Hyperspectral image, video compression using sparse tucker tensor decomposition

Hyperspectral image and videos provide rich spectral information content, which facilitates accurate classiﬁcation, unmixing, temporal change detection, and so on. However, with the rapid improvements in technology, the data size has increased many folds. To properly handle the enormous data volume, efﬁcient methods are required to compress the data. This paper proposes a multi-way approach for compression of the hyperspectral image or video sequence. In this approach, a differential representation of the data is ﬁrst obtained. In the case of hyperspectral images, the difference between consecutive bands is obtained and in case of videos, the difference between consecutive frames is computed. In the next step, a sparse Tucker tensor decomposition is performed and the sparse core tensor obtained. Finally, the core tensor and the corresponding factor matrices are truncated and the data encoded to obtain the compressed version for transmission. The compression method utilises the multi-way structure of the data and hence can be extended for hyperspectral videos. Experimental results on several real data imply that the proposed compression approach obtains better efﬁciency in terms of compression ratio, signal to noise ratio.


INTRODUCTION
Hyperspectral images record the reflectance pattern of a ground scene of interest at several narrow spectral bands at different wavelengths in the electromagnetic spectrum [1]. Normally, the images acquire detailed, high-resolution spectral information from hundreds of spectral bands at both visible and near infrared regions. The wealth of information makes the data size enormous and thereby causes difficulties in transmission, processing, and storage. With the advent of high-end MEMS-based sensor technology, the spatial, as well as the spectral resolution of the data, have improved. This improvement in technology facilitates accurate classification and object detection. However, the resulting volume of the data becomes huge, and it poses a challenge in processing and transmission. Proper handling of data calls for efficient compression means. In this work, we propose a tensor decomposition-based approach to compress a hyperspectral image/video in a unified method. A hyperspectral image/video contains significant redundancy in both spatial and spectral domains. In hyperspectral videos, This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2021 The Authors. IET Image Processing published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology successive frames convey similar information and add a temporal redundancy with the spatio-spectral redundancy. Hyperspectral image/video compression methods account for either spatial or spectral redundancy and obtain a reduced representation of the data by encoding. The encoding approach ensures that the encoded version of the data can be represented by lower numbers of bits or parameters. The encoding process aids transmission of the huge volume of data over narrow bandwidth transmission channel. On the other hand, the decoder reconstructs the transmitted data, such that it leads to minimal loss or no loss of information content. Data compression methods can be categorised into-lossless and lossy category. The lossless compression methods preserve the information contained in the data. On the other hand, lossy compression techniques allow negligible reconstruction error. In practice lossy compression is extensively used as very few lossless compression methods can achieve a high compression ratio. An image or video data possess a high degree of spatial and/or temporal redundancy, which can be encoded by lossless compression efficiently without significant loss in information. Hence, we present a lossy compression method that obtains a decent compression performance.
Any natural image contains a large amount of spatial redundancy and homogeneous regions. Traditional image compression approaches mostly exploit these spatial redundancies and obtain an encoding that represents the image in a compact form. A remotely sensed hyperspectral image contains spatial redundancy like any natural image and a significant spectral redundancy. Some hyperspectral image compression approaches apply these image-based compression approaches to grayscale images corresponding to the individual band of the hyperspectral image. On the other hand, some hyperspectral compression approaches exploit the spectral redundancy, somewhat disregarding the spatial redundancy of a hyperspectral image. However, an efficient compression approach should ideally take both the spatial redundancy and the spectral correlation into account. Rate control is another important attribute of image compression. The three major parameters rate, distortion, and perception have some relation [2]. Valsesia et al. [3] discussed the effect of rate distortion on hyperspectral compression. Although rate distortion is an important characteristics while considering image compression, it is difficult to establish direct relationship between rate control and compression.
To this aim, we introduce a multi-way approach for tensor compression, that considers a hyperspectral image/video as a higher-order tensor. The proposed approach first obtains the differential version of the data by computing the difference of consecutive bands of the data. Next, the approach performs a fast, sparse Tucker tensor decomposition and obtains the core tensor, which assumes sparse structure. Next, the method truncates the resulting core tensor, encodes the core tensor, and the factor matrices and transmits the data. While the decoding block decodes the factors, and compute the reconstructed tensor. The proposed compression approach handles spatio-spectral redundancies simultaneously without loss of generality and can be extended for compression of hyperspectral video sequences. This paper proposes a tensor-based compression approach which offers the following advantages over the prevalent approaches: 1. The proposed approach is a multi-way compression method, which exploits the sparsity structure of the hyperspectral image/video and the spatio-spectral redundancy. In the case of hyperspectral images, the data first computes the difference between successive bands of the data and thereby utilise the spectral redundancy. Next, the tensor decomposition considers the spatial redundancy. The hyperspectral video compression method first computes the differential frame and accounts for the temporal redundancy. The tensor decomposition performed in the next step accounts for the spatio-spectral redundancy. 2. The proposed approach obtains a higher compression ratio, and lower reconstruction error as it mostly preserves the spatial consistency of the visual data and the correlation between the spectral bands or the frames. 3. The proposed compression method is more noise-robust compared to the previous approaches. Unlike prevalent com-pression approaches, our proposed tensor-based compression method is generalised to both image and video data, and can easily adapt to compress hyperspectral image, video.

Paper organisation
This paper is organised in the following sections: Section 2 discusses the existing approaches for hyperspectral data compression. Section 3 presents a brief introduction of the widely used tensor decomposition methods. Section 4 illustrates our proposed compression method, while Section 5 demonstrates the results obtained on synthetic as well as real image experiments.

EXISTING APPROACHES
Hyperspectral data compression methods can be classified into-transform coding methods, dimensionality reduction approaches, compressive representation approach, image coding-based approach, and tensor-based multi-dimensional approach. Researchers have explored different philosophies and proposed various techniques for hyperspectral compression. We briefly explain these approaches in the following section: Transform coding-based approaches include: 3D Set Partitioning in Hierarchical Trees (SPIHT) [4] and Set Partitioned Embedded bloCK (SPECK) algorithms [5], the progressive 3-D coding algorithm [6], the 3-D reversible integer lapped transform [7], and HV 2.4 video compression approach [8]. These approaches encode either the 3D data or the individual band images using traditional encoding methods. The imagebased 2D encoding methods do not exploit the high correlation between the spectral band of any hyperspectral image. Among these approaches, 3D Set Partitioning in Hierarchical Trees (SPIHT) [4] and Set Partitioned Embedded bloCK (SPECK) algorithms [5], the progressive 3-D coding algorithm [6], the 3-D reversible integer lapped transform [7] encoded the hyperspectral data cube using 3-dimensional encoding and transmitted the obtained compact version of the data. Wang et al. [9] introduced 3D lapped transformation approach for compression.
Other approaches such as the discrete wavelet transform coupled with tucker decomposition [10], vector quantisation with PCA and JPEG compression method [11] used dimensionality reduction. Rucker et al. [12] proposed a wavelet-based compression approach that transforms the data after decorrelating the spectral bands. Lastly, the method employed the JPEG2000 algorithm to compress the transformed data in the spatial domain. Du et al. [13] modified the aforementioned method and used discrete wavelet transformation in conjunction with the JPEG2000 method [12] and transformed the data thus obtained by the conventional principal component analysis (PCA) algorithm for decorrelating the data in the spectral domain. In the work [14] Huber et al. presented a new approach, which integrated discrete cosine transformation with PCA. The method aimed to obtain efficient compression for target detection. Chen et al. [15] proposed a compression method, that iteratively modified eigenvalues and eigenvectors of the covariance matrix until it achieved desired compression performance. Sparse representation has found some applications in image compression tasks, both in specific and general image compression. Bryt and Elad [5] proposed a method for compressing facial images using the K-SVD dictionary learning algorithm [16]. The learned dictionary was deployed in the sparse approximation of image patches following a pre-processing geometric alignment of facial images. This method showed evidence that a sparse representation-based compression scheme could outperform the state-of-the-art wavelet-based compression method, such as JPEG2000, for some specific types of images. Jifara et al. [17] presented a dictionary learning-based approach, which learned a spectral library and represented the data as a sparse combination of the library.
Mielikainen [18] introduced a new approach where it represents an image pixel in terms of the neighborhood pixels from the closer bands. Skretting and Engan [6] presented a dictionary-based approach that learned dictionaries in pixel and wavelet domains during the training phase and resulted in superior performance compared to JPEG.
Zhang et al. [19] introduced a tensor decomposition approach, which obtained a core tensor that is considered to be the representative of the original tensor. Another work introduced by Wang et al. [20] presented a compressed sensing approach that identifies a sparse coding to learn the data in terms of a dictionary and encoded the sparse coefficients using quantisation approach. Valesseia et al. [21] implemented an onboard compression method based on coding. Karami et al. [22] proposed a compression method that is had designed for facilitating unmixing. In that work, they used Tucker decomposition to obtain a compact form of the data and minimised the abundance between the original and the compressed version of the data. Diaz et al. [23] implemented hyperspectral compression methods on embedded GPUs and enabled their realtime operation.

FUNDAMENTALS OF TENSOR ALGEBRA
Tensors are multi-dimensional arrays containing two or more dimensions. The dimension of a tensor is termed as the order of a tensor. Although tensors are a multi-dimensional extension of vectors and matrices, tensor algebra is not a straightforward extension of matrix algebra. Hence, we briefly outline the important properties of a tensor in the following subsection. To differentiate between scalar, vectors. matrices and tensors we follow common notation. We denote scalars as small case letters, for example, u, vectors as lowercase italic letters, for example, u, matrices as bold font uppercase letters U and tensors as the uppercase calligraphic letter  .

Fundamental tensor operations
A tensor of order n consists of n dimensions and is denoted as- ∈  I 1 ×I 2 ×⋯×I n . The individual elements of the tensors are referred to as- i 1 ,i 2 ,…,i n [24]- [25]. Higher order tensors can be reshaped into-lower-order tensors, matrices or vectors by different operations such as-unfolding, matricisation, vectorisation or slicing operations [24]. The vectorisation (vec()) operation converts n-order tensors into vectors by varying the nth index within its range (1, 2, … , I n ) and keeping all other indices fixed. The unfolding operation converts a tensor into a matrix. Mode n unfolding converts tensor to a matrix by by considering the n-th dimension as columns and rearranging all other elements into rows according to- obtains the matrix X (n) by considering the n-th dimension as columns and rearranging all other elements into rows. The tensor matrix multiplication also follow some specific rules, which are dissimilar from traditional matrix multiplication methods. The mode r product of a tensor  ∈ ℝ I 1 ×I 2 ×⋯×I n and a matrix U ∈ ℝ J r ×I n obtains a resultant tensor  of dimension I 1 × I 2 × ⋯ × J r × ⋯ × I n by multiplying the n-mode unfolded version of  with the left-hand side of U according to The individual elements of the tensor are represented as Two tensors ,  ∈  I 1 ,I 2 ,…,I n of the same size can also be multiplied according to- We can compute tensor norm such as Frobenius norm, l 0 norm, l 1 norm similar to matrix norms. l 1 norm of the tensor is computed according to The l 0 norm of the tensor  is defined as the number of nonzero elements,which is computed according to where, denotes the number of components. On the other hand, the Frobenious norm of the tensor  is calculated from the inner product according to the formulation Tensor algebra supports different types of multiplication operations such as Kronecker product or Khatri-Rao product and outer product. Kronecker product is denoted as ⊗ operator, whereas Khatri-Rao product is expressed using • operator. Kronecker product of a matrix A ∈ ℝ n 1 ×n 2 and B ∈ ℝ p 1 ×p 2 obtains a matrix C ∈ ℝ n 1 p 1 ×n 2 p 2 whose entries are C (n 1 −1)P 1 +p 1 ,(n 2 −1)P 2 +p 2 .

Tensor decomposition
Tensor decomposition is at the heart of multi-way data analysis. Researchers have proposed different approaches for factorising tensors. Among these approaches, PARAllel FACtorisation (PARAFAC) or Canonical Polyadic decomposition (CPD) and Tucker decomposition are the most fundamental decompositions [25]. PARAFAC or CPD is generally employed for blind signal separation, dictionary learning tasks. On the other hand, Tucker decomposition finds application in data compression, dimensionality reduction, and other tasks [24]. Our proposed hyperspectral image, video compression scheme relies on Tucker tensor decomposition. Hence, we describe the fundamental of Tucker tensor decomposition approaches in the following section:

Tucker decomposition
Tucker decomposition is regarded to be a higher order extension of singular value decomposition. Tucker decomposition method factorises a tensor of n-th order,  ∈  I 1 ×I 2 ×⋯I n into a core tensor and some factor matrices according to Here,  ∈  I 1 ×I 2 ×⋯I n is the core tensor, and F 1 , F 2 , … , F n denotes the factor matrices corresponding to different modes. The core tensor is essentially a compressed version of the data. On the other hand, the factor matrices contain rows which are orthogonal to each other. The factors matrices can be considered to be analogous to singular values. Traditionally, tucker decomposition is used for data compression, dimensionality reduction, as the core tensor can be truncated to represent the data in a compact and reduced form. According to CP or PARAFAC decomposition, a tensor of order n  ∈ ℝ I 1 ×I 2 ×⋯×I n is written as a sum of product of N rank1 vectors according to The tensor can also be represented in terms of factor matrices in an alternative formulation according to where, n ] ∈  I k ×r represents the factor matrix/loading matrix in mode-k and  n,r is the n-dimensional identity tensor or diagonal tensor of size R × R × ⋯ × R.

PROPOSED TENSOR BASED COMPRESSION APPROACH
This paper proposes a tensor decomposition based compression approach. The main advantage of this compression method is that it is a generic approach, which can be applied to both hyperspectral images or videos. To compress a hyperspectral image, we first compute the difference between individual bands of the image and obtain a sparse representation of the data. We encode the first band of the hyperspectral image using standard JPEG compression and compress the differential sparse 3-rd order data by Tucker tensor decomposition. In the decoder side, we add the decoded version of the first band image with the decoded version of the differential 3-dimensional data and thus obtain the decoded data of the hyperspectral image. The sparse data can be compressed more efficiently compared to the original data. In the case of video, we compress the first frame using the 3-rd order tensor compression approach, obtain a 4-th order differential tensor by computing the difference between the successive frames of the video. We compress this data by our proposed Tucker tensor decomposition approach. While decoding, we add the decoded data corresponding to the first frame and differential data corresponding to the other frames.
We assume the hyperspectral data to be a tensor of order n and is denoted as-Y ∈  N 1 ×N 2 ×⋯×N n . We compute the difference between consecutive spectral bands and obtain the differential data Y ∈  N 1 ×N 2 ×⋯N n −1 using the approach Y :,:,…,i = X :,:,…,i − X :,:,…,i+1 .
The consecutive spectral bands in a hyperspectral image and consecutive frames of a hyperspectral video are highly correlated. As a consequence, the differential data Y assumes sparse representation. Besides, the hyperspectral image or video is a low-rank data because it is a latent mixture of few endmembers [26]. The sparsity of the data and the inherent low-rank structure creates redundancy, which is particularly useful in compression of the data. We exploit the spatial and spectral redundancy in the compression and factorise the tensor using Tucker tensor decomposition.

Sparse tucker decomposition
The factor matrices obtained by Tucker decomposition have orthogonal rows. The core tensor obtained by the decomposition leads to a compact representation of the data [27]. Tucker decomposition of a sparse tensor produces a core tensor that contains a certain level of sparsity. Although Tucker tensor is widely used, the decomposition is not always unique and does not guarantee that the core tensor will be sparse. The nonuniqueness property of Tucker decomposition has proved to be a major stumbling block in the signal processing task, as these tasks require repeatability. Ensuring uniqueness is a primary objective in a tucker decomposition-based compression task. According to studies [28,29], the use of regularisation terms constrain the core tensor and factor matrices and make the decomposition unique. When the core tensor is sparse, it rarely interacts with the factor matrices. The core tensor sparsity enforces the decomposition makes the decomposition unique. In our case, we aim to decompose differential image/video, which in itself is sparse. Hence, we can safely assume that the core tensor will be sparse in our case. Yokota et al. [30] maximised the sparsity of the obtained core tensor, while minimising the reconstruction error and maintaining the orthogonality of the individual factor matrices, and overcame the limitation of non-uniqueness. Motivated by this work, we perform Tucker decomposition with an additional regularisation term, which makes the solution of the decomposition unique and the decomposition repeatable. We mathematically represent the aforementioned optimisation formulation as Here, we have utilised pq-norm-based sparsity [31] instead of the traditional l 1 or l 1 2 norm, as it obeys most the desired attributes of an efficient sparsity measure, in the first term. The second term in Equation (11) enforces the reconstructed tensor to be within close to the original tensor . On the other hand, the final term ensures that the factor matrices obtained by the decomposition are orthogonal. In signal processing applications the factor matrices often correspond to the signal sources. The orthogonality of the factor matrices synonymous with the assumption that the signal sources are statistically independent.

Proposed compression by truncation of tucker decomposition
Tucker tensor decomposition depends on the size of the core tensor. The core tensor size is identified by the multi-linear rank or Tucker rank estimation process. We decompose the differential hyperspectral data into a core tensor and factor matrices by Tucker's alternative least square algorithm. We consider the size of the core tensor to be R 1 , R 2 , … , R n , which is the estimated multi-linear rank of the tensor. We identify the mutilinear rank using higher-order singular value decomposition and information-theoretic criteria [32]. This approach reshapes the tensor into mode matrices and computes the eigenvalues of these reshaped mode matrices. Finally, the method computes the multi-linear rank by minimum description length (MDL), which is an information-theoretic measure.
The core tensor obtained by the decomposition contains some redundant elements due to the sparse nature of the original data. The core tensor and the corresponding factor matrices can be further truncated to achieve a higher compression ratio. However, the truncation process should identify the redundancy efficiently to prevent loss of information. To this aim, we exploit the properties of the core matrix that states that the norm of the hyper-slices of the core tensor in analogous with the higher-order singular value of the corresponding mode (n) k = ‖(:, :, … , k, :, … , :)‖ 2 .
Here, the l 2 norm of the n-th slice of the core tensor denoted by (:, :, … , k, :, … , :) equals the k-th singular value of the n-th mode (n) k . We reshape the data into matrices across different modes and compute the singular values. We attempt to identify the optimum size of the factor matrices using the singular values in each mode and thereby estimate the size of the optimum core.
The singular values are arranged in non-increasing value The multi-linear singular values also show a similar pattern as singular values of low-rank matrices often used in signal separation. The first few singular values are higher and highly significant. Whereas the lower singular values correspond to noise or non-signal components. Many works attempted to identify the number of signal components from the eigenvalues. The singular value plot contains a knee region that corresponds to the threshold separating signal and non-signal components. The differential singular value displays the same pattern as the original singular value plot. The plot shows a sharp decline around the higher singular values and contains a sharp cutoff point. Hence, we first compute the difference between successive singular values of each mode according tô We identify the optimum number of singular values to obtain a compact representation of core tensor using a measure termed gap index. This index computes the ratio between the standard deviation of singular values and is calculated according to where,̃i (17) represents the difference between the consecutive eigenvalues. The optimum number of singular values are computed from the minima of the gap index according to P = arg min k=1,2,.,L−3 According to the random matrix theory, the higher differential singular values correspond to signal components and have higher standard deviation. On the other hand, the lower singular values consists of noise components and have significantly lower standard deviation. When, k equals the actual number of signal components the numerator becomes low and hence, the ratio term attains a global minima. Most of the previous approaches to identify the optimum number signal component relied on calculating the optimum threshold. Unlike those, our proposed gap index measure calculates standard deviation of the eigenvalues, as a consequence perturbation of eigenvalues due to noise effects both the numerator and denominator simultaneously. The overall effect of noise is compensated.

Hyperspectral image compression
In the case of hyperspectral images, we compress the grayscale image corresponding to the first spectral band using the JPEG compression method. Next, we obtain a differential tensor by subtracting the consecutive spectral bands of the image and compress this differential image tensor. On the decoding end, we first encode the first band image and in the next stage, we decode the differential tensor. Finally, we obtain the reconstructed data by adding the first band image with the encoded version of the differential image obtained at the receiver side.
Due to the use of optimal core truncation, and efficient encoding the data transmission can be performed using only a few bits. Hence, the compression method achieves a high compression ratio.

Hyperspectral video compression
A hyperspectral video is essentially a tensor of order 4. A hyperspectral video contains a very high temporal correlation as the consecutive frames provide similar information. Besides an individual frame contains a high correlation between consecutive spectral bands of the hyperspectral image. Because of the high temporal similarity, when the consecutive frames of the video are subtracted, it produces a sparse tensor of order 4. Since sparse tensor is easier to compress, we first obtain the differential tensor. In the next stage, we compress the differential tensor corresponding to the video and the hyperspectral image corresponding to the first frame of the video using 3-rd order tucker decomposition. In the decoder side, we sum individual frames of the differential video with the decoded version of the first frame and obtain the original video. The multi-linear rank estimation strategy identifies the optimum size of the truncated tensor and thereby performs efficient compression.

Hyperspectral images
We have implemented the compression algorithm on three real hyperspectral images and some other hyperspectral video data. We present a brief description of these data below: • Jasper ridge This image (shown in Figure 4a) has a spatial dimension of 512 × 614 and contains 224 spectral bands covering the wavelengths ranging from 380 to 2500 nm. The spectral resolution is 10 nm. As the original hyperspectral data set is too complex to get the Ground Truth, we consider a sub-image with 100 × 100 pixels like other approaches. The first pixel of the reduced image corresponds to the pixel (105, 269) in the original image. We also remove the noisy bands 1-3, 108-112, 154-166 and 220-224, and use a 198 band version of the image. The ground truth reports informed that the image consists of four endmembers.

• DC mall
Washington DC Mall image (available at http://www.tec. army.mil/Hypercube) displayed in Figure 4b), is extensively used for analysis. The image was recorded using Hyperspectral Digital Imagery Collection Experiment (HYDICE) in an urban area at Copperas Cove, TX, US. The image has a spatial and spectral dimension of 307 × 307 pixels and 210 bands respectively. The 210 spectral bands for the wavelength range 400 to 2500 nm, and has a spectral resolution of 10 nm. Like most other works, we also exclude the noisy and water absorption bands 1-4, 76, 87, 101-111, 136-153, and 198-210 before processing. The image is known to contain four endmembers.

• Moffett Field
Moffett Field image (displayed in Figure 4c) was captured in 1997 from Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) over the Moffett field area in CA, USA. The image has been commonly used to validate unmixing performance.
The original image consists of 224 spectral bands, which also include some water absorption bands and noisy bands. We remove the noisy bands and utilise a 189 band version of the data as employed in [33]. Like other works, we apply unmixing on a 60 × 60 sub-image of the whole image. This subimage is known to be composed of three endmembers.

Hyperspectral video
We also apply our compression study on the hyperspectral video recorded by Mian et al. [34]. The hyperspectral video contains a total of 31 frames, each containing a hyperspectral image of spatial size. The images consist of 33 spectral bands in the range of 400 to 720 nm.

Performance measures
We compare the compression algorithm in terms of signal to reconstruction error ratio and compression ratio. Besides, we also verify classification as well as unmixing performance after performing compression of the data. The performance of the compression algorithm is to evaluate using three measures that quantify the similarity between the decompressed data and the original data. To this aim, we use three measures: signal-to-noise ratio (SNR).
• SNR: This measure is computed according to- Here, var (X ) represents the variance of the data X.

Compression performance
We evaluate compression performance and report the obtained SNR on the three real hyperspectral images. We vary the compression rate in terms of bit per pixel per band (bpppb). The SNR values tabulated in Tables 1-3 suggest that our proposed sparse tucker tensor decomposition approach yields higher SNR, which corresponds to better performance. We also analysed the noise performance of the compression methods. To this aim, we corrupted the images with additive white Gaussian noise with varying SNR and computed the compression  performance. The noise performance tabulated in Tables 4-6 proves the superiority of our proposed compression approach. We applied the above-mentioned compression methods frame by frame in the hyperspectral video and obtained SNR performance by varying the compression ratio. From Table 7, we conclude that our proposed compression achieves SNR as desired.
Since we compressed the sparse data using an efficient core truncation method, we obtain better compression performance compared to the other approaches.

Performance of classification after compression
Hyperspectral classification [35,36] assigns label on each pixels using supervised or unsupervised approach. We classify the compressed version of the images using three standard classifiers: (a) support vector machine (SVM) [37], (b) classification and regression tree (CART) [38], and (c) convolutional neural network (CNN) method [39]. We study the effects of compression on the above-mentioned classification approaches. Hyperspectral image classification is an important task in earth monitoring where the presence of different endmember materials is detected from the remotely sensed image. The result of classification on the Moffett Field image shown in Figure 4c

Performance of unmixing after compression
We carry out linear unmixing after compression. The unmixing process identifies the reflectance pattern of the macroscopic endmembers present in the image scene. We assume linear mixing model while performing unmixing, and estimate endmembers by Harsanyi-Ferrand-Chang virtual dimensionality process. identify the library endmembers by covariance similarity approach [40] and compute the abundance of the endmembers by sparse unmixing by split augmented Lagrangian (SUnSAL) method [41]. In the spectral unmixing process, we assume the spectral library as an over-complete collection of endmembers. We quantify the performance of the unmixing method using the signal to reconstruction ratio. The term signal to reconstruction ratio computes the ratio between the signal power and reconstruction error in the logarithmic scale. The term is computed using-SRE = 10 log 10 where, X is the original hyperspectral data, andX is the hyperspectral data reconstructed by the unmixing or dictionary pruning algorithm. A compression method with higher SRE is considered to be more efficient as it obtains a better approximation.

CONCLUSION
This paper proposes a novel multi-way approach for hyperspectral image/video compression. The proposed method that performs efficient compression without affecting the performance of succeeding applications. The proposed approach obtains a sparse data by computing the difference between the consecutive spectral bands of hyperspectral images. In the case of video data, we compute the difference between the consecutive timeframes of hyperspectral video. Unlike other approaches, the use of sparsity and multidimensional tensor decomposition incorporates both spatial as well as spectral information into account, and thus capture the inherent structural properties of the data more effectively. We demonstrate the proficiency of our proposed algorithm on several real as well as synthetic images and present the unmixing as well as classification performance.