Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson 1 - - PowerPoint PPT Presentation

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Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson 1 - - PowerPoint PPT Presentation

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Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson 1 1 ETH Zurich Tino Weinkauf 2 2 MPI Saarbrcken Olga Sorkine 1 August 9, 2012 Real-time deformation relies on smooth, shape-aware functions input shape + handles August 9,

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Smooth Shape-Aware Functions with Controlled Extrema

Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1

August 9, 2012

1ETH Zurich 2MPI Saarbrücken

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Real-time deformation relies on smooth, shape-aware functions

August 9, 2012 Alec Jacobson 2

input shape + handles

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Real-time deformation relies on smooth, shape-aware functions

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precompute weight functions

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Real-time deformation relies on smooth, shape-aware functions

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deform handles à deform shape

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Real-time deformation relies on smooth, shape-aware functions

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Real-time deformation relies on smooth, shape-aware functions

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unconstrained [Botsch & Kobbelt 2004]

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local max local min

Spurious extrema cause distracting artifacts

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unconstrained [Botsch & Kobbelt 2004]

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local max local min

Spurious extrema cause distracting artifacts

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bounded [Jacobson et al. 2011]

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local max local min

Bounds help, but don’t solve problem

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bounded [Jacobson et al. 2011]

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local max local min

Bounds help, but don’t solve problem

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bounded [Jacobson et al. 2011]

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local max local min

Gets worse with higher-order smoothness

  • scillate too much
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bounded [Jacobson et al. 2011]

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local max local min

Gets worse with higher-order smoothness

  • scillate too much
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  • ur

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local max local min

We explicitly prohibit spurious extrema

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  • ur

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local max local min

We explicitly prohibit spurious extrema

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Same functions used for color interpolation

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Same functions used for color interpolation

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unconstrained [Finch et al. 2011]

Same functions used for color interpolation

Image courtesy Mark Finch

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unconstrained [Finch et al. 2011]

Same functions used for color interpolation

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unconstrained [Finch et al. 2011]

Same functions used for color interpolation

Our

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Want same control when smoothing data

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Want same control when smoothing data

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Exact, but sharp geodesic

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Want same control when smoothing data

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Exact, but sharp geodesic

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Want same control when smoothing data

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Exact, but sharp geodesic Smooth, but extrema are lost

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Want same control when smoothing data

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Exact, but sharp geodesic Smooth and maintain extrema

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Ideal discrete problem is intractable

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Interpolation functions:

arg min

f

E(f)

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Ideal discrete problem is intractable

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Data smoothing:

arg min

f

E(f)

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Ideal discrete problem is intractable

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arg min

f

E(f)

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Ideal discrete problem is intractable

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arg min

f

E(f) s.t. fmax = known fmin = known

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linear

Ideal discrete problem is intractable

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fj fmax

arg min

f

E(f) s.t. fmax = known fmin = known fj < fmax fj > fmin

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nonlinear linear

Ideal discrete problem is intractable

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fi

fj

arg min

f

E(f) s.t. fmax = known fmin = known fj < fmax fj > fmin fi > min

j∈N (i) fj

fi < max

j∈N (i) fj

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nonlinear linear

Assume we have a feasible solution

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interior handles

u

“Representative function”

uj < umax uj > umin ui > min

j∈N (i) uj

ui < max

j∈N (i) uj

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interior handles

Assume we have a feasible solution

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“Representative function” u

uj < umax uj > umin ui > min

j∈N (i) uj

ui < max

j∈N (i) uj

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linear

Copy “monotonicity” of representative

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At least one edge in either direction per vertex

arg min

f

E(f) s.t. fmax = known fmin = known (fi − fj)(ui − uj) > 0 ∀(i, j) ∈ E

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Rewrite as conic optimization

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Optimize with MOSEK

QP Conic

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We always have harmonic representative

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arg min

u

1 2 Z

Ω

kruk2dV

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We always have harmonic representative

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arg min

u

1 2 Z

Ω

kruk2dV s.t. umax = 1

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We always have harmonic representative

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arg min

u

1 2 Z

Ω

kruk2dV s.t. umax = 1 s.t. umin = 0

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We always have harmonic representative

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arg min

u

1 2 Z

Ω

kruk2dV s.t. umax = 1 s.t. umin = 0

Works well when no input function exists

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Data energy may fight harmonic representative

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Anisotropic input data

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Data energy may fight harmonic representative

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Anisotropic input data Harmonic representative

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Data energy may fight harmonic representative

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Anisotropic input data Harmonic representative

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Data energy may fight harmonic representative

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Anisotropic input data Harmonic representative

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Data energy may fight harmonic representative

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Anisotropic input data Resulting solution with large

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If data exists, copy topology, too

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Anisotropic input data [Weinkauf et al. 2010] representative

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If data exists, copy topology, too

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Anisotropic input data Resulting solution with large

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Final algorithm is simple and efficient

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  • Data smoothing: topology-aware representative

§ Morse-smale + linear solve ~milliseconds

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Final algorithm is simple and efficient

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  • Data smoothing: topology-aware representative

§ Morse-smale + linear solve ~milliseconds

  • Interpolation: harmonic representative

§ Linear solve ~milliseconds

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Final algorithm is simple and efficient

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  • Data smoothing: topology-aware representative

§ Morse-smale + linear solve ~milliseconds

  • Interpolation: harmonic representative

§ Linear solve ~milliseconds

  • Conic optimization

§ 2D ~milliseconds, 3D ~seconds

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Final algorithm is simple and efficient

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  • Data smoothing: topology-aware representative

§ Morse-smale + linear solve ~milliseconds

  • Interpolation: harmonic representative

§ Linear solve ~milliseconds

  • Conic optimization

§ 2D ~milliseconds, 3D ~seconds

Interpolation: functions are precomputed

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We preserve troublesome appendages

Bounded Our

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We preserve troublesome appendages

Bounded Our

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We preserve troublesome appendages

Bounded Our

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Our weights attach appendages to body

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Our method [Botsch & Kobbelt 2004, Jacobson et al. 2011]

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Extrema glue appendages to far-away handles

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[Botsch & Kobbelt 2004, Jacobson et al. 2011]

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Extrema glue appendages to far-away handles

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[Botsch & Kobbelt 2004, Jacobson et al. 2011]

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Our weights attach appendages to body

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Our method

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Our weights attach appendages to body

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Our method

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Extrema distort small features

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Unconstrained [Botsch & Kobbelt 2004]

weight of middle point

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Extrema distort small features

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Unconstrained [Botsch & Kobbelt 2004]

weight of middle point

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Extrema distort small features

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Bounded [Jacobson et al. 2011]

weight of middle point

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“Monotonicity” helps preserve small features

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Bounded [Jacobson et al. 2011] Our

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Spurious extrema are unstable, may “flip”

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slightly larger region

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Spurious extrema are unstable, may “flip”

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slightly larger region

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Spurious extrema are unstable, may “flip”

Unconstrained [Botsch & Kobbelt, 2004]

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Spurious extrema are unstable, may “flip”

Unconstrained [Botsch & Kobbelt, 2004]

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Spurious extrema are unstable, may “flip”

Unconstrained [Botsch & Kobbelt, 2004]

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Spurious extrema are unstable, may “flip”

Bounded

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Spurious extrema are unstable, may “flip”

Bounded

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Lack of extrema leads to more stability

Our

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Lack of extrema leads to more stability

Our

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Even control continuity at extrema

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Original Direct extension of [Botsch & Kobbelt 2004]

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Even control continuity at extrema

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Direct extension of [Botsch & Kobbelt 2004] [Botsch & Kobbelt 2004] + data term Original Direct extension of [Botsch & Kobbelt 2004]

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Even control continuity at extrema

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[Botsch & Kobbelt 2004] + data term Our method without data term Original Direct extension of [Botsch & Kobbelt 2004]

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Even control continuity at extrema

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Our method without data term Original Direct extension of [Botsch & Kobbelt 2004]

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Even control continuity at extrema

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Our method with data term

Original Direct extension of [Botsch & Kobbelt 2004]

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Reproduces results of Weinkauf et al. 2010…

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Original noisy data

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Reproduces results of Weinkauf et al. 2010…

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Original noisy data

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Reproduces results of Weinkauf et al. 2010…

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Original noisy data Simplified and smoothed

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Reproduces results of Weinkauf et al. 2010…

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Original noisy data Simplified and smoothed

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… but 1000 times faster

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30K vertices 5 seconds per solve

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… but 1000 times faster

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30K vertices 5 seconds per solve

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… but 1000 times faster

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30K vertices 5 seconds per solve

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  • Copy “monotonicity” of harmonic

functions

  • Reduces search-space, but optimization is

tractable

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Conclusion: Important to control extrema

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  • Larger, but still tractable subspace?

§ Consider all valid harmonic functions?

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Future work and discussion

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  • Larger, but still tractable subspace?

§ Consider all valid harmonic functions?

  • Continuous formulation?

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Future work and discussion

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We thank Kenshi Takayama for his valuable

  • feedback. This work was supported in part

by an SNF award 200021_137879 and by a gift from Adobe Systems.

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Acknowledgements

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Alec Jacobson (jacobson@inf.ethz.ch) Tino Weinkauf Olga Sorkine

Smooth Shape-Aware Functions with Controlled Extrema

MATLAB Demo: http://igl.ethz.ch/projects/monotonic/