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CP-based HSR
In [86], the HSR problem was expressed as a coupled rank-N CP approximation. Under this assumption, the general model (1.28) becomes YH rrP1A;P2B;Css EH; YM rrA;B;P3Css EM: The matrices A P RIN, B P RJN, C P RKN are the factor matrices of the CPD. With this notation, the SRI admits a CPD Y rrA;B;Css: (2.2)
The case of known spatial degradation
If the degradation matrices P1, P2 and P3 are known, then (2.1) can be recast into the following optimization problem: minimize pA ;pB;pC fCPppA;pB;pCq; (2.3) where fCPppA;pB;pCq }YH rrP1pA;P2pB;pCss} 2 F }YM rrpA;pB;P3pCss} 2 F ; and is a balance parameter3 that controls the respective weights of the HSI and MSI in the above coupled CP approximation problem. Problem (2.3) is a special case of (1.29).
In the noiseless case (EH; EM 0), the coupled CP model (2.1) is generically identifiable if N ¤ mint2tlog2pKMJqu2; IHJHu: (2.4) In the proof for [86, Theorem 3], it is specified that condition (2.4) only requires the CPD of the MSI to be unique. An alternating least squares (ALS) algorithm called Super-resolution TEnsor REconstruction (STEREO) is proposed for solving (2.3). It is summarized in Algorithm 1. The updates of the factor matrices in Algorithm 1 can be computed by using efficient solvers for the (generalized) Sylvester equation [8], [61]. For example, the total cost of one iteration (updating pAk; pBk; pCk) in Algorithm 1 becomes:
OpIJKMN IHJHKNq flops for computing the right hand sides in the least-squares subproblems;
OpI3J3K3N3q flops for solving Sylvester equations. For more details on solving Sylvester equations for STEREO, see4 [86, Appendix E].
The case of unknown spatial degradation
In some cases however, the spatial degradation operators are unknown, therefore blind algorithms are needed. A first solution called Blind-STEREO was also proposed in [86]. It adresses the following coupled CP model: YH rrrA;rB;Css EH; YM rrA;B;P3Css EM:
The matrices rA P RIHN and rB P RJHN are degraded versions of the CP factors A P RIN and B P RJN by unknown spatial degradation matrices. Model (2.5) can also be reformulated Conditions on exact noiseless recovery of the SRI by Blind-STEREO were provided in [86] and require both the CPD of the HSI and MSI to be unique. In [87], an alternative approach was proposed. It is based on a single CPD of YM together with a SVD of Yp3q H , and a least squares problem. This approach, that does not necessary need separability of the spatial degradation operation, is summarized in in Algorithm 3. As noted in [87], the true SRI Y rrA;B;Css can be uniquely recovered only if ranktCu R3 does not exceed the number KM of spectral bands in the MSI. To overcome this limitation, in [87] it was proposed to apply Algorithm 3 to corresponding non-overlapping subblocks of the MSI and HSI (based on the hypothesis that only a small number of materials are active in a smaller block). This procedure is summarized in Algorithm 4, called Super-resolution CUBe Algorithm (SCUBA).
Coupled multilinear model and approximation problem
Let the SRI admit a Tucker decomposition with multilinear ranks R pR1;R2;R3q as Y rrG; U;V;Wss; (2.7) where G P RR1R2R3 is the core tensor and U P RIR1 , V P RJR2 and W P RKR3 are the factor matrices. Under this assumption, the degradation model (1.28) can be expressed a YH rrG; P1U;P2V;Wss EH; YM rrG; U;V;P3Wss EM;
thus the HSR task can be performed by estimating the factor matrices U, V, W and the core tensor G in (2.7). However, it should be emphasized that our goal here is not to recover the low-rank factors themselves, but rather the whole SRI tensor. Thus, the following reformulation of the HSR task as an optimization problem is only one possible option. In a similar spirit as in [86], we consider the following optimization problem: minimize pG ;pU;pV;xW fTppG; pU; pV;xWq; where (2.9) fTppG; pU; pV;xWq }YH rrpG ; P1pU;P2pV;xWss} 2 F }YM rrpG; pU; pV;P3xW ss} 2
Table of contents :
List of Figures
List of Tables
List of Algorithms
List of notations
List of acronyms
Chapter 1 General introduction
1.1 Multimodal data fusion
1.1.1 Principles
1.1.2 Challenges of data fusion
1.1.3 Advantages and limitations of matrix low-rank models
1.2 Hyperspectral super-resolution
1.2.1 Spectral imaging
1.2.2 Natural tradeoff in resolutions
1.2.3 Hyperspectral unmixing
1.2.4 Hyperspectral super-resolution
1.3 Tensor algebra preliminaries
1.3.1 Matrix operations
1.3.2 General definitions
1.3.3 Low-rank tensor decompositions
1.4 Tensor-based hyperspectral super-resolution
1.4.1 Basic observational model and optimization problem
1.4.2 Comparison of different matrix and tensor decompositions
1.4.3 Variations of the basic observational model
1.4.4 Overview and timeline of tensor-based HSR
1.5 Manuscript outline and contributions
1.6 List of publications
1.7 Code
Part I Algorithms for hyperspectral super-resolution
Chapter 2 Hyperspectral super-resolution with coupled Tucker approximation: recoverability and SVD-based algorithms
2.1 Introduction
2.2 CP-based HSR
2.2.1 The case of known spatial degradation
2.2.2 The case of unknown spatial degradation
2.3 HSR problem as a coupled multilinear approximation
2.3.1 Coupled multilinear model and approximation problem
2.3.2 Proposed algorithms
2.4 Recoverability of the Tucker model
2.4.1 Deterministic exact recovery conditions
2.4.2 Exact recoverability for generic tensors
2.4.3 Recoverability in the blind case
2.5 Numerical experiments
2.5.1 Experimental setup
2.5.2 Semi-real data: comparison with other methods
2.5.3 Synthetic examples
2.5.4 Choice of multilinear ranks in the presence of noise
2.5.5 Recovery of underlying spectra
2.6 Conclusion of Chapter 2
Chapter 3 Hyperspectral super-resolution accounting for spectral variability: LL1-based recovery and blind unmixing
3.1 Introduction
3.2 Proposed model
3.2.1 Degradation model and indeterminacies
3.2.2 LL1-BTD mixing model for the underlying SRIs
3.2.3 Modeling spectral variability
3.3 Recoverability analysis
3.4 Algorithms
3.4.1 Unconstrained optimization
3.4.2 Constrained optimization
3.4.3 Initialization
3.5 Experiments for image recovery
3.5.1 Degradation model
3.5.2 Recovery of the SRI and variability tensor
3.5.3 Recovery without variability
3.6 Blind unmixing experiments
3.6.1 Experiments setup
3.6.2 Unmixing of the SRI with exact LL1 model
3.6.3 Unmixing for semi-real datasets
3.6.4 Choice of the ranks
3.7 Conclusion of Chapter 3
Conclusion of Part I
Appendix A Solving normal equations as generalized Sylvester equations
Appendix B Constructing the degradation matrices
Appendix C Comparison metrics for HSR and unmixing performance
Appendix D Updates in Algorithms 8 and 9
Part II Performance analysis for coupled tensor models
Chapter 4 Constrained Cramér-Rao bounds for reconstruction problems formulated as coupled canonical polyadic decompositions
4.1 Introduction
4.2 Observation model and estimation
4.2.1 Observation tensor model for the reconstruction problem
4.2.2 Reformulation as a coupled CP model
4.2.3 Estimation
4.2.4 Link between uniqueness and identifiability
4.3 Cramér-Rao lower bounds for coupled models
4.3.1 Background on standard CRBs
4.3.2 Coupled model with constraints
4.3.3 Uncoupled CRB
4.3.4 Expression for CCRB
4.3.5 Reparameterized CRB
4.3.6 Lehmann-unbiased CCRB
4.4 Different parameterizations and estimation scenarios
4.4.1 Model parameters
4.4.2 General framework for the fusion problem
4.4.3 Scenario 1 – Assessing performance for the fully-coupled model
4.4.4 Performance on the reconstructed tensor
4.4.5 Scenario 2 – Comparing performance bounds in the uncoupled and blind cases
4.5 Deriving performance bounds based on the coupled CP model
4.5.1 Uncoupled case
4.5.2 Partially coupled case
4.5.3 Fully-coupled case
4.6 Computer results
4.6.1 Simulations setup
4.6.2 Equivalence between CCRB and reparameterized CRB
4.6.3 Asymptotic values for constrained FIM
4.6.4 Choice of the rank
4.6.5 Assessing the efficiency of the estimators
4.6.6 Impact of on the performance and a modified STEREO scheme
4.6.7 Performance of STEREO without identifiability of Y1
4.6.8 A modified ALS algorithm accounting for non-linear constraints
4.7 Design of the hyperspectral measurements system
4.7.1 Influence of the filter size
4.7.2 Influence of the downsampling ratio
4.7.3 Optimal values of q and d
4.8 Conclusion of Chapter 4
Chapter 5 Performance bounds for coupled tensor LL1 models
5.1 Introduction
5.2 A randomly constrained Cramér-Rao bound
5.2.1 Random Equality Constraints
5.2.2 CRBs with Random Equality Constraints
5.2.3 Further considerations
5.3 Cramér-Rao bounds for coupled LL1 models
5.3.1 Basic observation model
5.3.2 Model parameters
5.3.3 Fisher information matrix
5.3.4 Standard constrained Cramér-Rao bound
5.3.5 Performance analysis in the case of non-random constraints
5.4 Degradation model accounting for uncertainties
5.4.1 Proposed model
5.4.2 Standard CCRB
5.4.3 Limitations of standard CCRB
5.4.4 Usefulness of the RCCRB for coupled LL1 models with uncertainties .
5.4.5 Performance loss for fully-coupled algorithm in case of constraints mismatch
5.5 A new efficient blind algorithm
5.5.1 Partially-coupled LL1 model
5.5.2 A new blind ALS algorithm for solving (5.42)–(5.43)
5.5.3 Numerical performance analysis
5.6 Conclusion of Chapter 5
Conclusion of Part II
Appendix E Closed-form expressions for deriving CP-based CRBs
E.1 Uncoupled CRB
E.2 Blind-CCRB for partially-coupled models
E.3 Fully-coupled CCRB
E.3.1 Scenario 1 with linear constraints
E.3.2 Scenario 2 with non-linear constraints
Appendix F LL1-based standard FIM and CCRB
General conclusion
1 Conclusion of the manuscript
2 Perspectives
Bibliography
