Rotation robust non-rigid point set registration with Bayesian student’s t mixture model

Abstract

Aiming to improve the performance of non-rigid point set registration, this paper proposes a probabilistic method with student’s t mixture model (SMM) under the Bayesian inference framework. In the proposed method, non-rigid point set registration is formulated as a probabilistic density estimation problem with SMM in Bayesian manner. In order to improve the robustness to rotation degradation, we consider the rotation transformation in modeling non-rigid displacement. Then, the hierarchical Bayesian model of non-rigid point set registration is constructed, and approximate posteriors of model parameters are derived by the variational Bayesian Expectation Maximization update rules, which can provide the uncertainty measurements of parameters. For those parameters without priors imposed, the updating formulae are obtained by directly maximizing variational lower bound. Finally, an empirical coarse-to-fine algorithm is designed to perform non-rigid point set registration process. The experimental results demonstrate that the proposed method can achieve higher matching performance compared with other several state-of-the-art registration methods.

Appendix

In this section, the posterior distributions of some model parameters are listed successively according to the VBEM update rules in Eq. (13) and Eq. (14).

• (1) Indicator variables ${\bf Z}$.

The approximate distribution of $Z$ is expressed as follows:

$$ q\left( {z_{nm} { = }1} \right){ = }\frac{{\left\langle {\pi_{m} } \right\rangle \left\langle {\Lambda_{m} } \right\rangle^{\frac{D}{2}} \Gamma \left( {\frac{{\upsilon_{m} { + }D}}{2}} \right)}}{{\left( {\upsilon_{m} \pi } \right)^{\frac{D}{2}} \Gamma \left( {\frac{{\upsilon_{m} }}{2}} \right)}}\left( {1{ + }\frac{{\varpi_{nm} }}{{\upsilon_{m} }}} \right)^{{ - \frac{{\upsilon_{m} { + }D}}{2}}} {,}\forall n{,}\forall m, $$

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where $\varpi _{{nm}} = \Big\langle \Lambda _{m} \big( {\bf x}_{n} - s{\bf Ry}_{m} - s{\bf Rv}_{m} - {\bf t} \big)^{T} ({\bf x}_{n} - s{\bf Ry}_{m} - s{\bf Rv}_{m} - {\bf t}) \Big\rangle$. After normalization, we obtain

$$ \left\langle {z_{nm} } \right\rangle { = }\frac{{q\left( {z_{nm} { = }1} \right)}}{{\sum\nolimits_{m = 1}^{M} {q\left( {z_{nm} { = }1} \right)} }}{,}\forall n{,}\forall m. $$

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• (2) Latent variables U.

Under the condition $z_{nm} { = }1$, the posterior of each element $u_{nm}$ of ${\bf U}$ is a Gamma distribution, where the shape parameter and the scale parameter are given as follows:

$$ \alpha_{nm} { = }\frac{{v_{m} { + }D}}{2}, $$

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$$ \beta_{nm} { = }\frac{{v_{m} { + }\varpi_{nm} }}{2}. $$

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• (3) Mixing coefficient $\boldsymbol{\pi}$

The posterior of mth element of $\boldsymbol{\pi}$ is updated according to $\xi_{m}$, which is expressed as follows:

$$ \xi_{m} = \sum\nolimits_{{n{ = }1}}^{N} {\left\langle {z_{nm} } \right\rangle } { + }\xi_{0}^{m} . $$

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Next, the updating equation of degree of freedom is derived by directly maximizing the variational lower bound ${\mathcal{L}}\left( q \right)$.

• (4) Degrees of freedom $\upsilon$.

By setting the derivatives of ${\mathcal{L}}\left( q \right)$ with respect to degree of freedom to zero, we have the following (independent) nonlinear equation:

$$ \begin{aligned} &\ln \left( {\frac{{\upsilon_{m} }}{2}} \right){ + }1 - \psi \left( {\frac{{\upsilon_{m} }}{2}} \right)\\ &\quad { + }\frac{1}{{\sum\nolimits_{{n{ = }1}}^{N} {\left\langle {z_{nm} } \right\rangle } }}\sum\nolimits_{{n{ = }1}}^{N} {\left\langle {z_{nm} } \right\rangle \left( {\left\langle {\ln u_{nm} } \right\rangle - \left\langle {u_{nm} } \right\rangle } \right)} = \,0. \end{aligned}$$

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In order to avoid solving the nonlinear equations, we adopt the Stirling’s formula [39] for $\ln \Gamma \left( \cdot \right)$, namely $\ln \Gamma \left( z \right){ = }z\ln z - z{ + }{{\ln \left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } z}} \right. \kern-\nulldelimiterspace} z}} \right)} \mathord{\left/ {\vphantom {{\ln \left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } z}} \right. \kern-\nulldelimiterspace} z}} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$, to obtain the closed form of degree of freedom parameters as suggested by Baldacchino et al. [40]. By the relationship that $\psi \left( z \right){ = }{{\partial \ln \Gamma \left( z \right)} \mathord{\left/ {\vphantom {{\partial \ln \Gamma \left( z \right)} {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z}}$, we have the resulting updating equation:

$$\begin{aligned} \upsilon_{m} &= - \frac{1}{{1{ + }\frac{1}{{\sum\nolimits_{{n{ = }1}}^{N} {\left\langle {z_{nm} } \right\rangle } }}\sum\nolimits_{{n = 1}}^{N} {\left\langle {z_{nm} } \right\rangle \left( {\left\langle {\ln u_{nm} } \right\rangle - \left\langle {u_{nm} } \right\rangle } \right)} }},\\ m &= 1, \ldots ,M. \end{aligned}$$

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