Surface Approximation Problem Research Articles

Arbitrary-Scale Point Cloud Upsampling by Voxel-Based Network with Latent Geometric-Consistent Learning

Recently, arbitrary-scale point cloud upsampling mechanism became increasingly popular due to its efficiency and convenience for practical applications. To achieve this, most previous approaches formulate it as a problem of surface approximation and employ point-based networks to learn surface representations. However, learning surfaces from sparse point clouds is more challenging, and thus they often suffer from the low-fidelity geometry approximation. To address it, we propose an arbitrary-scale Point cloud Upsampling framework using Voxel-based Network (PU-VoxelNet). Thanks to the completeness and regularity inherited from the voxel representation, voxel-based networks are capable of providing predefined grid space to approximate 3D surface, and an arbitrary number of points can be reconstructed according to the predicted density distribution within each grid cell. However, we investigate the inaccurate grid sampling caused by imprecise density predictions. To address this issue, a density-guided grid resampling method is developed to generate high-fidelity points while effectively avoiding sampling outliers. Further, to improve the fine-grained details, we present an auxiliary training supervision to enforce the latent geometric consistency among local surface patches. Extensive experiments indicate the proposed approach outperforms the state-of-the-art approaches not only in terms of fixed upsampling rates but also for arbitrary-scale upsampling. The code is available at https://github.com/hikvision-research/3DVision

Geometric Optimization for Developable Panel Approximation

ABSTRACTThis paper presents a type of surface approximation problem that commonly arises in relation to design for manufacturing of complex architectural surfaces. The engineering context of the problem is introduced by a real-world example. The problem requires approximation of each facet of a surface subdivision by a simpler surface that is economical to fabricate into a panel. However, apart from the fitting accuracy, it is also required to manage the gaps between neighboring panels. We show that the concept of shape features is useful in modeling such problems into a global optimization problem. When such non-linear optimization formulations can be solved, we obtain better solutions that the two stage method currently used by the industry.

An Algorithm for Approximating Convex Pareto Surfaces Based on Dual Techniques

We consider the problem of approximating Pareto surfaces of convex multicriteria optimization problems by a discrete set of points and their convex combinations. Finding the scalarization parameters that optimally limit the approximation error when generating a single Pareto optimal solution is a nonconvex optimization problem. This problem can be solved by enumerative techniques but at a cost that increases exponentially with the number of objectives. We present an algorithm for solving the Pareto surface approximation problem that is practical with 10 or less conflicting objectives, motivated by an application to radiation therapy optimization. Our enumerative scheme is, in a sense, dual to a family of previous algorithms. The proposed technique retains the quality of the best previous algorithm in this class while solving fewer subproblems. A further improvement is provided by a procedure for discarding subproblems based on reusing information from previous solves. The combined effect of the enhancements is empirically demonstrated to reduce the computational expense of solving the Pareto surface approximation problem by orders of magnitude. For problems where the objectives have positive curvature, an improved bound on the approximation error is demonstrated using transformations of the initial objectives with strictly increasing and concave functions.

A condition for isotopic approximation

In this paper, we give a very simple and purely topological condition for two surfaces to be isotopic. This work is motivated by the problem of surface approximation. Applications to implicit surfaces are given, as well as connections with the well-known concepts of skeleton and local feature size.

Surface Approximation Is Sometimes Easier than Surface Integration

The approximation and integration problems consist of finding an approximation to a function $f$ or its integral over some fixed domain $\Sigma$. For the classical version of these problems, we have partial information about the functions $f$ and complete information about the domain $\Sigma$; for example, $\Sigma$ might be a cube or ball in $\reals^d$. When this holds, it is generally the case that integration is not harder than approximation; moreover, integration can be much easier than approximation. What happens if we have partial information about $\Sigma$? This paper studies the surface approximation and surface integration problems, in which $\Sigma=\Sigma_g$ for functions $g$. More specifically, the functions $f$ are $r$ times continuously differentiable scalar functions of $l$ variables, and the functions $g$ are $s$ times continuously differentiable injective functions of $d$ variables with $l$ components. The class of surfaces considered is generated as images of cubes or balls, or as oriented cellulated regions. Error for the surface approximation problem is measured in the $L_q$-sense. These problems are well defined, provided that $d\le l$, $r\ge 0$, and $s\ge 1$. Information consists of function evaluations of $f$ and $g$. We show that the $\e$-complexity of surface approximation is proportional to $(1/\e)^{1/\mu}$ with $\mu=\mrs/d$. We also show that if $s\ge 2$, then the $\e$-complexity of surface integration is proportional to $(1/\e)^{1/\nu}$ with $$ \nu=\min\left\{ \frac{r}{d},\frac{s-\delta_{s,1}(1-\delta_{d,l})}{\min\{d,l-1\}}\right\}. $$ (This bound holds as well for several subcases of $s=1$; we conjecture that it holds for all $r\ge0$, $s\ge1$, and $d\le l$.) Using these results, we determine when surface approximation is easier than, as easy as, or harder than, surface integration; all three possibilities can occur. In particular, we find that if $r=s=1$ and $d<l$, then $\mu=1/d$ and $\nu=0$, so that surface integration is unsolvable and surface approximation is solvable; this is an extreme case for which surface approximation is easier than surface integration.

Evolutionary computation applied to mesh optimization of a 3-D facial image

We apply evolutionary algorithms to the approximation of a three-dimensional image of a human face using a triangular mesh. The problem is how to locate a limited number of node points such that the mesh approximates the facial surface as closely as possible. Two evolutionary algorithms are implemented and compared. The first does selection and reproduction in the population of node points in a single triangulation. The second is a genetic algorithm in which a set of different triangulations is regarded as a population. We expect that such evolutionary computation can be used in other engineering applications which share the same problem of surface approximation.

APPROXIMATION OF NOISY BIVARIATE TRAVERSE DATA FOR PRECISION MAPPING

The special problem of surface approximation from noisy traverse data was examined using three differentclasses of algorithms: global function minimization, triangulation, and weighted sum approximations. Tests utilized bothan analytical function and field data collected from a real-time, ground-based organic matter sensor. Thin plate splinesand natural neighbor approximation were shown to generally outperform Shepards method in the generation of surfacesfrom noisy traverse data. Thin plate splines were recommended for surface generation of data from ground-based soilproperties sensors.

Vector field approximation: a computational paradigm for motor control and learning.

Recent experiments in the spinalized frog (Bizzi et al. 1991) have shown that focal microstimulation of a site in the premotor layers in the lumbar grey matter of the spinal cord results in a field of forces acting on the frog's ankle and converging to a single equilibrium position. These experiments suggested that the neural circuits in the spinal cord are organized in a set of control modules that "store" a few limb postures in the form of convergent force fields acting on the limb's end-point. Here, we investigate how such postural modules can be combined by the central nervous system for generating and representing a wider repertoire of control patterns. Our work is related to some recent investigations by Poggio and Girosi (1990a, b) who have proposed to represent the task of learning scalar maps as a problem of surface approximation. Consistent both with this view and with our experimental findings in the spinal frog, we regard the issue of generating motor repertoires as a problem of vector-field approximation. To this end, we characterize the output of a control module as a "basis field" (Mussa-Ivaldi 1992), that is as the vectorial equivalent of a basis function. Our theoretical findings indicate that by combining basis fields, the central nervous system may achieve a number of goals such as (1) the generation of a wide repertoire of control patterns and (2) the representation of these control patterns with a set of coefficients that are invariant under coordinate transformations.