Shape Modeling with Point-Sampled Geometry Mark Pauly, Richard Keiser, Leif P. Kobbelt, Markus Gross (ETH Zurich and RWTH Aachen)

Abstract Modeling framework with point-sampled geometry Hybrid representation –Point clouds –Implicit surface with MLS General operations on models –Booleans operations –Deformations

Surface representations Implicit surfaces: level sets, RBF +Topology defined –Non-intuitive to control –Rendering is slow Parametric surfaces: splines, subdivision surfaces, triangle meshes +Simplicity –Extreme deformation –Connectivity information Introduction

Hybrid surface Unstructured points Implicit surface –Signed distance –Normal defined Boolean operations –Preserve shard edges

Freeform modeling Global deformation –Preserve the sampling density Tools: –Push, pull, twist and etc. Topology control

A component in a modeling system Takes scanner inputs Rendering techniques –QSplat [Zwicker 01] –[Rusinkiwicz 00] –[Botsch 02] Fast with free LOD

Related work Points primitive –Szeliski and Tonnesen 92 Oriented particles Physical simulation –Witkin and Heckbert 94 Blend operation Limited deformations Freeform modeling –Chang and Rackwood 94 Wires system Dynamically sampling –Welch and Witkin 94 Trangle mesh Vertex split and edge collapse –Kobbelt 00 Multiresolution Dynamic mesh connectivity

Hybrid surface model Input points with attributes Moving least squares (MLS)

Moving least squares 1.Local reference domain 2.Minimize to find H 3.Minimize to find g 4. proj(r)=q+g(0,0)n

MLS kernel function Small h cause Gaussian decay faster Small h means approximation more local MLS act as an low pass filter

Adaptive MLS If the sampling of points is adaptive k ranges [6:20]

Boolean operations CSG in a binary tree. Computing the boolean operation 1.Find 2.New set of intersection curves 3.Crisp curves

Inside outside classification

Optimization 90% can be approximated Else use MLS projection to find y Local coherence –If x’ is within the sphere center at x, radius is

Classification table Points in Q1 is picked from P1 only if –p is outside of surface defined by P2 –Similarly for Q2

Intersection curves A set of points for intersection curves 1.Find points near the intersection using the distance function 2.Closest point pairs (q1 in Q1, q2 in Q2) 3.r: in the intersection of the tangent planes 4.Project r to new q1 and q2 5.Repeat step 3 to 5 for 3 iterations

Intersection curves diagram

Adaptive refinement Match sample density Use a simple subdivision New p for the Newton iteration

Rendering sharp creases Surface splatting [Zwicker 01] Surfel: elliptical splat –Intersection curve points –Clip against two normals

Sharp creases results

Freeform deformation Bending, twisting, stretching, compressing Interactive speed Intuitive control Global operation

Deformable regions Region zero: not selected Region one: handle d0=0, d1=x, t=b(0/(0+x))=0 d0=dist(p and x0)=x, d1=0, t=b(x/(x+0))=1 d0=dist(p and x0)=x, d1=min(p and x1)=y, t=b(x/(x+y))

Applying translation and rotation Demo

Blending function

Rotation and twisting results

Topology control Self intersections Collision detection –Nearest point for each point in Xd –Within the sphere: collision free

Collision handling 1.Undo (disallow self intersections) 2.Union (sharp edges) 3.Blending –Inter particle potential [Szeliski 92] –Define a local neighborhood

Topology results

Dynamic sampling Cause: Distortion and insufficient sampling Goal: insert and remove points Feature: Interpolate attributes –Color –Texture value

Measuring surface stretch u and v are on the tangent plane –Orthogonal –Unit length Local anisotropy –Ratio of the two eigenvalues Split a point into two

Dynamic sampling results

Filter operations Problem: what to do with the new points Relaxation [Turk 92] Confined radius of influence Projection back to the tangent plane

Interpolation (with Zombies) Scalar values Drifting happens when split happens Zombies are fixed –Only used for the attributes –Delete after each edit operation

Interpolation results

Downsampling Special case: –Shrink then grow –Lost values cause blur Preserve old value Garbage collection

Results and discussion Pointshop3D Multiresolution surface modeling –Detail vectors [Zorin 97] –Spectral decomposition [Guskov 99]

69,706295,220

69,268222,955

25,020non-uniform

100,269

Implementation Closest points query: –Kd-tree –Building 300,000 points in 0.23 s –Querying 10 points in 4.5e-6 and 6.2e-6 s

Dynamic updates Boolean classification –Two static structure Free-form deformation –No update until an edit session is done Collision detection –Only for deformable region with the zero region –Cannot handle collision of deformable regions Dynamically sampling –[Linsen 02] –Dynamic update at insertions and relaxation

Performance No connectivity to update Handling one million points Pointshop3D software rendering –50% of the total computation is rendering Software renderer [Botsch 02] Hardware renders [Rusinkiewicz 00]

Conclusion and future work A solid framework for modeling Provides basic operations Extends to traditional surface modeling techniques Efficiency and performance Future work: –Hairy or furry models, plants –Dynamic simulation