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Ilmenau University of Technology Communications Research Laboratory 1 Tensor-Based Signal Processing with Applications in Biomedical Signal Analysis Technische.

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Präsentation zum Thema: "Ilmenau University of Technology Communications Research Laboratory 1 Tensor-Based Signal Processing with Applications in Biomedical Signal Analysis Technische."—  Präsentation transkript:

1 Ilmenau University of Technology Communications Research Laboratory 1 Tensor-Based Signal Processing with Applications in Biomedical Signal Analysis Technische Universität Ilmenau Fachgebiete Nachrichtentechnik & Biosignalverarbeitung

2 Ilmenau University of Technology Communications Research Laboratory 2 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

3 Ilmenau University of Technology Communications Research Laboratory 3 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

4 Ilmenau University of Technology Communications Research Laboratory 4 Why tensors?  Well, why even matrices?  Matrix equations are usually more compact  insights, manipulations  Example: DFT  Not a different data model but a more compact representation  More than two dimensions: tensors even more compact  new insights d n = ? d n = ?? 

5 Ilmenau University of Technology Communications Research Laboratory 5 “Classical” Communications Research  Description of the Mobile Radio Channel  resolve, characterize individual propagation paths of the mobile radio channel

6 Ilmenau University of Technology Communications Research Laboratory 6 Biomedical engineering  For example: EEG data  diagnostics (neurology, ophthalmology)  human-machine interface

7 Ilmenau University of Technology Communications Research Laboratory 7 Automotive engineering  Wind tunnel analysis  find sources of disturbance to optimize aerodynamic behavior Audio Source Localization

8 Ilmenau University of Technology Communications Research Laboratory 8 Motivation  More applications  Signal Processing (sensor arrays, blind multi-user detection, source separation, CDMA, SONAR and seismo-acoustic signal processing)  Computer vision (Face and facial expression recognition, handwritten text recognition)  Data mining (weblink analysis, personalized web search, cross-language information retrieval)  Neuroscience (Multisubject fMRI anlaysis, concurrent EEG/fMRI)  Chemical engineering (food industry, NIR spectroscopy)  Geophysics (moment tensor inversion)  Data compression (image coding, video coding)

9 Ilmenau University of Technology Communications Research Laboratory 9 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

10 Ilmenau University of Technology Communications Research Laboratory 10 The term “tensor” Here: Intuitive definition: Here: Intuitive definition: ”A tensor of order p is a collection of elements referenced by p incides“  multi-way array ”A tensor of order p is a collection of elements referenced by p incides“  multi-way array Mathematics: 1846: W. Voigt Mathematics: 1846: W. Voigt Physics: 1915: M. Grossmann and A. Einstein Physics: 1915: M. Grossmann and A. Einstein  very abstract definition  describe physical quantities ScalarsVectorsMatricesOrder-3-tensorsOrder-4-tensors ? …

11 Ilmenau University of Technology Communications Research Laboratory 11 Notation  Symbols  Matrix operations

12 Ilmenau University of Technology Communications Research Laboratory 12 Matrix unfoldings  n-mode matrix unfoldings  vary the n-th along rows, the others along columns  e.g., R = 3:  n-rank of. In general, 1-, 2-, and 3-rank can differ. M1M1 M2M2 M3M3 “1-mode vectors” “2-mode vectors” “3-mode vectors”

13 Ilmenau University of Technology Communications Research Laboratory 13 n-mode products i.e., all the n-mode vectors multiplied from the left-hand- side by  n-mode product between a tensor and a matrix 1 2  outer product between two tensors: all pair-wise products between elements   

14 Ilmenau University of Technology Communications Research Laboratory 14 The tensor rank  Definition of the tensor rank  A tensor is rank one, iff  A tensor is rank r iff it can be decomposed into a sum of r and not less than r rank one tensors  (Only) connection to the n-ranks:  The rank of a tensor can exceed its size (which is a good thing and a bad thing) 2 x (maximum rank, cf. [Kolda08])

15 Ilmenau University of Technology Communications Research Laboratory 15 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

16 Ilmenau University of Technology Communications Research Laboratory 16 The Higher-Order SVD (HOSVD)  Singular Value Decomposition  Higher-Order SVD (Tucker3) “Full HOSVD”“Full SVD” [Tucker: 1966] [de Lathauwer: 2000]

17 Ilmenau University of Technology Communications Research Laboratory 17 Computing the HOSVD  Computing the HOSVD  The core tensor  not necessarily any zero elements (only if n-rank-deficient)  all-orthogonality condition: also three sets of singular values: n-mode singular values 4 3 5

18 Ilmenau University of Technology Communications Research Laboratory 18 The Higher-Order SVD (HOSVD)  Singular Value Decomposition  Higher-Order SVD (Tucker3) “Full HOSVD” “Economy size HOSVD” Low-rank approximation (truncated HOSVD) “Full SVD” “Economy size SVD” Low-rank approximation

19 Ilmenau University of Technology Communications Research Laboratory 19 Summary HOSVD  The HOSVD …  is an extension of the SVD to tensors.  generalizes the concept of row-space and column-space to r-spaces. the r-mode singular vectors are an orthonormal basis for the r-space of the tensor.  is very easy to compute (Matrix-SVD of the unfoldings).  the remaining core tensor is not diagonal, it may be full of non-zero elements (same size as original data) not a decomposition into rank-one components

20 Ilmenau University of Technology Communications Research Laboratory 20 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

21 Ilmenau University of Technology Communications Research Laboratory 21  Another way to look at the SVD  decomposition into a sum of rank one matrices  also referred to as principal components (PCA)  Tensor case: PARAFAC: Motivation ++ = ++ = Canonical Decomposition (CANDECOMP) Parallel Factor Analysis (PARAFAC) [Carroll, Chang 1970] [Harshman 1970]

22 Ilmenau University of Technology Communications Research Laboratory 22 PARAFAC expressions  Many equations to express the same model:      HOSVD:

23 Ilmenau University of Technology Communications Research Laboratory 23 HOSVD vs. PARAFAC  Example: HOSVD PARAFAC  “Core tensor” diagonal  Not easy to find the factors  Factors may be flat (underdetermined)  Reveals the tensor rank  Often used for analyzing data  Core tensor not necessarily diagonal, can be full.  Direct, easy computation via matrix-SVDs  “Factors” always tall or square  Reveals the n-ranks  Often used for “compressing” data

24 Ilmenau University of Technology Communications Research Laboratory 24 Uniqueness  When is the PARAFAC decomposition of X into A,B,C unique?  Let be the Kruskal-rank of A.  Then, given that …  … the PARAFAC decomposition is unique up to scaling and permutation. [Kruskal, 1966]  scaling and permutation can be removed if additional constraints are imposed or prior knowledge is used scaling: permutation:

25 Ilmenau University of Technology Communications Research Laboratory 25 Finding the parallel factors  Since we only have the noisy tensor, we restate the goal:  “Plain vanilla” approach: ALS  Many years of research to improve convergence speed  smart initializations  smart updates: Enhanced line search …  “PARAFAC”, “COMFAC” works, but very slow convergence requires good initial solution “Least Squares Fit”

26 Ilmenau University of Technology Communications Research Laboratory 26 Finding the parallel factors  Closed-form solutions?  GRAM (generalized rank annihilation method): An exact closed-form solution if either of M 1, M 2 or M 3 = 2 (only two slices). Can be used as initialization for other methods.  DTLD (direct trilinear decomposition): A suboptimal approximation, mostly as initialization to PARAFAC. Based on Tucker3 (HOSVD) and GRAM. Very fast though.  Ours: Reduced the problem onto joint diagonalization of matrices (which by itself is a very well studied problem).

27 Ilmenau University of Technology Communications Research Laboratory 27 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

28 Ilmenau University of Technology Communications Research Laboratory 28

29 Ilmenau University of Technology Communications Research Laboratory 29 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

30 Ilmenau University of Technology Communications Research Laboratory 30 PARAFAC via Joint Diagonalization  First, consider the case where  The transform matrices diagonalize the core tensor:  We can estimate the transform matrices via joint diagonalization  the fundamental link between the HOSVD and PARAFAC

31 Ilmenau University of Technology Communications Research Laboratory 31 PARAFAC via Joint Diagonalization  One slide on the six diagonalization problems

32 Ilmenau University of Technology Communications Research Laboratory 32  One slide on the R-D extension

33 Ilmenau University of Technology Communications Research Laboratory 33 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

34 Ilmenau University of Technology Communications Research Laboratory 34 Processing Chain  Time-Frequency-Analysis Time Frequency Channel Time Frequency Channel Time Channel Wavelet-basiert Wigner-basiert Biomedical process Measurement Time-Frequency- Analysis Component Analysis

35 Ilmenau University of Technology Communications Research Laboratory 35 Component Analysis  Given a three-way tensor (time, frequency, channel), we decompose it into a predefined number of components  for each component: time-, frequency-, and spatial characteristics Zeit Frequenz Raum ≈ ++

36 Ilmenau University of Technology Communications Research Laboratory 36 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

37 Ilmenau University of Technology Communications Research Laboratory 37 Outline  Motivation: Applications of multi-linear signal processing  Introduction to multi-linear algebra  Tensor decompositions  Multilinear extensions of the SVD HOSVD PARAFAC/CANDECOMP  Other decompositions  PARAFAC via Joint Diagonalization  3-way PARAFAC for EEG data  Methodology and current status  Open issues and questions  Discussion  Status of the project proposals

38 Ilmenau University of Technology Communications Research Laboratory 38 Geplante Folgeprojekte (1) BMBF - Innovationswettbewerb Medizintechnik  Modul I „Innovationswettbewerb - BASIS“ Schlüsselexperiment zum Nachweis der Machbarkeit:  „Tensor-basierte Analyse von polygraphischen Biosignalen zur Anfallsvorhersage bei Epilepsie“  Projektpartner TU Ilmenau: FG BSV & FG NT GJB Datentechnik GmbH, Langewiesen Zentralklinik Bad Berka GmbH, Klinik für Neurologie  Status Projektskizze eingereicht Hauptantrag vorzubereiten im Sommer/Herbst 2008  Modul II „Innovationswettbewerb – Transfer“ F&E-Vorhaben  „Neue Methoden der Tensor-basierten Analyse von polygraphischen Biosignalen“  Projektpartner TU Ilmenau: FG BSV & FG NT GJB Datentechnik GmbH, Langewiesen Zentralklinik Bad Berka GmbH, Klinik für Neurologie Psychotherapeutische und Neurologische Praxen  Status Projektskizze vorzubereiten im Herbst 2008

39 Ilmenau University of Technology Communications Research Laboratory 39 Geplante Folgeprojekte (2) BMBF - Innovationswettbewerb Medizintechnik  „Frühdiagnostik und Intervention von Essanfällen mittels Polygraphie bei Patienten mit Bulimia Nervosa“ (BuPoly)  Projektpartner TU Ilmenau: FG BSV & FG NT NeuroConn GmbH, Ilmenau Praxis Dr. Braun, Gotha  Status Projektskizze eingereicht Hauptantrag vorzubereiten im Sommer/Herbst 2008  „Zeitvariable Niederfeldmagnetstimulation in der Therapie depressiver Erkrankungen und deren Wirkung auf die Herzratenvariabilität“ (DeNFMagS)  Projektpartner TU Ilmenau: FG BSV & FG NT Neurologische Praxis Henkel/Müller, Ilmenau  Status Projektskizze eingereicht Hauptantrag vorzubereiten im Sommer/Herbst 2008

40 Ilmenau University of Technology Communications Research Laboratory 40 Geplante Folgeprojekte (3) BMBF - Ernährungsforschung  „Polygraphiebasierte methodische und experimentelle Untersuchung der Reizreaktion auf lebensmittelbezogene visuelle Stimulationen bei Personen mit und ohne psychogene Essstörungen“  Projektpartner TU Ilmenau: FG BSV & FG NT Praxis Dr. Braun, Gotha Neurologische Praxis Henkel/Müller, Ilmenau  Status Projektskizze eingereicht Hauptantrag vorzubereiten im Sommer 2008

41 Ilmenau University of Technology Communications Research Laboratory 41 Geplante Folgeprojekte (4) LUBOM – Thüringen  „Zeitvariable Niederfeldmagnetstimulation in der Therapie depressiver Erkrankungen und deren Wirkung auf die Herzratenvariabilität“  Projektpartner in der ersten Phase TU Ilmenau: FG BSV & FG NT in der nächsten Phase zusätzlich neurologische und psychotherapeutische Praxen  Status Projektbeginn bei Bewilligung Anfang 2009  „Die Wirkungsweise von Eye Movement Desensitization and Reprocessing analysiert anhand polygraphischer Untersuchungen multimodaler Biosignale“  Projektpartner in der ersten Phase TU Ilmenau: FG BSV & FG NT in der nächsten Phase zusätzlich neurologische und psychotherapeutische Praxen  Status Projektbeginn bei Bewilligung Anfang 2009

42 Ilmenau University of Technology Communications Research Laboratory 42 Geplante Folgeprojekte (5) LUBOM – Thüringen  „EKG-Analyse zur Bestimmung der anaeroben Schwelle anhand der Absenkung des ST-Komplexes“  Projektpartner TU Ilmenau: FG BSV & FG NT  Status Projektbeginn bei Bewilligung Herbst 2008 TAB – Thüringen  „Erkennung von psychischen Verarbeitungsprozessen angstgestörter Patienten zur Unterstützung der lnterventionstherapie mittels mobiler onIinefähiger Biofeedbackgeräte“  Projektpartner TU Ilmenau: FG BSV & FG NT GJB Datentechnik GmbH, Langewiesen Psychotherapie Dr. Wilms, Erfurt  Status Projektbeginn bei Bewilligung Herbst 2008

43 Ilmenau University of Technology Communications Research Laboratory 43 Geplante Folgeprojekte (6) European Research Council: Advanced Investigators Grants im FP 7  „Mikrosensoren zur Erfassung wichtiger Lebensfunktionen“  Projektpartner TU Ilmenau: FG BSV & FG NT & IMN & FG NIKR IDMT Fraunhofer, Ilmenau  Status Projektskizze in Vorbereitung einzureichen im Winter 2008

44 Ilmenau University of Technology Communications Research Laboratory 44 Geplante Folgeprojekte (7) DFG  Forschungsprojekt zur dynamischen tensorbasierten Analyse von nichtlinearen zeitvariablen Prozessen  Projektpartner TU Ilmenau: FG BSV & FG NT Zentralklinik Bad Berka GmbH, Klinik für Neurologie GJB Datentechnik GmbH, Langewiesen Psychotherapeutische und Neurologische Praxen  Status Projektskizze vorzubereiten im Herbst 2008


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