Variational Shape Approximation

Transcription

1 Movie, anyone? Variational Shape Approximation Scanned vs. Concise Geometry Abundant repositories of surface meshes with subtle details exquisitely sampled but so as all the less interesting parts From verbose to concise meshes simplification [Garland-Heckbert 97, Lindstrom-Turk 98] segmentation/clustering [Katz-Tal 03, Sheffer 01] remeshing [Alliez et al. 03, Marinov-Kobbelt 04] optimization [Hoppe et al. 93, Alliez et al. 99] Goal: faithful approximation of original geometry 3 Approximating Surfaces Artists use and respect local symmetries capture shape efficiently! know-how hard to put into words or equations Sharp contrast with automatic remeshers usually, careful placement of vertices but only isotropic elements still very verbose 4 Approximation Theory 101 Lots of work dedicated to approximation fctnal setting, depends on class of fcts/norm Best PL approximation of height field if: elements stretched in min. curv. direction aspect ratio κ κ (for basically any L p 1 ) NOT EFFICIENT see [Nadler 86, Simpson 94, Heckbert- Garland 99] κ 1 κ 1 κ κ EFFICIENT Importance of Orientation and aspect ratio 5 Approximation Theory 10 When you read the fine prints best only in the infinitesimal limit! therefore, not really applicable for coarsening plus, requires estimation of curvatures reputedly delicate for scanned meshes A good approximation algorithm should: satisfy optimal orient. & aspect ratio in limit but not use this as golden rule at coarse scales 6 1

2 A Discrete Approach Variational Shape Approx. Shape Approximation: 100% discrete all the way no estimation of diff. quantities like curvatures 100% variational (error-driven) not greedy 100% direct no parameterization no splitting in genus-0 patches, no cutting We can recast surface approximation as a variational k-partitioning problem from partition, derive a polygonal mesh 7 8 Variational Shape Approx. Variational Shape Approx. We can recast surface approximation as a variational k-partitioning problem from partition, derive a polygonal mesh We recast surface approximation as a variational k-partitioning problem from partition, derive a polygonal mesh for each of k regions, find best-fit linear proxy for each of k regions, find best-fit linear proxy best fit for a given metric (say, L ) best fit for a given metric (say, L ) distortion = error btw proxy and region E = = 1.. k x R d( x, proxy ) dx best k-partition has minimum distortion 9 10 Example of Proxies / Meshing Initial Mesh + Partition Associated Proxies (blueprint for mesh) Proxy-based Remeshing Problem: how to find the optimal partition? 11 Partition Optimization (I) Key Idea: extending Lloyd s clustering in its usual form: finds optimal point sampling of a flat space alternates: partition w.r.t. points (e.g. Voronoi tiling) move points to centroids (i.e. local min) optimizes compactness i.e., minimizes: E = = 1.. k x R d( x, x ) dx demo 1

3 Partition Optimization (II) Lloyd s algo ideal as a minimization tool proxy-based, not point-based partition w.r.t. distance to proxies find "best fit" proxy equivalent to: move to centroid minimizes distortion for a given # of proxies Algorithm Overview Input: initial mesh, #proxies, metric 1. Init. Lloyd-type Iterative Minimization a: finding new partition b: finding new best fits (proxies) for partition c: possibly, teleport proxies 3. Meshing demo on trivial examples Init a. New Partition - Pick K random seed triangles T i - Initialize each proxy (X i,n i ) X i = barycenter(t i ) N i = normal(t i ) Region-growing instead of Voronoi ensures connected regions closed form expression Global priority queue adacent triangles pushed in the queue ordered by error btw triangle and proxy b. New Best Fit c. Convergence/Teleportation For each region: best fit plane w.r.t. chosen metric closed form expression, too Good convergence properties guaranteed convergence for convex obects small oscillations in the worst case May need to tunnel out of local minima often on flat regions Teleportation of a proxy! similar to [Kanungo et al. 03] better bounds on closeness to optimality

4 3. Meshing Straightening the edges Discrete approximation of CDT multi-source Disktra's shortest path algorithm Proection of vertices on proxies Metric Trouble Optimality for L not satisfactory hyperbolic regions troublesome no unique minimum due to asymptotic lines convergence in L does not guarantee H 1 example: Schwarz s chinese lantern [Shewchuck 04]: gradient bounds harder than interpolation x R d( x, proxy ) dx triangles convex polygons 19 0 A New Metric Expressions (from paper) Introducing L,1 a normal-based shape metric: asymptotically, aspect ratio is κ 1 κ hyperbolic regions ok! captures normal field hence, lighting information x R n n x proxy dx 1 Comparison Between Metrics Other Results L L,1 L L,1 (based on normals) 3 4 4

5 Approximation Zoo Bunny 5 6 Vase-lion David I 400KV 5KV 7 8 David II Comparison w/ Greedy Methods CAD 04 but much faster!

6 Discussion Varitional shape approximation clustering method, error-driven, totally discrete extension of Lloyd s algo to proxies a study on shape metrics, asymptotic behavior Implementation: clustering easy to code in less than hours! Generality: can be applied to lots of datatypes! 31 6