements in the initial blur map. In the literature, a wide variety of blur detection techniques and blur measure operators have been  
Figure 1: Sample images and their ground-truth from the data set. Row 1: First 4 images are defocus blurred while last 2 images are motion blurred. Row 2 displays ground-truth blur maps for row 1 images.  
proposed to measure the level of blur of either the whole image or neighborhood window or individual pixels. In this work, blur measure operators have been grouped into four broad categories according to their working principle. This categorization is to recognize the similarities and differences, if any, in their performances and then rank them accordingly. A brief description of each of the categories is presented in this section. The interested reader is referred to Appendix A for more detailed description of the blur measure operators which have been analyzed in this work. Abbreviations have been used for the blur measure operators as described in Table 1. This is to refer them conveniently, as well as to signify the category to which they belong.  
The four categories of blur measure operators analyzed in this work are:  
1. Derivative-based operators [DER*]: The blur measure operators in this category are based on the derivative of the image. These operators are based on the assumption that non-blurred images present sharp edges as compared to blurred images. First and second order derivatives of the image neighborhood windows provide the base to distinguish between blurred and non-blurred regions of the image.
2. Statistical-based operators [STA*]: The blur measure operators of this category utilize several statistical measures which are computed on image neighborhood windows to differentiate between blurred and non-blurred neighborhood windows in the image.
3. Transform-based operators [TRA*]: The blur measure operators within this category are based on the transform domain representations of the image content. These frequency domain representations offer to be the true replica of the same information as in the spatial domain and thus this frequency content of the image can be utilized to differentiate between blurred and non-blurred regions of the image.
4. Miscellaneous operators [MIS*]: These operators do not belong to any of the previously mentioned categories.  
Figure 2: Framework for Blur Segmentation.