一. 论文总结

1.1 核心贡献

提出了一种改进的图卷积操作，称为语义图卷积(SemGConv)，它源自cnn。其关键思想是学习图中暗示的边的信道权值，然后将它们与核矩阵结合起来。这大大提高了图卷积的能力。
其次，我们引入了SemGCN，其中SemGConv层与非局部[65]层交叉。该体系结构捕获节点之间的本地和全局关系。
第三，我们提出了一个端到端学习框架，以表明SemGCN还可以合并外部信息，如图像内容，以进一步提高3D人体姿态回归的性能。

1.2 语义图卷积(SemGConv)

1.2.1 ResGCN

$\vec{{x_{i}^{l}}}$ ∈ $\mathbb{R}^{D_{i}}$ 和 $\vec{x_{i}^{l+1}}$ ∈ $\mathbb{R}^{D_{i+1}}$ 分别是节点i在第l个卷积之前和之后的表示。
可学习参数矩阵W∈ $\mathbb{R}^{D_{l+1}\times D_{i} }$
其中 $\bar{A}$ 是在常规GCNs中对A进行对称归一化。A∈ $\sqsubset 0,1\sqsupset ^{K\times K}$ 是G的邻接矩阵，对于节点j∈N (i)有 $a_{ij}$ = 1， $a_{ii}$ = 1

两个明显的缺点：

首先，为了使图卷积在具有任意拓扑结构的节点上工作，所有边共享学习的核矩阵W。结果，相邻节点的关系或图中的内部结构，不能很好地利用。
其次，之前的工作只收集每个节点的第一个序列上的（应该是最近邻的意思）邻居的特征。因为接受野被固定为1，这也是存在不足的。

1.2.2 SemGConv

在传统的图卷积中添加了一个可学习的权重矩阵M∈ $\mathbb{R}^{K\times K}$ (按我的理解就是上图中的 $a_i$ )。然后转化为:

其中 $\rho _{i}$ 是Softmax非线性，它在节点 $^{i}$ 的所有选择上归一化输入矩阵;
⊙是一个元素级运算，如果 $a_{ij}$ = 1则返回mij，或者在 $\rho _{i}$ 后饱和到零的大指数为负数;
A作为一个掩码，它迫使图中的节点i，我们只计算其相邻节点j∈N (i)的权值。

学习每一个通道的加权权重，将公式进一步扩展：

其中 || 表示对所有层的输入进行拼接，
$\vec{w_{d}}$ 是变换矩阵w的第d行。

1.3 网络架构

整体结构

SemGCN网络架构是通过SemGConv和 Non-Local层交错以捕获节点的局部和全局语义关系

Non-Local来自2018年cvpr论文Non-local Neural Networks

（翻译和解析后面会更新）

1.3.1 基本结构

在这项工作中，所有块中的SemGCN具有相同的结构，由两个具有128通道的SemGConv层构建的一个残差块[20]组成，然后再跟随一个非本地层。

二.代码实现

2.1 生成邻接矩阵

adj = adj_mx_from_skeleton(dataset.skeleton())#建立对称邻接矩阵

以human36M为例：

dataset.skeleton()返回的就是

h36m_skeleton = Skeleton(parents=[-1,
                                  0, 1, 2, 3, 4,
                                  0, 6, 7, 8, 9,
                                  0, 11, 12, 13, 14, 12,
                                  16, 17, 18, 19, 20, 19, 22, 12, 24, 25, 26, 27, 28, 27, 30],
                         joints_left=[6, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 22, 23],
                         joints_right=[1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31])

具体来看一下adj_mx_from_skeleton这个生成邻接矩阵的函数：

from __future__ import absolute_import

import torch
import numpy as np
import scipy.sparse as sp


def normalize(mx):
    """Row-normalize sparse matrix"""
    rowsum = np.array(mx.sum(1))
    r_inv = np.power(rowsum, -1).flatten()
    r_inv[np.isinf(r_inv)] = 0.
    r_mat_inv = sp.diags(r_inv)
    mx = r_mat_inv.dot(mx)
    return mx


def sparse_mx_to_torch_sparse_tensor(sparse_mx):
    """将scipy稀疏矩阵转换为torch稀疏张量。Convert a scipy sparse matrix to a torch sparse tensor."""
    sparse_mx = sparse_mx.tocoo().astype(np.float32)
    indices = torch.from_numpy(np.vstack((sparse_mx.row, sparse_mx.col)).astype(np.int64))
    values = torch.from_numpy(sparse_mx.data)
    shape = torch.Size(sparse_mx.shape)
    return torch.sparse.FloatTensor(indices, values, shape)


def adj_mx_from_edges(num_pts, edges, sparse=True):
    edges = np.array(edges, dtype=np.int32)
    data, i, j = np.ones(edges.shape[0]), edges[:, 0], edges[:, 1]
    adj_mx = sp.coo_matrix((data, (i, j)), shape=(num_pts, num_pts), dtype=np.float32)
    #sp.coo_matrix() 的作用是生成矩阵
    #sp.coo_matrix((data, (row, col)), shape=(4, 4))用指定数据生成矩阵

    # 建立对称邻接矩阵 build symmetric adjacency matrix
    adj_mx = adj_mx + adj_mx.T.multiply(adj_mx.T > adj_mx) - adj_mx.multiply(adj_mx.T > adj_mx)
    adj_mx = normalize(adj_mx + sp.eye(adj_mx.shape[0]))
    if sparse:
        adj_mx = sparse_mx_to_torch_sparse_tensor(adj_mx)
    else:
        adj_mx = torch.tensor(adj_mx.todense(), dtype=torch.float)
    return adj_mx


def adj_mx_from_skeleton(skeleton):
    num_joints = skeleton.num_joints()
    edges = list(filter(lambda x: x[1] >= 0, zip(list(range(0, num_joints)), skeleton.parents())))
                                                        #盆骨父节点为-1
    return adj_mx_from_edges(num_joints, edges, sparse=False)

2.2 SemGCN

2.2.1 SemGraphConv

class SemGraphConv(nn.Module):
    """
    Semantic graph convolution layer
    """

    def __init__(self, in_features, out_features, adj, bias=True):
        super(SemGraphConv, self).__init__()
        self.in_features = in_features
        self.out_features = out_features

        self.W = nn.Parameter(torch.zeros(size=(2, in_features, out_features), dtype=torch.float))
        #作为nn.Module中的可训练参数使用，与torch.Tensor的区别就是nn.Parameter会自动被认为是module的可训练参数

        nn.init.xavier_uniform_(self.W.data, gain=1.414)

        self.adj = adj#邻接矩阵
        self.m = (self.adj > 0)
        self.e = nn.Parameter(torch.zeros(1, len(self.m.nonzero()), dtype=torch.float))
        nn.init.constant_(self.e.data, 1)

        if bias:
            self.bias = nn.Parameter(torch.zeros(out_features, dtype=torch.float))
            stdv = 1. / math.sqrt(self.W.size(2))
            self.bias.data.uniform_(-stdv, stdv)
        else:
            self.register_parameter('bias', None)

    def forward(self, input):
        h0 = torch.matmul(input, self.W[0])
        h1 = torch.matmul(input, self.W[1])

        adj = -9e15 * torch.ones_like(self.adj).to(input.device)
        adj[self.m] = self.e
        adj = F.softmax(adj, dim=1)

        M = torch.eye(adj.size(0), dtype=torch.float).to(input.device)
        output = torch.matmul(adj * M, h0) + torch.matmul(adj * (1 - M), h1)

        if self.bias is not None:
            return output + self.bias.view(1, 1, -1)
        else:
            return output

    def __repr__(self):
        return self.__class__.__name__ + ' (' + str(self.in_features) + ' -> ' + str(self.out_features) + ')'

2.2.2 GraphNonLocal

from __future__ import absolute_import, division

import torch
from torch import nn


class _NonLocalBlock(nn.Module):
    def __init__(self, in_channels, inter_channels=None, dimension=3, sub_sample=1, bn_layer=True):
        super(_NonLocalBlock, self).__init__()

        assert dimension in [1, 2, 3]

        self.dimension = dimension
        self.sub_sample = sub_sample

        self.in_channels = in_channels
        self.inter_channels = inter_channels

        if self.inter_channels is None:
            self.inter_channels = in_channels // 2

        assert self.inter_channels > 0

        if dimension == 3:
            conv_nd = nn.Conv3d
            max_pool = nn.MaxPool3d
            bn = nn.BatchNorm3d
        elif dimension == 2:
            conv_nd = nn.Conv2d
            max_pool = nn.MaxPool2d
            bn = nn.BatchNorm2d
        elif dimension == 1:
            conv_nd = nn.Conv1d
            max_pool = nn.MaxPool1d
            bn = nn.BatchNorm1d
        else:
            raise Exception('Error feature dimension.')

        self.g = conv_nd(in_channels=self.in_channels, out_channels=self.inter_channels,
                         kernel_size=1, stride=1, padding=0)
        self.theta = conv_nd(in_channels=self.in_channels, out_channels=self.inter_channels,
                             kernel_size=1, stride=1, padding=0)
        self.phi = conv_nd(in_channels=self.in_channels, out_channels=self.inter_channels,
                           kernel_size=1, stride=1, padding=0)

        self.concat_project = nn.Sequential(
            nn.Conv2d(self.inter_channels * 2, 1, 1, 1, 0, bias=False),
            nn.ReLU()
        )

        nn.init.kaiming_normal_(self.concat_project[0].weight)
        nn.init.kaiming_normal_(self.g.weight)
        nn.init.constant_(self.g.bias, 0)
        nn.init.kaiming_normal_(self.theta.weight)
        nn.init.constant_(self.theta.bias, 0)
        nn.init.kaiming_normal_(self.phi.weight)
        nn.init.constant_(self.phi.bias, 0)

        if bn_layer:
            self.W = nn.Sequential(
                conv_nd(in_channels=self.inter_channels, out_channels=self.in_channels,
                        kernel_size=1, stride=1, padding=0),
                bn(self.in_channels)
            )
            nn.init.kaiming_normal_(self.W[0].weight)
            nn.init.constant_(self.W[0].bias, 0)
            nn.init.constant_(self.W[1].weight, 0)
            nn.init.constant_(self.W[1].bias, 0)
        else:
            self.W = conv_nd(in_channels=self.inter_channels, out_channels=self.in_channels,
                             kernel_size=1, stride=1, padding=0)
            nn.init.constant_(self.W.weight, 0)
            nn.init.constant_(self.W.bias, 0)

        if sub_sample > 1:
            self.g = nn.Sequential(self.g, max_pool(kernel_size=sub_sample))
            self.phi = nn.Sequential(self.phi, max_pool(kernel_size=sub_sample))

    def forward(self, x):
        batch_size = x.size(0)  # x: (b, c, t, h, w)

        g_x = self.g(x).view(batch_size, self.inter_channels, -1)
        g_x = g_x.permute(0, 2, 1)

        # (b, c, N, 1)
        theta_x = self.theta(x).view(batch_size, self.inter_channels, -1, 1)
        # (b, c, 1, N)
        phi_x = self.phi(x).view(batch_size, self.inter_channels, 1, -1)

        h = theta_x.size(2)
        w = phi_x.size(3)
        theta_x = theta_x.expand(-1, -1, -1, w)
        phi_x = phi_x.expand(-1, -1, h, -1)

        concat_feature = torch.cat([theta_x, phi_x], dim=1)
        f = self.concat_project(concat_feature)
        b, _, h, w = f.size()
        f = f.view(b, h, w)

        N = f.size(-1)
        f_div_C = f / N

        y = torch.matmul(f_div_C, g_x)
        y = y.permute(0, 2, 1).contiguous()
        y = y.view(batch_size, self.inter_channels, *x.size()[2:])
        W_y = self.W(y)
        z = W_y + x

        return z


class GraphNonLocal(_NonLocalBlock):
    def __init__(self, in_channels, inter_channels=None, sub_sample=1, bn_layer=True):
        super(GraphNonLocal, self).__init__(in_channels, inter_channels=inter_channels, dimension=1,
                                            sub_sample=sub_sample, bn_layer=bn_layer)