weighted minimal surface model for surface reconstruction from scattered points, curves, and/or pieces of surfaces.

Transcription

1 weighted minimal surface model for surface reconstruction from scattered points, curves, and/or pieces of surfaces. joint work with (S. Osher, R. Fedkiw and M. Kang) Desired properties for surface reconstruction: has the ability to handle complicated topology and geometry as well as noise and non-uniformity of the data. the constructed surface is a good approximation of the data set and has some smoothness. has a data structure that is good for static rendering as well as dynamic deformation and other operations. extends to high dimensions. 34

2 previous approaches explicit representation: find f s.t. Γ = f(u, v). triangulated surfaces using Voronoi diagram and Delaunay triangulations. parametric surfaces, such as NURBS. variational and PDE formulation. membrane model: min f 2 dx subject to: f(x i )=f i implicit representation: find φ(x) s.t. Γ={x φ(x) =const.} the scalar function is a combination of basis functions. the scalar function is a signed distance function. 35

3 our approach Define a minimal surface model and a convection model on the continuous level using variational formulation and PDEs. Develop efficient numerical algorithms based on rectangular grids for processing data, computing distance and defining the initial surface. Construct the final surface Γ, as an implicit surface by continuous deformation using the level set method, i.e., construct a signed distance function φ(x) s.t. Γ ={x : φ(x) = 0}. 36

4 the weighted minimal surface model Let S denote a general data set and d(x) =dist(x, S). Variational formulation: Minimize E(Γ) = [ ]1 p Γ dp (x)ds. 1 p, The energy functional: uses distance which is independent of parametrization and is invariant under rotation and translation. if p<, E(Γ) Γ dp (x)ds, a weighted surface area. if p =, E(Γ) = maxx Γ d(x) p = 1, scale invariant E(Γ) = Γ d(x)ds 37

5 geodesic model for image segmentation Denote I(x) to be the intensity function. Define the boundary to be the curve Γ that minimizes Γ g( I)ds where g( I) is an edge detection function, e.g. The gradient flow is: g( I) =(1+ I p ) 1 v n = (g( I)n) = g n gκ where n is the unit normal and κ is the curvature. 38

6 the gradient descent flow [ ]1 dγ dt = p 1 Γ dp (x)ds d p 1 (x)[ d(x) n+ 1p ] d(x)κ n, balance between attraction d n and surface tension dκ scaled by distance, where κ is the mean curvature. [ ]1 p 1 Γ dp (x)ds d p 1 (x) scales the motion of Γ, where if p 1, [ ]1 [ p 1 Γ dp (x)ds d p 1 d(x) (x) d max =max x Γ d(x) [ Γ dp (x)ds] 1/p d max ] p 1 In practice, we often use p = 1 (scale invariant) or p =2 (least square). 39

7 the Euler-Lagrange equation The minimizer satisfies the weighted minimal surface equation: d p 1 (x) [ d(x) n + 1p ] d(x)κ =0, The rigidity is adapted to the local sampling density. d(x) n + 1 pd(x)κ =0 local sampling condition. 40

8 mathematical questions if d(x) =dist(x, Γ 0 ), Γ 0 is a smooth surface, E(Γ) = 0 when Γ=Γ 0 or Γ =. if d(x) =dist(x, S), S is a discrete set, E(Γ)=0whenΓ=. Questions: 1. Existence and uniqueness of a local minimum, i.e., a weighted minimal surface attached to the data set? 2. How to find it? 41

9 2D results If p 1, Lemma1: Proof: Polygon with vertices in S is a local minimum. 1. d p 1 (x)[ d(x) n + 1 pd(x)κ] =0 a.e. 2. it is a local minimum by perturbation analysis. Lemma2: If a curve passes a data point it will not leave the point. Remark: It is more complicated in 3D and depends on the sampling density in general. 42

10 continuous deformation Deform an initial surface that encloses S following the gradient flow (a weighted mean curvature flow): [ ]1 dγ dt = p 1 Γ dp (x)ds d p 1 (x) [ d(x) n+ 1p ] d(x)κ n, Numerical Challenges: 1. topological changes during the deformation. 2. convergence and CFL condition t = O( x 2 ). Solutions: 1. the level set method provides a powerful numerical tool for the deformation and construction of implicit surfaces. 2. Starting with a good initial guess. 43

11 local analysis in the continuous limit If Γ 0 beasmoothclosedsurface,letd(x) be the signed distance function to Γ 0 (d is negative inside Γ 0 )and Γ d = {x d(x) =d}, then and F (d) def = E(Γ d )= [ Γ d d p ds ]1 p = d [ Γ d ds ]1 p def L (d) = κds = χ(d). Γ d So sign(d)f (d) =L 1 p 1 (d) [L(d)+ dp ] χ(d) = L 1 p 1 (d) = d L 1 p(d) Γ d ( 1 dκ p ) ds Remark: dκ < 1 until singularity develops. In 2D χ(a) = winding number. 44

12 aconvectionmodel The surface is convected in the potential field defined by the distance function d(x). Γ t = d(x) Properties: 1. v n = d n Each point on Γ is attracted by its closest data point except those equal distant points. 3. Linear convection equation, t = O( x) in numerical computations. The final construction is like piecewise linear. 45

13 x 2 3 x 5 x x 6 x 1 x 4

14 the level set method Two key steps for the level set method: 1. Embed the surface: find a level set function φ s.t. Γ={x : φ(x) =0}. Geometric properties of the surface Γ can be easily computed using φ, e.g. n = φ φ, κ = φ φ. 2. Embed the motion: derive the time evolution PDE for φ s.t. Γ(t) ={x : φ(x,t)=0}. dφ(γ(t),t) = φ t + v φ =0 dt where v is a velocity field that agrees with the motion of Γ. Remark: Reinitialization is usually needed to keep φ close to the signed distance function, i.e., φ 1. 47

15 advantages of the level set method Geometric problem becomes a PDE problem. Capture the moving interface on a fixed Cartesian grid. Handle topological changes easily. Efficient numerical schemes are available. Local level set method reduces the computation cost. Remark: The level set method and implicit surfaces provide a general framework for the modeling, analysis and simulation of surfaces. 48

16 the level set formulation for our problems embed the motion: gradient flow for the minimal surface model: [ φ t = φ ]1 [ d p p 1 1 δ(φ) φ dx d p 1 d φ φ + 1 φ d p φ ] the convection model φ t = d(x) φ standard finite difference scheme for the level set method. reinitialization. 49

17 numerical algorithms Three most important ingredients: Computing the unsigned distance function to an arbitrary data set. Finding a good initial surface. Solving the PDEs for the level set method. 50

18 finding a good initial surface 1. Start with the outer distance contour d = ɛ, 2. Push the initial surface even closer to the data set using a fast tagging algorithm. marching boundary distance contour data point Remark: The fast tagging algorithm can be regarded as a crude upwind scheme for the convection model. 51

19 a fast tagging algorithm the final surface = an appropriate boundary between the exterior and the interior regions. We start with an exterior region and use the following fast tagging algorithm to move the temporary boundary inward. 1. Heapsort the boundary points according to the distance. 2. for the furthest boundary point, (a) if it has an interior neighbor that has a larger distance, tag this boundary point as a final boundary point and remove it from the temporary boundary point. (b) otherwise move the point to the exterior and put its neighboring interior points to the temporary boundary. Repeat this process until either the temporary boundary is empty or all points in the temporary boundary have distance less that some tolerance, which are then put in the final boundary. Signed distance function to the final boundary is then computed using the previous fast sweeping algorithm. 52

20 Tagging algorithm Hongkai Zhao

21 A good initial surface is important for the efficiency of the PDE based method. On a rectangular grid, we view an implicit surface as an interface with some regularity thatt separates the exterior grid points from the interior grid points. In other words, volumetric rendering requires identifying all exterior (interior) grid points correctly. A novel, extremely efficient tagging algorithm that tries to identifyif asmanycorrectexterior grid points as possible and hence provide a good initial implicit surface.

22 Start from any initial exterior region that is a subset of the true exterior region. All grid points that are not in the initial exterior region are labeled as interior points. Those interior grid points that have at least one exterior neighbor are labeled as temporary boundary points. N th f ll i d t h th Now we use the following procedure to march the temporary boundary inward toward the data set.

23 We put all the temporary boundary points in a heapsort binary tree structure t sortingaccording to distance values. Take the temporary boundary point that has the largest distance (which is on the heap top) and check to see if it has an interior neighbor that has a larger or equal distance value. If it does not have such an interior neighbor, turn this temporary boundary point into an exterior point, take this point out of the heap, add all this point s interior neighbors into the heap and resort according to distance values. If it does have such an interior neighbor, we turn this temporary boundary point into a final boundary point, take it out of the heap and re sort the heap. None of its neighbors are added to the heap.

24 We repeat this procedure on the temporary boundary points until the maximum distance of the temporary boundary points is smaller than some tolerance, e.g. the size of a grid cell, which means all the temporary boundary points in the heap are close enough to the data set, which are then put in the final boundary. or the temporary boundary is empty. Finally, we turn these temporary boundary points into the final set of boundary points and our tagging procedure is finished. Now we have the final sets of interior, exterior and boundary points. Signed distance function to the final boundary is then computed using the previous fast sweeping algorithm.

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26 multi-resolution There are two resolutions, i.e., the data set and the grid. The best scenario is when the two resolutions are comparable. Remark: The computation cost depends mainly on the grid resolution. Low resolution may be desirable when there are redundancies/noises and speed or memory limit. Hierarchical multi-resolution algorithm can be used to improve the efficiency. 53

27 Efficient storage and reconstruction Storage: Store the values and locations (indices) of those grid points that are next to the surface. (a thin shell around the surface) No connections and orderings are needed. For a reconstruction on a grid of size 290x206x134, storage of the whole grid: 118 MB, (44 MB compressed), storage of the thin shell: 15 MB (5 MB compressed). Reconstruction: Use the fast sweeping and tagging algorithm to reconstruct the signed distance function in O(N) operations. 54

28 results All computations are done on a PentiumIII 600Mhz PC with 1GB memory. The used data sets are from The Stanford 3D Scanning Repository, Georgia Institute of Technology s Large Geometric Models Archive. Model Data Grid CPU CPU points size (initial) (min) Rat brain x77x Torus x80x Buddha x350x Buddha x150x Dragon x212x Dragon x71x Hand x141x

29 (a) data points (b) starting surface (c) final reconstruction (a) data points (b) front view (c) side view (a) data curves (b) reconstruction

30 (a) high resolution reconstruction (b) low resolution reconstruction 57

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