这篇文章记录下下面两篇文章
Point Pair Features Based Object Detection and Pose Estimation Revisited
PPFNet: Global Context Aware Local Features for Robust 3D Point Matching

1.Point Pair Features Based Object Detection and Pose Estimation Revisited

1.1 摘要

摘要讲了之前的做什么，怎么做，有什么缺点，文章的方法，
摘要先讲了之前的某一篇论文的做法，1.using self-similar-point pairs represent $3D$ target object; 2. Hough-like on reduced pose parameter space 3.match $3D$ model to $3D$ scene.然后提出这些做法有几个缺点：比如对建立的 3D correspondences 敏感，还有模型稀疏或者outlier太多的时候效果不好。
然后文章提出自己的方法：1、couple object detection with a coarse-to-fine segmentation; 2、match: a weighed Houghing voting; an interpolated recovery of pose

1.2 Introduction

对于本文的做法

• object:depth only; extract feature relating pairs of 3D points and their normals: store in hash table
• scene: extract feature: query hash table, hough-like voting; multi instances:multi Hough peak

1.3 Method

1.3.1 Contribution

• an enhanced model representation
• voting with segmentation
• a fast hypotheis verification

1.3.2 Model representation

• computing the surface normals and the weights
• downsample the points
• hash-table is created, storing the quantized pair features as well as the weights and the rotation angles to the ground plane

1.3.2.1 Surface Features

$F(m_1,m_2)=(||d|{|}_2,\angle (n_1,d),\angle (n_2,d),\angle (n_1,n_2))$
上代码比较好理解

1 ppf(point1,point2)
2     d = point1.Location - point2.Location
3     d_unit = d/norm(d)
4     apha1 = acos(point1.Normal*d_unit')
5     apha2 = acos(point2.Normal*d_unit')
6     apha3 = acos(point1.Normal*point2.Normal')
7     return [norm(d),apha1,apha2,apha3] 

Also compute the angle between two vectors as follows:
$\angle (v_1,v_2)=ta{n}^{-1}\frac{||v_1\times v_2||}{v1\cdot v2}$

1.3.2.2 Computing Model Normals

上面的特征是需要法线的，这里能得到越准确的法线越好
use $2^{nd}$ order term
目标是在给定局部参考坐标系的情况下，求出二阶多项式的参数，逼近相邻点的高度场

Given a point $p_i$ on the set $P\in R^3$, MLS operates by fitting a surface of order $m$ in a local $K$-neighborhood $p_k$ and projecting the point on this surface. Fitting is essentially a standard weighted least squares estimation of the polynomial surface parameters.（加权最小二乘估计）
The closer the neighbors are, the higher the contribution is. This is controlled by the weighting function : $w(p_i)=exp(-||p_i-p_k||/2{\sigma }_{mls}^2)$
${\sigma }_{mls}$ can be selected adaptively

1.3.2.3 Weighting Model Points

做这个的原因是作者认为不同的点对于match来说有不同的重要性。文章的做法是将焦点放在物体的可见表面上，因为那些点法线是准确的好。从文章看这个权重是用于hashbin上吗。

• focus on the visible surfaces of the object (accuracy)
• base weighting strategy on ambient occlusion maps

给一个半球，通过积分公式可以算出一个点$p$上面的遮挡
a hemisphere $\Omega$, the occlusion $A_p$ at point $p$ on a surface with normal $n$ can be obtained by computing the integral of the visibility function $V$ :
$A_p=\frac{1}{\pi }{\int }_{\Omega }V(p\cdot w)dw$
$V$ is a dirac delta function, defined to be $1$ if $p$ is occluded in the direction of $w$ and $0$ otherwise. Based on $A_p$, we propose to weigh the entries of the hashtable. Thus, given the hashtable bins, our weights are nothing but a normalized, geometric mean of $A_{m_r}$ and $A_{m_i}$

1.3.2.4 Global Model Description

Given the extracted PPF, the global description is implemented as a hash table mapping the feature space to the space of point pairs.
给定提取的PPF，全局描述被实现为将特征空间映射到点对空间的哈希表
sample distance and angles
a careful downsample a Poisson DiskSampling algorithm
all the points to have at least $d_{dist}$ distances
This algorithm consists of generating samples from a uniform random distribution where the minimum distance between each sample is $2r$.

1.3.3 Online Matching

input : depth
the required normals are computed using SRI method

1.3.3.1 Hough Voting

给定一个固定场景的点对$(s_r,s_i)$,寻找一个最优的模型对应的$(m_r,m_i)$去匹配，计算$6D$ pose.
当找到scene pair相对应的model pair，就会建立一个中间坐标系，其中$m_i$和$s_i$通过围绕法线旋转对象来对齐。模型和场景的平面旋转角有先进行预计算。
Whenever a model pair, corresponding to a scene pair is found, an intermediate coordinate system is established, where $m_i$ and $s_i$ are aligned by rotating the object around the normal. The planar rotation angle ${\alpha }_m$ for the model is precomputed, while the analogous for the scene point ${\alpha }_s$ is computed online. The resulting plana r otation angle aroundx-axis is found by a simple subtraction, $\alpha ={\alpha }_m-{\alpha }_s$.

1.3.3.2 Matching Disjoint Segments

显示segment场景,然后filter掉一些场景

深度图看成是无向图，顶点和边，每条边都有非负权重。然后找到一个set $C$，计算$C$之间的相似性。
treat the depth image as an undirected graph G={V,E}G = \{V,E\}G={V,E}, vertices $v_i\in V$ and edges $(v_i,v_j)\in E$, each edge has a non-negative weight $w(v_i,v_j)$, We then seek to find a set of components $C\in S$, where S isthesegmentation. The component-wise similarity is achieved via the weights of the graph.
A pair-wise comparison predicate $(P)$
$$P(C_1,C_2)=\{\begin{array}{r}1,ifD(C_1,C_2)>M_{int}(C_1,C_2)\le 0\\ 0,others\end{array}$$
$D(C_1,C_2)$ is the difference between components, defined as the minimum weight edge:
$D(C_1,C_2)={\min }_{v_1\in C_1,v_j\in C_2,(v_i,v_j)\in E}w((v_i,v_j))$
and minimum internal difference $M_{int}$ equals :
$M_{int}(C_1,C_2)=min(Int(C_1)+\tau (C_1),Int(C_2)+\tau (C_2))$

This approach generates a descent segmentation. But do not need to process every segment. Use three method to filter.

• undefined depth values
• not obeying the size constraints
• evaluate the linearity of the segments(normal)

1.3.3.3 Pose Clustering and Averaging

1.3.3.4 Hypotheses Verification

Categorize the visible space into : Clutter(outlier) $S_c$, occluders $S_o$ and points on the model $S_m$ according to the following projectin error function:
$E_h(p,m)=D_p-\Phi (p|M,{\Theta }_h,K)$
$\Phi$ selects the projection of the model points $M$ corresponding to pixel $p$, given a camera matrix $K$ and the pose parameters ${\Theta }_h$ for hypothesis $h$. The classification for a given valid point $p$ is conducted as :
$$p\in \{\begin{array}{r}{S}_m,if|E_h(p,m)|\le {\tau }_m\\ S_o,if|E_h(p,m)|\ge {\tau }_o\\ S_c,otherwise\end{array}$$
the Score for a given hypothesis is:
$S_h=(1-\frac{p\in S_o}{N_m})\ast \frac{p\in S_m}{N_m-|S_o|}$
$N_m$ the number of model points on valid region of the projection $\Phi (p|M,{\Theta }_h,K)$, ${\tau }_m$ and ${\tau }_o$ depend on the sensor and are relaxed

1.4 Result

evaluate on synthetic and real datasets
real datsets: ACCV3D dataset

1.2 PPFNet: Global Context Aware Local Features for Robust 3D Point Matching

持续更新。。