Edge-Based Prediction for Lossless Compression of Hyperspectral Images*

Sushil K. Jain and Donald A. Adjeroh VIP Lab, Lane Department of Computer Science

and Electrical Engineering, West Virginia University Morgantown WV 26506, USA [sushilj, don]@csee.wvu.edu

Abstract– We present two algorithms for error prediction in lossless compression of hyperspectral images. The algorithms are context-based and non-linear, and use a one-band look-ahead, thus requiring a minimal storage buffer. The first algorithm (NPHI) predicts the pixel in the current band based on the information from its context. Prediction contexts are defined based on the neighboring causal pixels in the current band and the corresponding co-located causal pixels in the reference band. EPHI extends NPHI using edge-based analysis. Prediction is performed by classifying the pixels into edge and non-edge pixels. Each pixel is then predicted using information from pixels in the same edge class within the context. Empirical results show that the proposed methods produce competitive results when compared with other state-of-the-art algorithms with comparable complexity. On average, the edge-based technique (EPHI) produced the best overall result, over the images in the test dataset. Keywords: hyperspectral images; lossless compression; non-linear prediction

1 Introduction With the advent of satellite imaging, huge amounts of data are captured daily and transmitted back to earth for further analysis and storage. Improved sensor technology in hyperspectral imagery has made it possible to capture images at very narrow wavelengths resulting in hundreds of spectral bands being captured for a scene. AVIRIS (Airborne Visible/Infrared Imaging Spectrometer) sensors capture 224 bands in the wavelength range of 400nm to 2500nm in steps of 10nm. With constraints on available bandwidth and storage space, there is a need for efficient representation and storage of this data. Standard algorithms such as JPEG-LS [14] and CALIC [15] perform poorly on hyperspectral images [13, 2, 16]. Hyperspectral images have a stronger spectral correlation than spatial correlation, hence spectral prediction algorithms work better than spatial prediction algorithms. The standard lossless image compression algorithms have been optimized to compress 8 bit images, while hyperspectral images have varying bit depth across the bands ranging from 6 to 16 bits per pixel† per band.

Hyperspectral images generally have a similar global structure across bands, but nearby bands could have different pixel intensities due to different absorption properties of the atmosphere and the material surface being imaged. Similarly, the general edge and non-edge features remain similar across bands, but the local pixel variation could be different. There is a need to capture this variation at the time of prediction to reduce the prediction error. Pixel correlation is higher along an edge then across edges, and hence appropriate edge classification of pixels could be used for improved compression. In this paper, we exploit this global structure and the stability in edge features across bands for improved prediction in hyperspectral image compression. In the next section we present a background to the problem and related work. Section 3 describes our basic algorithm NPHI - Non-

* This work is partly supported by an NSF grant: IIS-0312484. † We use the term “pixel” to refer to one sampled position in a band, rather than one position across all bands.

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linear Prediction for Hyperspectral Images. This algorithm is extended for edge-based prediction in Section 4. Results are presented in Section 5. The paper is concluded in Section 6.

2. Background and Related Work 2.1 Background Hyperspectral imaging involves the acquisition of spatially co-registered images, in hundreds of spectrally-contiguous bands. The basis of hyperspectral imaging is the fact that all materials reflect, absorb, and emit electromagnetic energy, at particular wavelengths, and in distinctive patterns determined by their molecular composition [7]. The sun is the basic source of light energy and hence dominates the spectrum of light captured. Every material has its own signature absorption and reflectance property which can be used to characterize the material. For instance, vegetation has higher reflectance in the infrared region than in the visible spectrum, since plants have a high chlorophyll content which absorbs visible light while reflecting infrared light. In comparison, soil has a higher absorption in the infrared region.

Atmospheric absorption is another important characteristic of hyperspectral images. Fig. 1 shows absorption properties for water and the dominant gases present in the atmosphere. We notice the high absorption of spectral energy around 1.4 µm and 1.9 µm due to carbon dioxide and water. This corresponds to bands 107 to 114 and bands 153 to 160 respectively in the AVIRIS dataset. The interaction of light with the atmosphere reduces the energy of light in these spectral ranges resulting in the capture of very little information about the scene. Thus, the incoming spectral information is dominated by sensor noise, resulting in poor spectral correlation. For compression, these bands are best predicted using spatial predictors, or may even be encoded without prediction.

Hyperspectral imaging has become an important tool in various remote sensing tasks, such as mineral exploration, environmental monitoring, agriculture, and resource management. AVIRIS‡ sensors capture 224 images/spectral bands in the range of 400nm to 2500nm, covering the visual, near infrared, and short wave IR spectra, with each band having a range of 10 nm. A typical AVIRIS scene consists of 512 lines and covers a ground area of 10 km diameter, yielding about 140 MB of data.

2.2. Lossless compression of hyperspectral images The huge data volumes involved in hyperspectral imagery calls for methods for efficient storage and representation. Popular lossless image compression schemes, such as JPEG-LS, CALIC, and JPEG2000 that perform well on natural images do not perform well on hyperspectral images [13, 2]. The major problem is the fact that these methods largely ignore the spectral correlation between bands. A key issue in hyperspectral image compression is how the predictor is constructed. A good prediction scheme should be able to model the pixels into smooth regions, edge regions and textured regions. Usually the smooth regions yields lower prediction errors compared to the edge or textured regions. For hyperspectral images, such modeling should consider the spatial and spectral redundancy observable in the data. ‡ AVIRIS dataset is available at: http://aviris.jpl.nasa.gov/html/aviris.freedata.html

Fig. 1: Spectral curves for atmospheric absorption [3]

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Rizzo et al [11] proposed two predictors – an interband linear predictor and a least square based approach for lossless compression of hyperspectral images. The linear prediction scheme (LP) computes the average difference between the context of the pixel ( , , )I x y t in the current band and the context of the pixel ( , , 1)I x y t − in a reference band. If the deviation between the minimum and maximum is greater than a threshold, then the prediction is corrected by adding the average prediction error of the two previous bands. In [11] it was observed that the first 8 bands are best predicted using intraband prediction. Thus, the linear prediction scheme operates in two modes, switching between interband (spectral) and intraband (spatial) prediction as needed. The standard median predictor was used for intraband prediction. The least square based prediction scheme proposed in [11] is fully interband. SLSQ (Spectral Oriented Least Square) utilizes 4 neighboring pixels from the current and the previous band, to optimize the prediction at each pixel in each band. Edge-based least square approaches have been used for lossless compression of natural images [1, 8]. They optimize the prediction parameters along the edge hence improving the prediction at edge pixels. One major advantage of SLSQ is its low complexity – requiring only 4 multiplications and 6 additions at each step of prediction.

Wang et al [13] proposed CCAP, (Correlation based Conditional Average Predictor) for hyperspectral images. The motivation was the fact that hyperspectral images have the same global structure, hence, the relationship between a pixel and its surrounding in one band should be similar to that of a co-located pixel in a nearby band. Thus, the correlation is computed between the context of the pixel in the current band and the context of the co-located pixel in the 5 previous bands. The context having the highest correlation with the context in the current band is used to predict the pixel in the current band. To improve the robustness of the prediction, rather than one single prediction value, the average prediction from all the pixels with a prediction context having correlation coefficient higher than 0.95 is computed. CCAP produced the best published result for Cuprite (compression ratio of 3.53), however, on average, it did not perform as well on other images (when compared with other published methods).

Given the performance of CALIC [15] in compressing natural images, there is an interest in extending this algorithm for hyperspectral images. Two extensions of the original CALIC algorithm, namely 3D CALIC and M-CALIC were proposed in [16] and [2] respectively. Overall, M-CALIC produced a better result than 3D CALIC. Other proposed approaches include vector quantization [12, 9] and clustering-based schemes [4]. We focus on the problem of prediction in lossless compression of hyperspectral images. In particular we propose a context-based non-linear prediction scheme aimed at exploiting the significant spatial and spectral correlation in hyperspectral images. We then extend this scheme to an edge-based predictor that exploits the edge structure in these images. While edge-based prediction has been used previously in lossless compression of natural images [1, 8], to our knowledge, this is the first time edge-based analysis is being used in non-linear prediction for lossless compression of hyperspectral images.

3. NPHI: Non-linear Prediction for Hyperspectral Images 3.1 Three-Mode Prediction Motivated by the nature of hyperspectral images (HSI), we propose a three-mode approach to their compression. Fig. 2 shows a plot of the prediction performance for spectral prediction and spatial prediction using Jasper Ridge, scene 3. Similar plots were obtained for the other AVIRIS images. We notice how the entropy plots closely follow the plot in Fig. 1. Simple spatial prediction leads to a significant reduction in the entropy of the prediction residuals (compare with the self entropy). Expectedly, on average, spectral prediction resulted in the least entropy for the prediction residues.

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An interesting observation, however, is that, for some band ranges, the self entropy is significantly lower than the entropy obtained after prediction, be it spatial or spectral. This is particularly the case for bands 150 to 160 (corresponding to around 1.9 µm). We also see that for bands 1 to 15, and 105 to 115, spatial prediction performs better than spectral prediction§. These bands have poor spectral correlation and hence perform poorly with spectral predictors. This observation motivates a three-mode prediction scheme for hyperspectral images: no prediction, spatial-only prediction, and spatio-spectral prediction (or simply spectral prediction). This not only improves the compression performance, but also efficiency.

3.2 Spatial Prediction and No Prediction For the bands that require spatial prediction, we observed a high correlation between the pixels and their vertical neighbors. Thus, we use a simple predictor: ˆ( , , ) ( 1, , )I x y t I x y t= − . This worked better than other spatial predictors, such as JPEG’s seven predictors, or JPEG-LS’s median predictor. Empirically we found that bands 1 to 8 and 107 to 114 are best predicted spatially. The bands that require no prediction have high sensor noise as the reflected light in these wavelengths is absorbed by the atmosphere, and thus the intensity captured by these sensors is heavily affected by the sensor noise. They usually have very low dynamic range, with pixel values in the range of -20 to 20. In AVIRIS images these bands are in the range of 153 to 160. These bands that do not require prediction, and those predicted spatially remain same over different scenes in different images and hence can be incorporated in the compression scheme without any overhead. 3.3 NPHI We propose a context-based non-linear spectral prediction for HSI. The context is defined in terms of 12 neighboring causal pixels in the current band and 12 neighboring causal pixels in a reference band. To predict the current pixels, weights are applied to the corresponding pixels in the context in the current band. To obtain the required weights, the coefficients calculated for the pixels in the reference band are modified according to the variance and scale between the two contexts. The computed weights are fine-tuned using a feedback mechanism wherein the error in the computed weights for the pixels in the current context is used to correct the weights calculated for the pixel being predicted. Fig. 3 shows a block diagram for the proposed prediction technique. The initial block for edge analysis is only relevant to the proposed edge-based prediction (see next section).

Hyperspectral imaging involves the acquisition of images from different surface materials at hundreds of spectral bands. Since the portion of the area being imaged remains the same in all the spectral bands, the global structure remains similar across all the bands. Different materials display different absorption and reflectance characteristics at different wavelengths, and hence the intensity of the acquired image could vary quite significantly from band to band. Spectral §Rizzo et al [11] made a similar observation, but only for bands 1 to 8, and hence compressed these bands using the spatial prediction mode. It appears that this is the fist report on the extensive nature of this phenomenon, (from bands 1 to 15, and bands 105 to 115), and also on the observation of no compression for bands 150 to 160.

Fig. 2: Entropy plot for different prediction schemes

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prediction schemes proposed till date utilize one band-look ahead schemes [11], more than one band [4, 13], or average of bands [2] to predict the band under consideration. The proposed prediction scheme (NPHI) utilizes a one-band look-ahead mechanism for band prediction. The band decoded last is used as the reference band to predict the next band. The context is formed by using the nearest causal pixels to the current pixel ( , , )I x y t being predicted, and for the context in the reference band. Too large a context could result in context dilution while too small a context may not capture enough information to minimize the prediction error. We use a context size of 12. Given that the dynamic range could vary significantly between bands, prediction using simple differences may not produce good results. Similarly, global normalization is not feasible given the large dynamic range of the images which introduces error in prediction. We use a scale-based prediction, where ratios are calculated instead of linear prediction. The prediction scheme works on the basis that even with sensor noise there is some correlation between co-located neighborhoods in nearby bands. Hence if the pixel variation between two contexts can be computed efficiently then the prediction error magnitudes can be reduced. Consider ( , , 1)I x y t − , the co-located pixel (in the reference band) to ( , , )I x y t , the current pixel being predicted. Let C denote the prediction context being used, cn = context size, and xW = weights computed for the pixels used to predict ( , , 1)I x y t − . The pixels within the context and in the reference band would have already been decoded by the time encoding reaches the current pixel. Thus we can compute xW :

( , , 1)( , , 1) ,( , , 1)x

I x y tW x i y j t i j CI x i y j t

−− − − = ∈

− − − (1)

( )∑∑ −−−⋅−−−=−i j

xc

tjyixItjyixWn

tyxI )1,,()1,,(1)1,,( Cji ∈, (2)

Let yW be the actual weight needed to predict ( , , )I x y t . Ideally, xW should be similar to yW .

Thus, we expect that x yW W∝ . However, since the bands could be at different intensity levels and the light absorbed by different materials is different at different wavelengths, there is typically some variation between the two weights. Let α be a variation factor between the two contexts. We relate the weights computed for the two contexts as follows:

y xW Wα≈ ⋅ (3)

Essentially, α serves as a factor that needs to be estimated to reduce the variation between xW and yW . We estimate the local variation between the bands using the ratio between the contexts:

( , , 1)( , , ) ,( , , )

I x i y j tx i y j t i j CI x i y j t

ρ − − −− − = ∈

− − (4)

Fig. 3: Block diagram for proposed prediction scheme

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Since the bands may have different pixel intensity levels, the entries in ρ could be affected by the scale factor between the two contexts. To nullify this, we determine the error incurred in computing the weights after predicting each pixel, and then use this to correct the ratios. Similar to (1) and (2) we compute the actual weights used for each pixel in the context of ( , , )I x y t :

( , , )( , , ) ,( , , )

cc

c

I x y tW x i y j t i j CI x i y j t

− − = ∈− −

(5)

( )1( , , ) ( , , ) ( , , ) ,c c cc i j

I x y t W x i y j t I x i y j t i j Cn

= − − ⋅ − − ∈∑∑ (6)

Subscript ‘c’ denotes that the calculations are made for each pixel in the context using their predicted values. The ratio between the actual weights of pixel ( , , )cI x y t in the current band and of pixel ( , , 1)cI x y t − in the reference band will give the estimate needed to modify xW to obtain

yW . The correction factor used to correct the ratio between the two contexts is then given by:

( , , )1 ( , , ) ,( , , 1)c

c cc ci j

W x i y j tf x i y i t i j Cn W x i y j t

ρ⎛ ⎞− −

= − − − ∈⎜ ⎟− − −⎝ ⎠∑∑ (7)

cf is computed for all the pixels after they have been predicted. We note that these only need to be computed once for each position in the image, since we can store and reuse previously computed values. The size of the buffer required for such storage is just the same as the context size. The final result will be a set of values ),,( tjif , Cji ∈, , with one entry for each position in the context. Using f, α is then computed as:

( )1( , , ) ( , , ) ( , , ) ,c i j

x i y j t x i y j t f x i y j t i j Cn

α ρ− − = − − + − − ∈∑∑ (8)

The weights for the pixels in the context of ( , , )I x y t are then estimated as follows: ˆ ( , , ) ( , , ) ( , , 1) ,y xW x i y j t x i y j t W x i y j t i j Cα− − = − − ⋅ − − − ∈ (9)

The prediction for pixel ( , , )I x y t and the corresponding prediction error are then given by:

( )1ˆ ˆ( , , ) ( , , ) ( , , ) ,yc i j

I x y t W x i y j t I x i y j t i j Cn

= − − ⋅ − − ∈∑∑ (10)

ˆ( , , ) ( , , ) ( , , )e x y t I x y t I x y t= − (11) 4. EPHI: Edge-based Prediction 4.1. EPHI Edge-based prediction has been shown to improve the performance of traditional prediction schemes [1, 8]. Pixels along an edge have higher correlation than pixels across the edge. In this section, we modify the previous prediction scheme to incorporate edge-based information. We call the new algorithm EPHI: Edge-based Prediction for Hyperspectral Images.

An important issue in edge-based prediction is how to determine the thresholds for classifying edge and non-edge pixels. Different threshold values work better with different images. Different edge classification schemes have been studied in computer vision. Canny edge detectors perform well in classifying the edges and non-edges in natural images but the computation time required to perform the edge detection is high. Hyperspectral images have high pixel gradients and large texture-like areas. Hence most of the edge detection algorithms identify almost all the pixels

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as edge pixels. Given the typical huge data sizes involved, computing the edge gradient for every pixel becomes computationally intensive. It was shown in [8] that computation time can be reduced at the expense of little loss in compression ratio when a global threshold is computed. In the case of HSI, each band has a different intensity range, with significant variation between bands. It is difficult to assign one global threshold value to a complete set of 224 bands. Local thresholds, (example, for each context in a band), on the other hand will be very time consuming, Thus we assign a global threshold to each band. Empirically we found that a threshold value of 20% of the mean of each band gives the best result.

To save the overhead of sending the threshold value for each band, and since the global structure in all the bands is similar, we predict the threshold using the reference band:

11 1

1 ( , , 1)*

m n

ty x

I x y tm n

μ −= =

= −∑∑ ; 12.0 −= tt μτ (12)

where, tτ is the edge threshold for the current band, 1tμ − is the average intensity in band 1t − . Another issue in the edge-based technique is how to efficiently isolate edge pixels in a

context. We may have multiple edge structures within a context. However, edge pixels sharing the same edge structure are expected to have a higher correlation.

Fig. 4: Context for edge pixels. Edge structures are shown as dotted lines

Consider Fig. 4. Here ( , , 1)I x y t − lies on a edge and the edge is formed by pixels )1,,1( −− tyxI ,

( 1, 1, 1)I x y t− + − , ( 1, 2, 1)I x y t− + − and ( 2, 2, 1)I x y t− + − . In the same context, we have a second edge structure formed by pixels ( 1, 2, 1)I x y t− − − , ( 2, 2, 1)I x y t− − − and

( 2, 1, 1)I x y t− − − . Two pixels belonging to different edge structures should not be combined in performing prediction. The pixel being predicted is part of the first edge structure, but not of the second. Thus, pE , ( CEp ∈ ), the set of edge pixels for predicting the current pixel ),,( tyxI is formed as: { )1,,1( −−= tyxIEp , ( 1, 1, 1)I x y t− + − , ( 1, 2, 1)I x y t− + − , })1,2,2( −+− tyxI . Since only connected pixels are tested for edge properties we save time as the edge calculations are made only on the reduced set of pixels.

The computations for the edge-based prediction are similar to those used for the context-based method. The major difference is that, now edge-pixels are predicted only based on edge pixels on the same edge structure, while non-edge pixels are predicted based only on non-edge pixels. That is, prediction is not allowed to use information from two pixels that lie on different sides of an edge. Below, we consider the case for a pixel classified as an edge pixel. Then we can compute the required weight at the reference band:

( , , 1)( , , 1) ,( , , 1)x p

I x y tW x i y j t i j EI x i y j t

−− − − = ∈

− − − (13)

where, CEp ∈ denotes for the edge-pixels in the context, which belong to the same edge structure as the current pixel. Analogous computations can be performed for (6) – (11). Since the edge

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calculations in the reference band should hold for the current band, co-located pixels classified as edge pixels in the reference band are selected from the current band to compute the ratio between the two contexts. The edge based approach leads to an improved performance over the non-edge based method. Gains are highest in the case of Low Altitude, which has more distinct edges. 4.2. Brief Complexity Analysis Let the image size for each band be NM × . Let bη = number of bands, and cη = context size. The complexity of the proposed schemes depends on the cost of band classification (into spatial, spectral, or no prediction), the cost for computing the weights, and the cost of edge-based analysis (for edge-based prediction). For band classification, this is pre-computed before compression, based on a selected scene from one image set, using a reduced image size (example 128×128 bands). The same band grouping is used for all the image sets, and for all scenes. Thus this does not really affect the time for on-line compression. For NPHI, we observe that, given a pixel position in a band, the cost of computing all the required weights is generally in )( cO η time on average. See (1) – (11). This gives a total of ( )NMO cη per band, or ( )NMO bcηη for an entire image set. The extra space required will be in )( cO η for the entire processing of an image set. This is because no more than an )( cO η pixels is required in performing the prediction for any given pixel position. Also some of the calculations within a context need not be repeated for each pixel. For EPHI, the computational cost depends on three components: cost of NPHI, cost of edge detection, and cost of edge-based classification. After edge-based analysis, the cost of EPHI is upper-bounded by the cost of NPHI, since some pixels within a context may not be involved in computing the required weights for a given pixel. Thus, the extra cost will be mainly due to the edge-based analysis. However, given the stability of the general image structure across bands, in this work, edge-based analysis is performed only once for each hyperspectral image, based on only a few bands. The cost of this analysis is ( )NMO for edge detection, and ( )NMO cη for edge-based classification of pixels in a context.

5. Results Experiments were performed on a PC with 1.53 GHz, AMD Athlon processor and 1 GB RAM using MATLAB 7. The tests were performed using AVIRIS 1997 dataset. The parameters used in the prediction method were as follows: context size – 12; edge threshold – 20 % of the mean of each band; bands for spatial predition: 1–8 and 107–114; bands encoded without prediction: 153–160; all other bands were spectrally predicted. For a perspective on the reported results, Table I shows the performance for standard lossless image compression algorithms on the hyperspectral images used in this work. All the algorithms have a compression ratio of less than 3.0 on average, with a best performance of about 2.84 on average.

TABLE I: PERFORMANCE OF STANDARD LOSSLESS IMAGE COMPRESSION ALGORITHMS

CALIC [2] JPEG LS JPEG 2000 Differential JPEG LS [11]

Differential JPEG 2000 [11]

bpp CR bpp CR bpp CR Bpp CR bpp CR Cuprite 6.99 2.29 7.66 2.09 8.38 1.91 5.50 2.91 5.48 2.92

Jasper Ridge 7.69 2.08 8 2.00 8.89 1.80 8.84 2.81 1.8 2.82 Moffet Field 6.53 2.45 8.38 1.91 8.99 1.78 5.63 2.84 5.65 2.83 Lunar Lake 6.48 2.47 8.04 1.99 8.79 1.82 5.46 2.93 5.44 2.94

Low Altitude - - 7.47 2.14 8.16 1.96 5.93 2.70 5.95 2.69 Average 6.90 2.32 7.62 2.03 8.65 1.85 5.63 2.84 5.63 2.84

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5.1 Effect of Algorithmic Parameters Fig. 6(a) shows the effect of context size on prediction performance for different image sets (using NPHI). This is based on image sizes of 256 256× from scene-1 of the respective image sets. We observe that the entropy does not change much beyond context size 12. Hence, considering the gain in compression and computation time required, the context size was fixed to 12. Fig. 6(b) shows the effect of edge thresholds on the prediction residual. The curves flatten out from around 0.02 (20 % of the mean intensity for each). Although the entropy does not vary much with the threshold for Lunar Lake and Cuprite, it changes significantly for Moffet Field and Jasper Ridge.

(a) (b)

Fig. 6: Effect of algorithmic parameters: (a) effect of context size; (b) effect of edge thresholds

5.2 NPHI versus EPHI Table II (see last 2 columns) shows the results obtained with the two proposed methods for prediction namely NPHI and EPHI. EPHI performed better than NPHI for all the image sets. Higher compression gains are obtained with EPHI over NPHI on Moffet Field and Low Altitude. 5.3 Comparative Performance Table II also shows comparative results with other state-of-the-art hyperspectral image compression algorithms with comparable complexity. The results for other proposed methods have been taken from published results as indicated in the table. CCAP proposed in [13] performs better for Cuprite while LUT proposed in [5] performs better on Lunar Lake. The table shows that, on average, the proposed methods provide a better performance than existing approaches, producing the best result on Jasper Ridge and Low Altitude. On average, the proposed edge-based method (EPHI) produced the overall best result on the AVIRIS dataset used in this study. Average 1 in the table is for average over the first 4 image sets, while the Average 2 includes results for Low Altitude. The results for [6] are for using optimal band ordering described in [6].

TABLE III: COMPARISON WITH OTHER PUBLISHED ALGORITHMS M-CALIC SLSQ-OPT [6]

(Opt)CCAP 3D CALIC LUT NPHI EPHI

Cuprite 3.21 3.24 3.23 3.53 3.06 3.44 3.34 3.36 Jasper Ridge 3.17 3.23 3.18 2.56 3.08 3.23 3.27 3.29 Moffet Field 3.38 3.21 3.13 2.57 3.25 3.17 3.22 3.28 Lunar Lake 3.28 3.23 3.21 3.21 3.09 3.40 3.34 3.36

Low Altitude - 3.04 3.03 2.81 - - 3.01 3.08 Average 1 3.26 3.23 3.19 2.97 3.12 3.31 3.29 3.32 Average 2 - 3.19 3.16 2.97 - - 3.24 3.27

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6. Conclusion We have proposed two non-linear context-based prediction algorithms for lossless compression of hyperspectral images. Each algorithm uses a one-band look-ahead, and hence does not require a huge amount of memory. The first algorithm (NPHI) is a non-linear technique that predicts the pixel in the current band based on information from its context. Prediction contexts are defined using a causal pixel neighborhood in the current band and the corresponding co-located causal pixels in the reference band. EPHI extends NPHI using edge-based analysis. Prediction is performed by classifying the pixels into edge and non-edge pixels, and each pixel is then predicted using information from pixels in the same class within the context. On average, the proposed prediction methods produced competitive results when compared with other state-of-the-art algorithms with comparable complexity. The edge-based technique (EPHI) produced the best overall result on the images in the dataset tested. Another key contribution of this work is the identification of bands for spatial prediction, those for spectral prediction, and those that require no prediction. References [1] Li X., and Orchard, M.T., “Edge-directed prediction for lossless compression of natural images” IEEE

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