# A Local Discrete Feature Histogram for Point Cloud Feature Representation
## 3 Methods  
### LRF and LMA  
The performance of feature descriptors depends on an efficient and reliable LRF. Nowadays, there are two primary categories of LRF construction methods [10]: covariance analysis-based and geometric attribute-based approaches. We use the stable LRF approach [15] to make the proposed descriptor invariant under rigid transformations. Additionally, we construct the LRF in the following manner to increase its efficiency and repeatability.  
The mathematical representation of LRF at key point \(p\) is  
\[\text{LRF}(p)=\{\text{x}(p),\text{x}(p)\times\text{z}(p),\text{z}(p)\}, \tag{1}\]  
where \(\text{x}(p)\) and \(\text{z}(p)\) refer to the x and z axes of the \(\text{LRF}(p)\).  
To compute the \(z\)-axis, we first obtain the set of local neighborhood points \(\text{Q}=\{q_{1},q_{2},\ldots,q_{k}\}\) within a support radius \(R\) of \(p\). We use all local neighborhood points to compute the \(z\)-axis. Specifically, we perform covariance matrix analysis on Q, and the covariance matrix \(Cov(\text{Q})\) is expressed as follows:  
\[Cov(\text{Q})=\begin{bmatrix}q_{1}-\overline{q}\\ \ldots\\ q_{k}-\overline{q}\end{bmatrix}^{T}\begin{bmatrix}q_{1}-\overline{q}\\ \ldots\\ q_{k}-\overline{q}\end{bmatrix}, \tag{2}\]  
where \(k\) is the number of neighboring points around key point \(p\) and \(\overline{q}\) denotes the centroid of Q. The minimum eigenvalue's corresponding eigenvector \(\text{v}(p)\) of \(Cov(\text{Q})\) is computed. To resolve sign ambiguity, all neighboring points within the support radius are used to determine the sign of the \(z\)-axis:  
\[z(p)=\left\{\begin{array}{l}\text{v}(p),\text{if}\ \text{v}(p)\cdot\sum_{i=1}^ {k}\text{q}_{i}\text{p}\geq 0\\ -\text{v}(p),\text{\emph{otherwise}}\end{array}\right., \tag{3}\]  
where \(\text{q}_{i}\text{p}\) refers to the direction vector from \(p\) to \(q_{i}\), \"\(\cdot\)\" represents the dot product between vectors.  
The construction of the \(x\)-axis follows a similar approach, as described in reference [15]. Specifically, the \(x\)-axis is computed as  
\[\mathrm{x}(p)\,=\,\sum_{i=1}^{k}w_{i1}w_{i2}v_{i}/\left\|\sum_{i=1}^{k}w_{i1}w_{ i2}v_{i}\right\|, \tag{4}\]  
where \(v_{i}\) denotes the projection of each neighboring point \(q_{i}\) onto a tangent plane \(T\) at the key point, i.e., \(w_{i1}\) refers to the weight associated with the distance of \(q_{i}\) to \(p\), given by \(w_{i1}=(R-\left\|\mathbf{p}-\mathbf{q}_{i}\right\|)^{2}\). \(w_{i2}\) is another weight based on the projection distance of \(q_{i}\) onto \(T\), given by \(w_{i2}=(\mathbf{p}\mathbf{q}_{i}\cdot\mathrm{z}(p))^{2}\).  
Finally, by calculating the cross-product of the \(z\)-axis and \(x\)-axis, the \(y\)-axis is obtained, and construction of the LRF is complete.  
For the calculation of the local minimum axis (LMA), the directions and symbols are determined by Equations (2) and (3). According to the validation of the literature [19], the direction of the LMA and the support radius used for disambiguation were uniformly set at a mesh resolution of 7 times (where the mesh resolution (mr) is generally defined as the average of the distance from each point to its nearest neighbor in a point cloud). Compared with traditional normal vector computation, the LMA offers two major advantages [20]. First, compared to a normal vector calculation, the support radius for an LMA calculation is larger. Normal vectors calculated using covariance matrix methods typically use a radius smaller than 3 mr. In contrast, when the radius is set to 7 mr (as defined in [19]), the computed LMA exhibits higher repeatability under conditions including noise and variations in mesh resolution. Second, the traditional normal vector disambiguation generally depends on the viewpoint information. When the viewpoint is uncertain, it needs to be handled manually. And the disambiguation of the LMA is realized via Equation (3), which is more efficient without depending on the viewpoint information.