# Haptic Texture Rendering of 2D Image Based on Adaptive Fractional Differential Method
## 3 Differential Order for Adaptive Selection Algorithm
We define \(CSGV=SGV1||SGV2||SGV3||SGV4\), where \(||\) is a \"superimpose\" operation. Therefore, CSGV can better describe the expression of image texture than the original gradient vector. Studying the human visual characteristics reveals that the sensitivity of the human eye to the gray value in the range of 0-255 in the gray image is not constant. When the gray value is particularly high or low, it is difficult for the human eye to perceive the grayscale change of the intensity value. In the vicinity of gray level 0, the human eye can only feel the change of gray level 8, while in the vicinity of gray level 255, the human eye can only feel the change of gray level 3, and when the gray level is 128, the human eye can perceive changes in 2 gray levels [37]. In digital image research, the gradient size of a certain point in the image is calculated by the change rate of the gray value of the point. The above-mentioned gradient coincidence vector CSGV represents the gray value change rate of the image at multiple scales. Therefore, we regard the pixels with CSGV less than 2 as a region with constant grayscale, and the differential order is 0; the pixels with CSGV in the range of 2-128 are regarded as regions with small grayscale changes, and appropriately increasing the differential order can enhance the human eye's perception of fine textures; while the pixels with CSGV greater than 128 are usually edge contour areas, it must have a correctly limited gradient interval, and the differential order should be appropriately reduced. From the above analysis, we have established the function of fractional derivative \(\gamma\) and CSGV:  
\[\gamma=\left\{\begin{array}{l}\frac{1}{\max^{\prime}(SGV)+\varepsilon_{1}} \cdot CSGV+\{\varepsilon_{2}\}\,\ CSGV>128\\ 0\ \  
order differential, five groups of comparative experiments were conducted, and their results are shown in Figure 1. These results of five sets of comparative experiments show the advantages of fractional difference in extracting complex texture information. Although the 0.3-order differential extraction effect can preserve the detailed texture well, the texture extraction effect is too weak to be clearly displayed on the picture (as shown in Figure 1(a1,b1,c1,d1,e1). However, as the differential order increases (between 0 and 1), image enhancement promotes \"large\" textures to sharper images, but \"small\" textures are lost. The adaptive differential order selection algorithm adopted in this paper can retain almost all texture details to achieve the best texture extraction among five groups (as shown in Figure 1(a4,b4,c4,d4,e4).  
Multi-scale structural similarity and information entropy (MS-SSIM) is used as an evaluation criterion to evaluate the effect of texture information extraction in images [38; 39]. Among them, information entropy can indirectly reflect the amount of information contained in grayscale images, and is very sensitive to images containing textures. Therefore, we adopt the calculation method of information entropy to analyze the amount of information in the image texture after extracting the image texture features. The information entropy is calculated as follows:  
\[E(p)=-\sum_{i=1}^{N}p_{i}\ln(p_{i}) \tag{5}\]  
Figure 1: Texture extraction comparison tests: (**a0–e0**) are the original image; (**a1–e1**) are the 0.3-order fractional derivative; (**a2–e2**) are the 0.55-order fractional derivative; (**a3–e3**) are the 0.7 th order fractional derivative; (**a4–e4**) are the adaptive fractional differentiation. Because the human vision system is very suitable for extracting structural information from scenes, measurement results with similar structures can provide a reference for whether image quality is good for perception. The MS-SSIM value of an image is usually used to compare the quality of two images and is an objective evaluation method that replaces the human subjective perception [39].