B. Descriptors by BRISK

The BRISK descriptor adopts the neighborhood sampling model, which takes the feature points as the center of the circle. The points on the concentric circles of several radii are selected as the sampling points. In order to reduce the effect caused by sample image grayscale aliasing, the Gauss function can be used for filtering. The Gauss function of standard deviation sigma is proportional to the distance between the points on each concentric circle. Selecting a pair from the point pairs formed by all sampling points, denoted as (*Pi, Pj*), the gray values after treatment are *I*(*Pi,* σ*i*) and *I*(*Pj,*σ*j*), respectively. Hence, the gradient between two sampling points is

$$\log(P\_{i\prime}P\_j) = (P\_j - P\_i) \cdot \frac{I(P\_{j\prime}, \sigma\_j) - I(P\_{i\prime}, \sigma\_i)}{\|P\_j - P\_i\|^2}. \tag{3}$$

Set *A* is a collection of all pairs of sampling points, S is a set containing all the short-range sampling pairs, and L is a set containing all the long-distance pairs of sampling points:

$$A = \left\{ (P\_{i'}P\_{j}) \in \mathbb{R}^2 \times \mathbb{R}^2 \, \middle| \, i < N \land j < i \land i, j \in N \right\},\tag{4}$$

$$S = \left\{ (P\_{\bar{1}}, P\_{\bar{j}}) \in A \Big| \| P\_{\bar{j}} - P\_{\bar{i}} \| < \delta\_{\max} \right\} \subseteq A,\tag{5}$$

$$L = \left\{ (P\_{\text{i}}, P\_{\text{j}}) \in A \middle\| \| P\_{\text{j}} - P\_{\text{i}} \| < \delta\_{\text{min}} \right\} \subseteq A. \tag{6}$$

The general distance thresholds δ*max* = 9.75 *t,* δ*min* = 13.67 *t*, and *t* are characteristic point scales. The main direction for each feature point is specified by the gradient direction distribution characteristics of neighboring pixels of the feature point. In general, the BRISK algorithm can be used to solve for the direction *g* of the overall pattern according to the gradient between two sampling points:

$$\log \mathbf{g} = \begin{pmatrix} \mathbb{g}\_x \\ \mathbb{g}\_y \end{pmatrix} = \frac{1}{L} \cdot \sum\_{(P\_i, P\_j) \in L} \mathbf{g}(P\_{i\nu} P\_j). \tag{7}$$

In order to achieve rotation and scale invariance, the sampling pattern is sampled again after the rotation angle θ = *arctan2* (*gy*, *gx*). The binary descriptor *b* can be constructed by performing Equation (8) on all pairs of points in set *S* by short-range sampling points.

$$b = \begin{cases} 1 & l(P^0\_{j'}, \sigma\_j) > l(P^0\_{j'}, \sigma\_i) \\ 0 & \text{otherwise} \end{cases} \qquad \forall (P^0\_{i'}, P^0\_{j'}) \in \mathbb{S} \tag{8}$$
