# Haptic Texture Rendering of 2D Image Based on Adaptive Fractional Differential Method
## 3 Differential Order for Adaptive Selection Algorithm  
Texture is one of the most essential image processing and analysis properties. Texture gives intuitive assessments of qualities, such as regularity, coarseness, and smoothness. The majority of texture analysis methods analyze the picture at a single scale. As revealed by [PERSON] et al. [31], the visual cortex may be modeled as multiple channels, where each channel can perceive a specific direction and frequency tuning. Multi-scale texture analysis techniques are propelled by multichannel processing. Several multichannel texture analysis systems have been suggested [32; 33]. In the past 10 years, the rapid development of wavelet theory has also brought new theories and methods to the field of image processing. [PERSON] suggested a discretization approach for wavelet transform [34]. The relationship between multiresolution theory and wavelet transforms was further developed by [PERSON] [35]. Since then, wavelet theory has developed into a multi-scale (multi-resolution) mathematical tool in image analysis. The use of multi-scale methods in texture image analysis is based on the premise that lower resolution channels can better record \"large\" textures, while higher resolution channels can better record \"small\" textures. The following describes the process involved. Apply wavelet and scaling filters to the image both horizontally and vertically, then sub-sample each output image by 2-1. This produces a coarse or approximate image \(C_{j}\) and three detail images with direction-selectivity \(D_{j,k}\), where \(k\) = 1,2,3 and \(j\) represents the decomposition level. The same method is utilized to construct the following level of hierarchy resolution. Therefore, the hierarchical wavelet decomposition of the image is expressed as:  
\[\left\{ \begin{aligned} & C_{j}=\left[H_{x}*\left[H_{y}*C_{j-1} \right]_{\downarrow 2,1}\right]_{\downarrow 1,2}\\ & D_{j,1}=\left[H_{x}*\left[G_{y}*C_{j-1}\right]_{\downarrow 2,1} \right]_{\downarrow 1,2}\\ & D_{j,2}=\left[G_{x}*\left[H_{y}*C_{j-1}\right]_{\downarrow 2,1} \right]_{\downarrow 1,2}\\ & D_{j,3}=\left[G_{x}*\left[G_{y}*C_{j-1}\right]_{\downarrow 2,1} \right]_{\downarrow 1,2}\end{aligned} \right. \tag{1}\]  
where C0 = I represents the original image, \(\downarrow\)1,2 means down-sampling every other pixel in the y direction, \(\downarrow\)2,1 means down-sampling every other pixel in the x direction, and * means the convolution operator. \(G_{x}\) and \(H_{y}\) and \(G_{y}\) and \(H_{y}\) represent high-pass and low-pass filters in the x and y directions, respectively. Therefore, the original image can be represented by a series of sub-images of multiple scales, \(\left\{C_{j},D_{j,k}\right\}\) (\(j\) = 1,..., \(j\); \(k\) = 1, 2, 3) is the multi-scale representation of image I at depth J. DAUB4 is used as the wavelet basis in this instance because of its superior average performance. Gradient direction image may provide texture analysis with useful properties [36]. After the image is processed by the gradient operator, the change amplitude and direction of the pixel gray value can be obtained, and the image gradient describes the change trend of the image in different directions. A combination of a series of low-pass (H) and high-pass (G) filters enables wavelet decomposition, using multiple sets of filters for sampling, each set at half the sampling frequency of the previous set. Therefore, the original image can be processed to obtain four sub-images, namely:  
* LL sub-image: low frequencies in both x and y directions. * LH sub-image: low frequencies in the x direction and high frequencies in the y direction. * HL sub-image: high frequencies in the x direction and low frequencies in the y direction. * HH sub-image: high frequencies in both x and y directions. LL, LH, HL, and HH are four sub-images obtained by wavelet decomposition. A gradient vector is constructed for each sub-image, denoted by \(SGV1\), \(SGV2\), \(SGV3\), \(SGV4\), respectively.