# Regressed Terrain Traversability Cost for Autonomous Navigation Based on Image Textures
## 3 Traversability Cost Prediction
### Texture and Vibration Features Extraction/Association  
In this paper, texture features are obtained using of segmentation-based fractal texture analysis (SFTA) [33]. The SFTA method operates in two steps. In the first step, the input grayscale image is split into binary images using the two-threshold binary decomposition (TTBD) [33]. In the second step, binary images are used to compute the fractal dimension from its region boundaries. TTBD returns a set of thresholds \(T\) calculated by the multilevel algorithm [34]. After obtaining the targeted threshold number \(n_{\rm t}\), pairs of contiguous thresholds \(T\cup\{n_{\rm t}\}\) with \(n_{\rm t}\) being the highest value in the input gray scale image, in combination with pairs of thresholds \(\{t,n_{\rm t}\}\) with \(t\in T\), are employed to obtain the binary images as follows  
\[l_{x,y}^{b}= \begin{cases}1,&\text{if }t_{\rm t}<I(x,y)\leq t_{\rm u}\\ 0,&\text{otherwise}\end{cases} , \tag{7}\]  
where \(l_{x,y}^{b}\) denotes the binary value of image pixel \((x,y)\), \(t_{\rm t}\) and \(t_{\rm u}\) denote the lower and the upper threshold values, respectively. Therefore, \(2n_{\rm t}\) binary images will be obtained. In this paper, the SFTA feature vector includes only the fractal dimension of the boundaries. Let \(\Delta(x,y)\) denote the border image of the binary image \(l_{x,y}^{b}\), and which is obtained by the following equation:  
\[\Delta(x,y)= \begin{cases}1,&\text{if }\exists(x^{\prime},y^{\prime})\in N_{x,y}\\ &\text{s.t. }l_{x^{\prime},y^{\prime}}^{b}=0\wedge l_{x,y}^{b}=1\\ 0,&\text{otherwise}\end{cases} , \tag{8}\]  
where \(N_{x,y}\) is the 8-connectivity of a pixel \((x,y)\). \(\Delta(x,y)\) takes on a value of if the pixel at location \((x,y)\) in the related binary image \(l_{x,y}^{b}\) has a value of one and has a minimum one neighbor pixel with a value of zero. The border image serves to compute the fractal dimension \(D\in\mathbb{R}\) by the box counting method as follows  
\[D_{0}(x,y)=\lim_{\varepsilon\to 0}\frac{\log N(\varepsilon)}{\log( \varepsilon^{-1})}, \tag{9}\]  
where \(N(\varepsilon)\) is the number of hyper-cubes of length \(\varepsilon\) that fill the object. In this paper, the obtained SFTA feature vector is referred to as \(\mathbf{x}\in\mathbb{R}^{2n_{\rm t}}\). Let \(a_{k},k=1,\cdots,K\) be the vertical acceleration signal representing vibrations generated from the interaction of the wheels with the terrain when traversing a short range distance, where \(k\) is the time  
Figure 11: Identification of terrain patches for image feature extraction. step and \(K\) is the total time steps. The amplitude distance is retained to describe the behavior of the mobile platform, and is measured by  
\[a_{k_{\mathrm{d}}}=\max_{k=1,\cdots,K}a_{k}-\min_{k=1,\cdots,K}a_{k}. \tag{10}\]  
According to [35], all motion features (amplitude distance, root mean square, kurtosis, skewness, and crest factor) have a similar level of correlation with the SFTA feature. The amplitude distance feature was chosen as it is easy to implement and to understand. The acceleration signal does not undergo any noise filtering operation, and it is clear whether any filtering is performed at the sensor level. To measure the noise contribution to the measurements, signal-to-noise ratio (SNR) of the acceleration signal at rest was computed as follows:  
\[SNR=\frac{\mu}{\sigma}, \tag{11}\]  
where \(\mu\) and \(\sigma\) are the mean and standard deviation of the acceleration signal, respectively. Results of SNR calculation for four signals are given in Table 1. SNR values are high which means that the noise does not impact severely our intended purpose. Two subsequent terrain segments serve for image feature extraction. Appropriately, the relevant acceleration segment for motion feature calculation is paired to the image features by means of the coordinate transformation and odometry. As shown in Figure 12, at time \(t=0\), the robot is located at the origin of the world reference frame.