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

Smooth Shape-Aware Functions with Controlled Extrema Alec Jacobson 1 1 ETH Zurich Tino Weinkauf 2 2 MPI Saarbrücken Olga Sorkine 1 August 9, 2012

Real-time deformation relies on smooth, shape-aware functions input shape + handles August 9, 2012 Alec Jacobson # 2

Real-time deformation relies on smooth, shape-aware functions precompute weight functions August 9, 2012 Alec Jacobson # 3

Real-time deformation relies on smooth, shape-aware functions deform handles à deform shape August 9, 2012 Alec Jacobson # 4

Real-time deformation relies on smooth, shape-aware functions August 9, 2012 Alec Jacobson # 5

Real-time deformation relies on smooth, shape-aware functions August 9, 2012 Alec Jacobson # 6

Spurious extrema cause distracting artifacts unconstrained [Botsch & Kobbelt 2004] local max local min August 9, 2012 Alec Jacobson # 7

Spurious extrema cause distracting artifacts unconstrained [Botsch & Kobbelt 2004] local max local min August 9, 2012 Alec Jacobson # 8

Bounds help, but don’t solve problem bounded [Jacobson et al. 2011] local max local min August 9, 2012 Alec Jacobson # 9

Bounds help, but don’t solve problem bounded [Jacobson et al. 2011] local max local min August 9, 2012 Alec Jacobson # 10

Gets worse with higher-order smoothness bounded [Jacobson et al. 2011] local max local min oscillate too much August 9, 2012 Alec Jacobson # 11

Gets worse with higher-order smoothness bounded [Jacobson et al. 2011] local max local min oscillate too much August 9, 2012 Alec Jacobson # 12

We explicitly prohibit spurious extrema our local max local min August 9, 2012 Alec Jacobson # 13

We explicitly prohibit spurious extrema our local max local min August 9, 2012 Alec Jacobson # 14

Same functions used for color interpolation August 9, 2012 Alec Jacobson # 15

Same functions used for color interpolation August 9, 2012 Alec Jacobson # 16

Same functions used for color interpolation unconstrained [Finch et al. 2011] Image courtesy Mark Finch August 9, 2012 Alec Jacobson # 17

Same functions used for color interpolation unconstrained [Finch et al. 2011] August 9, 2012 Alec Jacobson # 18

Same functions used for color interpolation unconstrained [Finch et al. 2011] Our August 9, 2012 Alec Jacobson # 19

Want same control when smoothing data August 9, 2012 Alec Jacobson # 20

Want same control when smoothing data Exact, but sharp geodesic August 9, 2012 Alec Jacobson # 21

Want same control when smoothing data Exact, but sharp geodesic August 9, 2012 Alec Jacobson # 22

Want same control when smoothing data Exact, but sharp geodesic Smooth, but extrema are lost August 9, 2012 Alec Jacobson # 23

Want same control when smoothing data Exact, but sharp geodesic Smooth and maintain extrema August 9, 2012 Alec Jacobson # 24

Ideal discrete problem is intractable arg min E ( f ) f Interpolation functions: August 9, 2012 Alec Jacobson # 25

Ideal discrete problem is intractable arg min E ( f ) f Data smoothing: August 9, 2012 Alec Jacobson # 26

Ideal discrete problem is intractable arg min E ( f ) f August 9, 2012 Alec Jacobson # 27

Ideal discrete problem is intractable arg min E ( f ) f s.t. f max = known f min = known August 9, 2012 Alec Jacobson # 28

Ideal discrete problem is intractable arg min E ( f ) f s.t. f max = known f min = known f j < f max linear f max f j > f min f j August 9, 2012 Alec Jacobson # 29

Ideal discrete problem is intractable arg min E ( f ) f s.t. f max = known f min = known f j f j < f max linear f j > f min f i f i > min j ∈ N ( i ) f j nonlinear f i < max j ∈ N ( i ) f j August 9, 2012 Alec Jacobson # 30

Assume we have a feasible solution “Representative function” u u j < u max linear handles u j > u min u i > min j ∈ N ( i ) u j nonlinear interior u i < max j ∈ N ( i ) u j August 9, 2012 Alec Jacobson # 31

Assume we have a feasible solution “Representative function” u u j < u max handles u j > u min u i > min j ∈ N ( i ) u j interior u i < max j ∈ N ( i ) u j August 9, 2012 Alec Jacobson # 32

Copy “monotonicity” of representative arg min E ( f ) f s.t. f max = known f min = known ( f i − f j )( u i − u j ) > 0 ∀ ( i, j ) ∈ E linear At least one edge in either direction per vertex August 9, 2012 Alec Jacobson # 33

Rewrite as conic optimization Conic QP Optimize with MOSEK August 9, 2012 Alec Jacobson # 34

We always have harmonic representative 1 Z kr u k 2 dV arg min 2 Ω u August 9, 2012 Alec Jacobson # 35

We always have harmonic representative 1 Z kr u k 2 dV arg min 2 Ω u u max = 1 s.t. August 9, 2012 Alec Jacobson # 36

We always have harmonic representative 1 Z kr u k 2 dV arg min 2 Ω u u max = 1 s.t. u min = 0 s.t. August 9, 2012 Alec Jacobson # 37

We always have harmonic representative 1 Z kr u k 2 dV arg min 2 Ω u u max = 1 s.t. u min = 0 s.t. Works well when no input function exists August 9, 2012 Alec Jacobson # 38

Data energy may fight harmonic representative Anisotropic input data August 9, 2012 Alec Jacobson # 39

Data energy may fight harmonic representative Anisotropic input data Harmonic representative August 9, 2012 Alec Jacobson # 40

Data energy may fight harmonic representative Anisotropic input data Harmonic representative August 9, 2012 Alec Jacobson # 41

Data energy may fight harmonic representative Anisotropic input data Harmonic representative August 9, 2012 Alec Jacobson # 42

Data energy may fight harmonic representative Anisotropic input data Resulting solution with large August 9, 2012 Alec Jacobson # 43

If data exists, copy topology, too Anisotropic input data [Weinkauf et al. 2010] representative August 9, 2012 Alec Jacobson # 44

If data exists, copy topology, too Anisotropic input data Resulting solution with large August 9, 2012 Alec Jacobson # 45

Final algorithm is simple and efficient ● Data smoothing : topology-aware representative § Morse-smale + linear solve ~milliseconds August 9, 2012 Alec Jacobson # 46

Final algorithm is simple and efficient ● Data smoothing : topology-aware representative § Morse-smale + linear solve ~milliseconds ● Interpolation : harmonic representative § Linear solve ~milliseconds August 9, 2012 Alec Jacobson # 47

Final algorithm is simple and efficient ● Data smoothing : topology-aware representative § Morse-smale + linear solve ~milliseconds ● Interpolation : harmonic representative § Linear solve ~milliseconds ● Conic optimization § 2D ~milliseconds, 3D ~seconds August 9, 2012 Alec Jacobson # 48

Final algorithm is simple and efficient ● 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 August 9, 2012 Alec Jacobson # 49

We preserve troublesome appendages Bounded Our August 9, 2012 Alec Jacobson # 50

We preserve troublesome appendages Bounded Our August 9, 2012 Alec Jacobson # 51

We preserve troublesome appendages Bounded Our #

Our weights attach appendages to body [Botsch & Kobbelt 2004, Our method Jacobson et al. 2011] August 9, 2012 Alec Jacobson # 53

Extrema glue appendages to far-away handles [Botsch & Kobbelt 2004, Jacobson et al. 2011] August 9, 2012 Alec Jacobson # 54

Extrema glue appendages to far-away handles [Botsch & Kobbelt 2004, Jacobson et al. 2011] August 9, 2012 Alec Jacobson # 55

Our weights attach appendages to body Our method August 9, 2012 Alec Jacobson # 56

Our weights attach appendages to body Our method August 9, 2012 Alec Jacobson # 57

Extrema distort small features Unconstrained [Botsch & Kobbelt 2004] weight of middle point August 9, 2012 Alec Jacobson # 58

Extrema distort small features Unconstrained [Botsch & Kobbelt 2004] weight of middle point August 9, 2012 Alec Jacobson # 59

Extrema distort small features Bounded [Jacobson et al. 2011] weight of middle point August 9, 2012 Alec Jacobson # 60

“Monotonicity” helps preserve small features Bounded [Jacobson et al. 2011] Our August 9, 2012 Alec Jacobson # 61

Spurious extrema are unstable, may “flip” slightly larger region August 9, 2012 Alec Jacobson # 62

Spurious extrema are unstable, may “flip” slightly larger region August 9, 2012 Alec Jacobson # 63

Spurious extrema are unstable, may “flip” Unconstrained [Botsch & Kobbelt, 2004] #

Spurious extrema are unstable, may “flip” Unconstrained [Botsch & Kobbelt, 2004] #