A PATH TO PROCESS GENERAL MATRIX FIELDS joint work with Bernhard Burgeth Workshop Data Science | January 30, 2019 Andreas Kleefeld J¨ ulich Supercomputing Centre, Germany Member of the Helmholtz Association
INTRODUCTION Mathematics Division Head: Prof. Dr. Johannes Grotendorst Methods, Algorithms & Tools Lab Daniel Abele Dr. Andreas Kleefeld Christof P¨ aßler Lukas Pieronek Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
INTRODUCTION & MOTIVATION Data processing (difficulty: easy) E.g. gray-valued image processing. Tools: mathematical morphology (discrete or continuous). PDE-based processing (e.g. Perona-Malik diffusion, coherence-enhancing anisotropic diffusion). Prerequisites: linear combinations, discretizations of derivatives, roots/powers, max/min. Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
INTRODUCTION & MOTIVATION Data processing (difficulty: medium) What about color images/multispectral images ( vector-valued data)? No standard ordering available. Channel-wise approach, lexicographic ordering, etc. Problem: false-colors phenomenon (interchannel relationships are ignored). Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
INTRODUCTION & MOTIVATION Data processing (difficulty: hard) What about matrix-valued data, e.g. positive semi-definite matrices (DT-MRI)? Linear combinations, roots/powers, discretization of derivatives ready for use. Max/min is available (Loewner ordering). Catch: only partial ordering. . Real DT-MRI data MCED In other applications: matrices of a matrix field are not symmetric! E.g. material science: stress/strain tensors can loose symmetry; diagonalization: rotation fields. Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
INTRODUCTION & MOTIVATION Data processing (difficulty: bring it on) Interpolation of rotation matrices? 1 1 ? 2 · 2 · + = Interpolation specific for rotation matrices (M. Moakher, SIAM, 2002). 1 1 2 ⊙ ⊕ 2 ⊙ = What about further operations? What about other classes of non-symmetric matrices? Idea: complexification, Hermitian matrices, Her ( n ) . Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
CALCULUS FOR HERMITIAN MATRICES Basic properties Her ( n ) = { H ∈ C n × n | H = H ∗ } is R -vector space. • ∗ stands for transposition with complex conjugation. H = Re ( H ) + Im ( H ) i , • Symmetric real part Re ( H ) . • Skew-symmetric imaginary part Im ( H ) . H unitarily diagonalizable: H = UDU ∗ , • U unitary: U ∗ U = UU ∗ = I . • D = diag ( d 1 , . . . , d n ) diagonal matrix with real-valued d 1 ≥ . . . ≥ d n . Loewner ordering: H 1 ≥ H 2 ⇐ ⇒ H 1 − H 2 positive semi-definite. Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
CALCULUS FOR HERMITIAN MATRICES Dictionary for Hermitian matrices Setting Scalar-valued Matrix-valued � R − � Her ( n ) − → R → Her ( n ) Function f : F : H �→ U diag ( f ( d 1 ) , . . . , f ( d n )) U ∗ x �→ f ( x ) � � Partial ∂ ω h , ∂ ω H := ∂ ω h ij ij , derivatives ω ∈ { t , x 1 , . . . , x d } ω ∈ { t , x 1 , . . . , x d } ∇ h ( x ) := ( ∂ x 1 h ( x ) , . . . , ∂ x d h ( x )) ⊤ , ∇ H ( x ) := ( ∂ x 1 H ( x ) , . . . , ∂ x d H ( x )) ⊤ , Gradient ∇ h ( x ) ∈ R d ∇ H ( x ) ∈ ( Her ( n )) d Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
CALCULUS FOR HERMITIAN MATRICES Dictionary for Hermitian matrices Setting Scalar-valued Matrix-valued | w 1 | p + · · · + | w d | p , | W 1 | p + · · · + | W d | p , p � p � � w � p := �| W |� p := Length �| W |� p ∈ Her + ( n ) � w � p ∈ [ 0 , + ∞ [ psup ( A , B ) = 1 Supremum sup ( a , b ) 2 ( A + B + | A − B | ) pinf ( A , B ) = 1 Infimum inf ( a , b ) 2 ( A + B − | A − B | ) Image processing tools for symmetric matrices carry over to Hermitian matrices. Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
CALCULUS FOR HERMITIAN MATRICES Embedding M R (n) into Her ( n ) Φ : M R (n) − → Her ( n ) Linear mapping → 1 2 ( M + M ⊤ ) + i 2 ( M − M ⊤ ) Φ : M �− Φ − 1 : Her ( n ) − Inverse mapping → M R (n) → 1 2 ( H + H ⊤ ) − i Φ − 1 : H �− 2 ( H − H ⊤ ) Processing strategy: IO Her ( n ) Her ( n ) • Operations on Hermitian matrices via operator IO . Φ Φ − 1 • IO represents averaging, psup, pinf, time-step in numerical scheme, etc. M R (n) M R (n) Φ − 1 ◦ IO ◦ Φ Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
PROCESSING ORTHOGONAL MATRICES Processing orthogonal matrices, Q ∈ O( n ) O ( n ) ⊂ M R (n) There is a problem. • Before processing: Q ∈ O ( n ) . (Φ − 1 ◦ IO ◦ Φ)( Q ) / • After processing: ∈ O ( n ) . There is a remedy. • Projection from M R (n) back to O ( n ) via best Frobenius norm approximation ˜ Q ∈ O ( n ) � (Φ − 1 ◦ O ◦ Φ)( Q ) − ˜ Q � 2 F − → min . This nearest matrix problem allows for explicit solution: • Orthogonal factor in polar decomposition of M . � − 1 / 2 . • ˜ M ⊤ M � Q = P O ( n ) ( M ) = M Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
PROCESSING ORTHOGONAL MATRICES Projection into O( n ) Augmented processing strategy IO Her ( n ) Her ( n ) Φ Φ − 1 O ( n ) ⊂ M R (n) M R (n) O ( n ) Φ − 1 ◦ IO ◦ Φ P O ( n ) General strategy allows for processing of • any square real matrix ∈ M R (n) . • any matrices from an “interesting” subset S ⊂ M R (n) . But P S needs to be calculated. Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
SUMMARY & OUTLOOK Summary Dictionaries Sym ( n ) Transition from scalar calculus to 1 st ed. calculus for symmetric matrices. Her ( n ) Proposed an extension to 2 nd ed. Hermitian matrices. 1-to-1 link to general square Φ − 1 R Φ matrices. Specialization to “interesting” M R (n) matrix subsets possible, for P S example S = O ( n ) . S Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
SUMMARY & OUTLOOK Outlook Extending the “dictionary”. Considering other interesting classes of matrices. Solving (numerically) nearest matrix problems. Looking for interesting fields of applications: Material science (crack formation), problem size: 10 3 × 10 3 × 10 3 -grid, 10 matrix entries, 10 3 -iterations. High resolution 10 7 , multispectral ( 10 2 ) 2 images, 10 3 -iterations. Visualization is a problem. Increasing the efficiency of computations. HPC for real applications is necessary. Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
REFERENCES Partial list B. B URGETH & A. K LEEFELD , Towards Processing Fields of General Real-Valued Square Matrices , Modeling, Analysis, and Visualization of Anisotropy, Springer, 115–144 (2017). B. B URGETH & A. K LEEFELD , A Unified Approach to PDE-Driven Morphology for Fields of Orthogonal and Generalized Doubly-Stochastic Matrices , Springer LNCS 10225, 284–295 (2017). A. K LEEFELD & B. B URGETH , Processing Multispectral Images via Mathematical Morphology , Mathematics and Visualization, Springer, 129–148 (2015). A. K LEEFELD , A. M EYER -B AESE , & B. B URGETH , Elementary Morphology for SO(2)- and SO(3)-Orientation Fields , Springer LNCS 9082, 458–469 (2015). Workshop Data Science | January 30, 2019 Member of the Helmholtz Association Andreas Kleefeld
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