# Haptic Texture Rendering of 2D Image Based on Adaptive Fractional Differential Method
## 1 Introduction  
Haptic texture is a crucial cue to inform users of an interaction state with the surface texture, i.e., fine geometric surface features of an object. Haptic textures have received substantial attention in various fields [1; 2; 3]. Previous researchers have shown that the addition of haptic texture cues can greatly improve the realism of the virtual environment [4]. Therefore, the haptic texture rendering technique has several potential applications, such as virtual surgical simulation, haptic feedback teleoperation, on-line e-commerce, and aiding the visual impaired [5; 6; 7]. In general, there are three basic methods to render haptic textures. The first is the sample-based approach. [PERSON] introduced a system which is based on a tactile probe and a visual tracker for scanning and synthesizing tactile textures [8]. [PERSON] estimated the surface texture according to the frequency spectrum of vertical perturbations by dragging the tip of the tactile device on the object surface [9]. [PERSON] developed a PVDF-based haptic texture sensor by imitating human active texture perception to measure real object surface texture for haptic texture rendering in virtual reality [10]. [PERSON] used holograms to record the surfaces and textures of objects in a holo-haptic system. The produced haptic images were felt and shaped by a handheld device [11]. The second is the procedural texture approach which uses mathematical functions to synthesize the surface texture of objects. Several related researchers were summarized in detail [12; 13; 14]. The third is the image-based approach. This method constructs a texture force field from 2D image data. Therefore, this kind of haptic texturing approach can also be considered as a type of VR [15]. [PERSON] computed an elevation map based on the luminance coefficient for Texel images using four different techniques. These techniques were based on the fact that the height value of the pixel is proportional to the luminance value. Then, the elevation map is used to generate 3D bumps on the surface of the detected object, and calculate the corresponding tactile force for haptic rendering [16]. [PERSON] and [PERSON] processed the 2D images with a Gaussian filter to obtain low-frequency components, and then subtracted the filtered image from the original image; the left components denote the texture information. The forces simulated from textures were applied to the user using a Delta haptic device [17]. [PERSON] proposed an image-based haptic texture generation approach by replacing the Gaussian filter with an improved switching vector median filter for modeling the textured force and simulating the haptic stimuli [18]. [PERSON] used the conventional edge detection algorithms and proposed a design method of haptic mask to allow the user to feel the contours and textures of the image using haptic devices [19]. [PERSON] and [PERSON] presented a haptic texture rendering method based on color temperature and luminance to construct 3D texture force fields of 2D color images [20]. [PERSON] and [PERSON] proposed a fixed size descriptor based on local strength for texture calculation, and further expanded texture information by multi-channel color data collection [21]. Owing to the fact that the image-based haptic texture rendering approach has potential advantages of cost-effective realization, it has attracted substantial attention from researchers. However, the existing haptic texture rendering methods have problems in dealing with the images with fine geometric texture features. In this paper, we propose a novel fractional differential method for image-based haptic texture rendering. Fractional differentiation is currently a new tool for image signal processing. Complex texture details will show highly self-similar fractal information in image signals, and the mathematical basis of fractal theory includes fractional differentiation [22; 23].