smooth shape-aware functions with controlled extrema...smooth shape-aware functions with controlled...

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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precompute weight functions

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deform handles à deform shape

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unconstrained [Botsch & Kobbelt 2004]


local max

local min

Spurious extrema cause distracting artifacts

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bounded [Jacobson et al. 2011]


local max

local min

Bounds help, but don’t solve problem

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local max

local min

Bounds help, but don’t solve problem

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local max

local min

Gets worse with higher-order smoothness

oscillate too much

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our


local max

local min

We explicitly prohibit spurious extrema

# August 9, 2012 Alec Jacobson 15

Same functions used for color interpolation


unconstrained [Finch et al. 2011]


Image courtesy Mark Finch




Our

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Want same control when smoothing data


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Exact, but sharp geodesic

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Exact, but sharp geodesic Smooth, but extrema are lost

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Exact, but sharp geodesic Smooth and maintain extrema

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Ideal discrete problem is intractable


Interpolation functions:

argmin

fE(f)

s.t. fmax

= gmax

fmin

= gmin

fj > fmax

fj < fmin

fi > min

j2N (i)fj

fi < max

j2N (i)fj

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Data smoothing:

argmin

fE(f)

s.t. fmax

= gmax

fmin

= gmin

fj > fmax

fj < fmin

fi > min

j2N (i)fj

fi < max

j2N (i)fj

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argmin

fE(f)

s.t. fmax

= known

fmin

= known

fj < fmax

fj > fmin

fi > min

j2N (i)fj

fi < max

j2N (i)fj

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linear



fj

fmax

argmin

fE(f)

s.t. fmax

= known

fmin

= known

fj < fmax

fj > fmin

fi > min

j2N (i)fj

fi < max

j2N (i)fj

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nonlinear

linear



fi

fj

argmin

fE(f)

s.t. fmax

= known

fmin

= known

fj < fmax

fj > fmin

fi > min

j2N (i)fj

fi < max

j2N (i)fj

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nonlinear

linear

Assume we have a feasible solution


interior

handles

u“Representative function”

uj < umax

uj > umin

ui > min

j2N (i)uj

ui < max

j2N (i)uj

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interior

handles

Assume we have a feasible solution


“Representative function” u

uj < umax

uj > umin

ui > min

j2N (i)uj

ui < max

j2N (i)uj

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linear

Copy “monotonicity” of representative


At least one edge in either direction per vertex

argminf

E(f)

s.t. fmax

= known

fmin

= known

(fi � fj)(ui � uj) > 0 8(i, j) 2 E

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Rewrite as conic optimization


Optimize with MOSEK

QP

Conic

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We always have harmonic representative


argminu

1

2

Z

⌦

kruk2dV

s.t. umax

= 1

s.t. umin

= 0

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argminu

1

2

Z

⌦

kruk2dV

s.t. umax

= 1

s.t. umin

= 0

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argminu

1

2

Z

⌦

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


Anisotropic input data

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Anisotropic input data Harmonic representative

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Anisotropic input data Resulting solution with large

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If data exists, copy topology, too


Anisotropic input data [Weinkauf et al. 2010] representative

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If data exists, copy topology, too


Anisotropic input data Resulting solution with large

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Final algorithm is simple and efficient


●  Data smoothing: topology-aware representative §  Morse-smale + linear solve ~milliseconds

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●  Interpolation: harmonic representative §  Linear solve ~milliseconds

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●  Conic optimization §  2D ~milliseconds, 3D ~seconds

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●  Conic optimization §  2D ~milliseconds, 3D ~seconds

Interpolation: functions are precomputed


We preserve troublesome appendages Bounded Our

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Our weights attach appendages to body


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

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Extrema glue appendages to far-away handles


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

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Our method

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Extrema distort small features


Unconstrained [Botsch & Kobbelt 2004]

weight of middle point

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Unconstrained [Botsch & Kobbelt 2004]


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Bounded [Jacobson et al. 2011]


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“Monotonicity” helps preserve small features


Bounded [Jacobson et al. 2011]

Our

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Spurious extrema are unstable, may “flip”


slightly larger region

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slightly larger region

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Unconstrained [Botsch & Kobbelt, 2004]

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Bounded

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Lack of extrema leads to more stability

Our

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Even control continuity at extrema


Original

Direct extension of [Botsch & Kobbelt 2004]

[Botsch & Kobbelt 2004] + data term

Our method without data term

Our method with data term

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Original





Original





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Reproduces results of Weinkauf et al. 2010…


Original noisy data

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Original noisy data

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Original noisy data Simplified and smoothed

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… but 1000 times faster

Alec Jacobson 80

30K vertices 5 seconds per solve

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Alec Jacobson 81


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Alec Jacobson 82


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●  Copy “monotonicity” of harmonic functions

●  Reduces search-space, but optimization is tractable


Conclusion: Important to control extrema

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●  Larger, but still tractable subspace? §  Consider all valid harmonic functions?


Future work and discussion

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●  Larger, but still tractable subspace? §  Consider all valid harmonic functions?

●  Continuous formulation?


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.


Acknowledgements

Alec Jacobson ([email protected])

Tino Weinkauf

Olga Sorkine

Smooth Shape-Aware Functions with Controlled Extrema

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

smooth shape-aware functions with controlled extrema...smooth shape-aware functions with controlled...

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