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Compression of hyper-spectral images using an accelerated nonnegative tensor decomposition

  • Jin Li EMAIL logo and Zilong Liu
Published/Copyright: December 29, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

Nonnegative tensor Tucker decomposition (NTD) in a transform domain (e.g., 2D-DWT, etc) has been used in the compression of hyper-spectral images because it can remove redundancies between spectrum bands and also exploit spatial correlations of each band. However, the use of a NTD has a very high computational cost. In this paper, we propose a low complexity NTD-based compression method of hyper-spectral images. This method is based on a pair-wise multilevel grouping approach for the NTD to overcome its high computational cost. The proposed method has a low complexity under a slight decrease of the coding performance compared to conventional NTD. We experimentally confirm this method, which indicates that this method has the less processing time and keeps a better coding performance than the case that the NTD is not used. The proposed approach has a potential application in the loss compression of hyper-spectral or multi-spectral images

Keywords: Multi/hyper-spectral image compression; nonnegative tensor decompositon (NTD); pairwise multilevel Tucker Decomposition (PM-TD)
PACS: 07.05.Pj; 42.30.Va

1 Introduction

HYPER-SPECTRAL camera plays a important role in earth observation applications because it can simultaneously acquire a three dimensional data with the spectral and spatial information. Unfortunately, the data amount of a hyper-spectral image is greatly increased with the improvement of the spatial and spectral resolution of hyper-spectral cameras. Moreover, on-board resources of a spacecraft, such as the storage capacity and the transmission band-width, are limited. Therefore, a compressor of hyper-spectral images is the key to hyper-spectral cameras.

Most compression algorithms of hyper-spectral images have a 3-D transform stage, such as 3-D Karhunen-Loeve transform (3-D KLT) [1, 2], 3-D discrete cosine transform (3-D DCT), and discrete wavelet transform (3-D DWT) [3]. In 3-D DWT domain, the wavelet coefficients at same spatial position can be organized into a 3D block. The 3D wavelet block can be used to remove the spatial and spectral redundancies through a 3D entropy encoder. Many well-known 3D entropy encoders of hyper-spectral images, such as 3-D EZW [4], 3-D EBCOT [5], 3-D SPIHT and 3-D SPECK [6], are proposed. The JPEG2000 [7] algorithm is also used in hyper-spectral images when it perform a 3-D mode, where a 3-D DWT is used as the transform tool and a 3-D EBCOT as the coefficient encoder. This method is usually taken as a reference technique. Fully different from the ground compressor, a hyper-spectral image compression algorithm requires a low-complexity and error resilience encoder because it usually works on board, where the energy and memory resources are limited. Moreover, an on-orbit compressor of hyper-spectral cameras usually has the requirements of convenient implementation on a hardware platform, such FPGA processors. The JPEG2000 and 3-D transform methods can provide high compression performance. However, they have the excessive implementation complexity [8]. The aforementioned methods usually need to perform the corresponding improvement and optimization based on the different applied cases.

Recently, some compression methods use a third-order tensor decomposition technology [9, 10] because a hyper-spectral image is a 3D data, including two spatial dimensions and one spectral dimension. This method can simultaneously remove redundancies of spatial and spectral information in hyper-spectral images. Currently, the tensor decomposition methods focus on the Tucker decomposition (TD) in hyper-spectral images because it can freely control the size of a core tensor in three directions, which is advantageous for the removing of spatial and spectral redundancies. In [11], Karam et al. perform the TD decomposition in a discrete wavelet transform domain. They use a (9/7) bi-orthogonal wavelet to perform the transform processing. In their method, wavelet sub-bands are organized into a 3D tensor, which is decomposed by a Tucker decomposition method. In our previous works [12, 13, 14], we also investigated the low-complexity NTD methods. In [15], a new hyper-spectral image compression approach using a hybrid scheme based on three dimensional discrete cosine transform (3D-DCT) and Tucker Decomposition (TD) is proposed. From the existing documents, the nonnegative tensor Tucker Decomposition (NTD) in transform domain (e.g., 2D-DWT, etc) has been shown a better performance for hyper-spectral image compression because of not only exploiting redundancies between bands but also using spatial correlations of every band image. However, the NTD suffers a very high computational cost.

In our previous works [16], we used the NTD to compress multispectral images with comparatively few bands. Because the specific multi-spectral images only have relatively few bands, the computation cost can be accepted. However, the computation complexity will be tremendously increased when the band number is gradually increased. To overcome this computational cost of the use of the NTD, we proposed a fast pair-wise multilevel NTD (FPM-NTD) structure for hyper-spectral image compression in this paper.

2 NTD hyperspectral compression theory

The key idea of hyper-spectral image compression using a NTD is to apply the NTD to the transform coefficients of hyper-spectral images, taking advantage of both spatial and inter-band redundancies. First, a 2D transform (such as 2D-DWT, 2D-DCT) is applied to each spectral band of hyper-spectral images to obtain some sub-images for each spectral band. The transform coefficients can be considered as a 3D tensor. The tucker decomposition (TD) can effectively process the tensor to achieve higher compression results.

A given third order TD tensor is denoted by Y ∈ RI1×I2×I3. The decomposition of the tensor is described into an unknown core tensor, denoted by G ∈ RJ1×J2×J3, multiplied by a set of three unknown component matrices, denoted by A(n)=[a1(n),a2(n),…,aJn(n)]∈RIn×Jn(n=1,2,3).Figure 1 demonstrates the decomposition process of a third order TD tensor. As the above NTD process, the third-order NTD tensor can be represented as:

A=∑j1=1J1∑j2=1J2∑j3=1J3gj1j2j3aj1(1)∘aj2(2)∘aj3(3)+E=G×1A(1)×2A(2)×3A(3)+E=G×{A}+E=Y^+E≈Y^(1)
Figure 1 Nonnegative tensor Tucker decomposition
Figure 1

Nonnegative tensor Tucker decomposition

where the tensor ŷ is an estimation of Y, and it depends on the (J1, J2, J3) values, which are dimensions of the core tensor G. Tensor E is the estimation error tensor. Equation (1) can be solved by the following optimal problem:

min12Y−G×1A(1)×2A(2)×3A(3)F2s.t.G∈RJ1×J2×J3,A(n)∈RIn×Jn,n=1,2,3(2)

To find optimal component matrices A and G, most algorithms are based on alternating least squares (ALS) and hierarchical nonnegative Tucker decomposition algorithm. However, the large amount of hyper-spectral images, having hundreds of spectral bands, increases the complexity and time of processing. In this paper, we proposed a fast pairwise orthogonal nonnegative tensor decomposition method for the hyper -spectral image compression.

3 Proposed FPM-NTD

Weighing the computational complexity and compression performance in this paper, we propose a fast pairwise orthogonal nonnegative tensor decomposition method for the hyper-spectral image compression. A FPM-NTD adopts a multiple-level structure of NTD. At every level, multiple sub-tensors firstly are organized. Each sub-tensor is composed of the pair of consecutive spectral-components in a hyper-spectral 3D image. After that, a two-component NTD is used to decompose each sub-tensor. The same process is applied to each level. In the first level, the components of each sub-tensor are from original 3D wavelet coefficients. In the successive levels, each sub-tensor is established using one component of the resulted core tensor of the previous level. Figure 2 shows an example of a FPM-NTD, where eight spectral-components are used. The number of original 3D tensor or core tensors may be an odd number in a decomposition level. When the case exists, the unpaired spectral-component is directly moved on the next level.

Figure 2 A proposed pairwise multilevel structure for an image of eight component
Figure 2

A proposed pairwise multilevel structure for an image of eight component

To define a spectral multilevel grouping TD structure, the notation F = {flevel,index} is used. Fl is the set of groups in level lof the structure F. For each group, let Size (fi,j) be the size of the group and Th(fi,j) be the size of component in said group that proceeds to the next level. That is, (fi,j) is the J1 × J2 × J3 in the level, and Th(fi,j) is the J1 × J2 in next level. At each level, the resulted core tensor of each of two-component NTDs operates has the same the size of J1, J2, which denotes the size of each band. J3, denoting the number of bands, may be different for each resulted core. For further decomposing, the one component for the next level can be obtained from the resulted core tensor. The selection is achieved by maximization the l1 norm of one component of the resulted core tensor. Maximizing the l1 norm, that is, the sum of the magnitude of the coefficients of the component, favors the redundancy of the selected component. The selected component is the one with the highest magnitude coefficients:

g∗=arg⁡maxgj1j2j3,j3∈[1,J3]gj1j2j31gj1j2j31=∑j1J1∑j2J2gj1j2j3(3)

This l1 norm maximization leads to a rate-distortion optimization adapted to hyper-spectral compression. The rate for compression the selected component g* is minimized since there is a high occurrence rate of large coefficients in that component and these high coefficients will be further decomposing to reduce the redundancy between coefficients.

In this paper, at each level, we apply HALS-NTD algorithm to each of the two-component NTDs. In the HALS-NTD algorithm, NTD factors A(n)(n = 1,2,3) and core tensor G can be summarized as follows:

ajn(n)←Y(n)jnG×−n{A}(n)jnT+,ajn(n)←ajn(n)/ajn(n)2gj1j2j3←gj1j2j3+E×1aj1(1)×2aj2(2)×3aj3(3),(jn=1,...,J3)(4)

To low-complexity, we use a fast method in [17] to perform alternative update rules as

ajn(n)←Y(n)jnG×−n{A}(n)jnT+,ajn(n)←ajn(n)/ajn(n)1,(jn=1,...,J3),G←G⊛Y⊘Y^×AT+(5)

The HALS-NTD algorithm can be shown as Algorithm 1 [17].

Algorithm 1

HALS-NTD Algorithm

Input:Y:inputdataofsizeI1×I2×I3,J1,J2,J3:numberofbasiscomponentsforeachfactorOutput:3factorsA(n)∈RIn×Jn(n=1,2,3)andacoretensorG∈R∘J1×J2×J3BeginInitializationA(n)andGforn=1,2,3.Normalizeallajn(n)forn=1,2,3tounitlengthE_=Y−Y^RepeatForn=1to3doForjn=1toJndoY(n)(jn)=E(n)+ajn(n)[G(n)]jnU⊗−nTajn(n)←ajn(n)/||ajn(n)||1En←Y(n)(jn)−ujn(n)[G(n)]jnA⊗−nTEndEndG←G⊛Y⊘Y^×AT+Foreachj(n)=1,2,⋯,J(n),Forn=1,2,3dogj1j2j3←gj1j2j3+E×1aj1(1)×2aj2(2)×3aj3(3)E←E+Δgj1j2j3aj1(1)∘aj2(2)∘aj3(3)EndUntilastoppingcriterionismetend

The size of the core tensor, i.e. (J1, J2, J3) can be determined by compression rations. For all level, each level has different size of the core tensor. The size can be obtained by minimization of the Lagrangian rate-distortion cost:

L(Yq)=D(Yq)+λR(Yq)(6)

where D(Yq) is the square error between Y and Yq after a quantization step q, λ is a Lagrangian multiplier and R(Yq)is an estimation of the bit-rate needed to encode the quantized Yq.

4 Results

We use hyper-spectral images to test the proposed method for verifying its feasibility. AVIRIS hyper-spectral images are used as the input testing images. The image size is 256×256×200. The bit depth of each pixel is 8bit. We use a computer, with the frequency of 1.87 GHz and two-core CPU, to perform the experiments. In our experiments, a wavelet transform is used as the removing of spatial redundancies of intra-band. We use a traditional NTD method and a compression method without NTD. The traditional NTD method firstly adopts the wavelet transform as the spatial transform. After that, all wavelet sub-bands of all spectral bands are considered into a 3D tensor. The 3D tensor is decomposed by our previous method. We also remove the NTD processing, only using wavelet transform in combination with a bit plane encoder, as a reference method. We use the peak signal to noise ratio (PSNR) as the evaluation parameters. Figure 3 shows the PSNR of three methods. The using of NTD can improve the PSNR because it can remove spectral redundancies. Our method is lower than conventional NTD method because our method adopts the multiple-level NTD and ignores the part of spectral redundancies. Figure 4 shows the comparison of compression time of three methods. The method without NTD has the shortest running time. Our method is obviously less than the conventional NTD method. Although our method has a small part of loss of compression performance, the compression time is obviously decreased. We also compare our method and other hyper-spectral compression methods in PSNR. Figure 5 shows the comparison of PSNR of five hyper-spectral image compression methods. The results shows the PSNR of our method is higher than other methods, which means the proposed method has the better compression performance because it can remove redundancies between spectrum bands and also exploit spatial correlations of each band. In future research, the computation complexity can be reduced further by introducing compressive sensing [18] into the proposed method, as well as improving processing speed by the use of graphics processing unit (GPU) accelerated computing [19, 20, 21].

Figure 3 Comparison of compression performance using conventional NTD, our method, and without NTD at different bit rates, PSNR is used as the evaluation parameters
Figure 3

Comparison of compression performance using conventional NTD, our method, and without NTD at different bit rates, PSNR is used as the evaluation parameters

Figure 4 Comparison of computational time using conventional NTD, our method, and without NTD at bands
Figure 4

Comparison of computational time using conventional NTD, our method, and without NTD at bands

Figure 5 Comparison of compression performance using other hyper-spectral image compression methods at different bit rates
Figure 5

Comparison of compression performance using other hyper-spectral image compression methods at different bit rates

5 Conclusion

In this paper, we propose a pair-wise multilevel grouping approach for the NTD, which can obtain a better coding performance and does not have high computational cost. The proposed approach might be a good replacement for the wavelets as a spectral de-correlator in many of situations, where the NTD is not a suitable option.

Acknowledgement

National Natural Science Foundation of China (NSFC) (61505093, 61505190); National Key Research and Development Plan (2016YFC0103600).

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Received: 2017-9-13
Accepted: 2017-11-7
Published Online: 2017-12-29

© 2017 Jin Li and Zilong Liu

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2017-0123
Keywords for this article
Multi/hyper-spectral image compression; nonnegative tensor decompositon (NTD); pairwise multilevel Tucker Decomposition (PM-TD)
Creative Commons
BY-NC-ND 4.0

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