# No-Reference Blurred Image Quality Assessment by Structural Similarity Index
## 3 The Proposed IQA Metric
### Feature Extraction  
Considering the abundant information that images possess, the complexity of images can be represented based on their structure, noise and diversity [22] or based on fuzzy measures of entropy [23] or based on discrete wavelet transform decomposition [24], etc. In this paper, to represent the inputted image and the re-blurred image, we extract luminance, contrast, structure and blurriness of both the down-sampled 2D images, respectively. Then, we ameliorate the traditional structural similarity by combining the blurriness with luminance, contrast and structure in Figure 3. We would like to emphasize _image blurriness_ in this subsection.  
Image histogram reports grayscale distribution. As seen in Figure 4, we find that sharp images have a broader grayscale range. In contrast, grayscale distributions of blurred images are narrower and tend to approach the mean value according to their histograms. As is shown in Figure 4, grayscales  
Figure 2: Re-blur Process.  
close to the mean value \(\mu\) of the blurred image (e.g., pixels whose grayscale close to the 132.92 in the right histogram in Figure 4) take the most proportion in the histogram. Thus we describe image blurriness by distributing different weights to different pixel values. We assign heavy weights to the pixels those close to image mean grayscale value and fewer weights to those away from that. Thus we define the _image blurriness_ as  
\[d=\sum_{g_{i}=0}^{L}p\left(g_{i}\right)w\left(g_{i}\right) \tag{7}\]  
where \(d\) represents image blurriness, \(g_{i}\) is gray value whose range varies from 0 to the dynamic range \(L\) (e.g., \(L=255\) for 8-bit image), \(p(g_{i})\) is the proportion of \(g_{i}\) on the whole image, and \(w(g_{i})\) represents the weight of \(g_{i}\), which can be calculated as  
\[w(g_{i})=\left\{\begin{array}{ll}\frac{g_{i}}{\mu},&g_{i}<\mu\\ \\ \frac{L-g_{i}}{L-\mu},&g_{i}\geqslant\mu\end{array}\right. \tag{8}\]  
According to the proposed image blurriness index, we test images in Figure 1 to verify the impact of re-blur process. Using SSIM and image blurriness as image quality indexes, the results are shown in Table 1. It is shown that the quality of a sharp image has a more drastic decline than that of a blurred image. Then we analyze the image quality trend activated by re-blur, i.e., we blur the sharp image of Figure 1 with blur times varying from 0 to 11. Experimental results are shown in Figure 5. The proposed blurriness index has an approximate linear growth while the SSIM index gradually declined slower with the increase of blur times. This demonstrated that a high-quality image has a relatively small SSIM index when taking its blurred image as reference.  
In order to verify the consistency between the proposed image blurriness metric and natural blur images, we selected five Gaussian blur images from each of the four datasets, including CSIQ [25], Live II [26], IVC [27] and TID2013 [28], shown in Figure 6. Table 2 indicates that the proposed _image blurriness_ metric has positive correlation with the difference mean opinion scores (DMOS), i.e., the subjective quality scores in CSIQ and LIVE II datasets, and negative correlation with DMOS those belong to IVC and TID2013 datasets. It can be concluded that our blurriness metric can precisely represent the blurriness of natural images and fit the subjective quality assessment of the HVS.  
Figure 3: Feature extraction and structural similarity measurement. We extract four features, i.e., luminance, contrast, structural and blurriness, of the down-sampled 2D mode images and score structural similarity index by computing four features together.  
Figure 4: Sharp (**left**) and Gaussian blurred (**right**) images with their grayscale histograms.  
Figure 5: Trend of image quality and blurriness with Gaussian blur times that varies from 0 to \(11\).  
Figure 6: Sample images in the four datasets together with their subjective DMOS scores. Each set of images contains five Gaussian blur images those have gradient DMOS scores.