# Haptic Texture Rendering of 2D Image Based on Adaptive Fractional Differential Method
## 6 Haptic Texture Rendering Model  
Haptic texture rendering is a method to reconstruct the surface attributes of virtual objects according to force fields or force vectors, so that users can feel the surface texture of virtual objects happically by using haptic devices (such as Phantom, Force-Dimension Delta hand controller, etc.). In this section, we use the adaptive fractional differentiation method to provide a new tactile texture model based on the image texture extraction results. The texture force vector \(\overset{\rightarrow}{F}(i)\) at each pixel of an image can be modeled as the combination of normal force vector \(\overset{\rightarrow}{F}_{N}(i)\) and tangential force vector \(\overset{\rightarrow}{F}_{T}(i)\) as  
\[\overset{\rightarrow}{F}(i)=\overset{\rightarrow}{F}_{N}(i)+\overset{ \rightarrow}{F}_{T}(i) \tag{9}\]  
The tangential force vector of the image is calculated based on the following assumption. There exists an interaction force between any two pixels in the gray image after processed by the proposed adaptive fractional differential method. The interaction force between any two pixels \(P_{i}\) and \(P_{j}\) is proportional to the absolute vale of the difference of the two pixel's gray values, and inversely proportional to the distance between two pixels. The direction of interaction force vector is defined as from pixel with high luminance value to the pixel with low luminance value. \[\overset{\rightarrow}{F}_{ij}=\frac{\left|G(p_{i})-G(p_{j})\right|}{\left\|p_ {i}-p_{j}\right\|}r_{i,j} \tag{10}\]  
where \(\left\|p_{i}-p_{j}\right\|\) is distance between two pixels \(p_{i}\) and \(p_{j}\), and \(G(p_{i})\), \(G(p_{j})\) are grey value of pixels \(p_{i}\) and \(p_{j}\), respectively. \(r_{i,j}\) denotes the direction from high luminance pixel to low luminance pixel. If the distance between two pixels \(p_{i}\) and \(p_{j}\) is small, the difference of the two pixel's gray values will cause a big interaction force, and vise versa. Thus, the force vector of \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Original Images (a0–a4)**} & \multicolumn{4}{c}{**Extracted Textures (e0–e4)**} \\ \hline \multirow{4}{*}{AMS} & [0.00015 & 0.00013 & 0.00013 & 0.00011] & [0.00038 & 0.00031 & 0.00034 & 0.00030] \\  & [0.00040 & 0.00028 & 0.00037 & 0.00029] & [0.00075 & 0.00060 & 0.00077 & 0.00061] \\  & [0.00011 & 0.00009 & 0.00011 & 0.00010] & [0.00011 & 0.00009 & 0.00010 & 0.00009] \\  & [0.00027 & 0.00023 & 0.00036 & 0.00023 & [0.00052 & 0.00048 & 0.00061 & 0.00047] \\  & [0.00066 & 0.00041 & 0.00061 & 0.00040] & [0.00273 & 0.00187 & 0.00245 & 0.00188] \\ \hline \multirow{4}{*}{Entropy} & [0.0120 & 0.0112 & 0.0115 & 0.0105] & [0.0195 & 0.0175 & 0.0183 & 0.0174] \\  & [0.0201 & 0.0167 & 0.0192 & 0.0169] & [0.0274 & 0.0246 & 0.0277 & 0.0246 \\  & [0.0105 & 0.0094 & 0.0103 & 0.0099] & [0.0104 & 0.0096 & 0.0102 & 0.0100] \\  & [0.0162 & 0.0153 & 0.0189 & 0.0151] & [0.0228 & 0.0219 & 0.0247 & 0.0217] \\  & [0.0258 & 0.0202 & 0.0247 & 0.0200] & [0.0522 & 0.0433 & 0.0495 & 0.0433] \\ \hline \multirow{4}{*}{Contrast} & [457.73 & 657.91 & 531.97 & 1177.25] & [238.31 & 479.46 & 341.27 & 547.85] \\  & [202.81 & 782.97 & 616.01 & 780.25] & [122.90 & 238.49 & 134.69 & 243.24] \\  & [555.80 & 1184.74 & 621.75 & 940.61] & [1117.87 & 2451.20 & 1251.64 & 1540.86] \\  & [738.38 & 970.86 & 544.27 & 1283.43] & [221.82 & 293.59 & 165.43 & 323.46] \\  & [383.83 & 601.95 & 247.00 & 644.16] & [89.14 & 135.43 & 54.82 & 136.63] \\ \hline \multirow{4}{*}{Correlation} & [0.843 & 0.774 & 0.817 & 0.596] & [0.608 & 0.213 & 0.439 & 0.101] \\  & [0.897 & 0.602 & 0.687 & 0.604] & [0.647 & 0.316 & 0.614 & 0.303] \\ \cline{1-1}  & [0.888 & 0.76 & 0.875 & 0.811] & [0.627 & 0.183 & 0.582 & 0.487] \\ \cline{1-1}  & [0.790 & 0.723 & 0.846 & 0.635] & [0.465 & 0.294 & 0.603 & 0.222] \\ \cline{1-1}  & [0.879 & 0.812 & 0.923 & 0.799] & [0.610 & 0.407 & 0.761 & 0.401] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical information of the texture image.