Morphology for Matrix-Fields: 5 6 7 8 Ordering vs PDE 9 10 11 - PowerPoint PPT Presentation

M I Properties of MI and MS A 1 2 3 4 5 6 Properties of the approach based on the Loewner ordering: 7 8 � rotationally invariant, 9 10 11 12 � preserves positive definiteness, 13 14 15 16 � continuous dependence on the input matrices A i , 17 18 � extendable to indefinite matrices, 19 20 21 22 � extendable to higher order matrices. 23 24 25 26 27 28 For more details on Loewner based matrix morphology: 29 30 B.B. et al., Mathematical Morphology for Tensor Data Induced by the Loewner Ordering in Higher 31 32 Dimensions. Preprint 2005 (to be published in IEEE, Sig. Proc.) 33 34 35 36 37 38 39 40 41

M I Experiments A 1 2 3 4 5 6 Dilation and erosion 7 8 of a 3D matrix field F with a ball-shaped structuring element B of radius 2. 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Original Dilation Erosion 35 36 F ⊕ B F ⊖ B 37 38 39 40 41

M I Experiments A 1 2 3 4 5 6 Opening and closing 7 8 of a 3D matrix field F with a ball-shaped structuring element B of radius 2. 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Original Opening Closing 33 34 35 36 F ◦ B = ( F ⊖ B ) ⊕ B F • B = ( F ⊕ B ) ⊖ B 37 38 39 40 41

M I Experiments A 1 2 Top Hats 3 4 5 6 of a 3D matrix-field F with a ball-shaped structuring element B of radius 2 . 7 8 9 10 11 12 13 14 White 15 16 Original top hat 17 18 F − ( F ◦ B ) 19 20 21 22 23 24 25 26 27 28 29 30 Black Self-dual 31 32 top hat top hat 33 34 ( F • B ) − F ( F • B ) − ( F ◦ B ) 35 36 37 38 39 40 41

M I Experiments A 1 2 Morphological derivatives 3 4 5 6 of a 3D matrix-field F with a ball-shaped structuring element B of radius 2 . 7 8 9 10 11 12 13 14 External 15 16 Original Gradient 17 18 ( F ⊕ B ) − F 19 20 21 22 23 24 25 26 27 28 29 30 Internal Beucher 31 32 Gradient Gradient 33 34 F − ( F ⊖ B ) ( F ⊕ B ) − ( F ⊖ B ) 35 36 37 38 39 40 41

M I Experiments A 1 2 3 4 Morphological Laplacian and shock filtering 5 6 7 8 of a 3D matrix field F with a ball-shaped structuring element of radius 2 . 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Original Morphological Laplacian Shock filtering 33 34 ( F ⊕ B ) − 2 · F + ( F ⊖ B ) 35 36 37 38 39 40 41

M I Outline A 1 2 Content 3 4 � Matrix-valued data 5 6 7 8 � Morphology for matrix-fields via Loewner ordering 9 10 • Basic idea in the 2 × 2 -matrix case 11 12 13 14 • Extensions to 3 × 3 - and larger matrices 15 16 • Experiments 17 18 19 20 21 22 � Mathematical Morphology via PDEs 23 24 • Matrix-valued morphological PDEs 25 26 27 28 • Matrix-valued solution schemes 29 30 • Experiments 31 32 33 34 � Concluding Remarks 35 36 37 38 39 40 41

M I Continous Morphology I A 1 2 Continuous Morphology 3 4 Basic Approach (Boomgaard/Dorst ‘92): Nonlinear partial differential equations that 5 6 mimic the process of dilation and erosion. 7 8 R 2 − � Situation: Original image f : Ω ⊂ I → I R , transformed version u 9 10 11 12 � Dilation with a ball-shaped structuring element: 13 14 15 16 ∂ t u = �∇ u � 17 18 19 20 � Erosion with a ball-shaped structuring element: 21 22 23 24 ∂ t u = − �∇ u � 25 26 27 28 with initial condition u ( x, y, 0) = f ( x, y ) . 29 30 31 32 33 34 Advantages of PDE framework: 35 36 37 38 � Sophisticated machinery of numerical solution methods for PDEs is available 39 40 � Continuous approach allows for sub-pixel accuracy 41

M I Continous Morphology II A 1 2 ( ∂ x u ) 2 + ( ∂ y u ) 2 + ( ∂ z u ) 2 � Scalar PDE: ∂ t u = �∇ u � = 3 4 5 6 7 8 How to find a PDE for matrix-valued data U = ( u ij ) ij ∈ Sym n (I R) ? 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

M I Continous Morphology II A 1 2 ( ∂ x u ) 2 + ( ∂ y u ) 2 + ( ∂ z u ) 2 � Scalar PDE: ∂ t u = �∇ u � = 3 4 5 6 7 8 How to find a PDE for matrix-valued data U = ( u ij ) ij ∈ Sym n (I R) ? 9 10 11 12 R) : If U = V ⊤ diag( λ 1 , . . . , λ n ) V and � Define functions on Sym n (I 13 14 h : I ⊂ I R → I R , 15 16 17 18 h ( U ) := V ⊤ diag( h ( λ 1 ) , . . . , h ( λ n )) V 19 20 21 22 � Generalise partial derivatives ∂ ω , with ω ∈ { t, x 1 , . . . , x d } : 23 24 25 26 ∂ ω U := ( ∂ ω u ij ) ij 27 28 29 30 � Generalise gradient ∇ : 31 32 33 34 ∇ U := ( ∂ x 1 U, . . . , ∂ x d U ) ⊤ ∈ � d 35 36 � Sym n (I R) 37 38 39 40 41

M I Matrix-Valued Morphological PDEs A 1 2 3 4 5 6 � ( ∂ x U ) 2 + ( ∂ y U ) 2 + ( ∂ z U ) 2 Matrix PDE: ∂ t U = |∇ u | 2 = 7 8 9 10 11 12 13 14 15 16 How to solve the morphological matrix PDE ? 17 18 19 20 21 22 23 24 Numerical solution through the matrix-valued counterparts of the scalar schemes for the scalar PDEs 25 26 27 28 29 30 31 32 Example: OS-scheme by Osher & Sethian (1997) 33 34 35 36 37 38 39 40 41

M I Matrix-Valued Solution Schemes I A 1 2 � Osher-Sethian scheme, scalar-valued numerical approximation in 1D: 3 4 5 6 u ( i ) ( n +1) − u ( i ) ( n ) = 7 8 τ 9 10 � 2 � 1 / 2 11 12 � 2 � u ( i ) ( n ) − u ( i − 1) ( n ) � u ( i + 1) ( n ) − u ( i ) ( n ) � � � � � = min , 0 + max , 0 13 14 h h 15 16 17 18 19 20 � matrix-valued counterpart, numerical approximation in 1D: 21 22 U ( i ) ( n +1) − U ( i ) ( n ) 23 24 = 25 26 τ 27 28 � 2 � 1 / 2 � 2 � U ( i ) ( n ) − U ( i − 1) ( n ) � U ( i + 1) ( n ) − U ( i ) ( n ) � 29 30 � � � � = min , 0 + max , 0 31 32 h h 33 34 35 36 37 38 39 40 41

M I Matrix-Valued Solution Schemes II A 1 2 Definition: Let A, B ∈ Sym n (I R) then 3 4 1 � � 5 6 max( A, B ) := A + B + | A − B | 2 7 8 1 � � min( A, B ) := A + B − | A − B | 9 10 2 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Maximal and minimal matrices ∈ Sym 2 (I R) 29 30 Remark: The maximal and minimal matrices are the one induced by the 31 32 Loewner ordering: A ≥ B : ⇐ ⇒ A − B positive semidefinite 33 34 35 36 For comparison of ordering- and PDE-based matrix-morphology: 37 38 B.B. et al., Morphology for Tensor Data: Ordering versus PDE-Based Approach. To be published in JMIV 2006. 39 40 41

M I Experiments: PDE-based Dilation and Erosion A 1 2 Erosion and dilation of matrix-valued images by matrix-valued OS-scheme 3 4 5 6 Top: Dilation 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Original t = 4 t = 10 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Bottom: Erosion 39 40 41

M I Experiments: Ordering vs PDE II A 1 2 √ Ordering based approach (ball-shaped structuring element BSE ( 2) ) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Dilatation Erosion 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 PDE based approach (stopping time 2) 41

M I Experiments: Ordering vs PDE III A 1 2 √ Ordering based approach (ball-shaped structuring element BSE ( 2) ) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Closing Opening 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 PDE based approach (stopping time 2) 41

M I Experiments: Ordering vs PDE IV A 1 2 √ Ordering based approach (ball-shaped structuring element BSE ( 2) ) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 White Top Hat Black Top Hat Self-Dual Top Hat 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 PDE based approach (stopping time 2) 41

M I Experiments: Ordering vs PDE V A 1 2 √ Ordering based approach (ball-shaped structuring element BSE ( 2) ) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Internal Gradient External Gradient Beucher Gradient 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 PDE based approach (stopping time 2) 41

M I Experiments: Ordering vs PDE VI A 1 2 √ Ordering based approach (ball-shaped structuring element BSE ( 2) ) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Morphological Laplacian Shock Filter 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 PDE based approach (stopping time 2) 41

M I Concluding Remarks A 1 2 � Two novel approaches to mathematical morphology for matrix fields: 3 4 • A novel notion for the supremum and infimum of a set of matrices based on 5 6 the Loewner ordering 7 8 9 10 • A truely matrix-valued counterpart for nonlinear morphological PDEs 11 12 � Numerical schemes for scalar PDEs can be transfered to symmetric matrices. 13 14 � 15 16 The properties of the proposed concepts allow for the application of 17 18 • basic morphological operations as well as 19 20 • morphological derivatives 21 22 23 24 to matrix-valued data 25 26 � However, matrix data are “high dimensional” data and some scalar concepts 27 28 might not be directly transferable (discontinuity, ordering, oscillation,...) 29 30 31 32 � Ongoing research concentrates on the development of more sophisticated 33 34 operations for matrix fields based on the above notions. 35 36 37 38 39 40 41

M I A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Thank you very much for your attention! 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

M I Experiments I A 1 2 Experiments in 1D: PDE-driven dilation 3 4 5 6 7 8 t = 0 9 10 11 12 13 14 15 16 t = 0 . 5 17 18 19 20 21 22 23 24 25 26 t = 5 27 28 29 30 31 32 33 34 t = 10 35 36 37 38 39 40 41

M I Experiments II A 1 2 Experiments in 1D: PDE-driven erosion 3 4 5 6 7 8 t = 0 9 10 11 12 13 14 15 16 t = 0 . 5 17 18 19 20 21 22 23 24 25 26 t = 2 . 5 27 28 29 30 31 32 33 34 t = 5 35 36 37 38 39 40 41