1. Introduction

In recent years, the use of hyperspectral images has become more and more widespread due to the current development of hyperspectral sensors technology. However, the growing scientific and technological demands in spatial and spectral resolutions have drastically increased the data volume of hyperspectral images. Hyperspectral images contain large amounts of information. Spectral information is quantized into 224 contiguous bands, of approximately 10 nm each, with a spatial resolution of 20 m at operational altitude. Spectral components are sampled with a 12-bit ADC and then represented with 16-bit precision after calibration and geometric corrections. The unit size of the recorded image is called a scene, a data cube of 512 lines by 614 columns by 224 bands, for a total of 140 MB. However, large data volume of hyperspectral images introduces difficulties in transmission and storage; there is the need of reducing the data size in order to match the available bandwidth. Undoubtedly, data compression is the appropriate solution to this problem. However, any distortion caused by lossy compression of the image data is unacceptable in many of the corresponding applications such as automatic feature extraction, classification, target detection, and object identification. As a result, only lossless compression of hyperspectral images can guarantee both the requirements of data reduction and the original quality of the data.

Unlike 2D images, hyperspectral images have two types of inherent correlation: intraband correlation between neighboring pixels in the same band and interband correlation between pixels in adjacent bands. Hyperspectral image coding schemes explore data correlations for compression. Traditional approaches to the hyperspectral images compression are mainly based on differential pulse code modulation (DPCM), direct vector quantization, or dimensionality reduction through principal component analysis (PCA) [1]. DPCM basically consists of a prediction followed by entropy coding of quantized differences between original and predicted values. A unit quantization step size allows reversible compression to be achieved as a limit case. Recently, an adaptation of the context based, adaptive, lossless image codec (CALIC) to on-board hyperspectral data compression has been presented in [2]. The look-up table (LUT) [3] approach exploits the calibration-induced data correlation that is specific to hyperspectral images. To predict a pixel, the pixel value of the collocated pixel in the previous band is used as a key to search an LUT.

In a recent publication [4], Mielikainen introduced a very simple prediction given by the value taken on the current band by its nearest neighbor (NN), i.e., the spatially closest pixel, previously encountered along the scan path, having the same value as that at the current position on the previous band. Such a prediction, which is computationally very simple, can be effectively implemented by means of a dynamically updated lookup table, which is indexed by the value at the current pixel position in the previous band and contains the value of the NN previously taken from the band being encoded. The rationale of prediction based on LUTs has been later extended by Huang and Sriraja in [5] by exploiting two LUTs, respectively containing the first and second NNs in the current band. To yield the current pixel prediction, the choice between the two values contained in the two LUTs, indexed by the radiance level of the current pixel in the previous band, is based on the similarity to a reference prediction, which takes into account the cross gain between the current and the previous bands, as indicated by its acronym locally average interband scaling (LAIS). There are many excellent wavelet-based three-dimensional image compression algorithms such as 3D-SPIHT and 3D-SPECK for lossy or lossless hyperspectral images compression. 3D-SPIHT is the benchmark for three dimensional image compressions. It has been applied on multispectral image compression by Dragotti. Xiaoli Tang and William A. Pearlman use vector quantization (VQ) and Karhunen Loève transform (KLT) on the spectral dimension to explore the correlation between multispectral bands [6].

In [7], Penna et al. apply a 3-D wavelet transform to decorrelate the image in the spectral and spatial domains and develop a progressive coding scheme that works from lossy to lossless coding and complies with the second part of the JPEG 2000 Standard. Transform-based schemes can yield excellent coding gain for lossy compression at low bit rates while their lossless coding performance is inferior to these specialized lossless compression schemes. Recently, Zhang and Liu [8] proposed a two-step adaptive spectral-band-reordering algorithm. First, the bands are classified into groups based on the correlation factor of adjacent bands, and then, a reordering algorithm based on the Prim algorithm is applied to each group. A prediction method called ABPCNEF is proposed to take advantage of the similarity of structure and pixel relationship between two neighboring spectral bands and the residual is coded using adaptive arithmetic coding. In [9], M.Weinberger, G. Seroussi, and G. Sapiro proposed a JPEG-LS algorithm. JPEG-LS is a simple and effective algorithm, but in its current form, it can only handle 2-D and not 3-D images. However, since JPEG-LS has been developed to work with two-dimensional (2-D) data and do not exploit redundancy in disjoint bands, their compression ratios are very poor. In [10], J. Mielikfiinen, A. Kaarna, and P. Toivanen proposed an interband linear prediction approach (LP) based on least-squares optimization. In [11], Rizzo et al. proposed Spectral-oriented Least Squares (SLSQ), in which spectral correlation is exploited using linear prediction, and the prediction error is then entropy coded.

As an extension to the LP algorithm, we propose a lossless compression method which consists of intraband prediction and hybrid context prediction followed by arithmetic coding. Our primary goal is to develop a fast and efficiency compression algorithm. Simulation results show that our algorithm achieves higher compression ratios than others. This letter is organized as follows: The algorithm based on hybrid context prediction is presented in Section II; Section III describes experiments with AVIRIS images, together with improvements to the baseline; Section IV concludes and discusses future research.

2. Proposed algorithm

Different from 2D images, hyperspectral images are treated as a three-dimensional (3D) data set for the purposes of compression. Besides structural correlation can be found in 2D images, hyperspectral images have two other types of intrinsic correlations: intraband correlation between neighboring pixels in the same band and interband correlation between pixels in adjacent bands. However, the dynamic range and noise level of AVIRIS data (instrument noise, reflection interference, vibrations of the imager, etc.) are higher than those in photographic images. Assuming that the behavior of every image pixel can be predicted according to the information provided by its causal neighborhood, the interband redundancy can be exploited using interband prediction. A salient property of hyperspectral images is that strong spectral correlation exists throughout almost all bands. Due to the feature of spectral correlation in hyperspectral images, interband (spectral) prediction outperforms intraband (spatial) prediction in most cases. For this reason, a spatial predictor like the median predictor in JPEG-LS tends to fail in this kind of data. Motivated by these considerations, a new algorithm for lossless compression of hyperspectral images using hybrid context prediction is proposed. The block diagram of the proposed compression scheme is shown in Fig. 1 . It consists of a decorrelation stage and an entropy coding module. The decorrelation prediction stage supports both intraband and interband predictions. The intraband (spatial) prediction uses the median prediction model, since the median predictor is fast and efficient. The interband prediction uses hybrid context prediction. The LP and context prediction composes the hybrid context prediction. Finally, the residual image of hybrid context prediction is coded by arithmetic coding.

 

Fig. 1 Block diagram of the proposed lossless hyperspectral images compression scheme using hybrid context prediction.

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2.1 Intraband prediction

The decorrelation prediction stage supports both intraband and interband predictions. Intraband prediction is applied only to the first image along the spectral line. The median predictor is employed here due to its simplicity and efficiency for still images. The median predictor is also the standard predictor for the first band marked for intraband coding (IB set). Let x be the current pixel and NW (northwest), N (north), and W (west) denote three neighboring pixels, respectively. The estimate of pixel x is given as follows:

2.2 Hybrid context prediction

In our algorithm, we propose a new approach that uses a new interband linear predictor for bands marked for interband coding. The interband predictor relies on a small causal data subset of the pixel x to compute the prediction. The interband prediction is formed by simply adding the average difference between the current band and the previous one to the value of x. LP assumes that the interband prediction is likely to perform poorly and corrects the prediction by adding to it the average prediction error.

Through comparing the value of the previous pixel and the predicted value of the current pixel, and analyzing the relations of the previous pixel and the current pixel, we propose the hybrid context prediction. In general, the lossless compression based on hybrid context prediction can be expressed in Eq. (2), in which the predictor is the linear combination of pixels of the previous band. Where y and $\widehat{y}$ are the actual and the predicted values of the current pixel, x is the previous pixel, x_a, x_b, x_c are the left, upper and upper left neighbor of x, y_a, y_b, y_c are the left, upper and upper left neighbor of y. If we have an accurate prediction, $\widehat{y}$ is expected to be close to y.

Suppose we have two adjacent bands. We use the previous band pixel value x_a, x_b, x_c, x and the current band value y_a, y_b, y_c to predict the value of y. We can get horizontal gradients (|x_b−x_c|, |x−x_a|, and |y_b−y_c|), vertical gradients (|x_c−x_a|, |x_b−x|, and |y_c−y_b|) and spectrum gradients (|x_a−y_a|, |x_b−y_b|, and |x_c−y_c|). Firstly, we compare the value of spectrum gradients |x_a−y_a|, |x_b−y_b|, with |x_c−y_c |.

If |x_c−y_c| ≥ max [|x_b−y_b|, |x_a−y_a|], it means that the tendency of interband pixel transform ratio of horizontal and vertical gradients will decrease. In this case we use [|x_a−y_a| + |x_b−y_b| + |x_c−y_c|]/3 to amend the prediction function.

If |x_c−y_c| ≤ min [|x_b−y_b|, |x_a−y_a|], it means that the tendency of interband pixel transform ratio of horizontal and vertical gradients will increase. Accordingly we use [|x_a−y_a| + |x_b−y_b|]/2 to amend the prediction function

Otherwise, if |x_c−y_c| is in the range of min [|x_b−y_b|, |x_a−y_a|], max [| x_b−y_b|, |x_a−y_a|], it means that the tendency of interband pixel transform ratio of horizontal and vertical gradients are different. Therefore we use [|x_a−y_a| + |x_b−y_b|− |x_c−y_c|] to amend the prediction function.

3. Experimental result and comparisons

In order to test the performance of the proposed algorithm, some experiments are carried out. The hyperspectral images for test are the four scenes of the sequences Cuprite Mine and Lunar Lake in Nevada, Moffett Field and Jasper Ridge in California. All images comprise of 224 bands recorded at different wavelengths in the range 380 to 2500 nm, with a nominal spectral separation of 10 nm between two adjacent bands. Each image is constituted by a variable number of scenes of size 512 lines by 614 columns. All data that have been considered for compression are in radiance units, 16-bit format.

We compare the proposed lossless compression algorithm with some of the existing algorithms for hyperspectral images. Besides 3D-CALIC, M-CALIC, LUT, LAIS-LUT, LUT-NN, the other methods compared are the clustered DPCM (C-DPCM), JPEG-LS, LP, and the spectral-oriented least squares (SLSQ) encoder. Table 1 shows the lossless bit rates that are produced by these algorithms. All scenes of each image have been compressed. The standard 3D-CALIC has been evaluated for the lossless and near-lossless compression of hyperspectral data. Because the standard 3D-CALIC algorithm switches between the interband and intraband predictor, it achieves a compression rate of 5.11 bpp. Moreover, the M-CALIC algorithm outperforms standard 3D-CALIC by more than 0.2 bpp. By testing M-CALIC with and without the optimized model parameters and quantization thresholds, the gain is the same due to the multiband predictor and to the optimizations. The LUT approach exploits the calibration-induced data correlation that is specific to hyperspectral images. To predict a pixel, the pixel value of the colocated pixel in the previous band is used as a key to search an LUT. The LUT approach outperforms M-CALIC by more than 0.14 bpp. The LAIS-LUT is an improved and optimized LUT approach, and it yields a slightly improved performance than LUT. The LUT-NN approach is slightly worse than the LUT.

Table 1. Bit Rate Comparision of All Scenes (in Bits Per Pixel)

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C-DPCM is the clustered DPCM and is a very complex predictor, consisting of cluster and the optimal LP, both of which will consume much time. The bit rate is about 4.68 bpp. Thus, LP and SLSQ are more complex prediction algorithms, consisting of spatial prediction and spectral prediction. LP is applied with intraband (IB) (usually noisy and less correlated with other bands) and prediction threshold. SLSQ uses the same IB set. But SLSQ compression algorithm is worse than C-DPCM and LUT, which are the most advanced algorithms. The proposed algorithm achieves a compression rate of 4.56 bpp. JPEG-LS is based on the predictive coding technique, where the main compression phases are prediction, context modeling, error encoding, and run mode. JPEG-LS encodes each of the scenes independently. A bit rate up to 6.62 bpp can be observed. From the results in Table 1, the proposed hybrid context prediction algorithm has the best bit-per-pixel performance. The hybrid context prediction achieves the lowest compression rate among all the tested algorithms.

In order to further verify the effectiveness of the proposed algorithm, we calculate the compression ratios. Table 2 shows the compression ratios of the proposed algorithm, as compared with other schemes. The results show that the hybrid context prediction algorithm is able to outperform in terms of compression ratio such state-of-the-art algorithms as LAIS-LUT and JPEG-LS by 2.2% and 25%, respectively. We can see that the proposed algorithm has the highest compression ratios, while the JPEG-LS algorithm is the lowest. Therefore, the proposed hybrid context prediction algorithm is effective.

Table 2. Comparision of CR Result Achieved Using these Algorithms

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Then we choose AVIRIS Lunar Lake images bands from 121 to 130 for testing. For convenience, we resample it as 512 × 614, 16 bpp precision. It can be observed that the hyperspectral images have strong interband correlation. We use LP, JPEG-LS, LUT, 3D-CALIC, C-DPCM, and our algorithm to compress the images respectively. Table 3 shows the entropy of several test images. Ten images from Lunar Lake B121, to Lunar Lake B130 have been used for examination. They are respectively denoted as B121 …B130 in Table 3 for abbreviation. The results clearly show that the proposed algorithm outperforms any other scheme examined here by an extent of 4% up to almost 12%. We can see that the proposed algorithm has the lowest entropy among all other examined algorithms, while the JPEG-LS algorithm has the highest. Therefore, the proposed hybrid context prediction algorithm is effective for hyperspectral images compression. Figure 2 shows that our algorithm outperforms the compared algorithms in reducing the spatial and spectral redundancy.

Table 3. Comparision of Bands' Average Entropy

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Fig. 2 Performance of different algorithms for AVIRIS lunar lake images bands.

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In order to evaluate the complexity of the proposed algorithm, we have run some of the algorithms on a workstation with Dual Pentium Xeon 2.4-GHz processor and Linux operating systems. We have measured the CPU time employed by each algorithm by using the clock () function and have averaged the obtained values over a number of trials. The results are reported in Table 4 . JPEG-LS was conceived with the aim of low computational complexity, however its performance is not state-of-the-art for hyperspectral image coding. Other approaches conceived specifically for hyperspectral image coding have better performances at a cost of increased computational complexity, such as C-DPCM or 3D-CALIC. Our approach outperforms all the other tested proposals, and still has a computational complexity comparable to the simple LP encoder.

Table 4. Encoding Time of Various Algorithms

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4. Conclusion

We have developed a new efficient lossless compression algorithm that consists of intraband prediction and hybrid context prediction followed by arithmetic coding. It has been tested on AVIRIS hyperspectral images and produced very high compression ratios due to the successful hybrid context prediction. The new algorithm has been evaluated by comparing the obtained results with results produced by a number of lossless compression algorithms tested using the same set of hyperspectral images. The primary goal of this work was to develop a fast and efficiency compression algorithm. Simulation results show that our algorithm achieved this goal. Promising direction for the future work would be to study new lossless compression algorithm through bands grouping and bands reordering for hyperspectral images. As the high sensor-data rates of present and future hyperspectral missions call for simple and fast compression techniques, the proposed algorithm proved to be a good option for lossless hyperspectral image compression.