smooth shape-aware functions with controlled extrema...smooth shape-aware functions with controlled...
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![Page 1: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/1.jpg)
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
August 9, 2012 Alec Jacobson 3
precompute weight functions
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#
Real-time deformation relies on smooth, shape-aware functions
August 9, 2012 Alec Jacobson 4
deform handles à deform shape
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#
Real-time deformation relies on smooth, shape-aware functions
August 9, 2012 Alec Jacobson 5
![Page 6: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/6.jpg)
#
Real-time deformation relies on smooth, shape-aware functions
August 9, 2012 Alec Jacobson 6
![Page 7: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/7.jpg)
#
unconstrained [Botsch & Kobbelt 2004]
August 9, 2012 Alec Jacobson 7
local max
local min
Spurious extrema cause distracting artifacts
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#
unconstrained [Botsch & Kobbelt 2004]
August 9, 2012 Alec Jacobson 8
local max
local min
Spurious extrema cause distracting artifacts
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#
bounded [Jacobson et al. 2011]
August 9, 2012 Alec Jacobson 9
local max
local min
Bounds help, but don’t solve problem
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#
bounded [Jacobson et al. 2011]
August 9, 2012 Alec Jacobson 10
local max
local min
Bounds help, but don’t solve problem
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#
bounded [Jacobson et al. 2011]
August 9, 2012 Alec Jacobson 11
local max
local min
Gets worse with higher-order smoothness
oscillate too much
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#
bounded [Jacobson et al. 2011]
August 9, 2012 Alec Jacobson 12
local max
local min
Gets worse with higher-order smoothness
oscillate too much
![Page 13: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/13.jpg)
#
our
August 9, 2012 Alec Jacobson 13
local max
local min
We explicitly prohibit spurious extrema
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#
our
August 9, 2012 Alec Jacobson 14
local max
local min
We explicitly prohibit spurious extrema
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# August 9, 2012 Alec Jacobson 15
Same functions used for color interpolation
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# August 9, 2012 Alec Jacobson 16
Same functions used for color interpolation
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# August 9, 2012 Alec Jacobson 17
unconstrained [Finch et al. 2011]
Same functions used for color interpolation
Image courtesy Mark Finch
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# August 9, 2012 Alec Jacobson 18
unconstrained [Finch et al. 2011]
Same functions used for color interpolation
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# August 9, 2012 Alec Jacobson 19
unconstrained [Finch et al. 2011]
Same functions used for color interpolation
Our
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#
Want same control when smoothing data
August 9, 2012 Alec Jacobson 20
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#
Want same control when smoothing data
August 9, 2012 Alec Jacobson 21
Exact, but sharp geodesic
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#
Want same control when smoothing data
August 9, 2012 Alec Jacobson 22
Exact, but sharp geodesic
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#
Want same control when smoothing data
August 9, 2012 Alec Jacobson 23
Exact, but sharp geodesic Smooth, but extrema are lost
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#
Want same control when smoothing data
August 9, 2012 Alec Jacobson 24
Exact, but sharp geodesic Smooth and maintain extrema
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#
Ideal discrete problem is intractable
August 9, 2012 Alec Jacobson 25
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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#
Ideal discrete problem is intractable
August 9, 2012 Alec Jacobson 26
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
![Page 27: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/27.jpg)
#
Ideal discrete problem is intractable
August 9, 2012 Alec Jacobson 27
argmin
fE(f)
s.t. fmax
= known
fmin
= known
fj < fmax
fj > fmin
fi > min
j2N (i)fj
fi < max
j2N (i)fj
![Page 28: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/28.jpg)
#
Ideal discrete problem is intractable
August 9, 2012 Alec Jacobson 28
argmin
fE(f)
s.t. fmax
= known
fmin
= known
fj < fmax
fj > fmin
fi > min
j2N (i)fj
fi < max
j2N (i)fj
![Page 29: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/29.jpg)
#
linear
Ideal discrete problem is intractable
August 9, 2012 Alec Jacobson 29
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
![Page 30: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/30.jpg)
#
nonlinear
linear
Ideal discrete problem is intractable
August 9, 2012 Alec Jacobson 30
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
![Page 31: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/31.jpg)
#
nonlinear
linear
Assume we have a feasible solution
August 9, 2012 Alec Jacobson 31
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
August 9, 2012 Alec Jacobson 32
“Representative function” u
uj < umax
uj > umin
ui > min
j2N (i)uj
ui < max
j2N (i)uj
![Page 33: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/33.jpg)
#
linear
Copy “monotonicity” of representative
August 9, 2012 Alec Jacobson 33
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
August 9, 2012 Alec Jacobson 34
Optimize with MOSEK
QP
Conic
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#
We always have harmonic representative
August 9, 2012 Alec Jacobson 35
argminu
1
2
Z
⌦
kruk2dV
s.t. umax
= 1
s.t. umin
= 0
![Page 36: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/36.jpg)
#
We always have harmonic representative
August 9, 2012 Alec Jacobson 36
argminu
1
2
Z
⌦
kruk2dV
s.t. umax
= 1
s.t. umin
= 0
![Page 37: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/37.jpg)
#
We always have harmonic representative
August 9, 2012 Alec Jacobson 37
argminu
1
2
Z
⌦
kruk2dV
s.t. umax
= 1
s.t. umin
= 0
![Page 38: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/38.jpg)
#
We always have harmonic representative
August 9, 2012 Alec Jacobson 38
argminu
1
2
Z
⌦
kruk2dV
s.t. umax
= 1
s.t. umin
= 0
Works well when no input function exists
![Page 39: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/39.jpg)
#
Data energy may fight harmonic representative
August 9, 2012 Alec Jacobson 39
Anisotropic input data
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#
Data energy may fight harmonic representative
August 9, 2012 Alec Jacobson 40
Anisotropic input data Harmonic representative
![Page 41: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/41.jpg)
#
Data energy may fight harmonic representative
August 9, 2012 Alec Jacobson 41
Anisotropic input data Harmonic representative
![Page 42: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/42.jpg)
#
Data energy may fight harmonic representative
August 9, 2012 Alec Jacobson 42
Anisotropic input data Harmonic representative
![Page 43: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/43.jpg)
#
Data energy may fight harmonic representative
August 9, 2012 Alec Jacobson 43
Anisotropic input data Resulting solution with large
![Page 44: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/44.jpg)
#
If data exists, copy topology, too
August 9, 2012 Alec Jacobson 44
Anisotropic input data [Weinkauf et al. 2010] representative
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#
If data exists, copy topology, too
August 9, 2012 Alec Jacobson 45
Anisotropic input data Resulting solution with large
![Page 46: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/46.jpg)
#
Final algorithm is simple and efficient
August 9, 2012 Alec Jacobson 46
● Data smoothing: topology-aware representative § Morse-smale + linear solve ~milliseconds
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#
Final algorithm is simple and efficient
August 9, 2012 Alec Jacobson 47
● Data smoothing: topology-aware representative § Morse-smale + linear solve ~milliseconds
● Interpolation: harmonic representative § Linear solve ~milliseconds
![Page 48: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/48.jpg)
#
Final algorithm is simple and efficient
August 9, 2012 Alec Jacobson 48
● Data smoothing: topology-aware representative § Morse-smale + linear solve ~milliseconds
● Interpolation: harmonic representative § Linear solve ~milliseconds
● Conic optimization § 2D ~milliseconds, 3D ~seconds
![Page 49: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/49.jpg)
#
Final algorithm is simple and efficient
August 9, 2012 Alec Jacobson 49
● 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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# August 9, 2012 Alec Jacobson 50
We preserve troublesome appendages Bounded Our
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# August 9, 2012 Alec Jacobson 51
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
August 9, 2012 Alec Jacobson 53
Our method [Botsch & Kobbelt 2004, Jacobson et al. 2011]
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Extrema glue appendages to far-away handles
August 9, 2012 Alec Jacobson 54
[Botsch & Kobbelt 2004, Jacobson et al. 2011]
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Extrema glue appendages to far-away handles
August 9, 2012 Alec Jacobson 55
[Botsch & Kobbelt 2004, Jacobson et al. 2011]
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Our weights attach appendages to body
August 9, 2012 Alec Jacobson 56
Our method
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Our weights attach appendages to body
August 9, 2012 Alec Jacobson 57
Our method
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Extrema distort small features
August 9, 2012 Alec Jacobson 58
Unconstrained [Botsch & Kobbelt 2004]
weight of middle point
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Extrema distort small features
August 9, 2012 Alec Jacobson 59
Unconstrained [Botsch & Kobbelt 2004]
weight of middle point
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Extrema distort small features
August 9, 2012 Alec Jacobson 60
Bounded [Jacobson et al. 2011]
weight of middle point
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“Monotonicity” helps preserve small features
August 9, 2012 Alec Jacobson 61
Bounded [Jacobson et al. 2011]
Our
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Spurious extrema are unstable, may “flip”
August 9, 2012 Alec Jacobson 62
slightly larger region
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Spurious extrema are unstable, may “flip”
August 9, 2012 Alec Jacobson 63
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
August 9, 2012 Alec Jacobson 71
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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Even control continuity at extrema
August 9, 2012 Alec Jacobson 72
Original
Direct extension of [Botsch & Kobbelt 2004]
[Botsch & Kobbelt 2004] + data term
Our method without data term
Our method with data term
Original
Direct extension of [Botsch & Kobbelt 2004]
[Botsch & Kobbelt 2004] + data term
Our method without data term
Our method with data term
![Page 73: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/73.jpg)
#
Even control continuity at extrema
August 9, 2012 Alec Jacobson 73
Original
Direct extension of [Botsch & Kobbelt 2004]
[Botsch & Kobbelt 2004] + data term
Our method without data term
Our method with data term
Original
Direct extension of [Botsch & Kobbelt 2004]
[Botsch & Kobbelt 2004] + data term
Our method without data term
Our method with data term
![Page 74: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/74.jpg)
#
Even control continuity at extrema
August 9, 2012 Alec Jacobson 74
Original
Direct extension of [Botsch & Kobbelt 2004]
[Botsch & Kobbelt 2004] + data term
Our method without data term
Our method with data term
Original
Direct extension of [Botsch & Kobbelt 2004]
[Botsch & Kobbelt 2004] + data term
Our method without data term
Our method with data term
![Page 75: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/75.jpg)
#
Even control continuity at extrema
August 9, 2012 Alec Jacobson 75
Original
Direct extension of [Botsch & Kobbelt 2004]
[Botsch & Kobbelt 2004] + data term
Our method without data term
Our method with data term
Original
Direct extension of [Botsch & Kobbelt 2004]
[Botsch & Kobbelt 2004] + data term
Our method without data term
Our method with data term
![Page 76: Smooth Shape-Aware Functions with Controlled Extrema...Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson1 Tino Weinkauf2 Olga Sorkine1 August 9, 2012 1ETH Zurich 2MPI](https://reader034.vdocument.in/reader034/viewer/2022050518/5fa2470031e88d15a8411338/html5/thumbnails/76.jpg)
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Reproduces results of Weinkauf et al. 2010…
August 9, 2012 Alec Jacobson 76
Original noisy data
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Reproduces results of Weinkauf et al. 2010…
August 9, 2012 Alec Jacobson 77
Original noisy data
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Reproduces results of Weinkauf et al. 2010…
August 9, 2012 Alec Jacobson 78
Original noisy data Simplified and smoothed
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Reproduces results of Weinkauf et al. 2010…
August 9, 2012 Alec Jacobson 79
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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… but 1000 times faster
Alec Jacobson 81
30K vertices 5 seconds per solve
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… but 1000 times faster
Alec Jacobson 82
30K vertices 5 seconds per solve
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● Copy “monotonicity” of harmonic functions
● Reduces search-space, but optimization is tractable
August 9, 2012 Alec Jacobson 83
Conclusion: Important to control extrema
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● Larger, but still tractable subspace? § Consider all valid harmonic functions?
August 9, 2012 Alec Jacobson 84
Future work and discussion
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● Larger, but still tractable subspace? § Consider all valid harmonic functions?
● Continuous formulation?
August 9, 2012 Alec Jacobson 85
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.
August 9, 2012 Alec Jacobson 86
Acknowledgements
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Alec Jacobson ([email protected])
Tino Weinkauf
Olga Sorkine
Smooth Shape-Aware Functions with Controlled Extrema
MATLAB Demo: http://igl.ethz.ch/projects/monotonic/