Regularizing Part Geometry Instructor - Simon Lucey 16-623 - - PowerPoint PPT Presentation

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 X � � Φ T i ( x − x 0 ) � R ( x ) = λ i i x − x 0   λ 1 Φ i λ i ... � Λ = 0   x −   � x T Φ i λ n i

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 X � � Φ T i ( x − x 0 ) � R ( x ) = λ i i x − x 0   λ 1 Φ i λ i ... � Λ = 0   x −   � x T Φ i λ n i Φ 0

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 X � Φ T � i ( x − x 0 ) � R ( x ) = λ i i x − x 0   λ 1 Φ i λ i ... � Λ = 0   x −   � x T Φ i λ n i Φ 0 Φ i

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 X � Φ T � i ( x − x 0 ) � R ( x ) = λ i i x − x 0   λ 1 Φ i λ i ... � Λ = 0   x −   � x T Φ i λ n i Φ 0 Φ i Φ n − 1

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 X � Φ T � i ( x − x 0 ) � R ( x ) = λ i i x − x 0   λ 1 Φ i λ i ... � Λ = 0   x −   � x T Φ i λ n i Φ 0 Φ i Φ n − 1 Φ n

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � � x − x 0 � R ( x ) = L

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � x − x 0 � � R ( x ) = L ⇢ � � 1 Gaussian Distribution: 2 k x � µ k 2 N ( x ) / exp Σ − 1

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � x − x 0 � � R ( x ) = L ⇢ � � 1 Gaussian Distribution: 2 k x � µ k 2 N ( x ) / exp Σ − 1 µ = x 0

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � x − x 0 � � R ( x ) = L ⇢ � � 1 Gaussian Distribution: 2 k x � µ k 2 N ( x ) / exp Σ − 1 µ = x 0 Σ ∝ L − 1

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � x − x 0 � � R ( x ) = L ⇢ � � 1 Gaussian Distribution: 2 k x � µ k 2 N ( x ) / exp Σ − 1 µ = x 0 Σ ∝ L − 1 = ΦΛ − 1 Φ T

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � x − x 0 � � R ( x ) = L ⇢ � � 1 Gaussian Distribution: 2 k x � µ k 2 N ( x ) / exp Σ − 1 µ = x 0 Σ ∝ L − 1 = ΦΛ − 1 Φ T = ΨΩΨ T

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � x − x 0 � � R ( x ) = L ⇢ � � 1 Gaussian Distribution: 2 k x � µ k 2 N ( x ) / exp Σ − 1 µ = x 0 Σ ∝ L − 1 = ΦΛ − 1 Φ T = ΨΩΨ T   σ 1 ... Ω =   σ i   σ n i

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � � x − x 0 � R ( x ) = L ⇢ � � 1 Gaussian Distribution: 2 k x � µ k 2 N ( x ) / exp Σ − 1 µ = x 0 X Σ ∝ L − 1 = ΦΛ − 1 Φ T = ΨΩΨ T σ i Ψ i Ψ T = i i   σ 1 ... Ω =   σ i   σ n i

Laplacian Regularisation x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 � 2 � � x − x 0 � R ( x ) = L ⇢ � � 1 Gaussian Distribution: 2 k x � µ k 2 N ( x ) / exp Σ − 1 µ = x 0 X Σ ∝ L − 1 = ΦΛ − 1 Φ T = ΨΩΨ T σ i Ψ i Ψ T = i i Ψ 0   σ 1 Ψ 1 ... Ω =   σ i   Ψ n σ n i

Example: Optical Flow ∆ x ||I 0 ( x ) + ∂ I 0 ( x ) ∆ x − I 1 ( x ) || 2 arg min 2 ∂ x T   ∆ x 1 . ∆ x =   . .   ∆ x N N = no. of pixels

General Topology x 0 6 = What if grid?

General Topology x 0 6 = What if grid?

General Topology x 0 6 = What if grid? X X [( x i − x 0 i ) − ( x j − x 0 j )] 2 = R ( x ) j ∈ N i i

General Topology x 0 6 = What if grid? X X [( x i − x 0 i ) − ( x j − x 0 j )] 2 = R ( x ) j ∈ N i i x = x 0 + Ψ α

Smooth Deformation Basis ~Frequency x y z Ψ 0 Ψ 1 Ψ 2 Ψ 3 Ψ 25 Ψ 50

Smooth Deformation Basis ~Frequency x y z Ψ 0 Ψ 1 Ψ 2 Ψ 3 Ψ 25 Ψ 50

Heuristic Regularisation: Recap - Regularisation is important because image measurements are not enough - Priors model the space of valid instances of object’s geometry - Regularisers penalise object geometry outside the space of valid instances. - The smoothness the assumption is a good heuristic - Laplacian regularisers enforce smoothness by penalising high frequency variations more heavily than lower frequency variations - The concept of frequency that is penalised can be specialised to the topology of the object though defining specialised graph- laplacian - But... is that the best we can do? Prior Regulariser

Today • Parts Based Registration • Regularizing Parts (Heuristic) • Regularizing Parts (Learned) 30

Data Driven (Learned) Regularisers What if we have annotated data? [3] Huang et al.’07

Topology of Samples vs. Parts Parts Samples Data

Topology of Samples vs. Parts Parts < 3 d Samples Data Sample Topology

Topology of Samples vs. Parts Parts < d < 3 d Samples Data Sample Topology Part Topology