As-Rigid-As-Possible Shape Manipulation - …david/Classes/ICG/Talks/rigid...As-Rigid-As-Possible...

IntroductionOverviewAlgorithm

ResultsResults

Demo

As-Rigid-As-Possible Shape Manipulation

T. Igarashi1, T. Mascovich2 J. F. Hughes3

1The University of Tokyo 2Brown University

3PRESTO, JST

SIGGRAPH 2005Presented by: Prabin Bariya

T. Igarashi, T. Mascovich, J. F. Hughes As-Rigid-As-Possible Shape Manipulation


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Introduction

Interactive shape manipulation technique for 2D shapes, withoutusing a skeleton or FFD.

Introduces internal model rigidity into shape manipulation.

Uses a quadractic error metric so the minimization problem isformulated as a set of simultaneous linear equations.

The problem is split into two least-squares minimization problems(rotation and scale) that are solved sequentially.



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Introduction

The user chooses handle points within the shape and moveseach handle to a desired location.The system the moves, rotates and deforms the shape to matchthe new handle locations while minimizing distortions.



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Triangulation

The boundary of the shape should be represented as a simpleclosed polygon.

The system generates a triangulated mesh inside the boundaryof the shape.

Near equilateral triangles are used for better results.

Mesh should not be too large.

Resulting triangulation is the rest shape.



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Registration and Compilation

The system performs a pre-computation, called registration, toaccelerate computation during the interaction.When handles are added or removed, additionalpre-computations are performed, called compilation.Compilation allows computation of resulting shape, given handleconfigurations.



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Step 1Step 2

Algorithm

Input: xy-coordinates of the handles.Output: xy-coordinates of the free vertices that minimizedistortion.Uses two independent quadratic error functions for the rotationand scaling.Final result is obtained by sequentially solving two least-squaresproblems.



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Step 1Step 2

Scale-Free Construction

Generates an intermediate result by minimizing an error functionthat allows rotation and uniform scaling.For a triangle in rest shape {v0,v1,v2},v2 = v0 + x01v0v1 + y01R90v0v1.Given a deformed triangle {v ′0,v

′1,v

′2}, vdesired

2 can be computedsimilarly.



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Step 1Step 2

Scale-Free Construction

Then, Ev2 = ‖vdesired2 − v ′2‖

2.

And, E{v0,v1,v2} = ∑i=1,2,3 ‖vdesiredi − v ′i ‖

2

Error for the entire mesh is sum of error for all triangles,expressed in matrix form as: E1{v ′} = v′TGv′

⇒ G′u+Bq = 0

Only q changes during manipulation.

G can be pre-computed during registration and G′−1B duringcompilation.



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Step 1Step 2

Scale Adjustment

Step 1 doesn capture changes in scale and is handled here.First each triangle in the rest shape is fitted in the intermediateresult, allowing rotation and translation.For a triangle {v ′0,v

′1,v

′2} in the intermediate result, we need to

find a new triangle {vfitted0 ,vfitted

1 ,vfitted2 } that is congruent to the

corresponding triangle {v0,v1,v2} in the rest shape.



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Step 1Step 2

Fitting triangles

The fitted triangle should minizeEf{vfitted

0 ,vfitted1 ,vfitted

2 }= ∑i=1,2,3 ‖vfitted

i − v ′i ‖2.

Approximate by first minimizing the error allowing uniform scaling.

Like before, we have:vfitted

2 = vfitted0 + x01vfitted

0 vfitted1 + y01R90vfitted

0 vfitted1

So the fitting functional is a quadratic in the four free variablesw = (v0xfitted ,v0yfitted ,v1xfitted ,v1yfitted)



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Step 1Step 2

Fitting triangles

Minimize Ef by setting partial derivaties over w to zero.

⇒ dEfdw = Fw+C = 0

F is fixed for a given mesh, so can be pre-computed and invertedduring registration.

C is defined by the result of Step 1 and is computed duringmanipulation.

Scale the fitted triangle obtained by solving this equation by afactor of ‖vfitted

0 − vfitted1 ‖/‖v0− v1‖

This makes the fitted triangle congruent to the correspondingtriangle in the rest shape.



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Step 1Step 2

Generating final result

Need to reconcile vertices of several triangles that correspond tothe same mesh vertex.For a triangle {v ′′0 ,v ′′1 ,v ′′2 } and corresponding fitted triangle{vfitted

0 ,vfitted1 ,vfitted

2 }, a quadratic edge error function is defined:

E2{v ′′0 ,v ′′1 ,v ′′2 }= ∑(i,j)∈{(0,1),(1,2),(2,0)} ‖v ′′i v ′′j − vfitted

i vfittedj ‖

2



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Step 1Step 2


The optimal position for a vertex is some average of the positionsdesired by each triangle in which it appears.

The error for the entire mesh is given by: E2v ′′ = v′′THv′′+ fv′′+c

H is defined by the connectivity of the original mesh and, f and care determined by the fitted triangles.



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Step 1Step 2


To minimize E2, setting partial derivatives over u to zero, we get:H′u+Dq+ f0 = 0

H can be precomputed during registration and, H ′ and D duringcompilation.

f0 is computed during manipulation using fitted triangles.

Final result obtained by solving the last equation usingpre-computed LU factorization of H ′.



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Pentium III 1GHz processor, 756MB memory and Javaimplementation.

Real-time performance.

Interaction deterioriates starting at around 300 vertices.

Fairly robust against uneven triangulation and irregularly spacedmesh.



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