# How Hyperspectral Image Unmixing and Denoising Can Boost Each Other

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Contribution

- investigate the effect of HSI denoising as a preprocessing step on the performance of spectral unmixing.
- study the relation between noise and unmixing for different strategies of obtaining endmembers.
- investigate the performance of spectral unmixing as a denoiser.
- compare the denoising performance of spectral unmixing with a number of state-of-the-art denoising techniques.
- study the effect of colored versus white noise on all the above research.

## 2. Hyperspectral Unmixing

**Y**,

**X**, and

**N**(all $p\times n$) denote the observed HSI, the noise-free HSI, and the noise in the HSI, respectively. p and n indicate the number of spectral bands and pixels, respectively. In spectral unmixing, (1) is rewritten as:

**E**($p\times r$) and

**A**($r\times n$) are low-rank matrices (i.e., $r\ll p$), respectively containing the endmembers and abundances of the HSI. r denotes the number of endmembers. Unmixing is the problem of separating the abundance fractions

**A**from the endmember sources

**E**in the mixing model (2). We will discuss the three relevant strategies of the unmixing problem and consequent algorithms: (1) abundance estimation methods that are supervised, i.e., that assume that the endmember spectra of the pure materials are provided; (2) methods that extract or estimate the endmembers from the data; and (3) blind methods that estimate the endmembers and abundances simultaneously. We should mention that the selection of the number of endmembers (or the dimension/rank of the subspace) is also an important topic, which is often discussed in relation to noise, and for this, we refer to [20,21]. Additionally, model (2) is often used for hyperspectral supervised/unsupervised feature extractions as well [41].

#### 2.1. Fractional Abundance Estimation

**I**, where

**I**is the identity matrix, the maximum-likelihood estimation $\widehat{\mathbf{a}}$ of the abundance vector of each pixel $\mathbf{y}$ (we will omit the subscript i) is given by:

#### 2.2. Endmember Extraction

**N**= 0, the hyperspectral dataset that follows the fully constrained linear mixing model lies within a (r − 1)-dimensional simplex of which ${\mathbf{e}}_{m}$ ($m=1,\cdots ,r$) are the vertices or extreme points. This inspired many methods and algorithms that effectively search for embedding or enclosing simplices in the data ([2]). For example, NFINDR randomly selects a set of r pixels from the dataset as initial endmembers and iteratively updates the endmembers at each time, replacing an endmember by a pixel from the dataset to find the largest volume simplex. The volume is calculated as:

#### 2.3. Simultaneous Estimation of Abundances and Endmembers

**E**and

**A**by solving the following optimization problem:

## 3. Hyperspectral Denoising

**X**based on the observation

**Y**. Hyperspectral denoising was comprehensively discussed in [12]. From a modeling point of view, linear HSI denoising can be divided into two main groups; techniques that are developed based on either low-rank models or full-rank models. Differences between those models were discussed in [24]. In this paper, we employ the full-rank model-based method first order spectral roughness penalty denoising (FORPDN) [16], the low-rank model-based HyRes [27], and the conventional multiple linear regression technique (MLR) [21]. All three methods are discussed below.

#### 3.1. Multiple Linear Regression

#### 3.2. First Order Spectral Roughness Penalty Denoising

#### 3.3. HyRes: Automatic Hyperspectral Restoration

## 4. Experimental Setup

#### 4.1. The Data

#### 4.1.1. Simulated Dataset

_{2}O

_{3}, SiO

_{2}, and CaO. The endmembers of these minerals were measured by an AgriSpec spectrometer (manufactured by ASD (Analytical Spectral Devices)). These endmembers contained 200 reflection values in the wavelength range [1000–2500] nm. Figure 1a depicts the endmembers. Twenty squares of 5 × 5 pixels were generated with different binary and ternary linear mixtures. The remaining background contained binary mixtures of 50% of Fe

_{2}O

_{3}and 50% of SiO

_{2}. The obtained ground truth fractional abundances are shown in Figure 1b.

#### 4.1.2. Samson Image

#### 4.1.3. Jasper Ridge Image

#### 4.2. The Noise

#### 4.3. The Endmembers

#### 4.3.1. Ground Truth Endmembers

#### 4.3.2. Extracted Endmembers

#### 4.3.3. Library Endmembers

#### 4.3.4. Estimated Endmembers

#### 4.4. The Experiments

#### 4.4.1. Denoising as Preprocessing for Unmixing

#### 4.4.2. Unmixing as a Denoiser

## 5. Simulated Dataset Experiments

#### 5.1. Denoising as Preprocessing for Unmixing

- Overall, one could observe the advantage of denoising as a preprocessing step for spectral unmixing. In general, the effects were minor for very high SNR and gradually increased when lowering the SNR.
- The largest improvements were observed on the reconstruction errors. This could be observed for all endmember strategies. Even for an SNR as high as 40 dB, this improvement was clearly observable.
- However, the performance on the abundance estimation did not improve proportionally. One could conclude that in noisy images, abundances were estimated reasonably well, irrespective of the high reconstruction errors, and that the spectral unmixing performance was not that much affected by the noise.
- When no denoising was performed (i.e., the red plots), the results with ground truth endmembers were superior (we should point out the different vertical scales in the graphs, which were selected to achieve a better visualization). From the other strategies, the one where endmembers were extracted performed the worst. The use of library endmembers performed well down to 20 dB, below which the abundance errors and the standard deviations became very large. We observed that at these noise levels, wrong endmembers were selected from the libraries. Furthermore, when endmembers were estimated, spectral unmixing performed worse for low SNR.
- In general, spectral unmixing performed worse with colored noise, compared to white noise, except when ground truth endmembers were available.
- After denoising, the performance of the abundance estimation generally improved. This was most obvious in the case of extracted endmembers, where better results were obtained with noise levels on the observed image up to 35 dB. In the case of colored noise, the improvements were more prominent than with white noise.
- In the case of white noise, MLR had the worst performance, and HyRes performed slightly better than FORPDN. Only for very low SNR, FORPDN performed better in all strategies, but the one with ground truth endmembers.
- In the case of colored noise, FORPDN was least performant, and MLR performed slightly better than FORPDN. HyRes showed by far the best performance for all endmember scenarios and all SNRs. The abundances were estimated almost correctly. This result strongly encouraged the use of a well-established low-rank denoising technique as a preprocessing step for unmixing.

#### 5.2. Unmixing as a Denoiser

- When comparing the solid lines, it could be observed that spectral unmixing was comparable to, or outperformed, the denoising techniques in all scenarios, except in the case of extracted endmembers. This could be attributed to the low-rank modeling ability of the unmixing.
- The more prior knowledge on the endmembers was provided, the better the denoising performance of the spectral unmixing. The best performance was obtained in Scenario (a), where the ground truth endmembers were given, and in Scenario (c), where a library of noise-free endmembers was provided. In Scenario (d), where CoNMF estimated smooth endmembers, the performance was good in the case of white noise, but became worse in the case of colored noise, where the middle bands were highly corrupted by noise. In Scenario (b), the extracted endmembers were noisy, which heavily deteriorated the denoising performance of the unmixing.
- In the case of colored noise, HyRes outperformed the other denoisers, as well as spectral unmixing. This again proved the superiority of this well-established low-rank denoising technique.
- When first denoising the images before unmixing, results generally improved over plain unmixing. Scenario (b) provided the lowest performance. The best results were obtained by the combination of HyRes and spectral unmixing. In the case of colored noise, the performance of HyRes was even further improved by spectral unmixing in Scenarios (a) and (c), but was not further improved by unmixing in Scenarios (b) and (d). This showed that the availability of prior knowledge on the endmembers, as in Scenarios (a) and (c), could further improve the denoising performance of a denoiser.

## 6. Real Dataset Experiments: The Samson Image

#### 6.1. Denoising as Preprocessing for Unmixing

- Similar to the simulated data, the reconstruction errors were high when unmixing noisy images, but drastically improved when performing denoising prior to spectral unmixing. A clear difference with the simulated image was that, even for very high SNR (50 dB), the reconstruction error was not zero. This could be attributed to the fact that the true endmembers were not known. As a consequence, a number of spectra fell outside of the simplex formed by the endmembers, which made a perfect reconstruction impossible.
- The abundance estimation did not improve proportionally. This confirmed that the spectral unmixing performance was not much affected by the noise even though large reconstruction errors were obtained.
- For Scenario (d), the abundance RMSE was about 5% higher than for the other scenarios, even for very high SNR. This phenomenon coincided with a lower reconstruction error than in the other scenarios. This could be attributed to the fact that the objective function of CoNMF entailed the minimization of the reconstruction error. Since the estimated endmembers did not necessarily correspond to the true endmembers, this came with a cost of poor abundance estimation.
- The effect of denoising on the abundance estimation was less pronounced than in the simulated case. In case of white noise, only in the scenario of extracted endmembers, a clear improvement could be observed when the noisy image was denoised prior to unmixing, particularly for low SNR. In Scenario (a), only HyRes could slightly improve the results; in Scenarios (c) and (d), denoising did not improve the abundance estimation.
- In the case of colored noise, the use of HyRes as preprocessing clearly improved the abundance estimation in all four scenarios, which once again confirmed the advantage of the using a well-established low-rank denoiser as preprocessing for unmixing.

#### 6.2. Unmixing as a Denoiser

- The results were quite different from the simulation results. In all scenarios, although the denoising performance of spectral unmixing was not bad at all, it was considerably lower than that of the denoising techniques.
- Even for very high SNR, the spectral RMSE after unmixing was not zero. This could partially be attributed to the same problem as with the reconstruction errors. The endmembers were not the true ones, leading to a reconstructed image that deviated from the observed image. In Scenario (b), this effect became more prominent for lower SNR, as the noise prevented extracting correct endmembers. In Scenario (d), the effect was smaller, since CoNMF estimated the endmembers by reducing the reconstruction error.
- One other possible reason of the large difference in denoising behavior between the real and simulated cases was that the real data did not exactly follow the linear model, e.g., due to nonlinearities and other nuisances, caused, e.g., by scattered light. This caused the endmembers and the abundance estimates to be incorrect. As a result, in real images, the reconstructed image after unmixing differed from the original image, already in noiseless situations (high SNR). Therefore, in real images, unmixing was not a good denoiser, compared to the denoising techniques. In the simulated image, linear data were simulated and therefore fit the model perfectly.
- Prior denoising before spectral unmixing did not improve the results. In Scenario (d), it even slightly deteriorated the results. Only in Scenario (b), the results were improved in the case of a low SNR, because the denoising reduced the chances to extract wrong endmembers.
- The denoising techniques themselves had a small nonzero spectral RMSE in the case of high SNR. This could be attributed to the fact that the Samson image in itself contained some small amounts of real noise.
- In the case of colored noise, the denoising techniques tended to perform slightly better and the unmixing slightly worse when compared to the white noise situation. Of the denoising techniques, HyRes performed the best.

## 7. Discussion

#### The Effect of Noise and Denoising on Endmember Estimation/Extraction

## 8. Conclusions

- Spectral unmixing was highly resilient to noise, allowing it to adequately estimate fractional abundances in noisy images and producing reconstructed images with good denoising performance.
- A crucial role was played by the way endmembers were provided. The more prior knowledge about the endmembers was available, the better the unmixing and denoising performance of an unmixing procedure.
- Prior denoising could boost the unmixing, as well as the denoising performance of an unmixing procedure. This became more prominent in high noise and colored noise situations, when less prior knowledge of the endmembers was available or when incorrect endmembers were applied. To validate these findings in real datasets, accurate endmember and fractional abundance ground truth information was invaluable.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ghamisi, P.; Yokoya, N.; Li, J.; Liao, W.; Liu, S.; Plaza, J.; Rasti, B.; Plaza, A. Advances in Hyperspectral Image and Signal Processing: A Comprehensive Overview of the State of the Art. IEEE Geosci. Remote. Sens. Mag.
**2017**, 5, 37–78. [Google Scholar] [CrossRef] [Green Version] - Parente, M.; Plaza, A. Survey of geometric and statistical unmixing algorithms for hyperspectral images. In Proceedings of the 2nd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, Reykjavik, Iceland, 14–16 June 2010; pp. 1–4. [Google Scholar]
- Miao, L.; Qi, H. Endmember Extraction From Highly Mixed Data Using Minimum Volume Constrained Nonnegative Matrix Factorization. IEEE Trans. Geosci. Remote. Sens.
**2007**, 45, 765–777. [Google Scholar] [CrossRef] - Li, J.; Bioucas-Dias, J.M. Minimum Volume Simplex Analysis: A Fast Algorithm to Unmix Hyperspectral Data. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Boston, MA, USA, 7–11 July 2008; Volume 3, pp. 250–253. [Google Scholar]
- Chan, T.; Chi, C.; Huang, Y.; Ma, W. Convex analysis based minimum-volume enclosing simplex algorithm for hyperspectral unmixing. In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Taiwan, 19–24 April 2009; pp. 1089–1092. [Google Scholar]
- Li, J.; Bioucas-Dias, J.M.; Plaza, A. Collaborative nonnegative matrix factorization for remotely sensed hyperspectral unmixing. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Munich, Germany, 22–27 July 2012; pp. 3078–3081. [Google Scholar]
- Iordache, M.; Bioucas-Dias, J.M.; Plaza, A. Sparse Unmixing of Hyperspectral Data. IEEE Trans. Geosci. Remote. Sens.
**2011**, 49, 2014–2039. [Google Scholar] [CrossRef] [Green Version] - Iordache, M.; Bioucas-Dias, J.M.; Plaza, A. Total Variation Spatial Regularization for Sparse Hyperspectral Unmixing. IEEE Trans. Geosci. Remote. Sens.
**2012**, 50, 4484–4502. [Google Scholar] [CrossRef] [Green Version] - Iordache, M.; Bioucas-Dias, J.M.; Plaza, A. Collaborative Sparse Regression for Hyperspectral Unmixing. IEEE Trans. Geosci. Remote. Sens.
**2014**, 52, 341–354. [Google Scholar] [CrossRef] [Green Version] - Boardman, J.W. Geometric mixture analysis of imaging spectrometry data. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Pasadena, CA, USA, 8–12 August 1994; Volume 4, pp. 2369–2371. [Google Scholar]
- Heinz, D.C.; Chein-I-Chang. Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery. IEEE Trans. Geosci. Remote. Sens.
**2001**, 39, 529–545. [Google Scholar] [CrossRef] [Green Version] - Rasti, B.; Scheunders, P.; Ghamisi, P.; Licciardi, G.; Chanussot, J. Noise Reduction in Hyperspectral Imagery: Overview and Application. Remote. Sens.
**2018**, 10, 482. [Google Scholar] [CrossRef] [Green Version] - Atkinson, I.; Kamalabadi, F.; Jones, D. Wavelet-based hyperspectral image estimation. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Toulouse, France, 21–25 July 2003; Volume 2, pp. 743–745. [Google Scholar]
- Qian, Y.; Ye, M. Hyperspectral Imagery Restoration Using Nonlocal Spectral-Spatial Structured Sparse Representation With Noise Estimation. IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens.
**2013**, 6, 499–515. [Google Scholar] [CrossRef] - Rasti, B.; Sveinsson, J.R.; Ulfarsson, M.O.; Benediktsson, J.A. Hyperspectral image denoising using 3D wavelets. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Munich, Germany, 22–27 July 2012; pp. 1349–1352. [Google Scholar]
- Rasti, B.; Sveinsson, J.R.; Ulfarsson, M.O.; Benediktsson, J.A. Hyperspectral Image Denoising Using First Order Spectral Roughness Penalty in Wavelet Domain. IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens.
**2014**, 7, 2458–2467. [Google Scholar] [CrossRef] - Rasti, B.; Sveinsson, J.R.; Ulfarsson, M.O.; Benediktsson, J.A. Wavelet based hyperspectral image restoration using spatial and spectral penalties. In Image and Signal Processing for Remote Sensing XIX; International Society for Optics and Photonics: Bellingham, WA, USA, 2013; Volume 8892, p. 88920I. [Google Scholar]
- Chen, S.L.; Hu, X.Y.; Peng, S.L. Hyperspectral Imagery Denoising Using a Spatial-Spectral Domain Mixing Prior. J. Comput. Sci. Technol.
**2012**, 27, 851–861. [Google Scholar] [CrossRef] - Yuan, Q.; Zhang, L.; Shen, H. Hyperspectral Image Denoising Employing a Spectral-Spatial Adaptive Total Variation Model. IEEE Trans. Geosci. Remote. Sens.
**2012**, 50, 3660–3677. [Google Scholar] [CrossRef] - Rasti, B.; Ulfarsson, M.; Sveinsson, J. Hyperspectral Subspace Identification Using SURE. IEEE Geosci. Remote. Sens. Lett.
**2015**, 12, 2481–2485. [Google Scholar] [CrossRef] - Bioucas-Dias, J.; Nascimento, J. Hyperspectral Subspace Identification. IEEE Trans. Geosci. Remote. Sens.
**2008**, 46, 2435–2445. [Google Scholar] [CrossRef] [Green Version] - Liu, X.; Bourennane, S.; Fossati, C. Denoising of Hyperspectral Images Using the PARAFAC Model and Statistical Performance Analysis. IEEE Trans. Geosci. Remote. Sens.
**2012**, 50, 3717–3724. [Google Scholar] [CrossRef] - Rasti, B.; Sveinsson, J.R.; Ulfarsson, M.O.; Benediktsson, J.A. Hyperspectral image restoration using wavelets. In Image and Signal Processing for Remote Sensing XIX; International Society for Optics and Photonics: Bellingham, WA, USA, 2013; Volume 8892, p. 889207. [Google Scholar]
- Rasti, B. Sparse Hyperspectral Image Modeling and Restoration. Ph.D. Thesis, University of Iceland, Reykjavík, Iceland, 2014. [Google Scholar]
- Rasti, B.; Sveinsson, J.; Ulfarsson, M. Wavelet-Based Sparse Reduced-Rank Regression for Hyperspectral Image Restoration. IEEE Trans. Geosci. Remote. Sens.
**2014**, 52, 6688–6698. [Google Scholar] [CrossRef] - Rasti, B.; Sveinsson, J.R.; Ulfarsson, M.O. Total Variation Based Hyperspectral Feature Extraction. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Quebec City, QC, Canada, 13–18 July 2014; pp. 4644–4647. [Google Scholar]
- Rasti, B.; Ulfarsson, M.O.; Ghamisi, P. Automatic Hyperspectral Image Restoration Using Sparse and Low-Rank Modeling. IEEE Geosci. Remote. Sens. Lett.
**2017**, 14, 2335–2339. [Google Scholar] [CrossRef] [Green Version] - Liu, X.; Bourennane, S.; Fossati, C. Reduction of Signal-Dependent Noise From Hyperspectral Images for Target Detection. IEEE Trans. Geosci. Remote. Sens.
**2014**, 52, 5396–5411. [Google Scholar] - Sun, L.; Jeon, B.; Zheng, Y.; Wu, Z. Hyperspectral Image Restoration Using Low-Rank Representation on Spectral Difference Image. IEEE Geosci. Remote. Sens. Lett.
**2017**, 14, 1151–1155. [Google Scholar] [CrossRef] - Zhang, H.; He, W.; Zhang, L.; Shen, H.; Yuan, Q. Hyperspectral Image Restoration Using Low-Rank Matrix Recovery. IEEE Trans. Geosci. Remote. Sens.
**2014**, 52, 4729–4743. [Google Scholar] [CrossRef] - Rasti, B.; Ghamisi, P.; Benediktsson, J.A. Hyperspectral Mixed Gaussian and Sparse Noise Reduction. IEEE Geosci. Remote. Sens. Lett.
**2020**, 17, 474–478. [Google Scholar] [CrossRef] - Kocakusaklar, B.; Kahraman, N. The Effect of Impulse Denoising on Geometric Based Hyperspectral Unmixing. Int. J. Nat. Sci. Res.
**2016**, 4, 83–91. [Google Scholar] [CrossRef] [Green Version] - Swarna, M.; Sowmya, V.; Soman, K.P. Effect of Denoising on Dimensionally Reduced Sparse Hyperspectral Unmixing. Procedia Comput. Sci.
**2017**, 115, 391–398. [Google Scholar] [CrossRef] - Qu, Y.; Qi, H. uDAS: An Untied Denoising Autoencoder With Sparsity for Spectral Unmixing. IEEE Trans. Geosci. Remote. Sens.
**2019**, 57, 1698–1712. [Google Scholar] [CrossRef] - Aggarwal, H.K.; Majumdar, A. Hyperspectral Unmixing in the Presence of Mixed Noise Using Joint-Sparsity and Total Variation. IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens.
**2016**, 9, 4257–4266. [Google Scholar] [CrossRef] - Cerra, D.; Müller, R.; Reinartz, P. Noise Reduction in Hyperspectral Images Through Spectral Unmixing. IEEE Geosci. Remote. Sens. Lett.
**2014**, 11, 109–113. [Google Scholar] [CrossRef] [Green Version] - Ertürk, A. Enhanced Unmixing-Based Hyperspectral Image Denoising Using Spatial Preprocessing. IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens.
**2015**, 8, 2720–2727. [Google Scholar] [CrossRef] - Ertürk, A. Sparse unmixing based denoising for hyperspectral images. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Beijing, China, 10–15 July 2016; pp. 7006–7009. [Google Scholar]
- Ince, T.; Dundar, T. Simultaneous Nonconvex Denoising and Unmixing for Hyperspectral Imaging. IEEE Access
**2019**, 7, 124426–124440. [Google Scholar] [CrossRef] - Yang, J.; Zhao, Y.; Chan, J.C.; Kong, S.G. Coupled Sparse Denoising and Unmixing With Low-Rank Constraint for Hyperspectral Image. IEEE Trans. Geosci. Remote. Sens.
**2016**, 54, 1818–1833. [Google Scholar] [CrossRef] - Rasti, B.; Hong, D.; Hang, R.; Ghamisi, P.; Kang, X.; Chanussot, J.; Benediktsson, J.A. Feature Extraction for Hyperspectral Imagery: The Evolution from Shallow to Deep (Overview and Toolbox). IEEE Geosci. Remote. Sens. Mag.
**2019**. [Google Scholar] - Heylen, R.; Burazerovic, D.; Scheunders, P. Fully Constrained Least Squares Spectral Unmixing by Simplex Projection. IEEE Trans. Geosci. Remote. Sens.
**2011**, 49, 4112–4122. [Google Scholar] [CrossRef] - Craig, M.D. Minimum-volume transforms for remotely sensed data. IEEE Trans. Geosci. Remote. Sens.
**1994**, 32, 542–552. [Google Scholar] [CrossRef] - Stein, C.M. Estimation of the Mean of a Multivariate Normal Distribution. Ann. Stat.
**1981**, 9, 1135–1151. [Google Scholar] [CrossRef] - Zhu, F.; Wang, Y.; Fan, B.; Xiang, S.; Meng, G.; Pan, C. Spectral unmixing via data-guided sparsity. IEEE Trans. Image Process.
**2014**, 23, 5412–5427. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 2.**Samson image. (

**a**): True-color image (red: 571.01 nm, green: 539.53 nm, blue: 432.48 nm); (

**b**): endmembers; (

**c**): abundance maps.

**Figure 3.**Jasper Ridge image. (

**a**): True-color image (red: 570.14 nm, green: 532.11 nm, blue: 427.53 nm); (

**b**): endmembers; (

**c**): abundance maps.

**Figure 5.**Simulated data (top two rows $\eta =\infty $ and bottom two rows $\eta =18$): The results of unmixing in terms of abundance RMSE and reconstruction error as a function of the noise level of the observed image (in SNR). (

**a**) Ground truth endmembers; (

**b**) extracted endmembers; (

**c**) library endmembers; (

**d**) estimated endmembers. FCLSU, fully constrained least squares unmixing; FORPDN, first order spectral roughness penalty denoising; CoNMF, collaborative nonnegative matrix factorization; HyRes, hyperspectral restoration.

**Figure 6.**Simulated data (top row $\eta =\infty $ and bottom row $\eta =18$): The denoising performance of spectral unmixing compared with the denoising techniques. (

**a**) Ground truth endmembers; (

**b**) extracted endmembers; (

**c**) library endmembers; (

**d**) estimated endmembers.

**Figure 7.**Real data (top two rows $\eta =\infty $ and bottom two rows $\eta =18$): The spectral unmixing results on the Samson image in terms of abundance RMSE and reconstruction error. (

**a**) Ground truth endmembers; (

**b**) extracted endmembers; (

**c**) library endmembers; (

**d**) estimated endmembers.

**Figure 8.**Real data (top row $\eta =\infty $ and bottom row $\eta =18$): The denoising performance on the Samson image in terms of Spectral RMSE. (

**a**) Ground truth endmembers; (

**b**) extracted endmembers; (

**c**) library endmembers; (

**d**) estimated endmembers.

**Figure 9.**The quality of the estimated/extracted endmembers in terms of spectral angle distance (SAD) in degrees on the simulated data (top row) and Samson data (bottom row) for colored noise ($\eta =18$): (

**b**) extracted endmembers; (

**c**) library endmembers; (

**d**) estimated endmembers.

**Figure 10.**The ground truth endmembers and the endmembers extracted by the simplex volume maximization technique (SiVM), without and with prior denoising of the Samson image using HyRes, in the cases of colored noise ($\eta =18$).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rasti, B.; Koirala, B.; Scheunders, P.; Ghamisi, P.
How Hyperspectral Image Unmixing and Denoising Can Boost Each Other. *Remote Sens.* **2020**, *12*, 1728.
https://doi.org/10.3390/rs12111728

**AMA Style**

Rasti B, Koirala B, Scheunders P, Ghamisi P.
How Hyperspectral Image Unmixing and Denoising Can Boost Each Other. *Remote Sensing*. 2020; 12(11):1728.
https://doi.org/10.3390/rs12111728

**Chicago/Turabian Style**

Rasti, Behnood, Bikram Koirala, Paul Scheunders, and Pedram Ghamisi.
2020. "How Hyperspectral Image Unmixing and Denoising Can Boost Each Other" *Remote Sensing* 12, no. 11: 1728.
https://doi.org/10.3390/rs12111728