OPTIMIZED RATE ALLOCATION OF HYPERSPECTRAL IMAGES INCOMPRESSED DOMAIN UNDER JPEG2000 PART 2

Silpa Attluri, Vikram Jayaram, Bryan Usevitch

Division of Computing and Electrical EngineeringThe University of Texas at El Paso

500 W. University Ave, El Paso, Texas 79968-0523{silpa, jayaram, usevitch }@ece.utep.edu ∗

Abstract

This paper studies the application of bit allocation usingJPEG2000 for compressing multi-dimensional remote sens-ing data. Past experiments have shown that the Karhunen-Loeve transform (KLT) along with rate distortion optimal(RDO) bit allocation produces good compression perfor-mance. However, this model has the unavoidable disadvan-tage of paying a price in terms of implementation complex-ity. In this research we address this complexity problem byusing the discrete wavelet transform (DWT) instead of theKLT as the decorrelator. Further, we have incorporated amixed model (MM) to find the rate distortion curves insteadof the prior method of using experimental rate distortioncurves for RDO bit allocation. We compared our results tothe traditional high bit rate quantizer bit allocation modelbased on the logarithm of variances among the bands. Ourcomparisons show that by using the MM-RDO bit rate al-location method result in lower mean squared error (MSE)compared to the traditional bit allocation scheme. Our ap-proach also has an additional advantage of using DWT asa computationally efficient decorrelator when compared tothe KLT.

1 Introduction

The Hyperion (Hyperspectral Imager) provides a highresolution hyperspectral image capable of resolving 220spectral bands (0.4 - 2.5 µm) with a 30 meters pixelresolution. The data used in these experiments is the Mt.Fitton scene in southern Australia. The layers or bandsof this Hyperion data are highly correlated to each otherin the spectral direction. Decorrelation of these bands

∗This work was supported by National Geospatial Intelligence AgencyNMA401-02-2017 and Texas Instruments Foundation.

allows a more compact representation of an image bypacking the energy into fewer numbers of bands. Part 2of the JPEG2000 standard recommends the use of eitherthe KLT or the DWT for decorrelation. Past results haveshown that the KLT RDO combination works very wellfor bit allocation [2]. Due to the good accuracy of theresults achieved in this method has become a very usefulalternative to the lossless compression for this capaciousdata set. Nevertheless, due to the high computational com-plexity KLT is not a very attractive option as a decorrelator.Therefore, we use DWT decorrelation instead of the KLT.Our research refers to past experiments [1] which haveimplemented the mixed model for the bit allocation usingthe KLT and no decorrelation at different rates.

The high bit rate quantizer model performs bit allocationusing the log of variances (of the bands). The major prob-lems of this high bit rate quantizer model are non inte-ger rates, negative allocations and when these are forced tonon-negative integer rates it leads to suboptimal solutions.Moreover the present day challenges lies in looking at lowand very low bit rates where this model fails. To overcomethis our past research [2] implemented the approach of ex-perimentally obtained rate distortion curves, which allowsoptimal bit allocation with minimized MSE. However, thismethod has an unavoidable complexity of generating ratedistortion curves for each band. This problem is addressedin our current research by using the mixed model whichmathematically models each band’s rate distortion curvesusing two distinct regions to get a distortion model that suitsboth high and low bit rates.

2 Description of Data

The Hyperion instrument can image a 7.5 km by 100 kmland area per image and provide detailed spectral mapping

2281-4244-0069-4/06/$20.00/©2006 IEEE.

across all 220 channels with high radiometric accuracy.The Hyperion instrument, in conjunction with AdvancedLand Imager (ALI) instrument which provides accurateradiometric calibration on orbit using precisely controlledamount of Solar irradiance, forms the standalone camerasystem on the Earth Observatory-1 (EO-1) satellite [15].The analysis of Hyperspectral Imagery by means of suchspectrometers (Hyperion) exploits the fact that each ma-terial radiates different amount of electromagnetic energythroughout all the spectra. This unique characteristic of thematerial is commonly known as spectral signature whichcan be read using such airborne or space borne-baseddetectors.

The data used for experiments were collected by the Hype-rion over the semiarid Mount Fitton area which is locatedin the Northern Flinders Ranges of South Australia. Cen-tered at −29◦55

′S and 139◦25

′E, about 700 kms NNW of

Adelaide. This Hyperion image consists of 194 atmospher-ically corrected contiguous spectral bands. Each band hasdimensions of 6702 x 256 [12].

Figure 1. Hyperion color composite image ofMount Fitton test site.

3 The JPEG2000 Standard

The JPEG2000 standard has many advantages comparedto the earlier compression standards. Such advantages in-clude superior low bit rate performance, bit rate scalabilityand progressive transmission by quality or resolution. Thecoder and decoder for JPEG2000 use a wavelet based com-pression method which implements the Embedded BlockCoding with Optimized Truncation (EBCOT) [4] scheme.Quality scalability is achieved by dividing the wavelet trans-formed image into code-blocks. After each code-blockis encoded, a post-processing operation determines where

each code-block’s embedded stream should be truncated inorder to achieve a pre-defined bit-rate or distortion boundfor the entire data. This bit-stream rescheduling module isreferred to as Tier 2. It establishes a multi-layered represen-tation of the final bit-stream, guaranteeing an optimal per-formance at several bit-rates or resolutions. Figure 2 illus-trates the general flowchart of the JPEG2000 architecture.The Tier 2 component optimizes the truncation process, andtries to reach the desired bit-rate while minimizing the in-troduced distortion (MSE), utilizing Lagrangian rate alloca-tion principles. This following procedure is known as Post-Compression Rate-Distortion (PCRD) optimization [4] andthe basic principles behind it are extensively discussed in[7].

4 Description of our approach

Due to the computationally inefficient nature of the KLTwe employed DWT for decorrelating the bands. The DWTcan be implemented by either a convolution or a liftingscheme. We have used the lifting scheme implementationof DWT due to its lesser computations and memory require-ments. Here the 9/7 Daubechies filter is used as we areperforming the lossy compression.

Figure 3. 2-D Wavelet Decomposition

Once the data is decorrelated by the use of the transform, allof the energy or information is packed into just a few bandsinstead of all the bands. Figure 4 is a hypothetical portrayalof energy distributions: before and after decorrelation. Theright plot clearly shows that we cannot just allocate the samenumber of bits for each band. Assigning the same bit ratefor every band neglects the information importance of eachband. To avoid this we allocate bits optimally to the sepa-rate slices or image bands. The individual bands were com-pressed and reconstructed depending on the different rateallocation strategies. We performed experiments using sev-eral different approaches to bit rate allocation for individual

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DC LevelShift

ComponentTransform Transform

Quantization

ORIGINALIMAGE

DECODEDIMAGE

COMPRESSEDIMAGE

Encoder Encoder

DecoderDecoder QuantizationTransformComponentTransform

DC LevelShift

Tier−2

Tier−2 Tier−1

Tier−1 Inverse Wavelet Inverse

Forward Wavelet

Forward

Inverse Inverse

Forward

Figure 2. JPEG2000 architectural flowchart

bands. One of them is the traditional bit rate allocation ap-proach based on logarithm of variances of individual bands(see [9] for more details). In this model the MSE’s in eachband can be approximated as function of rate R and varianceσ2. Assumptions of a high bit rate model lead to a quantizerdistortion given by:

MSE(R) ∼= σ2 · 2−2R (1)

where σ2 is the variance (of the bands) and R being theaverage rate for each band.

Figure 4. Original band energy Vs. Trans-formed band energy

The traditional or the log of variances bit allocation schemeis not fully compatible, since this high bit rate quantizerdistortion model breaks down at low bit rates. Instead weuse the mixed model and the RDO [8], which is based onpost compression rate distortion (PCRD) extensively usedin JPEG2000, for bit allocation. Our research implementedthe RDO by using RD curves that have been obtained byusing the mixed model. The mixed model is essentially ahybrid splicing of the low and the high bit rate models. Thelow bit rate of the model follows the results of Mallat andFalzon [5] . Past research [1] tried modifying this so thatwe can make it work for our problem incorporating lower

bit rates [1]. The relation between the MSE and the bit ratecorresponding to the information about the data can still bedescribed by a modified Mallat-Falzon model,i.e.,

MSE(R) =

{A 1

(R+Ro)α if R ≤ R,

B · 2−2R if R ≥ R(2)

where A, Ro, α come from the low rate model and Bwhich is nothing but the variance of each band is the pa-rameter from the high rate model (same as 1). To find Ro,we use the fact that the MSE of each band is nothing butits entropy when Ro = 0. R is the cross over bit rate be-tween the low and the high rate models. The determinationof these mixed model parameters are derived in [1].

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2000

4000

6000

8000

10000

12000

14000

Bit rates

mea

n s

qu

ared

err

or

(MS

E)

Low bit rate model

High bit rate model

Experimental RD curve

Figure 5. Comparison of experimental RDcurve with Mixed model. Notice that the hy-brid splicing of mixed model shown as 2 dot-ted curves is close to the RD curves (solidlines) obtained experimentally.

Advantages of the mixed model approach are that it can give

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performance nearly identical to the RDO approach with amuch lower implementation complexity. Here we have gen-erated this approach for the data at bit rates in the range0.0375 bpppb to 4 bpppb (where bpppb stands for bits perpixel per band) and further compute the corresponding dis-tortions (MSE) at each average bit rate using JPEG2000coder and decoder. Results show that the mixed model ap-proach gives distortion results that are nearly identical tothe RDO approach on this data, while significantly reduc-ing complexity of implementation. We notice from Figure

Figure 6. 2-D band-wise compression.

5 that the mixed model approach gives distortion results thatare nearly identical to the experimental RDO approach onthis data, while significantly reducing the complexity. Fur-ther, our experiments are performed as shown in Figure 6.Each band of the Hyperspectral cube were compressed at16 bpppb and decompressed at bit-rates obtained from thetwo bit allocation schemes. Next we computed the MSE’sby comparing the original set of bands to the compresseddecompressed bands for the given set of average bit rates.

5 Results

Figure 7 shows a comparison of MSE’s obtained for eachaverage bit rates corresponding to traditional and the MM-RDO bit allocation schemes using DWT as a decorrela-tor. Our results confirm that MM-RDO has lower MSE’scompared to the traditional model. Moreover, MM-RDOwhen used in conjunction with DWT as decorrelator fur-ther produces lower MSE’s compared to KLT decorrelationwith traditional bit allocation for lower bit rates. Figure 8shows a comparison our past experimental results [2] thatemployed KLT docorrelation followed by RDO bit alloca-tion with the present DWT mixed model. In spite of KLT’s

advantage of an optimum compaction transform, the DWTmanages to match up with its MSE performance and alsodoes slightly better compared to the KLT traditional bit al-location in the low bit rate region as seen in Figure 8.

10−2

10−1

100

101

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

mea

n s

qu

ared

err

or

(MS

E)

Compression rate (bpppb) on Log Scale

DWT & MM RDO

DWT & MM Traditional

Figure 7. MSE comparison of mixed modelRDO and the high rate quantizer model.

10−2

10−1

100

101

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

Compression Rate on (bpppb) on Log Scale

mea

n s

qu

ared

err

or

(MS

E)

DWT & Mixed Model−RDO

DWT & Traditional Model

KLT & RDO

KLT & Traditional

Figure 8. MSE comparisons of DWT mixedmodel, DWT traditional model, KLT RDO andKLT traditional model.

6 Conclusions

In the case of KLT the computation time and memoryrequirements is very high (about 8 times higher) whencompared to DWT. One other main advantage of DWT, thatit allows to divide a huge data set into parts and pre-processeach of these independent of the other. This enables parallel

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processing which cannot be done with KLT. At higher bitrates we observe that both rate allocation strategies producesimilar MSE results. The rate distortion curves obtainedfrom the Mixed Model are very close to the R-D curvesobtained experimentally. The Mixed Model and RDOapproach gives lower MSE than the traditional model andalso slightly out performs our past KLT traditional methodat lower rates.

However, we do see that KLT-RDO provides lower MSE’sin most cases compared to the present DWT approach. Thisis because of the fact that the KLT is a better decorrela-tor than a DWT. The KLT is theoretically the ideal trans-form and is expected to give better results for both clas-sification performance and compression. Nevertheless, itsbiggest disadvantage is that it is computationally expensive.

7 Acknowledgment

We would like to thank Dr. Olga Kosheleva and AlbertoAguirre for their assistance with this research. We alsowould like to thank Mary Williams (EOIR) for perform-ing atmospheric corrections on our data and Raed Aldouri(PACES) for providing us ENVI software package.

References

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[2] Vikram Jayaram, Bryan Usevitch and OlgaKosheleva, “Detection from Hyperspectral imagescompressed using rate distortion and optimiza-tion techniques under JPEG2000 Part 2,” in Proc.IEEE 11th DSP Workshop & 3rd SPE Workshop(cdrom), August 2004.

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[12] T. J. Cudahy, R. Hewson, J. F. Huntington, M. A.Quigley and P. S. Barry, “The performance of theSatellite-borne Hyperion Hyperspectral VNIR–SWIR imaging system for mineral mapping atMount Fitton, South Australia,” in Proceedings ofIEEE International Geoscience and Remote Sens-ing Symposium IGARSS01, Sydney, July 2001.

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[15] http://eo1.gsfc.nasa.gov/

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