Proximal approach to denoising hyperspectral images under mixed‐noise model

IET Image ProcessingVolume 14, Issue 14 p. 3366-3372 Research ArticleFree Access Proximal approach to denoising hyperspectral images under mixed-noise model Hazique Aetesam, Hazique Aetesam Department of Computer Science, Engineering, IIT Patna, Bihta, Patna, IndiaSearch for more papers by this authorKumari Poonam, Kumari Poonam Department of Computer Science, Engineering, IIT Patna, Bihta, Patna, IndiaSearch for more papers by this authorSuman Kumar Maji, Corresponding Author Suman Kumar Maji smaji@iitp.ac.in orcid.org/0000-0003-4019-0980 Department of Computer Science, Engineering, IIT Patna, Bihta, Patna, IndiaSearch for more papers by this author Hazique Aetesam, Hazique Aetesam Department of Computer Science, Engineering, IIT Patna, Bihta, Patna, IndiaSearch for more papers by this authorKumari Poonam, Kumari Poonam Department of Computer Science, Engineering, IIT Patna, Bihta, Patna, IndiaSearch for more papers by this authorSuman Kumar Maji, Corresponding Author Suman Kumar Maji smaji@iitp.ac.in orcid.org/0000-0003-4019-0980 Department of Computer Science, Engineering, IIT Patna, Bihta, Patna, IndiaSearch for more papers by this author First published: 08 October 2020 https://doi.org/10.1049/iet-ipr.2019.1763AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract The authors present a proximal approach to hyperspectral image denoising adapted to the mixed noise behaviour of hyperspectral data; named hyperspectral image proximal denoiser (HSIProxDenoiser). A combination of Gaussian-impulse noise has been handled under maximum a posteriori framework using two data fidelity terms. They have incorporated prior information about the data in the form of two regularisation terms, namely Tikhonov–Miller (TM) and total variation (TV). Since TV possesses feature selection capability by setting some of the coefficients to zero, it works well when there are a small number of significant features. On the other hand, TM works well if there are a large number of similar features. Hence, including both regularisation terms can help achieve the desired denoising performance. The resultant optimisation problem is solved using a variant of primal-dual hybrid gradient by splitting the former into different functions and calculating their proximal operators individually. Experimental results over both synthetic as well as real hyperspectral image data validate the potential of the proposed technique both visually and in terms of quantitative metrics. 1 Introduction Hyperspectral images (HSIs) contain 100 or more contiguous narrow spectral bands based on the wavelengths of light (400–2500 nm) used. HSIs find their applications in areas including remote sensing, military surveillance [1], mineral exploration, agriculture, medical diagnosis [2], food processing, forensic science [3], pharmaceuticals and environmental monitoring, forgery detection [4], to name a few. More recent developments in HSI along with state-of-the-art application areas can be found in [5]. However, various types of noises contaminate the HSI data during their acquisition, transmission, and processing stages. Noise occurs due to the atmospheric turbulence and absorption of light during propagation from the source to the sensors. Other factors include faulty memory locations, fluctuations in power supply, or some sensor noise (instrumental noise) etc. All these factors lead to the inevitable introduction of noise patterns such as Gaussian noise, impulse noise (both fixed valued impulse noise in the form of salt and pepper noise and random-valued impulse noise), dead pixels, stripes (due to the change in the position of cameras) and dark current, which lead to deterioration in the visual quality of the acquired HSI. Denoising (noise removal) is, therefore, a crucial preprocessing step for HSI before performing any further analysis such as classification or spectral unmixing of data. Multiple techniques have been proposed for the restoration of HSI. In [6], the authors propose a dual step image restoration technique for HSI based on principal component analysis over local image structures. Split-Bregman has been used in many works on denoising of hyperspectral data [7] under a mixed noise scenario. Spatio-spectral correlation is exploited in [8] by performing two-dimensional (2D) total variation (TV) along the spatial dimension and 1D TV along the spectral dimension. A wavelet shrinkage approach to denoising is explored by taking into consideration the dissimilarity of the signal regularity along spatial and spectral dimensions [9] while working in the spectral derivative domain where the effect of noise is more pronounced. The authors in [10] extend the notion of bivariate wavelet thresholding for denoising high-dimensional data cube as observed in hyperspectral data. A decentralised evolutionary approach for temporal hyperspectral image denoising called cellular automaton filtering is used in [11] to continuously update the local pixels so that after a few iterations, we obtain a denoised image. A Gaussian process guided Bayesian inference model is used to extract clusters of red–green–blue (RGB) hyperspectral data and model their spectral information [12]. Extraction of end members and abundance matrices are carried out to estimate noise using the simplex volume optimisation technique [13]. Global redundancy and correlation (RAC) is exploited along the spatial dimension and local RAC along the spectral dimension in [14] using a sparse representation of data. A compressed sensing approach for non-local sparse representation of 3D data cube to exploit spatial and spectral information in coded aperture snapshot spectral imaging is explored in [15]. Similarly, a patch-based sparse representation technique to maintain correlation along a local spectral dimension and a non-local technique to obtain spatial correlation is proposed in [16]. The resultant non-convex problem is solved using the alternating direction method of multipliers (ADMM). A three-step procedure for denoising hyperspectral data is applied in [17] using the whitening filter and multi-dimensional Wiener filter. A somewhat similar approach is used in [18], which relies on the separability between signal subspace and noise subspace. Denoising methods based on the low-rank matrix have been used in a multitude of works. Rank-1 tensor decomposition technique is used in [19] by considering data cube as an order 3 tensor to accommodate spatio-spectral information. In [20], a low-rank matrix recovery-based denoising technique optimised using the augmented Lagrangian multiplier (ALM) is used on separate image patches to preserve local image details. A low-rank Tucker decomposition is applied for separating the clean image from noisy observation [21]. Furthermore, this TV-regularised Tucker decomposition model is solved using the ALM. Here, a higher-order orthogonality iteration is used to achieve tucker decomposition. Low rank matrix factorisation-based optimisation framework [low rank TV (LRTV)] using TV for spatially piecewise smooth data and nuclear norm for spectral low rank property is proposed in [22]. A low-rank dictionary learning approach using the non-local redundant spatial structure under the Gaussian noise assumption is discussed in [23]. The authors in [24] use bilinear low-rank matrix factorisation for handling the low-rank property of HSI data cube. Also, for handling sparse and Gaussian noise, Frebonius and -norm are used as regularisation constraints. A denoising framework guided by the group low-rank property to detect local similarity within the patch and non-local similarity among patches is utilised in [25]. Furthermore, spatio-spectral TV is used for removing mixed Gaussian-sparse noise. A multi-level feature representation and extraction for learning spatio-temporal features using a deep residual convolutional network is used in [26]. In the following sections, we formulate our problem in a Bayesian framework exploiting the underlying distribution of noise in HSI data. Here, a maximum a posteriori (MAP) estimator for Gaussian noise gets reduced to -norm data fidelity term, and sparse/impulse noise is modelled as -norm data fidelity term. Tikhonov–Miller (TM) regularisation term works for preserving a large amount of locally similar features. Similarly, a TV-based regularisation term is added to handle noise in piecewise smooth regions along with its edge-preserving property. The resultant optimisation problem is solved using a modified form of the primal-dual hybrid gradient (PDHG). 2 Preliminaries Assuming that our observed hyperspectral data is represented as a 3D data cube of dimension , the image formation model can be given by (1)where is the original image, is the noisy observation, g is the Gaussian noise with zero mean and variance , denoted by . s is the sparse noise (also known as impulse noise), approximated as Laplacian noise [27, 28], denoted by with 0 as the location parameter and as shape parameter corresponding to Laplacian distributed data. To exploit spectral correlation, in our present work, we have represented our data cube as shown in Fig. 1, with where each is a column vector obtained by vertical concatenation with dimension to give resultant data as . Each column represents a spectral dimension. This representation, also known as Casorati matrix representation, promotes low-rank representation of HSI data cube [29, 30] and facilitate image recovery using spatial and spectral similarity. Fig. 1Open in figure viewerPowerPoint Vectorised representation of hyperspectral data 3 Problem formulation We seek to estimate the original data from its noisy observation by formulating an optimisation problem based on the MAP criterion. Considering that individual pixels n are conditionally independent, we can write (2)Minimising the negative logarithm of (2), we obtain (3) Assuming that the underlying distribution is Gaussian with zero mean, (3) takes the following form: (4)Assuming Gibbs model for as and using TM and TV prior as the regularisation terms, respectively, and . Equation (4) becomes (5a)and is equivalent to minimising the functional: (5b) Similarly, assuming additive white Laplacian noise following Laplacian distribution [31, 32], (3) can be reduced to (6)When is the same as previously defined, upon simplification, we get (7a)which is the same as minimising the functional (7b)Combining the MAP estimator of (5b) and (7b), we therefore propose the following optimisation problem: (8)where and are the penalty terms corresponding to and data fidelity terms and is the hyperparameter corresponding to TM regularisation term. is the TV regularisation term used to approximate piecewise smooth solutions, due to its ability to preserve edges and suppress random oscillations during noise removal. Values of hyperparameters are chosen in a way to obtain best results; both visually and in terms of the quantitative metrics considered. 4 Proposed method In this paper, we propose to re-formulate (8) under the framework of a PDHG and solve the same by deriving the proximal operators associated with each of its cost terms. Equation (8) can be re-written in a generalised form as (9) where is an operator and is a function. Considering , (8) can be written as the summation of four terms (10)where operators and functions will have the following values: • , • , • , • , with being the identity matrix and the finite difference operator. Energy functionals of the class defined by (9) can be minimised using primal dual minimisation algorithms such as PDHG [33]. This necessitates defining the proximal operators of the function , whose mathematical form is (11)with . The parameter plays the same role as step size in a gradient method [34]. The proximal operators associated with different are as follows. 4.1 Proximal operator associated with function Considering , and solving for u, we get (12a) (12b) (12c) 4.2 Proximal operator associated with function Considering , , and solving for u, we get (13a) (13b)Since is a separable function, its proximal operator is given by piecewise soft-thresholding on data (chapter 6) [35, 36] (13c) 4.3 Proximal operator associated with function Considering and , where and are the first-order derivative operator along horizontal and vertical directions, and solving for u we get (14a) (14b)with . Here, L is the discrete Laplacian operator. 4.4 Proximal operator associated with function The proximal operator corresponding to the fourth term in the objective function is given by with denoted by where and . Here, and are the first-order derivative operators along horizontal and vertical directions concatenated as . We minimise (10) using a modified form of PDHG [33] shown in Fig. 2. Fig. 2Open in figure viewerPowerPoint Algorithm: PDHG u is the solution to the primal part of the formulation while v is the solution to the dual formulation of the problem at iteration r. The initial estimate is given by and respectively. k denotes the number of terms in the optimisation problem as shown in (8); in our case, it is 4. and are the proximal parameters and is the relaxation parameter. R is the number of iterations. Since we are handling the denoising problem, the linear operators in data-fidelity terms are set to identity I. 5 Experimental results To show the effectiveness of our proposed technique–HSIProxDenoiser, we have conducted experiments on two sets of data. First set consists of two synthetic data and the second set consists of real data. We have compared our proposed method using two state-of-the-art techniques for HSI noise removal: LRTV [37] and spatio-spectral TV (SSTV) [8]. LRTV makes use of low-rank penalty along the spectral dimension and TV along the spatial dimension. SSTV uses 2D TV along with the spatial dimension and 1D TV along with the spectral dimension; further solved using the split-Bregman technique. Parameters in comparing techniques have been selected to obtain best results for every noise case. To quantitatively evaluate the performance benefits obtained, we use the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) [38] and their mean values (averaged over all bands); mean PSNR (MPSNR) and mean SSIM (MSSIM). The weight parameters () and regularisation hyperparameter values of HSIProxDenoiser have been chosen to achieve the best visual and metric results. 5.1 Synthetic data We have conducted experiments on two artificially corrupted synthetic data with varying levels of Gaussian and impulse noise. For our first set of experiments, we have taken the Washington DC (WDC) Mall [39] data obtained from the hyperspectral digital imaging collection experiment(HYDICE) sensor. The dimension of the data is , where is the spatial dimension and 191 is the number of spectral bands captured in the visible and infrared regions of the electromagnetic spectrum in the wavelength range of 400–2400 nm. We conducted the second set of experiments on University of Pavia [40] data obtained from the reflective optics system imaging spectrometer (ROSIS). The dimension of the image is . All experiments are performed on Matlab-2016b. 5.2 WDC dataset For the WDC dataset, an image patch of size is used for experimentation purpose. Not all bands were taken because they were heavily corrupted by noise and hence could not be used as ground truth in our test experiments. The image has been changed to greyscale format prior to experimentation. The chosen levels of degradation of Gaussian noise and impulse noise are as follows: • Gaussian noise (SNR, dB): 21, 18, 12, 10, and 5. • Impulse noise, %: , 1, 5, 10 and 20. PSNRs under (impulse, Gaussian) noise = for all the layers are shown in Fig. 3a. Similarly, SSIM is shown in Fig. 3b. It can be seen that for both PSNR and SSIM, our HSIProxDenoiser performs considerably well for most of the layers with negligible deterioration near layers 140 and 180 for PSNR and layers 80 and 120 for SSIM. This is attributed to a large magnitude of noise in ground truth itself but our method still achieves better average results. To support this argument, MPSNR and MSSIM for five different levels of noise are tabulated in Table 1. Here, our technique shows better overall performance against all the competing methods by a considerable order of magnitude. The benefit obtained in PSNR is 2.83 dB for the least noisy case to 4.91 dB for the noisiest case. Thus, it can be concluded that performance benefits are more for more noisy data. Only in the case of LRTV under (Gaussian, impulse) noise, the PSNR value is slightly higher than our technique. The same pattern can be observed for SSIM, where maximum performance benefits are obtained for the noise case (Gaussian, impulse) noise. Here, gain in SSIM is . Fig. 3Open in figure viewerPowerPoint Quantitative evaluation of (a) PSNR for the noise level in WDC Mall dataset, (b) SSIM for the noise level ) in WDC Mall dataset, (c) PSNR for the noise level in the University of Pavia dataset, (d) SSIM for the noise level in the University of Pavia dataset Table 1. Mean PSNR and mean SSIM of WDC mall and University of Pavia dataset WDC Mall Noise MPSNR MMSIM Impulse, % Gaussian (SNR, dB) Noisy LRTV SSTV HSIProxDenoiser Noisy LRTV SSTV HSIProxDenoiser 0.5 21 33.10 31.16 36.35 35.93 0.901 0.898 0.949 0.948 1 18 30.14 30.35 33.61 34.12 0.831 0.872 0.912 0.929 5 12 23.35 26.41 26.82 27.86 0.564 0.717 0.733 0.789 10 10 20.49 23.92 23.54 25.33 0.423 0.599 0.605 0.728 20 5 14.76 20.42 17.35 19.67 0.178 0.375 0.302 0.487 University of Pavia Noise MPSNR MMSIM Impulse, % Gaussian (SNR, dB) Noisy LRTV SSTV HSIProxDenoiser Noisy LRTV SSTV HSIProxDenoiser 0.5 20 33.06 32.17 36.32 37.70 0.850 0.901 0.921 0.949 1 16 30.10 31.36 33.57 35.57 0.755 0.866 0.867 0.923 5 10 23.31 27.28 26.79 29.45 0.454 0.681 0.640 0.817 10 8 20.39 24.64 23.50 25.88 0.323 0.562 0.503 0.742 20 4 14.83 21.04 17.36 19.42 0.123 0.361 0.224 0.441 SNR, signal-to-noise ratio. We present the denoising results of layer 80 for WDC dataset for the noise level in Fig. 4. It can be seen that the visual results obtained in our case are better than the comparing techniques. However, results for SSTV is comparable to our case. We further perform experiments to obtain spectral signatures of original, noisy, and restored results of HSIProxDenoiser and comparing techniques (spectral signature is the intensity values of a single-pixel as recorded by the sensor along with the entire range of the spectrum for all layers). Fig. 5 shows the spectral signature of the results shown in Fig. 4. The plot is illustrated as a function of band number against the digital number (DN). (The intensity of a pixel in remote sensing is radiometrically calibrated, known as DN). The resemblance of spectral signature between the original data and our proposed technique shows its more accurate restoration capability. Fig. 6 shows the difference in the spectral signature between the pixel spectrum of noise-free data and restored results. A smooth representation in graphs (with a lesser amount of glitches) is an indicator of better restoration quality. We have also calculated the time taken to complete the denoising process for each of the techniques. It is found that HSIProxDenoiser takes the least amount of time of min compared to and min, respectively, for LRTV and SSTV. Fig. 4Open in figure viewerPowerPoint Restoration results for WDC dataset for the band 80 for noise level (I,G)=; (a) Original, (b) Noisy, (c) LRTV, (d) SSTV, (e) HSIProxDenoiser Fig. 5Open in figure viewerPowerPoint Spectral signature at pixel location for WDC dataset for the noise level (I,G)= (a) Original, (b) Noisy, (c) LRTV, (d) SSTV, (e) HSIProxDenoiser Fig. 6Open in figure viewerPowerPoint Difference in the noise-free spectrum with noisy and restored results for noisy data, LRTV, SSTV and proposed technique for WDC (for the pixel present at location ) dataset 5.3 Pavia University dataset Similar to the previous dataset, in the Pavia University dataset, the cropped image size of is considered to mitigate the computational burden. This is further converted to greyscale format corrupted by the same level of impulse noise and Gaussian noise varied in the range of 20–4 dB. PSNR and SSIM for layers 1–103 are shown in Fig. 3 for the noise level (Gaussian, impulse) =. It can be observed that the PSNR is low for initial layers but it rises gradually for the same reason as stated in the previous section. There is a huge margin in the results obtained between comparing techniques and the proposed one. Table 1 summarises the result for five different cases of noise for both the metrics; mean PSNR and mean SSIM. In the case of mean PSNR, minimum advantage is offered for the noise case where difference in mean PSNR is and maximum advantage is offered for the noise case where difference in mean PSNR is when compared with noisy mean PSNR. For mean SSIM, these values are and for noise cases and , respectively. We show visual results of the University of Pavia dataset for layer 100 at noise level in Fig. 7. Noticeable difference can be seen in our visual results juxtaposed with comparing techniques, which is basically attributed to the explicit data fidelity term for handling impulse noise. The same trend can be observed for spectral signature and difference in spectral signature as shown in Figs. 8 and 9, respectively. Less glitches in HSIProxDenoiser is the characteristic of better reconstruction quality in comparison with the noisy data. Computational time for the execution of HSIProxDenoiser is min, while it is min for LRTV and min for SSTV. Fig. 7Open in figure viewerPowerPoint Restoration results for University of Pavia dataset (row 2) for the band 100 for noise level (I,G)= (10%; 8dB); (a) Original, (b) Noisy, (c) LRTV, (d) SSTV, (e) HSIProxDenoiser Fig. 8Open in figure viewerPowerPoint Spectral signature at pixel location for the University of Pavia dataset for the noise level (I,G)= (a) Original, (b) Noisy, (c) LRTV, (d) SSTV, (e) HSIProxDenoiser Fig. 9Open in figure viewerPowerPoint Difference in the noise-free spectrum with noisy and restored results for noisy data, LRTV, SSTV, and proposed technique for the University of Pavia (for the pixels present at the location (50, 81)) dataset 5.4 Real data To justify our argument of better qualitative and quantitative results for synthetic hyperspectral data, we conduct experiments with the same comparing techniques on a real dataset. For our real dataset, we have considered Indian Pines [41] data gathered by the airborne visible/infrared imaging spectrometer (AVIRIS) at a test site in North-western Carolina. The data consists of pixels along the spatial dimension and 224 layers in the spectral dimension in the wavelength range of 400–2500 nm. The data cube comprises large sections of piecewise smooth regions. It is in these sections that improvement in restoration quality is more evident. Figs. 10 and 11 show the visual results for layers 164 and 170, respectively. Since data is noisier for layer-164, the difference in reconstruction quality is more evident for layer-170. Also, for layer 170, we can observe contrast preserving property of our technique as also highlighted in [42]. Fig. 10Open in figure viewerPowerPoint Restoration results of experiments conducted on real data. Visual results for layer-164 on Indian Pines dataset (a) Noisy, (b) LRTV, (c) SSTV, (d) Proposed Fig. 11Open in figure viewerPowerPoint Restoration results of experiments conducted on real data. Visual results for layer-170 on Indian Pines dataset (a) Noisy, (b) LRTV, (c) SSTV, (d) HSIProxDenoiser 6 Conclusion In this paper, we have designed an optimisation problem for denoising hyperspectral images corrupted by mixed Gaussian-impulse noise. Designing an objective function exclusive for removal of both noise provides us significantly better results. The resultant optimisation framework is solved using a variation of PDHG. Experimental results on both synthetic data and real hyperspectral data reveal the considerably better performance of our proposed method than two state-of-the-art techniques in terms of PSNR, SSIM, and execution time. 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