# Hyperspectral Image Denoising via Framelet Transformation Based Three-Modal Tensor Nuclear Norm

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Notations and Definitions

#### 2.2. Framelet

#### 2.3. Problem Formulation

#### 2.4. DFT-Based Tensor Fibered Rank

**Definition**

**1.**

## 3. Proposed Model

#### 3.1. Framelet-Based Tensor Fibered Rank and Corresponding Three-Modal TNN

**Definition**

**2.**

**Definition**

**3.**

#### 3.2. Proposed Denoising Model

#### 3.3. Optimization Procedure

Algorithm 1. Framelet-besed kth-modal singular value shrinkage operator. |

Input:$\mathcal{A}\in {R}^{{n}_{1}\times {n}_{2}\times {n}_{3}}$, $\tau $, $W\in {R}^{w{n}_{k}\times {n}_{k}}$, w and k |

Output:${D}_{\tau}\left(\mathcal{A}\right)$ |

1: ${\widehat{\mathcal{A}}}_{k}=W{\mathcal{A}}_{k}$ |

2: for $i=1,...,w{n}_{k}$do |

3: $[\widehat{U},\widehat{\Sigma},\widehat{{V}^{T}}]=SVD\left({\widehat{{\mathcal{A}}_{k}}}^{\left(i\right)}\right)$ |

4: ${\widehat{{\mathcal{A}}_{k}}}^{\left(i\right)}=\widehat{U}\xb7{(\widehat{\Sigma}-\tau )}_{+}\xb7\widehat{{V}^{T}}$ |

5: end for |

6: Compute ${D}_{\tau}\left(\mathcal{A}\right)={W}^{T}\widehat{{\mathcal{A}}_{k}}.$ |

Algorithm 2. HSI Denoising via the F-3MTNN minimization. |

Input: The observed HSI $\mathcal{X}\in {R}^{{n}_{1}\times {n}_{2}\times {n}_{3}}$, ${w}_{k}(k=1,2,3)$, ${\lambda}_{1}$, ${\lambda}_{2}$, $\rho $, $\tau $ and $\epsilon $. |

Output: The denoised HSI $\mathcal{L}$ |

1: Initialize: ${\mathcal{L}}^{0}$ = ${\mathcal{N}}^{0}$ = ${\mathcal{S}}^{0}$ = ${\mathcal{Z}}_{k}^{0}$, ${\mu}_{k}=\beta =0$, ${\mathcal{Y}}_{k}=\mathcal{M}=\mathcal{O}$, $\rho =1.2$, $\epsilon ={10}^{-6}$ |

2: Repeat until convergence: |

Update ${\mathcal{L}}^{p+1}$ by (9) |

Update ${\mathcal{Z}}_{k}^{p+1}$ by (11) |

Update ${\mathcal{N}}^{p+1}$ by (13) |

Update ${\mathcal{S}}^{p+1}$ by (15) |

Update ${\mathcal{Y}}^{p+1}$ and ${\mathcal{M}}^{p+1}$ by (16) |

Update ${\mu}^{p+1}$ and ${\beta}^{p+1}$: ${\mu}^{p+1}=\rho {\mu}^{p}$, ${\beta}^{p+1}=\rho {\beta}^{p}$; $p=p+1$ |

3: Check the convergence conditions |

$max\{\parallel {\mathcal{L}}^{p+1}-{\mathcal{L}}^{p}{\parallel}_{\infty},\parallel \mathcal{X}-{\mathcal{L}}^{p+1}-{\mathcal{N}}^{p+1}-{\mathcal{S}}^{p+1}{\parallel}_{\infty},\parallel {\mathcal{L}}^{p+1}-{\mathcal{Z}}_{k}^{p+1}{\parallel}_{\infty}\}\u2a7d\epsilon $ |

## 4. Experimental Results

#### 4.1. Experiments on Simulated Datasets

#### 4.1.1. Pavia City Center Dataset

#### (A) Visual Quality Comparison

#### (B) Quantitative Comparison

#### 4.1.2. USGS Indian Pines Dataset

#### 4.2. Experiments on Real Datasets

#### 4.2.1. AVIRIS Indian Pines Dataset

#### 4.2.2. HYDICE Urban Dataset

#### 4.3. Ablation Experiment

## 5. Discussion

#### 5.1. Parameter Analysis

#### 5.2. Convergence Analysis

#### 5.3. Running Time

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The distribution of singular values on each frontal slice of the two different transformed tensors. (

**a**) The first mode, (

**b**) the second mode, (

**c**) the third mode.

**Figure 3.**(

**a**) Original image, (

**b**) noisy image, image denoised by (

**c**) LRTA, (

**d**) BM4D, (

**e**) LRMR, (

**f**) LRTDTV, (

**g**) L1HyMixDe, (

**h**) LRTDGS, (

**i**) 3DTNN, (

**j**) ours of band 65 in dataset-1, noise case 3.

**Figure 4.**(

**a**) Original image, (

**b**) noisy image, image denoised by (

**c**) LRTA, (

**d**) BM4D, (

**e**) LRMR, (

**f**) LRTDTV, (

**g**) L1HyMixDe, (

**h**) LRTDGS, (

**i**) 3DTNN, (

**j**) ours of band 60 in dataset-1, noise case 8.

**Figure 5.**The PSNR values of each band in dataset-1 after denoising by eight different methods under (

**a**) case 1, (

**b**) case 2, (

**c**) case 3, (

**d**) case 4, (

**e**) case 5, (

**f**) case 6, (

**g**) case 7, (

**h**) case 8.

**Figure 6.**The SSIM values of each band in dataset-1 after denoising by eight different methods under (

**a**) case 1, (

**b**) case 2, (

**c**) case 3, (

**d**) case 4, (

**e**) case 5, (

**f**) case 6, (

**g**) case 7, (

**h**) case 8.

**Figure 7.**The reflectance of pixel (30, 30) in (

**a**) noisy dataset-1, dataset-1 denoised by (

**b**) LRTA, (

**c**) BM4D, (

**d**) LRMR, (

**e**) LRTDTV, (

**f**) L1HyMixDe, (

**g**) LRTDGS, (

**h**) 3DTNN, (

**i**) ours under noise case 4.

**Figure 8.**(

**a**) Original image, (

**b**) noisy image, image denoised by (

**c**) 3DTNN, (

**d**) ours of band 26 in the dataset-2, noise case 2.

**Figure 9.**(

**a**) Original image, (

**b**) noisy image, image denoised by (

**c**) 3DTNN, (

**d**) ours of band 4 in the dataset-2, noise case 5.

**Figure 10.**The PSNR values of each band in the dataset-2 after denoising by 3DTNN and our model under (

**a**) case 1, (

**b**) case 2, (

**c**) case 3, (

**d**) case 4, (

**e**) case 5, (

**f**) case 6, (

**g**) case 7, (

**h**) case 8.

**Figure 11.**The SSIM values of each band in the dataset-2 after denoising by 3DTNN and our model under (

**a**) case 1, (

**b**) case 2, (

**c**) case 3, (

**d**) case 4, (

**e**) case 5, (

**f**) case 6, (

**g**) case 7, (

**h**) case 8.

**Figure 12.**The reflectance of pixel (100,30) in the dataset-2 denoised by (

**a**) 3DTNN and (

**b**) ours under noise case 3.

**Figure 14.**(

**a**) Original image, image denoised by (

**b**) LRTA, (

**c**) BM4D, (

**d**) LRMR, (

**e**) LRTDTV, (

**f**) L1HyMixDe, (

**g**) LRTDGS, (

**h**) 3DTNN, (

**i**) ours of band 106 in dataset-3.

**Figure 15.**(

**a**) Original image, image denoised by (

**b**) LRTA, (

**c**) BM4D, (

**d**) LRMR, (

**e**) LRTDTV, (

**f**) L1HyMixDe, (

**g**) LRTDGS, (

**h**) 3DTNN, (

**i**) ours of band 163 in dataset-3.

**Figure 16.**(

**a**) Original image, image denoised by (

**b**) LRTA, (

**c**) BM4D, (

**d**) LRMR, (

**e**) LRTDTV, (

**f**) L1HyMixDe, (

**g**) LRTDGS, (

**h**) 3DTNN, (

**i**) ours of band 104 in dataset-4.

**Figure 17.**(

**a**) Original image, image denoised by (

**b**) LRTA, (

**c**) BM4D, (

**d**) LRMR, (

**e**) LRTDTV, (

**f**) L1HyMixDe, (

**g**) LRTDGS, (

**h**) 3DTNN, (

**i**) ours of band 109 in dataset-4.

**Figure 18.**PSNR values concerning different values of (

**a**) $\theta $ (controls $\omega $), (

**b**) ${\lambda}_{1}$, (

**c**) ${\lambda}_{2}$ and (

**d**) $\psi $ (controls $\tau $).

Data | $\mathit{\tau}$ | Transformation | The First Mode | The Second Mode | The Third Mode |
---|---|---|---|---|---|

Pavia | 0.04 | FFT | 78 | 78 | 188 |

Framelet | 13 | 13 | 70 | ||

0.05 | FFT | 75 | 75 | 185 | |

Framelet | 10 | 10 | 65 | ||

Indian | 0.04 | FFT | 16 | 16 | 106 |

Framelet | 3 | 3 | 42 | ||

0.05 | FFT | 16 | 16 | 105 | |

Framelet | 3 | 3 | 40 |

Noise | Gaussian Noise | Impulse Noise | Deadline Noise | Stripe Noise |
---|---|---|---|---|

Case 1 | mean value = 0, variance = 0.1 | percentage = 0.2 | \ | \ |

Case 2 | mean value = 0, variance = 0.15 | percentage = 0.2 | \ | \ |

Case 3 | mean value = 0, variance = 0.1 | percentage = 0.1 | \ | \ |

Case 4 | mean value = 0, variance = 0.1 | percentage = 0.3 | \ | \ |

Case 5 | mean value = 0, variance ∼ U(0.05, 0.15) | percentage = 0.2 | \ | \ |

Case 6 | mean value = 0, variance ∼ U(0.1, 0.2) | percentage = 0.2 | \ | \ |

Case 7 | mean value = 0 | percentage = 0.3 | 10% of the bands | \ |

variance = 0.1 | number ∼ U(1, 4) | \ | ||

Case 8 | mean value = 0 | percentage = 0.3 | 10% of the bands | 10% of the bands |

variance = 0.1 | number ∼ U(1, 4) | number ∼ U(20, 40) |

Noise Case | Level | Evaluation Index | LRTA | BM4D | LRMR | LRTDT | L1HyMixDe | LRTDGS | 3DTNN | Our |
---|---|---|---|---|---|---|---|---|---|---|

Case 1 | MPSNR | 29.4396 | 29.7014 | 31.2593 | 32.2970 | 32.9077 | 33.2535 | 32.1996 | 32.9243 | |

G = 0.1 | MSSIM | 0.9048 | 0.9203 | 0.9045 | 0.9138 | 0.9177 | 0.9253 | 0.9307 | 0.9256 | |

P = 0.2 | SAM | 6.8049 | 5.8404 | 6.8244 | 4.9305 | 4.4064 | 4.3546 | 3.4856 | 3.7368 | |

Case 2 | MPSNR | 27.0258 | 27.4204 | 29.0133 | 30.1107 | 30.6235 | 30.9044 | 29.8947 | 30.9274 | |

G = 0.15; | MSSIM | 0.8480 | 0.8787 | 0.8494 | 0.8669 | 0.8745 | 0.8833 | 0.8854 | 0.8857 | |

P = 0.2 | SAM | 7.8247 | 6.6710 | 7.6899 | 5.8480 | 4.9587 | 5.4792 | 4.2935 | 4.4098 | |

Case 3 | MPSNR | 30.2658 | 30.3936 | 32.3389 | 33.1557 | 34.3770 | 34.2453 | 32.9942 | 33.9806 | |

G = 0.1 | MSSIM | 0.9190 | 0.9281 | 0.9237 | 0.9267 | 0.9421 | 0.9374 | 0.9423 | 0.9477 | |

P = 0.1 | SAM | 6.4482 | 5.5052 | 6.4019 | 4.6077 | 3.5472 | 4.0836 | 3.1475 | 3.6220 | |

Case 4 | MPSNR | 28.4836 | 28.8640 | 30.1731 | 31.1878 | 32.1109 | 31.9825 | 31.1434 | 32.1440 | |

G = 0.1 | MSSIM | 0.8868 | 0.9098 | 0.8814 | 0.8972 | 0.9068 | 0.9057 | 0.9088 | 0.9099 | |

P = 0.3 | SAM | 7.2045 | 6.2362 | 7.2500 | 5.3489 | 4.7865 | 5.4171 | 4.3108 | 4.1209 | |

Case 5 | MPSNR | 28.9336 | 29.1991 | 30.4432 | 31.6456 | 33.2988 | 33.5211 | 32.1036 | 33.8946 | |

G = (0.05,0.15) | MSSIM | 0.9003 | 0.9161 | 0.8880 | 0.9062 | 0.9283 | 0.9293 | 0.9293 | 0.9302 | |

P = 0.2 | SAM | 7.1954 | 6.0548 | 7.1967 | 5.2504 | 4.2118 | 4.2629 | 3.6644 | 3.8739 | |

Case 6 | MPSNR | 26.0073 | 26.4507 | 28.0358 | 29.0656 | 30.0129 | 30.8340 | 29.9707 | 30.7688 | |

G = (0.1,0.2) | MSSIM | 0.8239 | 0.8621 | 0.8188 | 0.8452 | 0.8640 | 0.8808 | 0.8859 | 0.8777 | |

P = 0.2 | SAM | 8.3887 | 6.9711 | 8.1535 | 6.4062 | 5.5859 | 5.5074 | 4.5080 | 4.2183 | |

Case 7 | G = 0.1 | MPSNR | 28.4375 | 28.8168 | 30.1479 | 31.1278 | 32.1303 | 31.9213 | 30.9232 | 32.1686 |

P = 0.3 | MSSIM | 0.8863 | 0.9095 | 0.8812 | 0.8951 | 0.9108 | 0.9079 | 0.9123 | 0.9167 | |

+deadline | SAM | 7.2212 | 6.2393 | 7.2828 | 5.3694 | 6.6968 | 6.7940 | 5.2777 | 4.1205 | |

Case 8 | G = 0.1 P = 0.3 | MPSNR | 28.3902 | 28.7718 | 30.0586 | 31.0073 | 31.9432 | 31.8225 | 30.2764 | 32.0232 |

+deadline | MSSIM | 0.8852 | 0.9088 | 0.8798 | 0.8939 | 0.9075 | 0.9068 | 0.9027 | 0.9063 | |

+stripe | SAM | 7.2192 | 6.2496 | 7.3609 | 5.6445 | 6.9130 | 7.1390 | 5.7186 | 4.3586 |

Noise Case | Level | MPSNR | 3DTNN MSSIM | SAM | MPSNR | Our MSSIM | SAM |
---|---|---|---|---|---|---|---|

Case 1 | G = 0.1 | 30.9073 | 0.9050 | 2.7583 | 32.1435 | 0.8964 | 2.7254 |

P = 0.2 | |||||||

Case 2 | G = 0.15 | 28.5602 | 0.8697 | 3.5259 | 31.3247 | 0.9013 | 2.5945 |

P = 0.2 | |||||||

Case 3 | G = 0.1 | 31.6945 | 0.9185 | 2.5888 | 33.6358 | 0.9170 | 2.2334 |

P = 0.1 | |||||||

Case 4 | G = 0.1 | 29.7186 | 0.8931 | 2.8884 | 32.5633 | 0.9080 | 2.3556 |

P = 0.3 | |||||||

Case 5 | G = (0.05,0.15) | 31.2729 | 0.9089 | 2.6462 | 33.7245 | 0.9329 | 2.1341 |

P = 0.2 | |||||||

Case 6 | G = (0.1,0.2) | 28.7189 | 0.8699 | 3.5476 | 31.2484 | 0.9076 | 2.6554 |

P = 0.2 | |||||||

Case 7 | G = 0.1 P = 0.3 | 29.9070 | 0.8747 | 3.1159 | 32.7050 | 0.9274 | 2.2178 |

+deadline | |||||||

Case 8 | G = 0.1 P = 0.3 | 29.7521 | 0.8653 | 3.2478 | 32.5152 | 0.9256 | 2.3006 |

+deadline +stripe |

Spatial Information | Spectral Information | PSNR | SSIM |
---|---|---|---|

✓ | 21.7398 | 0.3938 | |

✓ | 25.8850 | 0.7504 | |

✓ | ✓ | 30.9274 | 0.8857 |

**Table 6.**The running time (in seconds) of the different methods in the real HSI dataset experiments.

HSI Data | LRTA | BM4D | LRMR | LRTDTV | L1HyMixDe | LRTDGS | 3DTNN | Our |
---|---|---|---|---|---|---|---|---|

AVIRIS Indian Pines | 35 | 261 | 412 | 131 | 4 | 123 | 87 | 1259 |

HYDICE Urban | 143 | 4076 | 6132 | 960 | 7 | 811 | 1503 | 6602 |

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## Share and Cite

**MDPI and ACS Style**

Kong, W.; Song, Y.; Liu, J.
Hyperspectral Image Denoising via Framelet Transformation Based Three-Modal Tensor Nuclear Norm. *Remote Sens.* **2021**, *13*, 3829.
https://doi.org/10.3390/rs13193829

**AMA Style**

Kong W, Song Y, Liu J.
Hyperspectral Image Denoising via Framelet Transformation Based Three-Modal Tensor Nuclear Norm. *Remote Sensing*. 2021; 13(19):3829.
https://doi.org/10.3390/rs13193829

**Chicago/Turabian Style**

Kong, Wenfeng, Yangyang Song, and Jing Liu.
2021. "Hyperspectral Image Denoising via Framelet Transformation Based Three-Modal Tensor Nuclear Norm" *Remote Sensing* 13, no. 19: 3829.
https://doi.org/10.3390/rs13193829