Density-adaptive registration of pointclouds based on Dirichlet Process Gaussian Mixture Models

Abstract

We propose an algorithm for rigid registration of pre- and intra-operative patient anatomy, represented as pointclouds, during minimally invasive surgery. This capability is essential for development of augmented reality systems for guiding such interventions. Key challenges in this context are differences in the point density in the pre- and intra-operative pointclouds, and potentially low spatial overlap between the two. Solutions, correspondingly, must be robust to both of these phenomena. We formulated a pointclouds registration approach which considers the pointclouds after rigid transformation to be observations of a global non-parametric probabilistic model named Dirichlet Process Gaussian Mixture Model. The registration problem is solved by minimizing the Kullback–Leibler divergence in a variational Bayesian inference framework. By this means, all unknown parameters are recursively inferred, including, importantly, the optimal number of mixture model components, which ensures the model complexity efficiently matches that of the observed data. By presenting the pointclouds as KDTrees, both the data and model are expanded in a coarse-to-fine style. The scanning weight of each point is estimated by its neighborhood, imparting the algorithm with robustness to point density variations. Experiments on several datasets with different levels of noise, outliers and pointcloud overlap show that our method has a comparable accuracy, but higher efficiency than existing Gaussian Mixture Model methods, whose performance is sensitive to the number of model components.

Appendix A: some derivations and positive semidefinite rotation matrix objective function

• 1. Derivation of $q^{\star }(v_k)$

Equation 10 in Sect. "Variational inference of Dirichlet Process Gaussian Mixture Model" is deduced as follows:

$$\begin{aligned} \begin{aligned} \ln q^{\star }(v_k)&=E_{{\varvec{V}}_{-k}}[\ln p(v_k\mid \varvec{\alpha })]\\&\quad +\sum _i\sum _j\sum _{l=1}^{\infty }E_{{\varvec{Z}}}[\ln p(z_{ij}=l\mid {\varvec{V}})]\\&=\ln \left[ \frac{\varGamma (\alpha _1+\alpha _2)}{\varGamma (\alpha _1)\varGamma (\alpha _2)}v_k^{\alpha _1}(1-v_k)^{\alpha _2}\right] \\&\quad +\sum _{i}\sum _{j}\sum _{l=1}^{\infty }q(z_{ij}=l\mid {\varvec{V}})\\&\propto \left( \alpha _1-1+\sum _i\sum _j q(z_{ij}=k)\right) v_k \\&\quad +\left( \alpha _2-1+\sum _i\sum _j\sum _{l=k+1}^{\infty }q(z_{ij}=l)\right) \\&\quad \times (1-v_k) +\textrm{const}. \end{aligned} \end{aligned}$$

(A1)

• 2. Derivation of $\rho _{ij,k}$ involved in the update of $q^{\star }({\varvec{Z}})$.

$$\begin{aligned} \begin{aligned} \rho _{ij,k}&=E_{V}[\ln p(z_{ij}=k \mid {\varvec{V}})]+E_{\varvec{\theta }_k}[\ln p({\varvec{x}}_{ij}\mid \varvec{\theta }_k)]\\&= E_{v_k}[\ln v_k]+\sum _{l=1}^{k-1}E_{v_l}[\ln (1-v_l)]\\&\qquad -E_{\Sigma _k}[\ln ((2\pi )^{d/2}\vert \Sigma _k\vert ^{1/2})]\\&\qquad -\frac{1}{2}E_{\varvec{\theta }_k}[({\varvec{x}}_{ij} -\varvec{\mu }_k)^{\textrm{T}}\Sigma _k^{-1}({\varvec{x}}_{ij}-\varvec{\mu }_k)]. \end{aligned} \end{aligned}$$

(A2)

• 3. The derivation of the objective function to solve the rigid transformation between input pointclouds and the global coordinate system.

$$\begin{aligned} \begin{aligned}&\varepsilon ({\varvec{R}}_i,{\varvec{t}}_i) =E_{\varvec{Z,\theta }}[ f({\varvec{X}}_i)\ln p({\varvec{X}}_i\mid \varTheta )]\\&\quad =\sum _{j=1}^{N_i}f ({\varvec{x}}_{ij})\left( \sum _{k=1}^{L}E_{q(z_{ij}=k)\delta _k(z_{ij})}E_{q(\varvec{\theta }_k)}[\ln {\mathcal {N}}({\varvec{x}}_{ij};\varvec{\theta }_k)]\right. \\&\qquad \left. +\left( 1-\sum _{l=1}^{L}\pi _l\right) E_{p(\varvec{\theta }_0 \mid \varvec{\lambda })}[\ln {\mathcal {N}}({\varvec{x}}_{ij} \mid \varvec{\theta }_0)]\right) \\&\quad =\sum _{j=1}^{N_i}f({\varvec{x}}_{ij})\left( \sum _{k=1}^{L}\gamma ^{\star }_{ijk}(\Phi ({\varvec{x}}_{ij})-\bar{\varvec{\mu }}_k)^{\textrm{T}}\bar{\Sigma }_k^{-1}(\Phi ({\varvec{x}}_{ij})-\bar{\varvec{\mu }}_k)\right. \\&\qquad +\left( 1-\sum _{l=1}^{L}\gamma ^{\star }_{ijl}\right) \left. (\Phi ({\varvec{x}}_{ij})-\varvec{\mu }_0)^{\textrm{T}}\Sigma _0^{-1}(\Phi ({\varvec{x}}_{ij})-\varvec{\mu }_0)\right) \end{aligned} \end{aligned}$$

(A3)

• 4. Positive semidefinite rotation matrix objective function. By replacing the rotation matrix ${\textbf{R}}_i$ with its vector representation ${\textbf{r}}_i=\textrm{Vec}({\textbf{R}}_i)$, ${\textbf{r}}_i \in {\mathcal {R}}^{9\times 1}$, the objective function Eq. 20 for solving the rotation matrix can be recast in the general form:

$$\begin{aligned} \begin{aligned} \varepsilon ({\textbf{r}}_i)={\textbf{r}}_i^{\textrm{T}}M{\textbf{r}}_i+2{\textbf{b}}^{\textrm{T}}{\textbf{r}}_i. \end{aligned} \end{aligned}$$

(A4)

The orthogonality constraints ${\textbf{R}}_i{\textbf{R}}_i^{\textrm{T}}={\textbf{I}}$ similarly become:

$$\begin{aligned} {\textbf{r}}_i^{\textrm{T}}\Delta _{kl}{\textbf{r}}_i=\delta _{kl}, k=1,2,3; l=1,2,3. \end{aligned}$$

(A5)

Here $\delta _{kl}=1$ when $k=l$, otherwise $\delta _{kl}=0$. $\Delta _{kl}$ are six symmetric $9\times 9$ matrices, that can be easily obtained from ${\textbf{R}}_i{\textbf{R}}_i^{\textrm{T}}={\textbf{I}}$.

To eliminate the first order term $2{\textbf{b}}^{\textrm{T}}{\textbf{r}}_i$ in ${\textbf{r}}_i$, we let $\mathbf {r'}_i=\{r_i[1],r_i[2],....,r_i[9],1\}$, so that Eq. A4 can be rewritten as

$$\begin{aligned} \varepsilon (\mathbf {r'}_i)=\mathbf {r'}_i^{\textrm{T}}M'\mathbf {r'}_i, \end{aligned}$$

(A6)

in which

$$\begin{aligned} M'=\begin{pmatrix} M&{}b\\ b^{T}&{}0 \end{pmatrix}. \end{aligned}$$

The constraints are then reformulated as:

$$\begin{aligned} \mathbf {r'}_i^{\textrm{T}}\Delta '_{kl}\mathbf {r'}_i=\delta _{kl}, k=1,2,3; l=1,2,3, \end{aligned}$$

(A7)

with

$$\begin{aligned} \Delta '_{kl}=\begin{pmatrix} \Delta _{kl}&{}{\textbf{0}}\\ {\textbf{0}}&{}1 \end{pmatrix}. \end{aligned}$$

Finally, the optimized problem of solving the rotation matrix ${\textbf{R}}_i$ can be formulated in the standard positive semidefinite optimization problem form as:

$$\begin{aligned} \begin{aligned} {{\rho }_i}&=\mathop {{{\,\mathrm{arg\,min}\,}}}_{{\rho }_i} tr(M', {\rho }_i)\\&s.t.\quad {\left\{ \begin{array}{ll} {{\rho }_i} =\mathbf {r'}_i\mathbf {r'}_i^{{\textbf{T}}}\\ {{\rho }_i}[10,10]=1\\ tr(\Delta '_{kl},{{\rho }_i})=\delta _{kl}.\\ \end{array}\right. } \end{aligned} \end{aligned}$$

(A8)

Once ${\rho }_i$ is solved, ${\textbf{r}}_i$ is obtained as ${\textbf{r}}_i={\rho }_i[1:9,10]$.