A scheme is developed for classifying the types of motion perceived by a humanlike robot. It is assumed that the robot receives visual images of the scene using a perspective system model. Equations, theorems, concepts, clues, etc., relating the objects, their positions, and their motion to their images on the focal plane are presented. >

http://ieeexplore.ieee.org/iel2/815/3148/00100651.pdf

Robot vision

We present a new shape representation, the multi-level partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape.Our approach gives us considerable flexibility in the choice of local shape functions, and in particular we can accurately represent sharp features such as edges and corners by selecting appropriate shape functions. An error-controlled subdivision leads to an adaptive approximation whose time and memory consumption depends on the required accuracy. Due to the separation of local approximation and local blending, the representation is not global and can be created and evaluated rapidly. Because our surfaces are described using implicit functions, operations such as shape blending, offsets, deformations and CSG are simple to perform.

/pdf/multi-level-partition-of-unity-implicits-141jvzwqia.pdf

Multi-level partition of unity implicits

This paper introduces a method to extract 'Shape-DNA', a numerical fingerprint or signature, of any 2d or 3d manifold (surface or solid) by taking the eigenvalues (i.e. the spectrum) of its Laplace-Beltrami operator. Employing the Laplace-Beltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach. Since the spectrum is an isometry invariant, it is independent of the object's representation including parametrization and spatial position. Additionally, the eigenvalues can be normalized so that uniform scaling factors for the geometric objects can be obtained easily. Therefore, checking if two objects are isometric needs no prior alignment (registration/localization) of the objects but only a comparison of their spectra. In this paper, we describe the computation of the spectra and their comparison for objects represented by NURBS or other parametrized surfaces (possibly glued to each other), polygonal meshes as well as solid polyhedra. Exploiting the isometry invariance of the Laplace-Beltrami operator we succeed in computing eigenvalues for smoothly bounded objects without discretization errors caused by approximation of the boundary. Furthermore, we present two non-isometric but isospectral solids that cannot be distinguished by the spectra of their bodies and present evidence that the spectra of their boundary shells can tell them apart. Moreover, we show the rapid convergence of the heat trace series and demonstrate that it is computationally feasible to extract geometrical data such as the volume, the boundary length and even the Euler characteristic from the numerically calculated eigenvalues. This fact not only confirms the accuracy of our computed eigenvalues, but also underlines the geometrical importance of the spectrum. With the help of this Shape-DNA, it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids. A patent application based on ideas presented in this paper is pending.

/pdf/laplace-beltrami-spectra-as-shape-dna-of-surfaces-and-solids-5djzm2hwdp.pdf

Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids

A method for concise, faithful approximation of complex 3D datasets is key to reducing the computational cost of graphics applications. Despite numerous applications ranging from geometry compression to reverse engineering, efficiently capturing the geometry of a surface remains a tedious task. In this paper, we present both theoretical and practical contributions that result in a novel and versatile framework for geometric approximation of surfaces. We depart from the usual strategy by casting shape approximation as a variational geometric partitioning problem. Using the concept of geometric proxies, we drive the distortion error down through repeated clustering of faces into best-fitting regions. Our approach is entirely discrete and error-driven, and does not require parameterization or local estimations of differential quantities. We also introduce a new metric based on normal deviation, and demonstrate its superior behavior at capturing anisotropy.

/pdf/variational-shape-approximation-30mutumbju.pdf

Variational shape approximation

In this paper, we introduce a novel approach to mesh editing with the Poisson equation as the theoretical foundation. The most distinctive feature of this approach is that it modifies the original mesh geometry implicitly through gradient field manipulation. Our approach can produce desirable and pleasing results for both global and local editing operations, such as deformation, object merging, and smoothing. With the help from a few novel interactive tools, these operations can be performed conveniently with a small amount of user interaction. Our technique has three key components, a basic mesh solver based on the Poisson equation, a gradient field manipulation scheme using local transforms, and a generalized boundary condition representation based on local frames. Experimental results indicate that our framework can outperform previous related mesh editing techniques.

/pdf/mesh-editing-with-poisson-based-gradient-field-manipulation-vcof4s6eef.pdf

Mesh editing with poisson-based gradient field manipulation

This paper presents frameworks to extend the mean and median filtering schemes in image processing to smoothing noisy 3D shapes given by triangle meshes. The frameworks consist of the application of the mean and median filters to face normals on triangle meshes and the editing of mesh vertex positions to make them fit the modified normals. We also give a quantitative evaluation of the proposed mesh filtering schemes and compare them with conventional mesh smoothing methods such as Laplacian smoothing flow and mean curvature flow. The quantitative evaluation is performed in error metrics on mesh vertices and normals. Experimental results demonstrate that our mesh mean and median filtering methods are more stable than conventional Laplacian and mean curvature flows. We propose thee new mesh smoothing methods as one possible solution of the oversmoothing problem.

/pdf/mesh-smoothing-via-mean-and-median-filtering-applied-to-face-15b8deoplw.pdf

Mesh smoothing via mean and median filtering applied to face normals

A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions

We propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed along a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate the sets starting from the coarsest level. We interpolate a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. An original point set and its coarse-to-fine hierarchy of interpolated sets is presented. According to our numerical experiments, the method is essentially faster than the state-of-the-art scattered data approximation with globally supported RBFs (Carr et al., 2001) and much simpler to implement.

A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions

3D scattered data approximation with adaptive compactly supported radial basis functions

We develop an adaptive RBF fitting procedure for a high quality approximation of a set of points scattered over a piecewise smooth surface. We use compactly supported RBFs whose centers are randomly chosen from the points. The randomness is controlled by the point density and surface geometry. For each RBF, its support size is chosen adoptively according to surface geometry at a vicinity of the RBF center. All these lead to a noise-robust high quality approximation of the set. We also adapt our basic technique for shape reconstruction from registered range scans by taking into account measurement confidences. Finally, an interesting link between our RBF fitting procedure and partition of unity approximations is established and discussed.

/pdf/3d-scattered-data-approximation-with-adaptive-compactly-3dxa9vxqme.pdf

Yutaka Ohtake

Papers

Mesh smoothing via mean and median filtering applied to face normals

A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions

A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions

3D scattered data approximation with adaptive compactly supported radial basis functions

3D scattered data approximation with adaptive compactly supported radial basis functions