# A Local Discrete Feature Histogram for Point Cloud Feature Representation
## 3 Methods
### Generation of the LDFH Descriptor  
After constructing the LRF, the LDFH descriptor is constructed based on the LRF. The process of generating the LDFH is illustrated in Figure 1.  
First, the local surface Q is transformed into the LRF to ensure rotational invariance, and the rotated surface is expressed as \(\mathrm{Q}^{\prime}=\left\{q_{1}^{\prime},q_{2}^{\prime},\ldots,q_{k}^{\prime}\right\}\). Next, the spatial and geometric information on the local surface is encoded. Specifically, we use radial distance to encode the local spatial information, and the normal deviation angle \(\theta\), polar angle \(\psi\), and normal lateral angle \(\phi\) to encode the local surface geometry information. They are computed as follows:  
\[\left\{ \begin{array}{l}r_{i}=\sqrt{q_{i}^{\prime}(x)^{2}+q_{i}^{\prime}(y)^{ 2}+q_{i}^{\prime}(z)^{2}}\\ \theta_{i}=\arccos\left(\frac{\mathrm{LMA}\left(q_{i}^{\prime}\right)\cdot \mathrm{z}(p)}{\left\|\mathrm{LMA}\left(q_{i}^{\prime}\right)\right\|\times \left\|\mathrm{z}(p)\right\|}\right)\\ \psi_{i}=\arccos\left(\frac{q_{i}^{\prime}(z)}{r_{i}}\right)\\ \phi_{i}=\arccos\left(\frac{\mathrm{LMA}\left(q_{i}^{\prime}\right)\cdot \mathrm{y}(p)}{\left\|\mathrm{LMA}\left(q_{i}^{\prime}\right)\right\|\times \left\|\mathrm{y}(p)\right\|}\right)\end{array} \right., \tag{5}\]  
where \(r_{i}\) is the radial distance of the neighboring point \(q_{i}^{\prime}\) from the key point \(p\), with a range of [0,R]. \(q_{i}^{\prime}(x),q_{i}^{\prime}(y)\), and \(q_{i}^{\prime}(z)\) are the x, y, z-coordinates of \(q_{i}^{\prime}\). \(\theta_{i}\) is the deviation angle between the LMA of \(q_{i}^{\prime}\) and the normal z(\(p\)) of the key point, with a range of [0,\(\pi\)]. \(\psi_{i}\) is the polar angle between \(q_{i}^{\prime}\) and the \(z\)-axis, with a range of [0,\(\pi\)]. \(\phi_{i}\) is the normal lateral angle between the LMA of \(q_{i}^{\prime}\) and the \(y\)-axis of the LRF, with a range of [0,\(\pi\)]. The number of partitions divided along the radial distance is \(N_{r}\). Three geometric features are computed in each subspace. And three corresponding feature histograms \(H_{\theta},H_{\phi},H_{\phi}\) are generated. During the statistical process, continuous feature values are discretized into a fixed number of bins to capture their spatial distribution characteristics. Suppose the ranges of the three geometric features are divided into \(N_{\theta},N_{\psi},N_{\phi}\) bins, the lengths of the corresponding feature histograms are \(N_{r}\times N_{\theta},N_{r}\times N_{\psi},N_{r}\times N_{\phi}\), respectively. The three histograms are normalized so that the sum of each histogram is 1, to achieve robustness against changes in point density. Additionally, considering the different contributions of the three features to the descriptiveness and robustness of the LDFH, three weighting coefficients--\(\lambda_{1},\lambda_{2},\lambda_{3}\)--are introduced to perform a weighted fusion of the feature histograms, resulting in \(f=\left\{\lambda_{1}H_{\theta},\lambda_{2}H_{\psi},\lambda_{3}H_{\phi}\right\}\). The procedure for generating the LDFH descriptor is described in Algorithm 1. Input: P (Point Cloud), K (Key Points), R, \(N_{r},N_{\theta},N_{\psi},N_{\phi}\), \(\lambda_{1},\lambda_{2},\lambda_{3}\), LRF, LMA  
Output: LDFH (LDFH descriptors)  
1. For each key point, \(p\in\) K:
2. Extract local surface around the key point \(p\);
3. Construct LRF at the key point \(p\);
4. Transform local surface of \(p\) to LRF;
5. Partition local space around \(p\) into radial bins \(N_{r}\);
6. For each neighboring point \(q^{\prime}\) in support radius:
7. Compute geometric attributes \((\theta,\psi,\phi)\) based on LRF;
8. Generate three feature histograms \(H_{\theta},H_{\psi},H_{\phi}\);
9. Apply a weighted fusion to generate the final LDFH descriptor \(\left\{\lambda_{1}H_{\theta},\lambda_{2}H_{\psi},\lambda_{3}H_{\phi}\right\}\);
10. Store the LDFH descriptor for key point \(p\);
11. Return LDFH (set of all LDFH descriptors for K). Figure 1: An illustration of the LDFH descriptor (**a**) 3D model.