Coherent Point Drift算法：图像点集非刚性注册详解

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"Point Set Registration: Coherent Point Drift 是一篇针对计算机视觉任务中关键组件——点集注册的研究论文。该研究由 Andriy Myronenko 和 Xubo Song 贡献，他们提出了一种名为 Coherent Point Drift (CPD) 的概率方法，旨在解决两个点集之间的对应关系建立以及两者之间的未知空间变换问题，无论是刚体（rigid）还是非刚体（nonrigid）情况。在点集注册中，目标是找出两组点之间的对应关系，并确定将一个点集映射到另一个点集的变换。然而，实际应用中存在多种挑战，如未知的非刚体空间变换、点集维度高、噪声和异常值等。CPD算法通过将点集对齐视为一种概率密度估计问题来解决这些难题。它利用高斯混合模型 (GMM) 来拟合第一个点集（即中心点）到第二个点集的数据，通过最大化似然函数来找到最佳匹配。在非刚体情况下，CPD算法强制GMM中心点以一致的方式移动，以保持点集的拓扑结构。这通过重新参数化GMM中心点的位置，使用刚体参数来实现，并且找到了一个优化问题的闭形式解。这种一致性约束确保了在变形过程中点集结构的连贯性。在处理刚体变换时，由于结构更简单，算法的计算过程相对直观，但同样注重保持对应关系的准确性和点集结构的稳定性。Coherent Point Drift算法提供了一个强大的工具箱，适用于那些需要精确点集对齐和变形估计的计算机视觉应用，如3D重建、医学图像分析和机器人导航等领域。通过其概率模型和优化策略，CPD算法能够在复杂场景下提供鲁棒的解决方案，提高了点集注册的准确性和效率。"

form solution with Euler angles parameterization is not

feasible. Luo and Hancock [6], [13] find the rotation matrix

through singular value decomposition (SVD). They ignore

some terms of the objective function, which leads to only an

approximate solution. We shall derive the exact closed form

solution (M-step) for the rigid point set registration and

discuss its difference from the related methods in Section 4.

2.2 Nonrigid Point Set Registration Methods

Earlier works on nonrigid point set registration include

Hinton et al. [19], [20], who used the probabilistic GMM

formulation. The GMM centroids are uniformly positioned

along the contour (modeled using splines), which allows for

nonrigid transformations. In practice, the method is applic-

able only to contour-like point sets. One of the most popular

nonrigid point set registration method is by Chui and

Rangarajan [8]. They proposed using Thin Plate Spline

(TPS) [21], [22] parameterization of the transformation,

following RPM, which results in the TPS-RPM method.

Similarly to the rigid case, they use deterministic annealing

and alternate updates for soft assignment and TPS para-

meters estimation. They also showed that TPS-RPM is

equivalent (with several modifications) to EM for GMM [9].

Tsin and Kan ade [ 23] proposed a correlation-ba sed

approach to point set registration, which was later

improved by Jian and Vemuri [24]. The method considers

the registration as an alignment between two distributions,

where each of the point sets represents the GMM centroids.

One of the point sets is parameterized by rigid/affine

parameters (in the rigid/affine case) or TPS (in the nonrigid

case). The transformation parameters are estimated to

minimize the L

norm between the distributions. These

methods all use explicit TPS parameterization, which is

equivalent to a regularization of second order derivatives of

the transformation [21], [22]. The TPS parameterization

does not exist when the dimension of points is higher than

three, which limits the applicability of such methods.

Huang et al. [25] proposed implicitly embedding the

shape (or point sets in our case) into distance transform

space and aligning them similarly to nonrigid image

registration algorithms. The authors use sum-of-squared-

differences similarity measure between the embedded

spaces and incremental free form deformation (FFD) to

parameterize the transformation. The method performs

well on relatively simple data sets.

Finally, in our previous work, we presented the Coherent

Point Drift (CPD) algorithm [26] for nonrigid point set

registration. The algorithm regularizes the displacement

(velocity) field between the point sets following the motion

coherence theory (MCT) [27], [28]. Using variational

calculus, we obtained that the optimal displacement field

has an elegant kernel form in multiple dimensions. In this

paper, we shall elaborate and analyze the CPD algorithm.

We also extend the general nonrigid registration framework

and show that CPD and TPS-RPM are its special cases.

Among other contributions, we estimate the width of GMM

components without the time-consuming deterministic

annealing and show a fast CPD implementation to reduce

the computational complexity to linear. We shall discuss

and compare o ur method to the works of Chui and

Rangarajan [8] and Jian and Vemuri [24] in Section 5.

3GENERAL METHODOLOGY

We consider the alignment of two point sets as a probability

density estimation problem, where one point set represents

the GMM centroids and the other one represents the data

points. At the optimum, two point sets become aligned and

the correspondence is obtained using the maximum of the

GMM posterior probability for a given data point. Core to

our method is to force GMM centroids to move coherently

as a group to preserve the topological structure of the point

sets. Throughout the paper, we use the following notations:

. D—dimension of the point sets,

. N;M—number of points in the point sets,

. X

ND

¼ðx

; ...; x

—the first point set (the data

points),

. Y

MD

¼ðy

; ...; y

—the second point set (the

GMM centroids),

. TðY;Þ—Transformation T applied to Y, where  is

a set of the transformation parameters,

. I—identity matrix,

. 1—column vector of all ones,

. dðaÞ—diagonal matrix formed from the vector a.

We consider the points in Y as the GMM centroids and the

points in X as the data points generated by the GMM. The

GMM probability density function is

pðxÞ¼

Mþ1

m¼1

PðmÞpðxjmÞ; ð1Þ

where pðxjmÞ¼

ð2

D=2

exp



xy

kk2

2

. We also added an

additional uniform distribution pðxjM þ 1Þ¼

to the

mixture model to account for noise and outliers. We use

equal isotropic covariances 

and equal membership

probabilities P ðmÞ¼

for all GMM components

(m ¼ 1; ...;M). Denoting the weight of the uniform

distribution as w, 0  w  1, the mixture model takes

the form

pðxÞ¼w

þð1  wÞ

m¼1

pðxjmÞ: ð2Þ

We reparameterize the GMM centroid locations by a set of

parameters  and estimate them by maximizing the

likelihood or, equivalently, by minimizing the negative

log-likelihood function

Eð; 

Þ¼

n¼1

log

Mþ1

m¼1

PðmÞpðx

jmÞ; ð3Þ

where we make the i.i.d. data assumption. We define the

correspondence probability between two points y

and x

as the posterior probability of the GMM centroid given the

data point: P ðmjx

Þ¼PðmÞpðx

jmÞ=pðx

Þ.

We use the EM algorithm [29], [30] to find  and 

. The

idea of EM is first to guess the values of parameters (“old”

parameter values) and then use the Bayes’ theorem to

compute a posteriori probability distributions P

old

ðmjx

Þ of

mixture components, which is the expectation or E-step of

the algorithm. The “new” parameter values are then found

2264 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 32, NO. 12, DECEMBER 2010

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