Error-Controlled Model Approximation for Gaussian Process Morphable Models

Abstract

Gaussian Process Morphable Models (GPMMs) unify a variety of non-rigid deformation models for surface and image registration. Deformation models, such as B-splines, radial basis functions, and PCA models are defined as a probability distribution using a Gaussian process. The method depends heavily on the low-rank approximation of the Gaussian process, which is mandatory to obtain a parametric representation of the model. In this article, we propose the use of the pivoted Cholesky decomposition for this task, which has the following advantages: (1) Compared to the current state of the art used in GPMMs, it provides a fully controllable approximation error. The algorithm greedily computes new basis functions until the user-defined approximation accuracy is reached. (2) Unlike the currently used approach, this method can be used in a black-box-like scenario, whereas the method automatically chooses the amount of basis functions for a given model and accuracy. (3) We propose the Newton basis as an alternative basis for GPMMs. The proposed basis does not need an SVD computation and can be iteratively refined. We show that the proposed basis functions achieve competitive registration results while providing the mentioned advantages for its computation.

A Appendix

1.1 A.1 Advanced Nyström Schemes

Nyström schemes are suitable if the eigenfunctions of the Karhunen–Loève expansion are only required in certain predetermined points $\mathbf {x}_1,\ldots ,\mathbf {x}_N$. For this purpose, the integral operator (2) is approximated by a quadrature formula

$$\begin{aligned} \int _\varOmega \mathbf {K}(\cdot ,\mathbf {x}) \mathbf {f}(\mathbf {x}) {\text {d}}\!\rho (\mathbf {x})\approx \sum _{i=1}^N\omega _i \mathbf {K}(\cdot ,\varvec{\xi }_i)\mathbf {f}(\varvec{\xi }_i) \end{aligned}$$

with quadrature points $\varvec{\xi }_i$ and weights $\omega _i$. The discrete eigenvalue problem then reads

$$\begin{aligned} \mathbf {C}_{\text {Nystr}}\hat{\varvec{\phi }}_{m,N}=\lambda _{m,N}\hat{\varvec{\phi }}_{m,N} \end{aligned}$$

with the system matrix

$$\begin{aligned} \mathbf {C}_{\text {Nystr}}=\big [\omega _j\mathbf {K}(\mathbf {x}_i,\mathbf {x}_j)\big ]_{i,j=1}^N \end{aligned}$$

and the point values

$$\begin{aligned} \hat{\varvec{\phi }}_{m,N}\approx \big [\varvec{\phi }_m(\mathbf {x}_i)\big ]_i,\quad i=1,\ldots ,N. \end{aligned}$$

Note that the system matrix $\mathbf {C}_{\text {Nystr}}$ is not symmetric in general. Assuming positive quadrature weights, i.e. $\omega _i>0$, defining

$$\begin{aligned} \mathbf {M}_{\text {Nystr}}={\text {diag}}(\sqrt{\omega _1},\ldots ,\sqrt{\omega _N}) \end{aligned}$$

and setting $\varvec{\phi }_{m,N}=\mathbf {M}_{\text {Nystr}}\hat{\varvec{\phi }}_{m,N}$ yields a symmetric, generalized eigenvalue problem

$$\begin{aligned} \mathbf {M}_{\text {Nystr}}\mathbf {C}\mathbf {M}_{\text {Nystr}}^{\intercal }\varvec{\phi }_{m,N}=\lambda _{m,N}\mathbf {M}_{\text {Nystr}}\varvec{\phi }_{m,N} \end{aligned}$$

with the matrix

$$\begin{aligned} \mathbf {C}=\big [\mathbf {K}(\mathbf {x}_i,\mathbf {x}_j)\big ]_{i,j=1}^N, \end{aligned}$$

(17)

see also [15]. As it turns out, the finite element scheme yields an eigenvalue problem with a similar structure.

1.2 A.2 Finite Element Scheme on a Rectangular Grid

Finite element schemes for functions with values in three dimensions rely on a finite dimensional subspace $\mathbf {V}_N\subset \big [L^2(\varOmega )\big ]^3$ with basis $\big \{\varvec{\varphi }_1,\ldots ,\varvec{\varphi }_N\big \}$ to represent the eigenfunctions of the Karhunen–Loève expansion. To construct such a finite dimensional space, we consider a uniform rectangular grid ${\mathcal {Q}}_h$ on $\varOmega$ where each cell has a size of $h_1\times h_2\times h_3$. To each vertex $\mathbf {x}_1,\ldots ,\mathbf {x}_N$ we assign a function $\varphi _i$ with the property

$$\begin{aligned} \varphi _i(\mathbf {x}_j)= {\left\{ \begin{array}{ll} 1,&{}i=j,\\ 0,&{}i\ne j, \end{array}\right. } \quad i,j=1,\ldots ,N, \end{aligned}$$

(18)

where on each cell $Q_h\in \mathcal {Q}_h$, the basis function $\varphi _i$ is a trilinear polynomial, i.e.

$$\begin{aligned} \varphi _i(\mathbf {y})\big |_{Q_h}&= a_1\,+a_2y_1+a_3y_2+a_4y_1y_2\\&\quad +\,a_5y_3+a_6y_1y_3+a_7y_2y_3+a_8y_1y_2y_3. \end{aligned}$$

Here, the coefficients are uniquely determined such that (18) holds. This means especially that the $\varphi _i$ are only non-zero in the eight cells with vertex $\mathbf {x}_i$. Note especially that all $\varphi _i$ are linearly independent, so we can define $V_h\subset L^2(\varOmega )$ as the vector space spanned by the basis $\varphi _1,\ldots ,\varphi _N$. A finite dimensional subspace of $\big [L^2(\varOmega )\big ]^3$ is then given by $\mathbf {V}_h=V_h\times V_h\times V_h$.

1.3 A.3 Advanced Finite Element Schemes

Having a finite dimensional subspace at hand yields, cf., e.g., [15], the generalized eigenvalue problem

$$\begin{aligned} \mathbf {C}_{\text {FEM}}\varvec{\phi }_{m,N}=\lambda _{m,N}\mathbf {M}_{\text {FEM}}\varvec{\phi }_{m,N} \end{aligned}$$

(19)

with system matrices

$$\begin{aligned} \mathbf {C}_{\text {FEM}}={}&\big [\big ({\mathcal {T}}_{\mathbf {K}}\varvec{\varphi }_j,\varvec{\varphi }_i\big )_{[L^2_{\rho }(D)]^3}\big ]_{i,j=1}^N,\\ \mathbf {M}_{\text {FEM}}={}&\big [\big (\varvec{\varphi }_j,\varvec{\varphi }_i\big )_{[L^2_{\rho }(D)]^3}\big ]_{i,j=1}^N, \end{aligned}$$

$\mathcal {T}_\mathbf {K}$ denoting the integral operator from (2), and the approximate eigenfunctions

$$\begin{aligned} \varvec{\phi }_m(\mathbf {x}) \approx \varvec{\phi }_{m,N}(\mathbf {x}) =\sum _{i=1}^N\big (\varvec{\phi }_{m,N}\big )_i\varvec{\varphi }_i(\mathbf {x}). \end{aligned}$$

It thus remains to explain how to assemble these matrices.

Since the basis functions $\varvec{\varphi }_i$ are non-zero only on a few elements, the mass matrix $\mathbf {M}_{\text {FEM}}$ is sparse. Inserting the definition of ${\mathcal {T}}_\mathbf {K}$ into the definition of $\mathbf {C}_{\text {FEM}}$, we obtain

$$\begin{aligned} \mathbf {C}_{\text {FEM}}=\bigg [\int _D\int _D\mathbf {K}(\mathbf {x},\mathbf {y}) \varvec{\varphi }_j(\mathbf {y})\varvec{\varphi }_i^\intercal (\mathbf {x}){\text {d}}\!{\rho }(\mathbf {y}){\text {d}}\!{\rho }(\mathbf {x})\bigg ]_{i,j=1}^N. \end{aligned}$$

In order to compute this integral, it is very common in finite element methods to replace $\mathbf {K}$ by its interpolation $\mathbf {K}_h$ in the finite element space, i.e. we approximate

$$\begin{aligned} \mathbf {K}(\mathbf {x},\mathbf {y})\approx \sum _{i,j=1}^N\mathbf {K}(\mathbf {x}_i,\mathbf {x}_j) \varvec{\varphi }_i(\mathbf {x})\varvec{\varphi }_j^\intercal (\mathbf {y}). \end{aligned}$$

Inserting this approximation into the definition of $\mathbf {C}_{\text {FEM}}$ yields

$$\begin{aligned} \mathbf {C}_{\text {FEM}}\approx \mathbf {M}_{\text {FEM}}\mathbf {C}\mathbf {M}_{\text {FEM}}^{\intercal }, \end{aligned}$$

with the matrix $\mathbf {C}$ defined as for the Nyström scheme in (17). The eigenvalue problem (19) thus turns into

$$\begin{aligned} \mathbf {M}_{\text {FEM}}\mathbf {C}\mathbf {M}_{\text {FEM}}^{\intercal }\varvec{\phi }_{m,N}= \lambda _{m,N}\mathbf {M}_{\text {FEM}}\varvec{\phi }_{m,N}. \end{aligned}$$

(20)

1.4 A.4 Connection Between the Two Schemes

The two schemes can lead to the very same eigenvalue problem. In implementations of finite element schemes, there are almost always quadrature formulas involved. Using piecewise linear ansatz functions and replacing the integrals by a trapezoidal rule yields a diagonal matrix $\mathbf {M}_{\text {FEM}}$ (this is also referred to as “mass lumping”). The definition of $\mathbf {M}_{\text {Nystr}}$ then amounts to quadrature weights to a quadrature formula with the vertices of the finite element mesh as evaluation points. The two schemes are thus equivalent in this specific case.

1.5 A.5 Computing Karhunen–Loève Expansions using Low-rank Approximations

Again, having a low-rank factorization $\mathbf {C}\approx \mathbf {L}_M \mathbf {L}_M^{\intercal }$ of rank M at hand, one can reduce the dimension of the eigenvalue problems (20). For ease of notation, we do not distinguish between $\mathbf {M}_{\text {FEM}}$ and $\mathbf {M}_{\text {Nystr}}$ and consider the eigenvalue problem

$$\begin{aligned} \mathbf {M}\mathbf {C}\mathbf {M}^{\intercal }\varvec{\phi }_{m,N}= \lambda _{m,N}\mathbf {M}\varvec{\phi }_{m,N}. \end{aligned}$$

(21)

By substituting the low-rank approximation $\mathbf {C}\approx \mathbf {L}_M \mathbf {L}_M^{\intercal }$ and $\mathbf {v}_{m,N}=\mathbf {M}^{1/2}\varvec{\phi }_{m,N}$ into (21), the eigenvalue problem becomes

$$\begin{aligned} \mathbf {M}^{1/2}\mathbf {L}_M\mathbf {L}_M^{\intercal }(\mathbf {M}^{1/2})^{\intercal }\mathbf {v}_{m,N}=\lambda _{m,N}\mathbf {v}_{m,N}. \end{aligned}$$

Exploiting the fact that $\mathbf {M}^{1/2}\mathbf {L}_M\mathbf {L}_M^{\intercal } (\mathbf {M}^{1/2})^{\intercal }$ has the same eigenvalues as $\mathbf {L}_M^{\intercal } (\mathbf {M}^{1/2})^{\intercal }\mathbf {M}^{1/2}\mathbf {L}_M=\mathbf {L}_M^{\intercal } \mathbf {M}\mathbf {L}_M$, we obtain an equivalent eigenvalue problem

$$\begin{aligned} \mathbf {L}_M^{\intercal } \mathbf {M}\mathbf {L}_M\tilde{\mathbf {v}}_{m,N}=\lambda _{m,N}\tilde{\mathbf {v}}_{m,N}. \end{aligned}$$

This modified eigenvalue problem has again dimension $M\ll N$ and can thus be solved by standard eigensolvers for dense matrices.