# Inverse Halftoning Methods Based on Deep Learning and Their Evaluation Metrics: A Review
## 4 Quality Evaluations of Inverse Halftone Images
### Objective Evaluation  
The most common used index of objective evaluation includes mean-square error (MSE), peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM) [41]. Besides these indexes, the color and clarity indexes are also introduced in this subsection. MSE is one of the most basic image quality evaluation indexes that can quantitatively evaluate the overall performance of inverse halftone images. It is a per-pixel comparison between the restored continuous-tone image \(I^{ih}\) and the original continuous-tone image \(I^{c}\), which is defined as follows. \[MSE=\frac{1}{mn}{\sum_{x=1}^{m}\sum_{y=1}^{n}\left(I^{ih}(x,y)-I^{c}(x,y) \right)^{2}} \tag{4}\]  
where \((x,y)\) is the pixel position, and \(m\) and \(n\) refer to the width and height of the image, respectively. PSNR is another commonly used image quality evaluation index. Recently, it has been widely used in the quality evaluation of inverse halftone image, focusing on the evaluation of image structure consistency. Theoretically, the higher the PSNR value, the less distortion of the inverse halftone image and the more similar the structure is to the original image. The specific expression is as follows. \[PSNR=10\log_{10}\left[\frac{255^{2}}{MSE}\right] \tag{5}\]  
SSIM is known as a perceptual-aware method for measuring the similarity between two images, and the expression of \(SSIM\) is shown as follows. \[SSIM=\frac{(2\mu_{c}\mu_{i}+c_{1})(2\sigma_{ci}+c_{2})}{(\mu_{c}^{2}+\mu_{i}^ {2}+c_{1})(\sigma_{c}^{2}+\sigma_{i}^{2}+c_{2})} \tag{6}\]  
where \(\mu_{c}\) and \(\mu_{i}\) refer to the mean pixel values of the original continuous-tone image and the inverse halftone image, acting as brightness estimates. \(\sigma_{c}\) and \(\sigma_{i}\) refer to the pixel value variance of the original continuous-tone image and the inverse halftone image, acting as contrast estimation. \(\sigma_{ci}\) refers to the covariance of the pixel values of the original continuous-tone image and the inverse halftone image, acting as a measure of the structural similarity. \(c_{1}\) and \(c_{2}\) are two constants for maintaining stability. The SSIM value ranges from 0 to 1, and the closer SSIM is to 1, the more similar the two images. For color halftone image, the color difference can objectively represent the color similarity between the original image and the inverse halftone image. In 2016, [PERSON] et al. [42] applied color difference to evaluate the quality of inverse color halftone image. Firstly, the image values in RGB color space are converted to the relative values in CIELAB color space. Then, the color difference between images is calculated as follows. \[\Delta E^{*}=\left[(\Delta L^{*})^{2}+(\Delta a^{*})^{2}+(\Delta b^{*})^{2} \right]^{1/2} \tag{7}\]  
where \(\Delta L^{*}\), \(\Delta a^{*}\), and \(\Delta b^{*}\) refer to the difference in color shades and hues between inverse halftone image and the original continuous-tone image, respectively. The smaller the color difference, the more similar the inverse halftone image is to the original continuous-tone image. It is well known that the human eyes perceive two images as being less different when the color tone changes analogously. However, the previous image quality evaluation methods did not take this into account, and only evaluated the image quality from the pixel level. To make up for this deficiency, [PERSON], [PERSON] et al. [43] proposed a new color image quality evaluation index \(D\). The smaller the value of \(D\), the more similar the tone changes of the inverse halftone image and the original image. \[D=||\text{RSD}_{\text{O}-I}|-\text{RSD}_{\text{O}}| \tag{8}\]where RSD\({}_{\text{O}-I}\) refers to the relative standard deviation of the pixel value difference between the continuous-tone image and the inverse halftone image, and RSD\({}_{\text{O}}\) refers to the relative standard deviation calculated by pixel values of the continuous-tone image.