Vector-lifting schemes for lossless coding and progressive archival of multispectral images

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 40, NO. 9, SEPTEMBER 2002 2011

Vector-Lifting Schemes for Lossless Coding andProgressive Archival of Multispectral Images

Amel Benazza-Benyahia, Member, IEEE, Jean-Christophe Pesquet, Senior Member, IEEE, and Mohamed Hamdi

Abstract—In this paper, a nonlinear subband decompositionscheme with perfect reconstruction is proposed for lossless andprogressive coding of multispectral images. The merit of thisnew scheme is to exploit efficiently the spatial and the spectralredundancies contained in the multispectral images related toa scene of interest. Besides, the proposed method is suitablefor telebrowsing applications. Experiments carried out on realscenes allow to assess its performances. The simulation resultsdemonstrate that our approach leads to improved compressionperformances compared with currently used lossless coders.

Index Terms—Filter banks, lifting schemes, lossless coding,multispectral images, progressive coding, satellite imaging.

I. INTRODUCTION

M ULTISPECTRAL images are of main interest for agreat number of applications such as target-imaging

and terrain-mapping applications. Generally, these imagesare supplied by specific satellites that observe the earth inseveral spectral channels. For instance, the “Satellite Pourl’Observation de la Terre” (SPOT 3) has two High Reso-lution Visible (HRV) imaging systems (HRV1 and HRV2).Each HRV is designated to operate in two sensing modes:a 10-m-resolution “panchromatic” (P) mode over the range[0.5, 0.73] m and a 20-m-resolution multispectral mode.For the multispectral mode, the XS1 channel is associatedwith the range [0.5, 0.59]m, the XS2 channel with therange [0.61, 0.78] m, and the XS3 channel with the range[0.79, 0.89] m. Therefore, for a given scene, the four bandsP, XS1, XS2, and XS3 are avalaible. Since March 1998, theSPOT family provides a service continuity with the upgradedsatellite SPOT 4. In addition to the XS1, XS2, and XS3channels, the SPOT 4 imaging system delivers images in afourth channel corresponding to a shortwave infrared spectralband ([1.58, 1.75] m). This XS4 channel was added in orderto enable early observation of plant growth through inspectionof vegetation water contents. Hence, a SPOT 4 scene containsfour multispectral components and a panchromatic image.Following SPOT 4, SPOT 5 was launched on May 3, 2002.

Manuscript received December 6, 2001; revised August 2, 2002.A. Benazza-Benyahia is with the Département Mathématiques Appliquées,

Signal et Communications, Ecole Supérieure des Communications de Tunis,Cité Technologique des Communications, 2083 El Ghazala, Ariana, Tunisia(e-mail: [email protected]).

J.-C. Pesquet is with the Institut Gaspard Monge and URA-CNRS 820, Uni-versité de Marne-la-Vallée, 77454 Marne la Vallee Cedex 2, France (e-mail:[email protected]).

M. Hamdi is with the National Digital Certification Agency, Tunisia.Digital Object Identifier 10.1109/TGRS.2002.803845

It offers new capabilities and performance to answer theincreasing demand in cartography, agriculture, planning, andenvironment, thanks to a new imaging instrument (high-res-olution geometry). As a consequence, a ground resolution of2.5 m (respectively, 10 m) is achieved for the panchromaticimages (respectively, XS1 to XS3). The XS4 component ismantained at a resolution of 20 m due to the limitations im-posed by the geometry of the sensors used in this intermediateinfrared band. Besides, since 1999, the satellite LANDSAT 7is producing great imagery of the planet more quickly and atlower cost than any previous part of the LANDSAT program.The earth-observing instrument on LANDSAT 7, the EnhancedThematic Mapper Plus (ETM+), replicates the capabilitiesof the Thematic Mapper (TM) instruments on LANDSAT 4and 5. However, the ETM+ includes additional features: apanchromatic band with 15-m spatial resolution and a betterradiometric calibration in a thermal infrared channel with60-m spatial resolution. A LANDSAT 7 scene is formedby seven spectral components and a panchromatic image.Larger and larger amounts of data are generated due to thecontinuous improvement and increased popularity of remotesensing systems. The problem of managing, transmitting, andarchiving such tremendous volumes of data is crucial. To givean idea of the importance of the problem, recall that a fullstandard SPOT 4 scene corresponds to 72 MB. A LANDSAT 7scene acquired by the ETM+ sensors corresponds to 475 MB.Furthermore, during a typical day of operations, approximately250 scenes are delivered to the U.S. archive. As a result, thewhole data flow collected daily amounts to 118.75 GB. Suchamounts of data require huge storage capabilities as well asa large bandwidth during downlinking. In this context, it isreally challenging to develop image compression techniquesso as to provide solutions to this problem. On-board andon-ground compression techniques may be distinguished forwhich constraints are different. On-board compression requiresreal-time encoding and transmission error robustness, whereason-ground compression (e.g., for archiving as discussed below)requires low-complexity decoders and, eventually, selectivereconstruction.

Two types of schemes can also be envisaged: lossy or loss-less. In this paper, we focus our attention on lossless codingtechniques for multispectral remotely sensed images, as we aremainly interested in archiving applications. Indeed, the storageof these images requires exact reproducibility of the data, sincethe introduction of any distortion may lead to an erroneous in-terpretation of the considered scene or to the computation ofincorrect ground parameters. Therefore, it must be possible toperfectly recover the original image.

0196-2892/02$17.00 © 2002 IEEE

2012 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 40, NO. 9, SEPTEMBER 2002

Furthermore, progressive reconstruction is a desirable fea-ture for telebrowsing through image databases. It consists inencoding the image in several layers: the first level correspondsto a highly compressed version, and each successive levelprovides more details until ultimately the image is completelyrecovered. Note that this approach actually corresponds to alossy-to-lossless compression method. Such gradual recon-struction requires a compact and nonredundant pyramidalrepresentation of the input images. Nonlinear subband decom-positions have been developed recently [1]–[4]. They providehierarchical and compact representations for gradual coding ofimages [5], and for this reason, some of these structures havebeen retained in the JPEG 2000 standard [6].

There are few publications in the context of progressiveand lossless coding of multispectral images. At this level, itis important to make a clear distinction between multispectralimages and hyperspectral images. Whereas multispectralremote sensing devices do not record more than ten imagesof the same scene, hyperspectral imaging techniques typicallygenerate several dozens of images at different wavelengths. Forexample, the Airbone Visible Infrared Imaging Spectrometer(AVIRIS) delivers images in 224 adjacent spectral channels[7]. Although the context is clearly different, a somehowsimilar situation frequently occurs in computed tomographyand magnetic resonance medical imaging: about 100 slicesare produced in a single examination. The large number of“bands” allows to consider the whole set of slices as a singlevolumetric dataset. Usually, monoresolution approaches forhyperspectral image coding make use of vector quantization[8]. In order to obtain a gradual reconstruction, it is possible toapply multiresolution decompositions using separable filters[9], [10] to this three-dimensional (3-D) signal. Such 3-Dmultiresolution transforms yield to better performances thanslice-by-slice two-dimensional (2-D) transforms [10]. Obvi-ously, this approach is not the most appropriate in the case ofmultispectral images because of the limited number of spectralbands. Consequently, specific decomposition techniques mustbe designed in this context.

To the best of our knowledge, the problem of extending 2-Dmultiresolution representations to multispectral images has notyet been addressed. Indeed, in existing methods, each spectralcomponent is very often coded independently of the others. Assequences of images of the same spatial area are recorded inmultiple bands, multispectral images are characterized by theexistence of both spatial and spectral redundancies. Spatial cor-relations are related to the similarities existing among neigh-boring pixels in thesamespectral band. Spectral correlationsexist between pixels that are approximately at the same spatiallocation but incontiguousspectral bands. One can expect a goodcompression algorithm to reduce substantially these two kindsof redundancies.

In this paper, we develop a new vector decomposition schemethat exploits the mutual dependencies among spectral bands,thanks to generalized nonlinear subband decompositions.The paper is organized as follows. In order to better situateour contribution, first we describe the related works in thefield of lossless compression by considering monoresolutioncoding techniques for multispectral images in Section II-A

and multiresolution intraband coders in Section II-B. We alsointroduce some useful notations in Section II. In Section III, wepropose a new hybrid hierarchical and reversible decompositionstructure that allows to take into account both the spectral andthe spatial redundancies. In Section V, problems concerningthe optimization and possible improvements of the proposedvector-lifting scheme are discussed. Finally, in Section VI, weprovide some experimental results and compare the proposedmethod to conventional approaches.

II. EXISTING METHODS FORLOSSLESSCODING

In this section, we give a rapid overview of existing methodsfor lossless coding of multispectral images. We put emphasis onthe two kinds of techniques at the origin of our work: mono-resolution predictive methods and nonlinear 2-D filter banksused in multiresolution decompositions.

A. Monoresolution Lossless Coders

Usually, spatial and spectral correlations are exploited byconventional monoresolution lossless coders that were initiallydeveloped for natural monoband images. Among the variousmonoresolution methods, differential predictive coding isprobably the most simple and efficient exact coding technique.As discussed below, the prediction can be purely spatial, purelyspectral, or hybrid. In the following,denotes a 2-D multiband signal, whereis the total number ofspectral bands; is the band index; and are the spatialcoordinates, whereas denotes the predicted value atthe current pixel .

1) Spatial Coders:There are various spatial coders basedon predictive techniques [11]. For instance, an optimal linearpredictor (OLP) based on the three nearest previous neighbors

, , andcan be applied. If the optimality is evaluated in terms of theprediction mean-squared error (MSE), the optimal coefficientsare solutions of the well-known Yule–Walker equations. Tocircumvent the resolution of these equations, it is possible touse predetermined coefficients for the predictors. For instance,the standard JPEG [12] provides eight fixed predictors fromwhich the user can select the best one in terms of lowestresidual entropy (BJPEG). An adaptation of coefficients ispossible (BJPEG*) by partitioning the image into nonover-lapping blocks of size 8 8. The “best” of the eight JPEGpredictors is chosen for each block. For an image ,the Median Adaptive Predictor (MAP) uses the median of thefollowing three predictors: , , and

[13]. An-other example of a nonlinear predictor is the gradient-adjustedpredictor used in the context-based adaptive lossless imagecodec (CALIC) proposed in [14].

As mentioned previously, there are few lossless compressionmethods specific to remote sensing images [15]–[17]. Recently,the Consultative Committee for Space Data Systems (CCSDS)has adopted a standard for lossless data compression [18], [19].The CCSDS lossless algorithm is a two-step compression algo-rithm characterized by the use of a predictor stage followed byRice entropy encoding [20], [21]. Initially, the applied predictor

BENAZZA-BENYAHIA et al.: LOSSLESS CODING AND PROGRESSIVE ARCHIVAL OF MULTISPECTRAL IMAGES 2013

TABLE IEXAMPLES OF MONORESOLUTIONPREDICTORS OFBAND b FROM BANDS b AND b

was a one-dimensional (1-D) nearest neighbor predictor (a basicpurely spatial predictor). An effort has been made in order topropose other predictors (spatial, spectral, or hybrid) [22]. It ispossible to select among them the predictor that yields the bestcompression ratio. For illustration, Table I gives the equationsof some of the predictors (denoted ID#where is their iden-tification number) recommended by the CCSDS.

2) Spectral Coders:It is possible to take into account exclu-sively the spectral redundancies by using only the pixels of thebands different from the considered one. A trivial example in-volves predicting by where . In[23], [24], a simple monoresolution spectral coding (MSC) ofSPOT 3 images ( ) was proposed. It involves coding theband with an OLP in a purely spatial mode. Then,the remaining images and are predictedaccording to

(1)

where denotes the rounding operation toward the nearest in-teger, and the coefficients, , and can be obtained throughthe resolution of the Yule–Walker equations. We will denote byMSC( ) the combination corresponding to the followingdecomposition order: decomposition of bandfollowed by

and then by . It is possible to define a generalized MSC(G-MSC) in the case . This can be realized by predictingthe additional bands from all the preceding bands. Note thatadaptive prediction methods can also be used to achieve spectraldecorrelation [25].

3) Hybrid Coders: It is more appropriate to capture simulta-neously the spectral and the spatial correlations by applying hy-brid predictors. A straightforward method consists in taking intoaccount the multidimensional nature of the multiband signals.In this manner, it is possible to generalize the OLP to the caseof a vector signal of dimension : the prediction of thevector

is realized based on the vector set

where denotes an appropriate template. For instance, in [26],. For SPOT 3 images, a similar ap-

proach consists in predicting (component by component) the

multiband signal. The first band is predicted by using the pre-vious samples of all the other bands

(2)In the prediction of band , the sample is used

(3)

Finally, both samples and are consid-ered in the prediction of the last band

(4)

The optimal coefficients (in the sense of the MSE) are solutionsof three linear systems [26]. In the sequel, we will denote thismonoresolution hybrid coder by the acronym MHC( ).Again, it is possible to use fixed or optimized predictors. Rogerand Cavenor [27] have applied five hybrid predictors [ID#4,SS1, SE-o1b, SE-o2b, and SS-o1 (given in Table I)] for hyper-spectral image compression. Recently, Aiazziet al. [28] havedescribed a space-spectrum-varying prediction followed by anefficient context-based classification of the prediction residuals.Gelli and Poggi [29] have described a new coding approach,but their scheme systematically leads to a lossy compression, asit involves the quantization of cosine-transformed coefficients.More recently, Wu and Memon [30] have proposed an interbandversion of CALIC using a gradient-adjusted interband predictor.Other sophisticated methods have been investigated in [31] and[32].


Fig. 1. M -band NL decomposition filter bank with exact reconstruction (derived from the linear case).

Fig. 2. M -band synthesis filter bank associated with the decomposition filter bank depicted in Fig. 1.

B. Multiresolution Intraband Decompositions

Nonlinear multiresolution decompositions constitute a suit-able tool for progressive coding of images. Some conditionsare required in order to obtain a complete and nonredundantmultiresolution representation [4]. For example, in [3], a gen-eralization of the matrix triangularization procedure to a linear1-D subband decomposition was shown to lead to the class ofnonlinear -band decompositions with perfect reconstructiondepicted in Fig. 1. Such a decomposition scheme will be de-noted by NL. The resulting subband signals can be obtained bythe following equations:

(5)

where are the polyphase components (of order)of the 1-D signal to be decomposed [33] (the polyphase com-ponents of a 1-D signalare defined by

). Furthermore, we have

(6)

(7)

Perfect reconstruction is guaranteed for arbitrary operators,, and injective operators for [3]. In par-

ticular, these operators can be linearor nonlinear filters. Theassociated synthesis filter bank is provided in Fig. 2. Often, theoperators are chosen equal to identity, whereas the operators

can be thought of as tools for decorrelating the input signals,and the operators can be considered as smoothing operators.They are not restricted to be infinite impulse response linear fil-ters as done in [34]. Indeed, any kind of nonlinear operators canbe used, such as integer-to-integer operators, rank order filters,or morphological operators [35].

As our objective is image coding, the extension to the 2-Dcase must be performed. For the sake of simplicity, this can behandled in a separable manner, but other subsampling lattices(such as quincunx lattices) can also be envisaged.

An important merit of this decomposition structure is toprovide a unifiying framework that encompasses the main loss-less compression schemes based on nonlinear multiresolutionanalyses [36].

The second generation of wavelets provides very efficienttransforms, based on the concept of lifting developed in [37].This structure is included in the general scheme NL where

and , and and are the so-called predic-tion and update operators. In the following, we will denote by


the wavelet transforms that map integers to integers wherethe number (respectively, ) corresponds to the number ofvanishing moments of the analyzing (respectively, synthesizing)high-pass filters [2]. Another compelling transform (called theS+P transform) was proposed in [1] to complete thetransformby an additional predictive step (P) .

All these multiresolution representations do not capture thespectral correlation. Our objective will be now to adapt the de-composition in Fig. 1 to the case of a multiband signal in orderto exploitboth the spatial and spectral correlations.

III. PROPOSEDINTERBAND DECOMPOSITION

A. Multiband Prediction-Update Operators

At first, we considered a 1-D multiband signalwhere denotes the total number

of bands, and corresponds to the bandof the multi-band signal. The index corresponds to the resolution level;the approximation of the bandat resolution is denoted by

; and the corresponding detail coefficients are denoted

by . The multispectral case is actually similar to anband decomposition. In the previously described

intraband decomposition, the input signals are polyphase com-ponents of asingleband approximation at the current resolution. Now, the even and odd samples ofseveralapproximation

bands ,constitute the 2 input coefficients

(8)

(9)

The 2 outputs will be split into two classes: the predictionerrors and the approximation coefficients

given by

(10)

(11)

Following the guidelines provided by the decomposition NL,the operators and must nowbe defined so as to realize hybrid (intraband and interband)prediction and update. Although the redundancies existingamong the different bands could be extracted by a wide classof nonlinear operators, they are more easily handled by linearfilters followed by rounding operations. In a previous work,we have already considered a very simplified case of such adecomposition involving linear predictors [38]. For a givenband , the general expression of the prediction error atresolution level is given by ,

(12)where the set is the support (or neighborhood) of the pre-

dictor of band from band at level . The coefficients

are the prediction weights. Similarly, it is possible to express theapproximations as follows:

(13)

where and are the supports of the considered up-

date filters, and and are the corresponding updatecoefficients. Similarly to conventional lifting schemes, the co-efficients allow to update the approximation coefficientsfrom the previously computed detail coefficients. The coeffi-cients are used to realize the update from the availableodd indexed values, as it is done in the S+P transform.

Finally, the case of a 2-D multiband signal (multispectralimages) is handled by means of a separable decomposition. Thedecomposition of line in band involves the lines in allthe other bands. The column of the resulting decompositioninvolves the columns of the associated intermediate results.

B. Conditions for Exact Recovery

Some care must be taken concerning the considered neigh-borhoods and the order of decomposition of the different bands[39]. We now provide simple sufficient conditions in order toguarantee that the inputs of the multiband decomposition struc-ture can be perfectly recovered from the outputs.

Proposition 1: Exact recovery is possible if there existand , two sequences of permutated values

of , such that

• ,

(14)

•

or (15)

and or (16)

• ,

(17)

and (18)

In the above, the sets of even integers, positive even integers,and negative even integers are denoted by, , and ,respectively.

Proof: We have to show that, subject to the stated condi-tions, it is possible to perfectly reconstruct the different spec-tral bands from the decomposition coefficients. According to(17), all the sets are empty, which means that theonly values from other bands involved in the computations of

are detail coefficients. The first reconstruction step con-

sists in calculating the coefficients from ,


and odd samples of by “reversing” (13). Ac-

cording to (15), either contains positive even integers(“past” samples) or negative even integers (“future” samples).This shows that can be reconstructed either by acausal processing or by an anticausal one. For example, in thecausal case, the inversion formula reads

(19)

For the remaining bands with , (14) showsthat the sets can contain even indices, which is consistentwith the fact that the odd samples of bandhave already beenreconstructed. Then, it is possible to recover the other samples

by proceeding similarly to band , in a recursive

way. To recover the even samples , any odd index may

be used in , as the odd samples of each band are availableafter the previous reconstruction steps. If we accept this fact, wehave basically the same kind of restrictions on as

those found for .Several remarks can be made at this point.

• The supports can be chosen arbitrarily.• The proof of the previous proposition shows that

(respectively, ) defines theorder of reconstruction for the odd (respectively, even)parts of the different bands. These orders may be different.

• If , the reconstruction of is real-ized by a 2-D recursive formula. This entails that stabilityand initialization problems may need to be addressed, eventhough they are not discussed in this paper.

In the sequel, we illustrate the above results by providing fourexamples of hybrid pyramidal decompositions.

IV. RETAINED DECOMPOSITIONS

The four examples that will be presented in this section cor-respond to level-dependent decompositions we have retainedin our simulations. These decompositions can be viewed asdifferent generalizations of the usual decompositions .The choices we have made after several trials are the result ofa tradeoff between computational complexity and the abilityto reduce spectral and spatial redundancies. Obviously, someimprovements of these structures may be expected by realizingmore intensive experiments. As we are mainly interested inSPOT images, only or band decompositions areconsidered subsequently. Note, however, that extensions to ahigher number of spectral bands can be easily realized.

In the following, we recall that the index is chosen inand corresponds to the resolution level (number of

decompositions) whereas is a spectral bandindex.

A. Examples of Three-Band Decompositions

1) Example 1: We first present a generalization of thedecomposition, which will be designated by DEC1. For

, image is predicted according to the following:

(20)

where

(21)

(22)

(23)

For images and , the update operations are similar

(24)where and .

However, the update of includes the sample

(25)

where .The considered decomposition is reversible. Indeed, the

conditions of Proposition 1 are satisfied withand

(26)

(27)

(28)

(29)

(30)

The odd samples and are easily

recovered by reversing (24). Then, is directlydeduced from (25). The even samples are recursively obtainedfrom (20), starting with , then calculating ,

and finally .


2) Example 2: A second decomposition (DEC2) has alsobeen considered. The predictions are given by (20), where

(31)

(32)

and

and

Note that the image is predicted in a purely spatial mode withthe intraband predictor of decomposition . All images areupdated as in the transform

(33)

This second decomposition is also reversible as a direct appli-cation of Proposition 1.

3) Example 3: A last decomposition (DEC3) is proposed inthe case of three-band images. The imageis predicted as inExample 2, using (31). The imagesand are predicted ac-cording to (20) where

(34)

(35)

For all images, the update operations are similar to those ofExample 1. As the two previous ones, it is easily checked thatthis decomposition is reversible [ ].

B. Four-Band Decomposition

Another decomposition (DEC4) has ben developed for thecase of four-band images ( ) like those provided bySPOT 4. Again, band is predicted as in Examples 2 and 3,using (31). The remaining bands are predicted according to(20) where

(36)

(37)

(38)

The update equations are similar to those of (24). The conditionsof Proposition 1 are also clearly satisfied in this example.

V. OPTIMIZATIONS AND IMPROVEMENTS

A. Optimizing the Operators

Obviously, the compression performances of the proposeddecomposition are closely related to the choice of the predictors.The most reliable performance measure for a lossless coder isprovided by the achievable average bit rate(in bpp). However,this criterion strongly depends on the entropy coder that is used[40], [41]. Another measure of the compactness of the resultingpyramidal representations is the sum of the zeroeth-orderentropies

(39)

where is the global entropy associated with band. For a-level decomposition, it is given by the weighted sum of the

zeroeth-order entropies of the approximation and detail subim-ages

(40)

where denotes the detail subimages of bandat decom-

position level , and denotes the approximation subimageof band at level . The more the entropy is decreased, themore compact the resulting representation is. The evaluation ofthe performances becomes independent of the coder by consid-ering simply the entropy rather than the bit rate. This is thereason why it is convenient to design the predictors so as to geta minimum entropy. Since the entropies are implicit functions


of the parameters of the decomposition, one must resort to nu-merical optimization methods such as the Nelder–Mead simplexalgorithm in order to find optimal parameters. Initial values ofthe prediction coefficients can be taken as the solutions ofthe Yule–Walker equations related to minimum variance predic-tions in (12). Due to the separable nature of the decomposition,the predictors are calculated for each line/column decomposi-tions and at each resolution level. In this way, some kind of adap-tivity may be introduced. At the same time, the update terms arechosen in order to minimize the global entropyby applying asimplex algorithm. To avoid a heavy computational load, otheralternatives could be used (e.g., in [42]–[44], the adaptations ofthe predictors are carried out on by a least mean-square algo-rithm).

B. Band Ordering

The prediction steps producing the detail coefficients takeinto account the correlation existing between the current bandand the remaining ones. But, certain bands may be more cor-related with bands other than the previous band. Therefore, animportant task is to determine the best ordering of the bandsat each decomposition stage. More precisely, the objective is tofind optimal orderings and that min-imize either the total zero-order entropy or the average bitrate . In the case of a low number of bands ( ), as inthe case of SPOT images, it is possible to examine exhaustivelyall the permutations, which is the method we have adopted.However, for TM images ( ), this approach becomes com-putationally intractable, as 25 401 600 possible decompositionsshould be tested! Hence, an efficient algorithm must be appliedfor computing a feasible band ordering. Since more than oneband is used for prediction, it is not straightforward to view theproblem as a graph-theoretic problem [32], [45], [46]. There-fore, the band-ordering problem becomes rather difficult, andheuristic solutions should be looked for. Specific approaches forTM sensors have been investigated in [46] and [47].

C. Pre- and Postprocessing

It is also possible to insert a processing stage of the multispec-tral image sequence before or after its multiresolution decom-position. In both cases (pre- or postprocessing), caution shouldbe taken in the choice of such operators in order to ensure a per-fect reconstruction. Inspired by works on color-image coding,the processing stage can be thought of as a spectral decorre-lating operator. Generally, a spectral decorrelation is performedbeforethe spatial decomposition. Therefore, it is a monoreso-lution operator. A postprocessing may also be envisaged. Forexample, a multiresolution intraband scheme of depthcan befirst applied separately on each band. Then, the spectral corre-lations existing between the resulting subbands can beexploited by a purely spectral coder such as the MSC describedby (1) [23], [24].

Alternatively, color transforms associated with the RGBmodel can be used as preprocessing operators before themultiresolution decompositions. Indeed, SPOT 3 images areobtained in the green, red, and near-red-infrared wavelengths.So, each image plane may keep some physical meaning

if an RGB model is considered. It is well known that theKarhunen–Loève transform (KLT) is the optimal linear trans-form that decorrelates the components of a random vector.Unfortunately, KLT is a lossy transform when a finite precisionis used for arithmetic operations. In [48], a lossless transformapproximating the KLT was presented, but it seems prefer-able to apply fixed (image-independent) integer-to-integerdecorrelating transforms in order to achieve the lowest compu-tational complexity. Such a commonly used transformation for

color-image coding is the original reversible colortransform (ORCT) [49]. It consists in transforminginto the triplets , defined as follows:

(41)

As can be noted, the ORCT is a transform that maps integersto integers. It is a reversible decomposition that approximatesthe widely used color transform mapping the RGB space tothe YCrCB color space. Its advantages are twofold: 1) ORCTreduces successfully the psychovisual redundancy in a colorimage, and 2) it is very easy to implement with a low compu-tational cost. For these reasons, ORCT has been adopted in theJPEG 2000 standard as a preprocessing stage. It is easy to ex-tend the ORCT transform to the-band case as follows:

...

(42)

where is the original multiband image, andis the resulting multicomponent image. This gen-

eralized ORCT will be denoted by G-ORCT. Note that theindices need to be defined, and in particular theband must be chosen. This choice can be made arbitrarily, orit can rely on physical arguments. For example, when ,the aforementioned analogy with color images can be used.

Other integer-to-integer decorrelating transforms have beeninvestigated for color images [50], [51].

VI. EXPERIMENTAL RESULTS

A. Test Images

Our simulations were carried on 512512 SPOT images witha radiometric precision of 8 bpp. The first set of images in Fig. 3corresponds to a three-band image of size 512512 depictingthe urban area of the city of Tunis (Tunisia). It is a complexcity scene that contains high-frequency details such as buildingstructures and roads. The second set of images corresponds to afour-band image provided by the SPOT 4 satellite. It concernsa rural region near the city of Kairouan (Tunisia) as shown inFig. 4. We define the correlation factor between bandsand as follows:

(43)


Fig. 3. Original spectral bands of “Tunis.”

Fig. 4. Original spectral bands of “Kairouan.” (a) XS1, (b) XS2, (c) XS3, and (d) XS4.

where is the cross-correlation function betweenand, and is the autocorrelation function of band. As

shown in Table II, high correlation exists between some pairs ofbands. Therefore, it seems quite reasonable to expect some gainin terms of compression by taking advantage of these correla-tions.

It is worth pointing out that, although the results providedsubsequently are concerned with these two specific scenes, tests

carried out with other images have led to the same conclusionsfor similar values of the optimized parameters.

B. Analysis Depth

For multiresolution decompositions, the numberof decom-position stages has to be selected by the user. In the context ofprogressive transmission, it is recommended to limit the numberof decomposition stages, as the receiver is not able to recognize


TABLE IICORRELATION FACTORS OFALL THE COUPLES OF THETEST IMAGES

TABLE III“TUNIS” IMAGE. RESULTING ENTROPIES OFPURELY SPATIAL APPROACHES.

THE DEPTH OFSUBBAND DECOMPOSITIONSIS J = 4 (H , H , H :ENTROPIES FOREACH BAND, H : CUMULATIVE ENTROPY)

rapidly and reliably images of low resolution. Hence, on eachset of test images, we performed only four-stage decomposi-tions ( ).

C. Comparison in Terms of Entropy

Several experiments have been carried out so as to tune theproposed coding scheme. First, we are interested in evaluatingthe capabilities of decorrelation of the proposed vector-liftingschemes. To be able to draw conclusions independent of thecoder, we begin to assess the performances in terms of the globalentropy .

In the case of the three-component “Tunis” image sequence,Tables III–VI allow to compare the performances of the pro-posed decompositions with those of conventional techniques.Table V provides the performance of the two-stage hybrid pro-cedure described in [24]: the transform decorrelates spa-tially the bands, and the MSC decorrelates the images spectrally.Compared to monoresolution coders, multiresolution represen-tations lead to a great decrease of entropy values, if we exceptthe monoresolution predictor ID#84. However, this scheme doesnot offer any scalable feature. As expected, we can concludefrom Tables III and V that it is preferable to take advantageof both spatial and spectral decompositions. Furthermore, whenspectral decorrelation is carried out, the band ordering is a keyfactor that can affect significantly the compactness of the rep-resentations. For instance, from Table VI, it can be noticed thata difference of 1.5689 bits exists between entropy values ob-tained with band ordering (3,2,1) and (2,3,1) for DEC2. Finally,

TABLE IV“TUNIS” IMAGE. RESULTING ENTROPIES OFHYBRID MONORESOLUTION

CODERS. THE REFERENCEBAND IS CODED BY AN OPTIMAL LINEAR

PREDICTOR (OLP) (H , H , H : ENTROPIES FOREACH BAND,H : CUMULATIVE ENTROPY, b = 1, MSC: MONORESOLUTION

SPECTRAL CODING)

TABLE V“TUNIS” IMAGE. RESULTING ENTROPIES OF THEHYBRID METHOD

PROPOSED BYMOZELLE et al.: A c DECOMPOSITIONWITH

J = 4 IS FOLLOWED BY SPECTRAL DECORRELATION WITH

A MONORESOLUTIONSPECTRAL CODING (MSC)

TABLE VI“TUNIS” IMAGE. RESULTING ENTROPIES FOR THEVECTOR-LIFTING SCHEMES

DEC1, DEC2,AND DEC3 DESCRIBED INSECTION IV

vector-lifting schemes (especially DEC3) outperform the ex-isting coders and are even better than the competitive two-stagehybrid procedure: compared to this latter technique, an improve-ment of 0.3326 bpp is obtained in terms of entropy with DEC3.

For the four-band image “Kairouan,” almost the sameconclusion as in the three-component case can be drawn fromTable VII. The DEC4 transform is clearly better than thetransform, since the difference in terms of entropy is equal to0.2684 bpp.

D. Potential Interest of a Preprocessing Stage

Table VIII indicates that our method DEC3 outperforms theexisting coders when no pre- or postprocessing is added. If apreprocessing stage is inserted, the compression ability of theconventional intraband decomposition such as is remark-ably enhanced: the entropy decrease is about 0.5 bpp with theORCT (for the “Tunis” image). At the same time, the gain forDEC3 is less important when an ORCT is applied. However,


TABLE VII“K AIROUAN” IMAGE. ENTROPIESCORRESPONDING TO THE

DIFFERENT DECOMPOSITIONSINTRODUCED IN SECTION IIAND THE FOUR-BAND VECTOR-LIFTING SCHEME DEC4

TABLE VIII“TUNIS” IMAGE. INFLUENCE OFPRE- AND POSTPROCESSING INTERMS OF

ENTROPYH (c : INTEGER-TO-INTEGERWAVELET TRANSFORM, DEC3:PROPOSEDVECTOR-LIFTING SCHEME, MSC: MONORESOLUTIONSPECTRAL

CODING, ORCT: ORIGINAL REVERSIBLE COLOR TRANSFORM)

it turns out that DEC3 remains the most compact representa-tion even when an ORCT is coupled with the transform.Note that the preprocessing and the spectral prediction realizedin the vector-lifting scheme may appear as redundant opera-tions, as their common objective is to decorrelate the spectralbands. This is the reason why the gain resulting from the useof the vector-lifting DEC3 is reduced when a preprocessing isapplied. In the case of the SPOT 4 image “Kairouan,” the per-formances are given in Table IX. Following the recommenda-tion of the JPEG 2000 standard, an ORCT is applied on the firstthree components, and the last one is not processed. Then, the re-sulting components are transformed separately by thetrans-form. Clearly, the G-ORCT preprocessing based on the decor-relation of all the components achieves better perfor-mances. However, DEC4, combined with G-ORCT, providesthe smallest value for the entropy. The third lines of Tables VIIIand IX indicate that a postprocessing is not the most adequatetechnique.

E. Comparison in Terms of Bit Rate

A comparison based only on the entropies is questionable.For real compression schemes, entropy coders must be appliedto the resulting hierarchical representations in order to exploit

TABLE IX“K AIROUAN” IMAGE. INFLUENCE OFPRE- AND POSTPROCESSING IN

TERMS OFENTROPYH (c : INTEGER-TO-INTEGER WAVELET

TRANSFORM, DEC4: PROPOSEDVECTOR-LIFTING SCHEME, G-MSC:GENERALIZED MONORESOLUTIONSPECTRAL CODING, ORCT: ORIGINAL

REVERSIBLE COLOR TRANSFORM, G-ORCT: GENERALIZED ORIGINAL

REVERSIBLE COLOR TRANSFORM)

Fig. 5. “Kairouan” image. PSNR versus exact bit rate (bpp) resulting from theEZW coder.

the residual redundancies. Therefore, the average bit rate of thegenerated bit stream needs to be computed. An entropy coderproducing an embedded bit stream enables a gradual reconstruc-tion of the original image. Indeed, only a small part of the bitstream is necessary to recognize the picture with a good quality.A well-known coder is the embedded-zerotree-of-wavelet coef-ficients (EZW) [40]. The success of the EZW coder is due to itsability to exploit simultaneously the sparseness of the waveletrepresentations and the interscale similarities. For the SPOT 3image, with the transform, the value of the actual bit rateusing this coder is 4.071 bpp, but with DEC3, it is 0.4037 bppless (3.668 bpp). Concerning the SPOT 4 image, the bit ratesobtained, respectively, with and DEC4 are 3.989 bpp and3.365 bpp. Here again, a substantial reduction of 0.6241 bpp ismade possible with DEC4.

As expected, a clear improvement in terms of bit rate re-sults from the use of such an efficient encoding algorithm. Itcan be noted that a greater savings in terms of bit rate can beachieved if the decoding process is stopped at an intermediatestep. Fig. 5 provides the sequence of the bit ratesversusthepeak-signal-to-noise ratio (PSNR) when the EZW encoding al-gorithm is applied on the different set of coefficients of eachspectral band, for the vector-lifting DEC4. However, this corre-sponds to a lossy compression of the images. Fig. 6 illustratesthe EZW coder by providing the last six reconstructed images atthe receiver side corresponding to a progressive transmission.


Fig. 6. “Kairouan” image (band XS4). Reconstructed images (a) at 0.0336 bpp (PSNR = 30.67 db), (b) at 0.0953 bpp (PSNR = 31.72 db), (c) at 0.4080 bpp(PSNR = 35.09 db), (d) at 1.2795 bpp (PSNR = 41.35 db), (e) at 2.6480 bpp (PSNR = 50.01 db), and (f) at 3.5019 bpp (PSNR=1 dB).

VII. CONCLUSION

In this paper, new vector-lifting schemes have been proposed,allowing to take into account the correlations existing betweenthe different components of a multispectral image. The providedresults show that these multiresolution decompositions lead toimproved performances for lossless image compression. In ourpreliminary results, a gain has also been observed for lossy com-pression of multispectral images.

The optimization techniques used in this paper in order to ob-tain the prediction/update coefficients are, however, quite inten-sive. This means that their use should be restricted to off-linedesign procedures. It should be useful, therefore, to carry outmore exhaustive tests on a larger dataset of satellite images soas to refine the values of a fixed set of coefficients.

The proposed framework offers a great number of degreesof freedom in the choice of the structure of the predictor/up-date operators, as well as the processing order of the different


bands at each decomposition stage. By varying these parame-ters, we think that some improvements of these decompositionscould be obtained. We also envisage the application of the re-sulting vector-lifting schemes to data collected by recent sys-tems (SPOT 5 and ENVISAT/MERIS).

In our future work, we plan to pay increased attention to thecoding part so as to better exploit the residual redundancies ex-isting for the resulting wavelet-like coefficients in the differentspectral bands. We will also study the possibility to compressimages with different resolutions (XS and P bands) by the pro-posed multiresolution analysis techniques.

ACKNOWLEDGMENT

The authors would like to thank the two reviewers, as wellas the associate editor, for their careful reading of the paper andtheir helpful suggestions. They are also grateful to R. Boussema(LTSIRS-ENIT) and Z. Belhadj (SUP’COM/LTSIRS-ENIT) forproviding the SPOT images.

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Amel Benazza-Benyahia(M ’02) received the B.S.degree in engineering from the National Institute ofTélécommunications, Evry, France, in 1988, and theDoctorat degree from the Université Paris-Sud (XI),Paris, France, in 1993.

In 1993, she joined the Ecole Supérieure desCommunications, Tunis, Tunisia, where she iscurrently an Assistant Professor with the Departmentof Applied Mathematics, Signal Processing andCommunications. Her research interest includelossless and progressive image compression, signal

denoising, and medical image analysis.

Jean-Christophe Pesquet (SM’ 99) receivedthe B.S. degree in engineering from Supélec,Gif-sur-Yvette, France, in 1987, and the Doctoratdegree from the Université Paris-Sud (XI), Paris,France, in 1990. He received the Habilitationà Diriger des Recherches from the UniversitéParis-Sud in 1999.

He is currently Professor at the Université deMarne-la-Vallée, France and a Research Scientistat the Laboratoire Traitement et Communicationde l’Information (URA CNRS 820), Paris, France.

From 1991 to 1999, he was a Maître de Conférences at the Université Paris-Sud,and a Research Scientist at the Laboratoire des Signaux et Systèmes, CentreNational de la Recherche Scientifique (CNRS), Gif sur Yvette, France.

Mohamed Hamdi received the B.S. degree in engi-neering from the Ecole Supérieure des Communica-tions, Tunis, Tunisia, in 2000.

Since 2000, he is with the National Digital Certifi-cation Agency, where he is currently Head of the Riskand Vulnerability Analysis Project.

Vector-lifting schemes for lossless coding and progressive archival of multispectral images

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