ISS 关键点检测

本教程展示如何检测三维形状的 ISS 关键点（keypoints）。其实现基于 Yu Zhong 在 2009 年论文《Intrinsic Shape Signatures: A Shape Descriptor for 3D Object Recognition》中提出的关键点检测模块。

ISS 关键点

ISS 显著性度量（saliency measure）基于点 $p$ 支撑域内各点的散布矩阵（scatter matrix） $\boldsymbol{\Sigma}(\mathbf{p})$ 的特征值分解（Eigenvalue Decomposition, EVD），即：

\begin{array}{l} \boldsymbol{\Sigma}(\mathbf{p})=\frac{1}{N} \sum_{\mathbf{q} \in \mathcal{N}(\mathbf{p})}\left(\mathbf{q}-\mu_{\mathbf{p}}\right)\left(\mathbf{q}-\mu_{\mathbf{p}}\right)^{T} \quad \text { with } \\ \mu_{\mathbf{p}}=\frac{1}{N} \sum_{\mathbf{q} \in \mathcal{N}(\mathbf{p})} \mathbf{q} \end{array}

给定 $\boldsymbol{\Sigma}(\mathbf{p})$ ，其特征值按从大到小的顺序记为 $\lambda_1$ 、 $\lambda_2$ 、 $\lambda_3$ 。在剪枝（pruning）阶段，会保留相邻两个特征值之比低于某阈值的点：

\frac{\lambda_{2}(\mathbf{p})}{\lambda_{1}(\mathbf{p})}<\gamma_{12} \wedge \frac{\lambda_{3}(\mathbf{p})}{\lambda_{2}(\mathbf{p})}<\gamma_{23}

这样做的理由是：避免在沿各主方向具有相似展布的点上检测关键点——在这些点上无法建立可重复的规范参考系（canonical reference frame），因此随后的描述阶段很难取得好的效果。在剩余的点中，显著性由最小特征值的大小决定：

\rho(\mathbf{p}) \doteq \lambda_{3}(\mathbf{p})

从而只保留沿每个主方向都有较大变化的点。检测之后，如果某点在给定邻域内具有最大的显著性值，则被视为关键点。

ISS 关键点检测示例

# Compute ISS Keypoints on ArmadilloMesh
armadillo = o3d.data.ArmadilloMesh()
mesh = o3d.io.read_triangle_mesh(armadillo.path)
mesh.compute_vertex_normals()

pcd = o3d.geometry.PointCloud()
pcd.points = mesh.vertices

tic = time.time()
keypoints = o3d.geometry.keypoint.compute_iss_keypoints(pcd)
toc = 1000 * (time.time() - tic)
print("ISS Computation took {:.0f} [ms]".format(toc))

mesh.compute_vertex_normals()
mesh.paint_uniform_color([0.5, 0.5, 0.5])
keypoints.paint_uniform_color([1.0, 0.75, 0.0])
o3d.visualization.draw_geometries([keypoints, mesh], front=[0, 0, -1.0])

输出：

ISS Computation took 666 [ms]

# This function is only used to make the keypoints look better on the rendering
def keypoints_to_spheres(keypoints):
    spheres = o3d.geometry.TriangleMesh()
    for keypoint in keypoints.points:
        sphere = o3d.geometry.TriangleMesh.create_sphere(radius=0.001)
        sphere.translate(keypoint)
        spheres += sphere
    spheres.paint_uniform_color([1.0, 0.75, 0.0])
    return spheres

# Compute ISS Keypoints on Standford BunnyMesh, changing the default parameters
bunny = o3d.data.BunnyMesh()
mesh = o3d.io.read_triangle_mesh(bunny.path)
mesh.compute_vertex_normals()

pcd = o3d.geometry.PointCloud()
pcd.points = mesh.vertices

tic = time.time()
keypoints = o3d.geometry.keypoint.compute_iss_keypoints(pcd,
                                                        salient_radius=0.005,
                                                        non_max_radius=0.005,
                                                        gamma_21=0.5,
                                                        gamma_32=0.5)
toc = 1000 * (time.time() - tic)
print("ISS Computation took {:.0f} [ms]".format(toc))

mesh.compute_vertex_normals()
mesh.paint_uniform_color([0.5, 0.5, 0.5])
o3d.visualization.draw_geometries([keypoints_to_spheres(keypoints), mesh])

输出：

ISS Computation took 57 [ms]