Graph Convolution Algorithm Based on Visual Selectivity and Point Cloud Analysis Application

The design of the graph convolution kernel and the design of the convolution calculation method are the two fundamental components of the graph convolution algorithm based on visual selectivity.

The near in proximity and sparse in distance elements of the point cloud, as well as the structural selectivity of the point cloud topology, are all congruent with the visual selectivity of primates. The distribution of the receptive field in the visual space of primates follows the visual selectivity trait of "close in the vicinity and sparse in the distance."

As a result, 3D point cloud data has similar characteristics and robust visual basis functions in the visual space. In contrast to the visual basis functions with different functions, which exist in regions far apart and form a relatively slow-changing structure, the visual basis functions with different functions exist in a fixed spatial region and have a consistent spatial topology structure. The point cloud data is a collection of point sets with stable spatial information and a spatial topology structure, which in turn has a significant data redundancy row.

As a result, the point cloud based on visual selectivity need to have the qualities listed below:

(1) The global point cloud space is made up of a number of point cloud subspaces, and its spatial topology is stable and consistent.

(2) Each point cloud subspace in the global space has a separate basis function (receptive field) and a topology that is either constant or slowly evolving. These properties can be calculated using probabilistic methods.

(3) Each independent subspace has modest topological structure change, spatial independence, and functional resemblance to visual basis functions; Each point subset in the independent space has comparable visual functions, comparable visual basis functions, and gradually changing visual space topology.

(4) Each point cloud space's qualities have statistical properties. Through statistical analysis, the lack of data collection-related spatial completeness is resolved, and the collection set can be increased through visually consistent selection.

(5) Each point cloud subset has a minimum generation structure, which can represent any point or structure within the subset. This structure is made up of the minimum generation point set and its spatial support structure.

2.1 Convolution kernel design

Based on the above theoretical analysis, the convolutional kernel $filter_p^r$ (receptive field of point p) of point p in the 3D point cloud space can be defined as:

$filter_p^r=\left\{M_p^r, e d g e_p^r\right\}$          (1)

where, $M_p^r$ is the point p and the points of its point cloud space with the radius of r; $e d g e_p^r$ is the set of edges between the points in set $M_p^r$ and point p, i.e., the effective visual topology of point p. The set has the following features:

(1) Set $filter_p^r$ is a finite set, i.e., sets $M_p^r$ and $e d g e_p^r$ are both finite and have redundancy.

(2) Since $filter_p^r$ is a finite redundant set, there exists at least one minimum support set  ${hold\text_filter }_p^r=\left\{\bar{M}_p^r, \overline{ { edge }}_p^r\right\} \subset filter _p^r$, where $\bar{M}_p^r \subset M_p^r$, $\overline{e d g e}_p^r \subset e d g e_p^r, \forall e \in \overline{e d g e}_p^r$ and $e \neq 0$. Thus, in the point cloud, ${hold\text_filter }_p^r$ expresses the topological information and the topological selectivity of the point cloud area with p as the circle center and r as the radius.

(3) ${hold\text_filter }_p^r$ is not unique.

(4) Inspired by visual selectivity, ${hold\text_filter }_p^r$ is an acquirable and illustrative quantity after training.

(5) Any point or edge in ${filter }_p^r$ can be expressed by the point or edge of ${hold\text_filter }_p^r$.

(6) Depending on the values of hyperparameter r, the support features of point p on different scales can be obtained, and the selective features of the global and local point cloud structures of that point can be obtained: $Global\text_faeture \subset hold \text_filter { }_p^r$ and $local \quad feature \subset hold \text_filter { }_p^r$.

(7) Maximum pooling convolution: When $\forall r \in R$, $\left\{\right. hold \left._{ {filter }_p^r}\right\}$ is the set of kernels at different scales r, $\forall p \in \bar{M}_p^r$. Then, the maximum pooling convolution z of point p can be expressed as:

Max_Pool_layer(q,hold_filter_p^r)= $\max _r$(convolution_coff($p$, hold $_{\text {filter }_p^r}$))          (2)

In the algorithm, the layer formed according to this formula is called the maximum pooling layer, which realizes feature aggregation at different scales.

Based on the above analysis, inspired by the theory of primate visual selectivity, combined with the distribution of points in the point cloud in space, the convolution kernel of the point cloud (point cloud space receptive field with p as the center and radius r) can be defined for:

hold_filter ${ }_p^r=\left\{\bar{M}_p^r, \overline{e d g e}_p^r\right\}=\left\{p_m, p_n \mid p_m \in \mathrm{N}\left(p_n, r\right)\right\}=\left(r, \theta, d_\theta, n_\theta, w_\theta,(x, y, z)\right)$          (3)

Here, the parameter is the super parameter of the basis function, which determines the observation space range of the receptive field, determines the size of the receptive field and the scale of the observation direction, and is a decisive factor; the parameter θ represents the observation direction in the r neighborhood, indicating that the basis function points in the direction are relatively dense; the parameter $d_\theta$ indicates that in the direction θ, the point cloud space topology selectivity strength is related to the number of point clouds in the direction and the distance between the point $p_n$ and the geometric center of the point in the area; the parameter $n_\theta$ indicates that in the direction In the upper r range, the number of receptive field basis functions is a concentrated expression of visual selectivity; the parameter $w_\theta$ indicates the update degree of the direction range after each training within the r range in the direction. Parameters: represented as the spatial position information of point $p_m$ position.

In the neighborhood of r, the convolution kernel design method of point P is show in Figure 1.

Figure 1. The design process diagram of the convolution kernel

Step 1. KNN clustering is adopted to divide hold_filter ${ }_p^r$ into K subsets:

hold_filter ${ }_p^r=\bigcup_{q=1}^K$ hold_filter $_p^r(q)$          (4)

Step 2. Compute the geometric center center(q) of each subset  ${{hold_{filter}}_p^r}(q)$:

$\operatorname{center}(q)=\left(x_q, y_q, z_q\right)=\left(\frac{1}{T} \sum_{j=1}^T x_q^j, \frac{1}{T} \sum_{j=1}^T y_q^j, \frac{1}{T} \sum_{j=1}^T z_q^j\right)$          (5)

Step 3. Parameter learning:

$\theta=\left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)=\left(\frac{p_m-p_n}{\left\|p_m-p_n\right\|}\right)$

$n_\theta=$ sizeof $\left(\right.$ hold_filter $\left.{ }_p^r(q)\right)$

$w_\theta^i=\frac{n_\theta^i}{n_\theta^{i-1}}$

$d_\theta^j=\frac{d_{(p, q)}^j \times n_\theta^j}{\sum_{j=1}^M d_{(p, q)}^j \times n_\theta^j} \times w_\theta^i$          (6)

Step 4. Forming convolutional kernels:

${hold_{ {filter }}}_p^r=\left(r, \theta^j, d_\theta^j, n_\theta^j, w_\theta^j, x, y, z\right)^r, j=1,2,3, \ldots, K$          (7)

where, $\theta^j, d_\theta^j, n_\theta^j, w_\theta^j, x, y$, and $z$ are learned through training

2.2 Graph convolution algorithm design

The graph convolution calculation is to calculate the direct similarity between the filter kernel and the analyzed point, therefore, the graph convolution calculation of point Q can be expressed as:

$convolution_{ {coff }}= similitude \left(\right.$  $\left.hold\text_filter_p^r, Q\right)+\operatorname{correct}\left(\right.$  $\left.hold\text_filter_p^r, Q\right)$          (8)

According to the requirements of similarity calculation in this paper, combined with the definition of vector inner product, the inner product of self-defined matrices A and B is defined as: if matrix A is expressed as $\alpha_i, i=1,2,3, \ldots, K_1$ by row vector, and matrix B as row vector $b_j, j=1,2,3, \ldots, K_2$, then the inner product is:

$A \odot B=\frac{1}{N} \sum_{\theta_i=\theta_j \text { and } \theta_i \neq \Phi, \text { and } \theta_j \neq \Phi} \quad a_i \odot b_j$          (9)

where, N represents the number of rows whose corresponding rows of matrices A and B are not zero vectors, and Φ represents an empty set. Therefore, similar calculations in graph convolution are expressed as:

similitude(hold_filter $\left.{ }_p^r, Q\right)=$ hold_filter ${ }_p^r \odot Q$          (10)

The modified part of the similarity calculation is expressed as:

$\operatorname{correct}\left(\right.$ hold_filter $\left.{ }_p^r, Q\right)=$

$\frac{1}{\max (\text { size }(\text { hold_filter }_p^r), \text { size }(Q))} \times$

jacard $($ hold_filterer $(\theta), Q(\theta))=$

$\frac{1}{\max (\operatorname{size}(\text { hold_filter }_p^r), \operatorname{size}(Q))} \times \frac{\text { hold_filter }(\theta) \cap Q(\theta)}{\text { hold_filter }(\theta) \cup Q(\theta)}$           (11)

where, correct $\left({{hold_{\text {filter}}}_p^r},  Q\right)$ is the matrix  ${hold_{filter}}_p^r$. The modification of Q graph convolution increases the similarity of graph convolution. To sum up, the graph convolutional coefficient is:

convolution $_{\text {coff }}=$ hold_filter $_p^r \odot Q+$

$\frac{1}{\max (\operatorname{size}(\text { hold_filter } \quad  _p^r), \quad  \operatorname{size}(Q))} \times \frac{\text { hold_filter }(\theta) \cap Q(\theta)}{\text { hold_filter }(\theta) \cup Q(\theta)}$          (12)

Thus, 3D_RFGCN,3D receptive field graph convolution algorithm can be expressed as:

2.3 Maximum pooling layer

Maximum pooling: When $\forall r \in R$, $\left\{\right.$ hold_filter $\left._p^r\right\}$ represents the set of kernels at different scales r, $\forall p \in \bar{M}_p^r$. Then, the maximum convolution $\mathrm{z}$ of point $\mathrm{p}$ is:

Max_Pool_layer $\left(q\right.$, hold_filter $\left.{ }_p^r\right)=$

$\max _r\left(\right.$ convolution $_{\text {coff }}\left(p\right.$, hold $\left.\left._{\text {filter }_p^r}\right)\right)$          (13)

In the algorithm, the layer formed according to this formula is called the maximum pooling layer, which realizes feature aggregation at different scales.

2.4 The overall structure of the algorithm

2.4.1 Classification algorithm design

In order to identify which known category the input 3D point cloud data s belongs to, the algorithm in this paper fuses the 3D_RFGCN layer and the Max_Pool_layer layer into one layer, and realizes the classification through a multi-layer perceptron (MLP). The standard soft_max loss function and gradient descent method are used for parameter learning of this algorithm. Therefore, the classification algorithm is designed as Figure 2:

2.png

Figure 2. Classification algorithm structure

2.4.2 Semantic segmentation algorithm design

The 3D point cloud data can be semantically segmented using the algorithm shown in this research. We propose a shared MLP method for point-by-point classification in order to accomplish this goal. The characteristics of 3D point cloud data between different layers do not match because the algorithm uses a pooling mechanism. Accordingly, this paper defines the following aggregation operations:

$p_{\bar{n}}^{j+1}=\operatorname{argmin}\left\{\left\|p-p_n^j\right\| \mid \forall p \in p^{j+1}\right\}$          (14)

Therefore, the generated pooled features are 2, and the corresponding semantic segmentation network is as Figure 3.

Figure 3. Segmentation algorithm structure