Multi-Level Surface Reconstruction with Implicit Functions

G. Patanè

IMATI-CNR, Genova, Italy

patane@ge.imati.cnr.it

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Contents

• Abstract
• 1. Introduction
• 2. Surface reconstruction with sparse implicits
• 3. Discussion
• 4. Conclusions and future work
• Acknowledgments
• References

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Abstract:

3D shape approximation and processing with implicitly defined surface primitives (Radial Basis Functions and more general kernel-based approximations, Partition of Unity approximations, Moving Least Squares, etc.) are currently a subject of intensive research in Geometric Modeling and Computer Graphics. In this paper, we propose an approach that combines two conflicting criteria: achieving a high approximation accuracy and obtaining an economical surface representation. We employ compactly-supported Radial Basis Functions and use Tikhonov Regularization to achieve a near optimal selection of their centers. An iterative approach, which defines a multi-level approximation, is used to cope with arising constrained optimization problems.

1. Introduction

Current scanning techniques are capable of acquiring large and redundant data sets, thus arising the problem of defining approximations that use only a small percentage of the available information. On discrete 3D surfaces, a common approach to approximating, reducing, and hierarchically organizing the input data is provided by simplification and multi-resolution techniques [3, 14, 19].

Implicit modeling [8] represents a valid and equally expressive alternative to the above-mentioned techniques. For instance, implicit reconstruction does not impose constraints on the topological complexity of the input data and guarantees that the reconstructed surface is water-tight. In this case, a 3D data set := {p_i : i = 1,…,N} is approximated by an implicit surface Σ := {x ³ : f(x) = 0}, where f(x) := ∑ _i=1^Nα _iφ_i(x) is a linear combination of the basis elements := {φ_i(x)}_i=1,…,N. The underlinying mathematical framework builds on numerical linear algebra and the degrees of freedom on the choice of (e.g. globally [10, 44] and compactly [25, 26] supported RBFs, Partition of Unity [28, 47], Moving-Least-Squares methods [2, 16, 22, 23, 31, 38]) enable to adapt the model parameters to specific problem constraints such as huge data sets with attributes, local accuracy, and degree of smoothness. Furthermore, multi-resolution techniques have been recently proposed [42] and a wide range of applications, including deformation, fast rendering, and collision detection [4, 27, 43], have been targeted by several authors.

Previous work on surface sparsification can be subdivided into three main groups: local, global, and clustering techniques. Local methods build a smooth surface through an iterative and multi-scale procedure based on a local polynomial approximation. In this case, the centers and radii are determined by a-posteriori updates of the model and guided by the local approximation error [10, 11, 21, 27, 38]. Global methods find a sparse representation by minimizing a constrained convex quadratic optimization problem [17, 30, 40, 46]. Since each φ_i is centered at a point p_i of , clustering techniques can also be used to select the centers of the sparse representation. The idea is to group those points which satisfy a common “property” and to center a basis function at a representative point of each cluster. Planarity and closeness, measured in the Euclidean space using k-means clustering [24], Principal Components Analysis (PCA) [20], and Voronoi diagrams [33], are possible criteria (see Fig. 1 ). These methods are quite stable with respect to outliers and noise but they do not take into account the kernel function used to construct the implicit surface. To overcome the limitations of Euclidean-based clustering, kernel-based methods [13] evaluate the correlation among points with respect to the scalar product induced by a positive defined kernel. Then, the PCA and the k-means algorithm lead to efficient clustering techniques such as the KPCA [34] (Ch. 1) and the Voronoi tessellation of the feature space [45]. An example is shown in Fig. 1 .

Overview and contribution. While previous work [37, 39] has been focused on Support-Vector Machine (SVM) [13, 34], in this paper we consider an equivalent formulation [17] provided by Tikhonov regularization theory [7, 41]. In this way, we introduce a novel approach to sparse approximation that replaces the large and constrained minimization problem related to SVM with an approximate formulation, which is defined by a system of non-linear equations. This choice enables to substitute heuristic and decomposition methods for constrained convex minimization problems with standard iterative solvers of non-linear equations. Therefore, we provide a multi-level approach to Sparse IMplicit approximations (SIMS for short [30]) and we prove that

• it achieves a sparse approximation while preserving properties of the input kernel and that correspond to sparsity and positive definiteness of its Gram matrix;
• it generates a multi-level approximation scheme represented by a sequence of nested spaces (V _n)_n, where each sparse approximation f⁽ⁿ⁾ of f^⋆ belongs to V _n.

We note that multi-resolution techniques such as [42] are not defined on the implicit representation of f^⋆ but on the sampling and polygonalization process; therefore, the resolution of the voxel grid determines the complexity of the reconstructed surface. In [27], a multi-scale representation of a smooth surface is achieved by applying a local polynomial approximation, whose number of centers and radii are adapted to the required resolution. This local approximation makes the approach best-suited for interactive visualization and feature extraction of large-size models.

The main difference with respect to our approach is that SIMS achieves a compact representation through a global procedure, which builds on well-defined properties such as approximation accuracy and smoothness. Therefore, it avoids a-posteriori updates of the model that are based on the evaluation of the reconstruction error. Since we use a global formulation of the sparsification problem, the center selection is achieved by a top-down procedure instead of a bottom-up structure as done by local methods [10, 15, 29, 32].

The paper is organized as follows: Section 2 defines our approach and its main features, Section 3 discusses numerical aspects and results, and future work is presented in Section 4 .

2. Surface reconstruction with sparse implicits

In our formulation, we consider a Reproducing Kernel Hilbert Space (RKHS, for short) [5] with kernel Φ(x,y); in this case, each basis function φ_i(x) := Φ(x,p_i), i = 1,…,N, is centered at a point of the input data set. Among the properties of RKHS, we remind the reproduction property

(1)

that will be used in the following discussion. Common choices are the Gaussian kernel Φ(x,y) := e^{(-∥x-y∥2∕2σ)} or the polynomial of degree d Φ(x,y) := (1- < x,y >)^d.

Among all the possible approximations of f with the same error, a sparse approximation method searches for the one f^⋆ that involves the smallest number of basis functions. In terms of the corresponding iso-surfaces, this is equivalent to approximating Σ with Σ^⋆ := {x ³ : f^⋆(x) = 0}.

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(a)	(b)
(c)	(d)

Fig. 1.

(a) Input points on a Bernoulli lemniscate and initial set of centers (yellow circles), (b-c) first and last iteration of the k-means clustering [24], (d) reconstructed curve and iso-contours of the associated scalar field.

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It follows that any sparsification scheme combines two conflicting criteria: achieving a high approximation accuracy and obtaining an economical surface representation. The approach we present builds on a global approximation method which does not require the use of heuristics. We employ compactly-supported radial basis functions and use Tikhonov regularization to achieve a near optimal selection of their centers; an iterative approach, which defines a multi-level approximation, is used to cope with arising constrained optimization problems.

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(a) N = 303	(b) 60% n = 68	(c) 34% n = 52
(d) 29% n = 73	(e) 22% n = 62	(f) 18% n = 58

Fig. 2.

(a) Input Bernoulli lemniscate with N centers depicted by black dots, (b-f) curve approximations with different percentages of selected centers: n is the number of iterations in (5 ).

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Following [17], the quality of the approximation of f with f^⋆ is measured by the quadratic misfit error ∥ f - f^⋆ ∥ ² and the selection of the basis functions which contribute to f^⋆(x) := ∑ _i=1^Na _iφ_i(x) is given by the coefficients a_i0, i = 1,…,N. The sparsification value, i.e. the number of basis functions used in f^⋆, is quantified by the l₁-norm ∥ a ∥_l1 := ∑ _i=1^N∣a _i∣ of the vector a := (a_i)_i=1^N. Then, we consider a compromise between these two terms and minimize the functional

(2)

where ε > 0 is the tradeoff between the misfit measure and sparsity. If ε = 0, we get the standard least-squares approximation scheme, while by increasing ε we neglect a greater number of basis functions and accept a lower approximation accuracy.

As shown in [17], replacing each unknown a_i of (2 ) with a pair of positive variables (a_i⁺,a _i^-), such that a _i = a_i⁺ - a _i^-, results in the following constrained convex quadratic optimization problem (i.e. Support Vector Machine) [17]:

(3)

with

f(p_i) = y_i, i = 1,…,N, and feasible set

This last formulation is equivalent to (2 ) and facilitates its numerical optimization by removing the absolute values. If is a large data set, defining a sparse representation of f as a solution of the quadratic minimization problem (3 ) with 2N unknowns is generally unfeasible due to the amount of input data, the possible ill-conditioning of the coefficient matrix L := (L_ij)_i,j=1,…,N, L_ij := Φ(p_i,p_j), and the unbounded feasible set . Therefore, all these factors badly affect the stability and convergence of iterative methods [12, 18]. In [36, 40], (3 ) is solved by applying a decomposition method and a heuristic coordinate descendent optimization scheme respectively. In the last case, at each iteration (2N - 1) variables are fixed and the objective function is minimized with respect to the remaining free parameter.

To avoid the formulation in (3 ), which is a consequence of the C⁰-regularity of the l₁-norm in (2 ), we use the smooth approximation

and we replace the functional (2 ) with

In the following, we prove that minimizing G is equivalent to solving a system of non-linear equations. Using the reproduction property (1 ), we can rewrite G as

where the term ∥ f ∥² is constant. Then, the critical points of G are the solutions of the following system of non-linear equations

(4)

with

and y := (y_i)_i=1^N. The matrix formulation shows the two factors of the sparsification scheme: the matrix L, which is associated to the kernel function, and the diagonal matrix Δ(a) related to the sparsification term.

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	t
(a)	(b)
(c)	(d)

Fig. 3.

Reconstructed curves using input centers (black dots) (a) affected by noise and (c) irregularly distributed. (b-d) Selected centers (black dots) and reconstructed curves.

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If we consider a Gaussian or compactly supported kernel, which corresponds to a positive/semi-positive definite matrix L, the Hessian matrix of G is positive definite and the convex functional G admits a unique minimum. The solution of (4 ) is evaluated by applying the iterative scheme

(5)

with a⁽⁰⁾ initial guess and n ≥ 1. The term a⁽ⁿ⁺¹⁾ is achieved from a⁽ⁿ⁾ by solving a linear system with direct or iterative solvers, e.g. the Gauss-Seidel or conjugate gradient method [18]. The iterative procedure stops when the solution becomes stationary, i.e. we do not improve the number of null coefficients and/or the residual error between two consecutive iterations is below a given threshold.

3. Discussion

Let us suppose that y_i := 0 and x_i is a point of the input data set, i = 1,…,N; then, the surface must interpolate the boundary constraints f(x_i) = 0 at . As done in [44] and in order to avoid the trivial solution f ≡ 0, we add positive (resp., negative)-valued normal constraints at x_i close to the boundary constraints and in the direction n_i (resp., -n_i), where n_i is the normal at x_i. If a polyhedral surface is given, surface normals are approximated by averaging the face normals; otherwise, we use those ones provided by the shape acquisition or achieved by a local fitting or clustering [1, 23]. Clearly, positive ⁺ and negative-valued ^- constraints can be set on a subset of where shape variations occur and selected by using the Principal Components Analysis [20]. Therefore, as centers of the basis functions we can consider the set ∪⁺ ∪^-, thus solving the 3N × 3N equation in (5 ).

Recently [46], the surface normal vectors have been incorporated in the regularization framework, thus avoiding off-set conditions to guarantee a non-null solution. This approach uses curvature and grid-based clustering on point clouds to guide the selection of the basis function centers and radii, and to achieve high-quality approximations.

Fig.s 2 and 3 show the curve reconstruction with several sparsification percentages (Φ is the Gaussian Kernel); we note that the local noise and irregular sampling, which affect the approximation in Fig. 3 (a), are attenuated in (b) where we use a minor number of basis functions. This property is due to the misfit error in (2 ) and to a lower conditioning number of the coefficient matrix related to the least-squares approximation. Furthermore, a smooth kernel function Φ and the induced norm ∥ ∥ provide a smooth solution, thus reducing the influence of noise and outliers of in the reconstructed surface. Finally, Fig. 3 (c-d) shows the center selection on a curve with a non-uniform sampling.

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(a) 84%	(b) 66%	(c) 50%	(d) 45%
L∞ = 3.2 × 10^-5	L ∞ = 1.3 × 10^-4	L ∞ = 4.2 × 10^-3	L ∞ = 4.6 × 10^-3

Fig. 4.

Surface reconstruction with a different percentage of selected centers and L∞ error; the number of input centers is 80K.

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Using a k-neighborhood of each center [6] requires a storage overhead of (3k + 1)N non-null entries for the matrix L. Then, each iteration (5 ) updates only the principal diagonal of L, preserves its sparsity and positive definiteness, and requires O(N) time to solve the linear system. The proposed sparsification scheme is equivalent to Support Vector Machine and it avoids the constrained convex quadratic minimization and the use of heuristics involved in SVMs. Since the sparsification starts from the full resolution with 3N basis functions, we get a fine-to-coarse approach and the iterative procedure (5 ) builds a multi-level approximation scheme based on a sequence of nested spaces; for more details, we refer the reader to [30]. If we get a complete sparsification, i.e. the iterative solver of the system of non-linear equations converges to the null solution, each approximation is achieved by using the intermediate iterations (see Fig. 4 ).

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k = 10 : (a) 52% L∞ = 4.4 × 10^-3	(b) 46% L ∞ = 5.3 × 10^-2	k = 20 : (c) 45% L ∞ = 4.2 × 10^-3	(d) 35% L ∞ = 7.8 × 10^-2

Fig. 5.

Reconstruction with compactly supported radial basis functions on a model with 240K input centers and different percentages of selected centers. In (c-d) row, the support of the RBFs is twice of the one used in (a-b); a greater support achieves a smoother reconstruction and a higher approximation accuracy with a minor number of basis functions.

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As the iterations in (5 ) proceed, the L_∞ error between the current approximation f⁽ⁿ⁾(x) = ∑ _i=1^Na _i⁽ⁿ⁾φ _i(x) and f, measured by

increases due to a minor number of basis functions. Unlike local approximation techniques, which are capable of adapting the center selection to local accuracy through an extensive use of function evaluations, we cannot use this error as a stop criterion due to the global formulation of our problem.

The fine-to-coarse structure requires a storage overhead and computational cost greater than local approximation [2, 27, 26]. For instance, if k = 20 we are able to handle at last 500K centers on a Pentium IV 2.80 GHz with 1 GB RAM; in this case, the evaluation of L and the sparsification scheme require approximately 90 - 180 seconds. Our method always converges to a global minimum and it can be used for those applications where center selection is mandatory to speed up surface sampling, interactive modeling and queries [9, 40]. However, it cannot deal with huge data sets, which can be handled efficiently by [28, 27, 46].

Setting the support of the basis functions is a delicate part of the approximation and sparsification scheme; in fact, its choice affects the size of the details that will be recovered as well as the maximum sparsification percentage, which avoids artifacts in the reconstructed model. In our framework, the support associated to the basis function φ_i is set equal to the minimum radius of the sphere centered at p_i that contains the k-nearest neighbor points of p_i, and k varies from 10 to 20 depending on the number of input points. Our tests have shown that these values lead to a good compromise between sparsification rate and approximation accuracy (see Fig. 5 ). Finally, in the examples of Fig.s 4 and 5 we used Φ(r) := (1 - r)₊⁴(4 + 16r + 12r² + 3r³) [25, 35] as sparse kernel, where r = and σ is its compact support.

4. Conclusions and future work

While there has been much work on implicit modeling and related applications, the study of implicit sparse approximations has been focused mainly on Support Vector Machine. Exploiting a formulation equivalent to SVM and based on Tikhonov regularization, the paper discussed an iterative method that defines a compact surface approximation while guaranteeing a good approximation accuracy. We introduced a novel multi-level sparsification scheme where a different resolution in the approximation of the input data is achieved by selecting the parameter ε or a different number of centers, which have been sorted in increasing order of importance by the iterative procedure.

The advantages of our approach are its efficiency and simplicity; it differs from previous work on the use of an approximated formulation of the sparsification problem, which avoids to use heuristic solvers of convex quadratic optimization problems. This aim is simply achieved by taking advantage of a smooth approximation of the l₁-norm and of robust iterative solvers of systems of non-linear equations.

Even thought the number of used centers can be set in a simple way trough the multi-level scheme, the main open issue is to control directly, and not through the parameter ε, the final error approximation.

Acknowledgments

Special thanks are given to Alexander Belyaev and the AG4-Computer Graphics Group at MPII-Saarbruecken for motivating and supporting the author to consider the problem discussed here. This work has been supported by the EC-IST FP6 Network of Excellence “AIM@SHAPE”; models are courtesy of the AIM@SHAPE repository (http://shapes.aimatshape.net).

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