Robust two-phase registration method for three-dimensional point set under the Bayesian mixture framework

Abstract

In order to establish effective correspondences, a two-phase registration method for three-dimensional point set is proposed under the Bayesian mixture framework. In the first phase, the mixture model consisted of student’s t distribution and von Mises-Fisher (vMF) distribution is designed to perform similarity point set registration for recovering rotation transformation, where both distributions are used to measure positional and directional errors, respectively. The second phase implements nonrigid (affine as a particular case) registration between data point set and transformed model point set obtained in the first phase, which is based on student’s t mixture model (SMM) using positional information only. In each phase, variational inference is used to obtain approximate posteriors of model parameters. The experimental results on various datasets demonstrate that our proposed method can achieve better registration performance in terms of robustness to rotation and outliers.

Appendices

Appendix A: The similarity point set registration

The updating formulae of partial parameters in the first phase are given as follows.

(1) Indicated variables ${\mathbf{Z}}$ and hidden variables ${\mathbf{U}}$.

Let $\varpi_{nm} \,{ = }\,\left\langle {\Lambda_{m} } \right\rangle \left( {{\mathbf{x}}_{n} - {\mathbf{w}}_{m} } \right)^{T} \left( {{\mathbf{x}}_{n} - {\mathbf{w}}_{m} } \right)$, where $\left\langle \cdot \right\rangle$ denotes the expectation with respect to $q\left( \cdot \right)$ with subscript omission. Considering the uncertainty of hidden variables $u_{nm}$, the expectation of indicated variable $z_{nm}$ is given as follows:

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

(20)

where $q\left( {z_{nm} { = }1} \right) \propto {{\left\langle {\pi_{m} } \right\rangle \left\langle {\Lambda_{m} } \right\rangle^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-\nulldelimiterspace} 2}}} \Gamma \left( {{{\left( {\tau_{m} { + }D} \right)} \mathord{\left/ {\vphantom {{\left( {\tau_{m} { + }D} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\mathcal{F} \left( {{\hat{\mathbf{x}}}_{n} \left| {{\hat{\mathbf{y}}}_{m} ,{\mathbf{R}}{,}\kappa } \right.} \right)} \mathord{\left/ {\vphantom {{\left\langle {\pi_{m} } \right\rangle \left\langle {\Lambda_{m} } \right\rangle^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-\nulldelimiterspace} 2}}} \Gamma \left( {{{\left( {\tau_{m} { + }D} \right)} \mathord{\left/ {\vphantom {{\left( {\tau_{m} { + }D} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)f_{vm} \left( {{\hat{\mathbf{x}}}_{n} \left| {{\hat{\mathbf{y}}}_{m} ,{\mathbf{R}}{,}\kappa } \right.} \right)} {\left( {\tau_{m}^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-\nulldelimiterspace} 2}}} \Gamma \left( {{{\tau_{m} } \mathord{\left/ {\vphantom {{\tau_{m} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\left( {1{ + }{{\varpi_{nm} } \mathord{\left/ {\vphantom {{\varpi_{nm} } {\tau_{m} }}} \right. \kern-\nulldelimiterspace} {\tau_{m} }}} \right)^{{{{\left( {\tau_{m} { + }D} \right)} \mathord{\left/ {\vphantom {{\left( {\tau_{m} { + }D} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\tau_{m}^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-\nulldelimiterspace} 2}}} \Gamma \left( {{{\tau_{m} } \mathord{\left/ {\vphantom {{\tau_{m} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\left( {1{ + }{{\varpi_{nm} } \mathord{\left/ {\vphantom {{\varpi_{nm} } {\tau_{m} }}} \right. \kern-\nulldelimiterspace} {\tau_{m} }}} \right)^{{{{\left( {\tau_{m} { + }D} \right)} \mathord{\left/ {\vphantom {{\left( {\tau_{m} { + }D} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right)}}.$

Given the condition $z_{nm} { = }1$, the posterior of hidden variable $u_{nm}$ is still a Gamma distribution with shape and scale parameters given as follows:

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

(21)

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

(22)

(2) Mixing coefficient ${{\varvec{\uppi}}}$.

For the mixing coefficient ${{\varvec{\uppi}}}$, the posterior is a Dirichlet distribution with component $\xi_{m}$ given by:

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

(23)

(3) Isotropic precision $\left\{ {\Lambda_{m} } \right\}$.

The approximate posterior of isotropic precision $\Lambda_{m}$ for the $m^{th}$ mixture component is still a Gamma distribution with updated shape and scale parameters:

$$\gamma_{m} \,{ = }\,\frac{D}{{2}}\sum\limits_{{n{ = }1}}^{N} {\left\langle {z_{nm} } \right\rangle } + \gamma_{0} ,$$

(24)

$$\delta_{m}\, { = }\,\frac{1}{2}\sum\limits_{{n{ = }1}}^{N} {\left\langle {z_{nm} u_{nm} } \right\rangle \left( {{\mathbf{x}}_{n} - {\mathbf{w}}_{m} } \right)^{T} \left( {{\mathbf{x}}_{n} - {\mathbf{w}}_{m} } \right)} + \delta_{0} .$$

(25)

(4) Degree of freedom $\left\{ {\tau_{m} } \right\}$.

By letting ${{\partial {\mathcal{L}}_{1} \left( q \right)} \mathord{\left/ {\vphantom {{\partial {\mathcal{L}}_{1} \left( q \right)} {\partial \tau_{m} }}} \right. \kern-\nulldelimiterspace} {\partial \tau_{m} }} = 0$ and using the stirling’s formula, we can obtain the below closed-form approximation:

$$\tau_{m} \approx - \frac{1}{{1{ + }{{\sum\nolimits_{n = 1}^{N} {\left\langle {z_{nm} } \right\rangle \left( {\left\langle {\ln u_{nm} } \right\rangle - \left\langle {u_{nm} } \right\rangle } \right)} } \mathord{\left/ {\vphantom {{\sum\nolimits_{n = 1}^{N} {\left\langle {z_{nm} } \right\rangle \left( {\left\langle {\ln u_{nm} } \right\rangle - \left\langle {u_{nm} } \right\rangle } \right)} } {\sum\nolimits_{n = 1}^{N} {\left\langle {z_{nm} } \right\rangle } }}} \right. \kern-\nulldelimiterspace} {\sum\nolimits_{n = 1}^{N} {\left\langle {z_{nm} } \right\rangle } }}}}.$$

(26)

Appendix B: The nonrigid point set registration

The updating formulae of partial parameters in the second phase are given as below.

(1) Indicated variables ${{\varvec{\Xi}}}$ and hidden variables ${{\varvec{\upchi}}}$.

Let $\omega_{nm} \,{ = }\,\left\langle {\left( {{\mathbf{x}}_{n} - {\mathbf{A\overline{w}}}_{m} - {\mathbf{B\Phi }}\left( {{\mathbf{y}}_{m} } \right)} \right)^{T} {{\varvec{\Delta}}}_{m} \left( {{\mathbf{x}}_{n} - {\mathbf{A\overline{w}}}_{m} { - }{\mathbf{B\Phi }}\left( {{\mathbf{y}}_{m} } \right)} \right)} \right\rangle$, we have

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

(27)

where $q\left( {\Xi_{nm} { = }1} \right) \propto {{\left\langle {\Pi_{m} } \right\rangle \left| {{\left\langle{{\varvec{\Delta}}}_{m}\right\rangle } } \right|^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \Gamma \left( {{{\left( {v_{m} { + }D} \right)} \mathord{\left/ {\vphantom {{\left( {v_{m} { + }D} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)} \mathord{\left/ {\vphantom {{\left\langle {\Pi_{m} } \right\rangle \left\langle {\left| {{{\varvec{\Delta}}}_{m} } \right|} \right\rangle^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \Gamma \left( {{{\left( {v_{m} { + }D} \right)} \mathord{\left/ {\vphantom {{\left( {v_{m} { + }D} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)} {\left( {v_{m} } \right)^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-\nulldelimiterspace} 2}}} \Gamma \left( {{{v_{m} } \mathord{\left/ {\vphantom {{v_{m} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\left( {1{ + }{{\omega_{nm} } \mathord{\left/ {\vphantom {{\omega_{nm} } {v_{m} }}} \right. \kern-\nulldelimiterspace} {v_{m} }}} \right)^{{{{\left( {v_{m} { + }D} \right)} \mathord{\left/ {\vphantom {{\left( {v_{m} { + }D} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}}} }}} \right. \kern-\nulldelimiterspace} {\left( {v_{m} } \right)^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-\nulldelimiterspace} 2}}} \Gamma \left( {{{v_{m} } \mathord{\left/ {\vphantom {{v_{m} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\left( {1{ + }{{\omega_{nm} } \mathord{\left/ {\vphantom {{\omega_{nm} } {v_{m} }}} \right. \kern-\nulldelimiterspace} {v_{m} }}} \right)^{{{{\left( {v_{m} { + }D} \right)} \mathord{\left/ {\vphantom {{\left( {v_{m} { + }D} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}}} }}.$ The posterior of hidden variable $\chi_{nm}$ follows a Gamma distribution with parameters:

$$\overline{\alpha }_{nm}\, { = }\,\frac{{v_{m} { + }D}}{2},$$

(28)

$$\overline{\beta }_{nm} \,{ = }\,\frac{{v_{m} { + }\omega_{nm} }}{2}.$$

(29)

(2) Mixing coefficient ${{\varvec{\Pi}}}$.

The posterior of mixing coefficient ${{\varvec{\Pi}}}$ follows a Dirichlet distribution with component $k_{m}$ given by:

$$k_{m} = \sum\limits_{{n{ = }1}}^{N} {\left\langle {\Xi_{nm} } \right\rangle } { + }k_{0}^{m} .$$

(30)

(3) Precision matrix ${{\varvec{\Delta}}}$

For the anisotropic precision matrix ${{\varvec{\Delta}}}_{m}$ of the $m^{th}$ mixture component, its posterior follows a Wishart distribution, i.e., $q\left( {\Delta_{m} } \right) \sim W\left( {\Delta_{m} \left| {r_{m} ,{\mathbf{S}}_{m} } \right.} \right)$, where

$$r_{m} { = }\sum\limits_{{n{ = }1}}^{N} {\left\langle {\Xi_{nm} } \right\rangle } { + }r_{0} ,$$

(31)

$${\mathbf{S}}_{m}^{ - 1}\, { = }\,{\mathbf{S}}_{0}^{ - 1} { + }\sum\limits_{{n{ = }1}}^{N} {\left\langle {\Xi_{nm} \chi_{nm} } \right\rangle \left\langle {\left( {{\mathbf{x}}_{n} - {\mathbf{A\overline{w}}}_{m} - {\mathbf{B\Phi }}\left( {{\mathbf{y}}_{m} } \right)} \right)\left( {{\mathbf{x}}_{n} - {\mathbf{A\overline{w}}}_{m} - {\mathbf{B\Phi }}\left( {{\mathbf{y}}_{m} } \right)} \right)^{T} } \right\rangle } .$$

(32)

When the precision is isotropic, the posterior degenerates to be a Gamma distribution.

(4) Precision ${{\varvec{\upupsilon}}}$

For the $l\text{th}$ element of precision parameter ${{\varvec{\upupsilon}}}$, the posterior follows a Gamma distribution with shape and scale parameters given as follows:

$$a_{l}\, { = }\,a_{0} { + }\frac{D}{2},$$

(33)

$$b_{l} \,{ = }\,b_{0} { + }\frac{1}{2}\sum\limits_{{q{ = }1}}^{D} {\left\langle {{\mathbf{A}}_{ql}^{2} } \right\rangle } .$$

(34)

(5) Precision ${{\varvec{\upeta}}}$

For the $l\text{{th}}$ element of precision parameter ${{\varvec{\upeta}}}$, the posterior follows a Gamma distribution with shape and scale parameters given as follows:

$$c_{l} \,{ = }\,c_{0} { + }\frac{D}{2},$$

(35)

$$d_{l}\, { = }\,d_{0} { + }\frac{1}{2}\sum\limits_{{q{ = }1}}^{D} {\left\langle {{\mathbf{B}}_{ql}^{2} } \right\rangle } .$$

(36)

(6) Regularization parameter $\lambda$

The posterior distribution of regularization parameter $\lambda$ is still a Gamma distribution, where scale and shape parameters can be updated as follows:

$$\sigma = \sigma_{0} + \frac{MD}{2},$$

(37)

$$\varsigma = \varsigma_{0} + \frac{1}{2}\sum\limits_{q = 1}^{D} {\left\langle {{\mathbf{B}}_{q \cdot } {\mathbf{\Phi B}}_{q \cdot }^{T} } \right\rangle } .$$

(38)

In this way, adaptively updating strategy can avoid inappropriate parameter settings.

(7) Degree of freedom $\left\{ {v_{m} } \right\}$.

Similar to the Eq. (26), we have the following approximate updating equation:

$$v_{m} \approx - \frac{1}{{1{ + }{{\sum\nolimits_{n = 1}^{N} {\left\langle {\Xi_{nm} } \right\rangle \left( {\left\langle {\ln \chi_{nm} } \right\rangle - \left\langle {\chi_{nm} } \right\rangle } \right)} } \mathord{\left/ {\vphantom {{\sum\nolimits_{n = 1}^{N} {\left\langle {\Xi_{nm} } \right\rangle \left( {\left\langle {\ln \chi_{nm} } \right\rangle - \left\langle {\chi_{nm} } \right\rangle } \right)} } {\sum\nolimits_{n = 1}^{N} {\left\langle {\Xi_{nm} } \right\rangle } }}} \right. \kern-\nulldelimiterspace} {\sum\nolimits_{n = 1}^{N} {\left\langle {\Xi_{nm} } \right\rangle } }}}}.$$

(39)