Andrew Nealen / Olga Sorkine / Mark Alexa / Daniel Cohen-Or SoHyeon Jeong 2007/03/02

Abstract Intuitive editing of surface meshes View-dependent sketching VS handle-based deformation Method By silhouette selection and cropping By sketching directly onto the surface Result Edit new view-dependent handle position Indirectly influencing differential properties along the sketch 2

Contents 1. Introduction 2. Mesh modeling framework 3. Silhouette sketching 4. Feature and contour sketching Geometry adjustment Sharp features Smooth features and suggestive contours 5. Discussion 3

A Sketch-Based Interface Detail-Preserving

Sketch-based shape Editing Editing silhouette & suggestive contours 5

Silhouette! The human visual system uses silhouettes as the first index into its memory of shapes Without color, shading or texture But by their contours Easily recognized silhouettes [D.Hoffman et al.] 6

Sketch-based Shape Modeling Creating 3D shape by sketching its Silhouette SKETCH : Zeleznik et al Teddy : Igarashi et al Karpenko et al Igarashi and Hughes 2003 Bourguignon et al [SKETCH][Teddy][Kerpenko et al.] [Bourguignon et al.] 7

Inverse NPR User can sketch a curve which becomes feature line Contour or suggestive contours 3D 2D NPR Sketch [DeCarlo et al. 2003] [Zeleznik et al. 1996] 8

Preserving Features Preserving the global and local geometric features of a model during editing The use of Laplacian/Poisson mesh models Constraints on the normals and the curvature Allows constraint to be placed on virtual vertices Users only suggest feature lines Properties of sketch can not always be accommodates exactly to preserve feature of the shape 9

Laplacian with Least Squares Method

Least Squares Method Solving an equation Approximating an equation 11

Least Squares Method Finding the closest of given sampled data (constraint) Minimizing the distances Solution Linear : the zero of differentiation, closed-form solution Non-linear : iterative method The equation form that is approximated 12

Least Squares – An Example Finding x which minimizes distances from sample data 12, 3, 8, 5, 24 Distances : Finding the zero of the differential equation 13

Laplacian Surface Editing Laplacian Relative coordinate of the center of neighbor-vertices Contains local intrinsic features of a shape Preserving local detail [Sorkine et al. 04] 14

Modeling Framework Laplacian in the least squares [Alexa 2003; Lipman et al. 2004] Linear modeling constraint Differential properties of the original geometry Solving linear system of the form in least squares According to normal equations : The Laplace operator : Vertices of original geometry : Constraint? : Deformed vertices 15

Modeling Framework Laplacian of : Weight : cotangent weight [Meyer et al. 2003] and is proportional to the mean curvature around vertex i is the degree of vertex i 16

Laplacian in the form of a Matrix Example v1 v2 v3 v4 v5 v0 17

Laplacian in the form of a Matrix Solving Pre-computed for each ROI (Factorized) 18

The idea 1. Define a Region of Interest on the surface and a camera viewpoint 2. Select one of the resulting silhouette 3. Sketch a new shape for this silhouette 20

Computation of Silhouette Object space silhouette + switch between edge silhouettes and smooth surface silhouettes Silhouette point on a edge its normal satisfy 21

Edge Detection [Hertzmann 1999] Normal  perpendicular view 22

Suggested new silhouette Segment Transforming the silhouette in 3D to 2D screen space  Mapping 2D silhouette with sketch [0,1] Transforming new position back to 3D (as position constraint) 23

Geometry adjustment Sharp features Smooth features and suggestive contours

Geometry Adjustment Adjust the mesh geometry to accommodate such a feature directly under the sketch Increase mesh complexity Preserve mesh topology 25

Finding Edge Path Weighted shortest path problem which minimizes distance from sketch screen v1v1 v2v2 n sketch vOvO (orthographic) viewer (schematic) cross-section 26

Adjust the Geometry under Sketch Move vertex along its tangent plane tangent plane vOCvOC n screensketch vSvS vSCvSC n v1v1 v2v2 vOvO (orthographic) viewer 27

Relax the Area around the Sketch Remove badly shaped triangle vOCvOC n screensketch v´ 1 v´ 2  cotangent x=  L fi x 28

vOCvOC n Feature Edit Edit: scale (or add to) Laplacians v´ 1 v´ 2  cotangent 29

Feature Edit Edit: scale (or add to) Laplacians v´ 1 v´ 2  cotangent n 30

Feature Edit v´ 1 v´ 2  cotangent n 31

Contour Edit n nv nrnr radial plane nrnr 32

Contour Edit n nv nrnr radial plane nrnr 33

Contour Edit radial curvature Inf lection line 34

Contour Edit 35

The Quality of shape editing Time required by system Update time is a potential bottleneck To solve linear systems (P4/2.0 GHz) How well the shape change This have improved 5.5K12K33K Factorization Substitution (Second) 37