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• Mesh Shape Editing

Computer Animation and Visualisation

Lecture 16

Taku Komura

• Today – Remeshing by the harmonic scalar field – Laplacian mesh editing – As-rigid-as possible shape interpolation – Deformation Transfer

• Remeshing & Simplification • Using the harmonic scalar field and the

isoparametric lines, we can re-mesh the object • By reducing the sampling rate, we can conduct

simplification

• Background • Points from range sensors are too dense • And the triangle shapes are usually random

and have undesirable shapes – Want to reduce the vertices – We may want the vertices to be sampled uniformly

• For simulation purpose, quad mesh is better • A method to remesh polygon data into uniform

quad mesh data Dong et al. 2005

• Harmonic Scalar Fields

 Laplace equation boundary conditions  A u that satisfies this equation is called harmonic

function  u produces a smooth transition between ui s.  When heat diffuses over some material, it follows the

Laplace equation

s.t.

• Laplace equation for a mesh

• After computing u

Once u is computed for all the vertices, we can interpolate and compute their values for anywhere in the triangles

We can compute the gradient lines and the isometric lines on the surface of the object

Then, we can uniformly sample points on the surface

• • Quadrangulation process

U = 0

U = 1

• More results

• Editing shapes

• For animating rigid/articulated objects, we can use previous techniques like rigging

• Let’s think of animating cloths, clay, rubber, soft tissues, dolls, etc

• Editing shapes • Usually, it is easier to edit a given shape rather

than modeling it from the beginning • What is important when editing shapes?

• Keeping the local information unchanged –For faces, the local details of parts such as

the eyes, nose, and mouth must be similar to the original mesh

–Laplacian coordinates –As-rigid-as possible

• Laplacian Mesh Editing [Sorkine ‘04]

• The user specifies the region of interest (ROI) (the area to be edited)

• The user directly moves some of the vertices • The rest of ROI is decided by minimizing the

error function

• Laplacian coordinates

• Assuming all polygons are split into triangles

• Every vertex vi is surrounded by a group of vertices Ni

• coordinate i will be represented by the difference between vi and the average of its neighbors

• Laplacian coordinates (2)

• Suppose the new position of the vertices are v'i • We want to keep the Laplacian coordinates the same after

the deformation

• T is a homogeneous transformation of scaling, rotation, and translation

• Laplacian coordinates (3)

• We also want to constrain the position of some points (keeping vi’ close to ui)

• After all, we want to compute v’ such that it minimizes the following function

• Some more … http://www.youtube.com/watch?v=Yn3P4EK8sYE&feature=channel_page

• Dealing with Volume Loss [Zhou ‘05]

• Sometimes the shape might shrink as we allow scaling for the transformation

• It might be better to keep the volume the same

• Creating a Volumetric Graph

• Add internal vertices and edges • Compute and preserve the details for the internal structure

• Some Results

• What if we want to interpolate different shapes?

– Not just editing the shapes but need to morph it to the target shape

• As-Rigid-As Possible Shape Interpolation [Alexa ’00]

• Interpolating the shapes of the two polygons so that each triangle (2D) / tetrahedron (3D) transformation appears as rigid as possible

• Interpolating the shapes of triangles Represent the interpolation of triangles by

rotation & scaling

• linear interpolation

• rotation & scaling

• Least-Distorting Triangle-to-Triangle Morphing

• Say the three vertices P=(p1,p2,p3) are morphed to Q=(q1,q2,q3)

• We want to compute an affine transform that produces Q=A P

• The intermediate vertice will be computed by V(t)=A(t) P

• A is decomposed into the rotation part R(t) and scaling part and S.

• • We can compute the rotation and scaling using

singular component decomposition

• Closed-Form Vertex Paths for a Triangulation

• As vertices are shared by many triangles, we cannot interpolate each triangle by the above mentioned method

• Let us assume the affine transformation we apply by B{i, j, k} : {i,j,k} are indices of vertices

original translation intermediate

• Keeping the transformation similar to A{i, j, k}

• A vertex configuration that minimizes the error between A and B

• Some 2D results

• 3D objects

• The method is applicable to polyhedra • Tetrahedralization is applied to polyhedra and

the tetrahedra are morphed so that they are as rigid as possible

• More results

• As-rigid-as possible shape manipulation [Igarashi et al. 05]

• interactive manipulation of characters based on the “as-rigid-as-possible” concept

• http://www-ui.is.s.u-tokyo.ac.jp/~takeo/research/rigid/index.html • http://www.youtube.com/watch?v=1M_oyUEOHK8

• Deformation Transfer

• Applying the motion of one character to another • For the corresponding triangle t and t’, we can

compute the affine transformation that transforms t to t’ – t’ = R t + d (R : rotation matrix, d : translation)

• When one character has moved, its movements can be mapped to the other character

• Use the method in slide 23 to keep the connectivity

• Some results of mapping the motion to another

• Summary

• Harmonic scalar fields (remeshing) • Deforming surfaces

– Laplacian Coordinates – As-rigid-as possible shape interpolation

• Deformation Transfer

• Readings • Alexa et al. “As-rigid-as-possible shape interpolation SIGGRAPH

’00 • Igarashi et al. “As-rigid-as-possible shape manipulation”,

SIGGRAPH ‘05 • O. Sorkine, D. Cohen-Or, Y. Lipman, M. Alexa, C.

Rössl and H.-P. Seidel, “Laplacian Surface Editing”, Eurographics Symposium on Geometry Processing (2004)

• Large Mesh Deformation Using the Volumetric Graph Laplacian – Kun Zhou, Jin Huang, John Snyder, Xinguo Liu, Hujun Bao, Baining Guo,

Heung-Yeung Shum. ACM SIGGRAPH 2005, 496-503. • Dong et al. Harmonic functions for quadrilateral remeshing

of arbitrary manifolds, Comput. Aided Geom. Des. 22, 5. 2005

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