# Hyperspectral and Multispectral Image Fusion Using Coupled Non-Negative Tucker Tensor Decomposition

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## Abstract

**:**

## 1. Introduction

- The application of non-negativity priors to the Tucker tensor decomposition of LRHSI and HRMSI, to estimate spectral and spatial mode-dictionaries in a Tucker model, respectively. To the best of our knowledge, this is the first time that a non-negative Tucker decomposition is used to represent hyperspectral images in a HSI fusion framework.
- The preservation of spatio-spectral joint structures of HSIs without prior knowledge requirements and much lower information losses than matrix frameworks.
- The construction of an algorithm with lower-order complexity than the state-of-the-art.

## 2. Preliminaries on Tensors

## 3. HSI-MSI Fusion Problem Formulation

#### 3.1. Matrix Factorization-Based Fusion Scheme

#### 3.2. Tensor Decomposition-Based Fusion Scheme

## 4. Proposed CNTD Approach

#### 4.1. Updating Mode-Dictionary Matrices

#### 4.2. Updating Core Tensor

**H**and width

**W**are initialized using the latter. The spectral mode-dictionary $S$ is initialized using the simplex identification split augmented Lagrangian (SISAL) algorithm [50], which efficiently identifies a minimum volume simplex containing the LRHSI spectral vectors. The spatial mode-dictionary matrices

**W**and

**H**are initialized from the mode-one and -two matricization of the HRMSI, respectively, via dictionary update cycles of the KSVD method [51]. The core tensor $\mathcal{C}$ is initialized using the ADMM framework presented in [9].

Algorithm 1: The proposed coupled non-negative tensor decomposition method. |

Input: LRHSI (${\mathcal{Y}}_{h}$), HRMSI (${\mathcal{Y}}_{m}$). Output: HRHSI ($\mathcal{Z}$) |

Estimate PSF (${P}_{1}$, ${P}_{2}$), SRF (${P}_{3}$), using method from [19]. |

Initialize the core tensor ($\mathcal{C}$) via ADMM [9], and mode-dictionaries ($W,\text{}H,\text{}S)$ via DUC KSVD [51]. |

NTD for ${\mathcal{Y}}_{h}$ Initialize ${W}_{h}$, ${H}_{h}\text{}by$ (20), (21), respectively. Update ${W}_{h}$, ${H}_{h}$, $S$ and $\mathcal{C}$ alternately by (27), (29), (30) and (35), respectively until convergence of |

NTD for ${\mathcal{Y}}_{m}$ Initialize ${S}_{m}$ by (22) Update $W$, $H$, ${S}_{\mathit{m}}$ and $\mathcal{C}$ alternately by (36)–(39) until convergence of the objective function in (24). Using the estimated $W$, $H$, $S$ and $\mathcal{C}$ to calculate the HRHSI ($\mathcal{Z}$) via Tucker tensor decomposition (16). |

## 5. Computational Complexity

## 6. Experimental Observations and Results

#### 6.1. Data Sets

#### 6.2. Evaluation Criteria

#### 6.3. Evaluation of the Parameters

#### 6.4. Comparison with State of the Art Fusion Methods

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Composite color images of (

**a**) HRMSI, (

**b**) LRHSI (

**c**) reference images of the Pavia University data set (top row) and Indian Pines data set (bottom row).

**Figure 4.**Spectral response functions; (

**a**) IKONOS like spectral response function, (

**b**) LANDSAT 7-like spectral response function.

**Figure 5.**The RMSE as function of the number of atoms ${n}_{w}$, ${n}_{h}$ and ${n}_{s}$ for the proposed CNTD approach: (

**a**) ${n}_{w}$; (

**b**) ${n}_{h}$; (

**c**) ${n}_{s}$.

**Figure 6.**Band 30 of the Pavia University (two top rows) and Indian Pines data sets (two bottom rows), respectively; the first and third rows (

**a**) LRHSI, (

**b**) CNMF, (

**c**) CSTF, (

**d**) NLSTF. The second and fourth rows (

**a**) reference image, (

**b**) CNN, (

**c**) STEREO, (

**d**) proposed CNTD.

**Figure 7.**The error images of the 30th band of Pavia University (two top rows) and Indian Pines data sets (two bottom rows), the first and third rows show the error image of (

**a**) LRHSI, (

**b**) CNMF, (

**c**) CSTF, and (

**d**) NLSTF. The second and fourth rows show the error images of (

**a**) reference error image, (

**b**) CNN, (

**c**) STEREO, and (

**d**) proposed CNTD method.

Notation | Description |
---|---|

$\mathcal{X}$ | Tensor |

X | Matrix |

$\U0001d4cd$ | Tensor element |

$\mathbf{\U0001d4cd}$ | Spectral vector of tensor |

X | Scaler |

${\times}_{n}$ | Mode-n product |

$\u2a02$ | Kronecker product |

$\u229b$ | Hadamard product |

${{\rm X}}_{\left(n\right)}$ | Mode-n matricization of tensor X |

${X}^{\left(n\right)}$ | n mode matrix in Tucker decomposition |

Method | Pavia University Data Set | ||||
---|---|---|---|---|---|

RMSE | SAM | DD | ERGAS | UIQI | |

Ideal value | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 |

CNMF [21] | 0.140 | 4.313 | 0.017 | 4.989 | 0.952 |

CSTF [9] | 2.160 | 2.390 | 1.055 | 1.230 | 0.991 |

NLSTF [23] | 1.452 | 0.964 | 0.846 | 0.520 | 0.993 |

CNN [27] | 0.016 | 2.203 | 0.103 | 1.447 | 0.976 |

STEREO [43] | 0.061 | 3.922 | 0.010 | 1.865 | 0.989 |

CNTD method | 0.008 | 1.963 | 0.005 | 1.169 | 0.996 |

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**MDPI and ACS Style**

Zare, M.; Helfroush, M.S.; Kazemi, K.; Scheunders, P. Hyperspectral and Multispectral Image Fusion Using Coupled Non-Negative Tucker Tensor Decomposition. *Remote Sens.* **2021**, *13*, 2930.
https://doi.org/10.3390/rs13152930

**AMA Style**

Zare M, Helfroush MS, Kazemi K, Scheunders P. Hyperspectral and Multispectral Image Fusion Using Coupled Non-Negative Tucker Tensor Decomposition. *Remote Sensing*. 2021; 13(15):2930.
https://doi.org/10.3390/rs13152930

**Chicago/Turabian Style**

Zare, Marzieh, Mohammad Sadegh Helfroush, Kamran Kazemi, and Paul Scheunders. 2021. "Hyperspectral and Multispectral Image Fusion Using Coupled Non-Negative Tucker Tensor Decomposition" *Remote Sensing* 13, no. 15: 2930.
https://doi.org/10.3390/rs13152930