Multi-level Partition of Unity Implicits

Transcription

1 Multi-level Partition of Unity Implicits Diego Salume October 23 rd, 2013 Author: Ohtake, et.al.

2 Overview Goal: Use multi-level partition of unity (MPU) implicit surface to construct surface models. 3 Key Concepts: Piecewise quadratic functions used as local estimates of the surface. Weighing functions that blend these local shape functions. Octree subdivision that adapts based on shape complexity. Flexible choice of shape functions Accurate representation of sharp features (edges, corners) Adaptive approximation based on required accuracy. Determines space/time complexity.

3 Advantages of Implicit Functions Data repairing capabilities from scattered data.

4 Advantages of Implicit Functions Edit surfaces using standard implicit modeling operations: shape blending, offsets, deformations.

5 Method Summary: Setup Given: set of points with normals to indicate surface orientation. Partition of unity: set of weighing functions that sum to one at all points in the domain. MPU implicit: adaptive error-controlled approximation of signed distance function from surface. Surface is zero-level of the distance function.

6 Method Summary: Algorithm To create implicit representation: Octree-based subdivision of bounding box for entire point set. At each cell, fit a piecewise quadratic function (local shape function). Signed distance function: 0 near points, positive inside, negative outside. If shape function isn t accurate enough, subdivide further until desired accuracy is achieved. In common boundary between cells, shape functions are blended together according to weights from partition of unity functions. Global implicit of function is given by blending of local shape functions at the leaves of the octree.

7 Partition of Unity Generate weight functions: For approximation: use quadratic B-spline b(t). For interpolation: use inverse-distance singular weights. Nonnegative compactly supported set

8 Partition of Unity Blend local functions using smooth, local weights that add up to 1. Partition of unity functions Define set of local shape functions V i Approximation of a function defined on domain

9 Adaptive Octree Points are rescaled so that an axis-aligned bounding cube has unitlength main diagonal. Octree-based subdivision. Each cell has center c and diagonal length d. Define the support radius for the cell s weight function: Bigger alpha -> smoother interpolation/approximation, slower computation Time complexity quadratic on alpha Must have at least N min points in the sphere to estimate shape function. If not enough, iteratively increase radius

10 Adaptive Octree Local max-norm approximation error estimated based on Taubin distance If error is larger than a threshold ɛ 0, subdivide the cell.

11 Algorithm: Pseudocode

12 Local Shape Functions A. General 3D quadric Larger parts of the surface: unbounded, more than one sheet B. Bivariate quadratic polynomial in local coordinates Local smooth patch C. Piecewise quadric surface to fit sharp features Edges, corners Let P be the points of P inside the ball associated with a cell. Associate normal n with the center c of cell. Compute n by taking the normalized weighted arithmetic mean of the normals of the points in the ball. Use the approximation weight functions. Let θ be the maximal angle between n and the normals N associated to points P.

13 Local Shape Functions: A. 3D Quadric Surface Larger parts of the surface: unbounded, more than one sheet If A: 3x3 symmetric, b: 3-vector, c: scalar To orient local shape function pick auxiliary points {q i }: corners and center of cell. Find 3 nearest neighbors p (i) from P and compute If not all 3 values have the same sign, discard the auxiliary point. Minimize (m is remaining number of q points)

14 Local Shape Functions: B. Bivariate quadratic polynomial Local smooth patch. If Local coordinates (u,v,w) at c such that (u,v) plane is orthogonal to n and positive direction of w axis is along direction of n. Quadratic shape function in new coordinate system (u,v,w) : Minimize

15 Local Shape Functions: C. Piecewise Quadratic Edges and corners If Automatic recognition of edges and corners based on normal clustering If, then surface has sharp feature. If where, then feature is a corner Subdivide N into 3 sets. First the two normal clusters N 1 and N 2. Then if add point to N 3. Else, surface has an edge Subdivide N into 2 sets N 1 and N 2. Quadratic fit applied to each P 1, P 2, separately. Construct non-smooth local shape function via max/min bool operations of Ricci (Misha!). Else go to B.

16 Application Visualization Bloomenthal s polygonizer. Hart sphere tracing. Compute MPU approximations on the fly during polygonization process. Combine a low resolution Bloomenthal polygonization with a postprocessing mesh optimization technique. Higher quality rendering can be achieved using ray tracing techniques. Interpolation requires more memory than approximation. Octree subdivision until each point has its own cell. Sphere centered at point. For overlapping range scans, use per-point measurement confidences.