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Contents lists available at ScienceDirect

Signal Processing

Signal Processing 90 (2010) 599–610

0165-16

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/sigpro

Satellite image compression by post-transforms in thewavelet domain

X. Delaunay a,�, M. Chabert b, V. Charvillat b, G. Morin b

a CNES/TeSA/NOVELTIS, Parc Technologique du Canal, 31520 Ramonville-Saint-Agne, Franceb IRIT/ENSEEIHT, 2 rue C. Camichel, 31071 Toulouse Cedex 7, France

a r t i c l e i n f o

Article history:

Received 5 August 2008

Received in revised form

23 July 2009

Accepted 27 July 2009Available online 3 August 2009

Keywords:

Still image coding

Discrete transforms

Wavelet transform

Satellite application

84/$ - see front matter & 2009 Elsevier B.V. A

016/j.sigpro.2009.07.024

responding author.

ail address: xavier.delaunay@noveltis.fr (X. De

a b s t r a c t

This paper proposes a novel compression scheme with a tunable complexity-rate-

distortion trade-off. As images increase in size and resolution, more efficient

compression schemes with low complexity are required on-board Earth observation

satellites. The standard of the Consultative Committee for Space Data Systems (CCSDS)

defines a strip-based compression scheme with the advantages of a low complexity and

an easy rate control [CCSDS, Image Data Compression Recommended Standard CCSDS

122.0-B-1 Blue Book, November 2005]. However, future mission specifications expect

higher performance in terms of rate-distortion. The scheme proposed in this paper

intends to perform better than the CCSDS standard while preserving low complexity and

easy rate control. Moreover, to comply with existing on-board devices, the proposed

core compression engine still uses the wavelet transform but in association with a linear

post-processing inspired from the bandelet transform. The post-transform decomposes

a small block of wavelet coefficients on a particular basis. This basis is adaptively

selected within a predefined dictionary by rate-distortion optimization. The computa-

tional complexity depends upon the dictionary size and of the basis structure. An

extremely simple dictionary, reduced to the Hadamard basis, is proposed. The post-

transform efficiency is illustrated by experiments on various Earth observation images

provided by the French Space Agency (CNES).

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

The decorrelation process is an important step in aclassical compression scheme based on transform coding.Indeed, images are highly spatially correlated. Thus, if nodecorrelation was applied, many pixels should be jointlyencoded to obtain a bit-rate close to the minimal entropy.However, high dimension vector coding requires hugecomputational capabilities and is thus impracticable on-board. Appropriate transforms allow to decorrelate thedata and to reduce low order entropies. The transformed

ll rights reserved.

launay).

coefficients can thus be quantized and then encodednearly independently by a low complexity entropy coder[2]. Currently, the best state-of-the-art compressionresults have been obtained with the discrete wavelettransform (DWT). This paper proposes to associate theDWT with an appropriate post-transform to improve thedecorrelation step with a tunable complexity increase.

Despite fair properties, the DWT presents somelimitations. First, many bits are required to encode strongDWT coefficients associated to edges and textures. Second,dependencies remain between local neighboring DWTcoefficients [3–5]. Future on-board image compressiontechniques should overcome these limitations. Currently,the remaining dependencies are exploited by smartlydesigned encoders. Coders from the zero-tree family [6,7]

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Table 1Resolutions and bit-rates of French Earth observation satellites.

SPOT4 (1998) SPOT5 (2002) PLEIADES (2010)

Swath (km) 60 60 20

Resolution (m) 10 2.5 0.7

Bit-rate 32 Mb/s 128 Mb/s 4.5 Gb/s

X. Delaunay et al. / Signal Processing 90 (2010) 599–610600

exploit inter- and intra-band redundancies by using atree-like encoding of the positions of large zero areas inthe bit-planes. The bit-plane encoder (BPE) of the CCSDS1

recommendation [1] which specially targets on-boardcompression may be classified into this family. Theembedded block coding with optimal truncation points(EBCOT) algorithm of JPEG2000 standard [8,9] usescontextual bit-plane coding. A clever adaptive scan ofthe bit-planes exploits intra-band redundancies alonggeometrical structures such as edges. Finally, morpholo-gical coders [10,11] build clusters of significantcoefficients and apply region growing techniques toexploit the redundancy associated to image structures.The storage of the significant bit lists requires a highmemory capacity. Moreover, the implementation is diffi-cult since the adaptive scan of the coefficients depends onsignificant bits.

Another way to go beyond the limitations of the DWTis to design more powerful transforms. Since 2000, manynew transforms have been proposed in the literature tocompete with the DWT [12–14]. Many, such as thecurvelets [12], introduce directionality and anisotropy inthe basis elements and are successful in restoration tasks.Unfortunately, few preserve the critical sampling feature,and the resulting redundancy is a huge penalty forcompression. The bandelet transform [14] is an orthogo-nal non-redundant post-transform. This article applies toon-board compression and generalizes the bandelettransform: the proposed compression method derivesfrom the bandelet transform and is called the post-transform compression scheme in the following. As thebandelet transform by groupings proposed by Peyre in[14,15], the post-transform applies on the DWT coeffi-cients. First a two-dimensional DWT of the image isderived using the wavelet recommended by the CCSDS.Second, small blocks of DWT coefficients are linearlytransformed by projection on an optimal basis selected ina predefined dictionary by rate-distortion optimization.Because it applies on small blocks, this process is notmemory demanding and could be implemented onhardware for strip-based on-board compression. More-over, this very simple processing only performs vectorialdot products. Finally, the dictionary size can be adapted tocomputational capacities.

The proposed post-transform is compared to theoriginal bandelet transform in terms of compressionresults and complexity for various dictionaries. In parti-cular, low complexity solutions are investigated for on-board implementation purpose. Indeed, the post-trans-form compression scheme has adjustable computationalcomplexity. Moreover, this scheme parameters (block size,rate-distortion criterion and dictionary composition) hasbeen optimized on a learning set of 12-bit high resolutionsatellite images provided by the French Space Agency(CNES). Finally, the proposed compression scheme iscompared to the CCSDS and JPEG2000 standards in terms

1 The CCSDS is an international consortium including the major

space agencies in the world such as the European Spatial Agency (ESA),

the National Aeronautics and Space Administration (NASA) and the CNES

(Centre National d’Etudes Spatiales).

of rate-distortion. Note however that, according to theCCSDS recommendation, JPEG2000 is too complex for on-board compression [16]. The CCSDS recommendationprovides many arguments against the use of JPEG2000in this context: JPEG2000 may be parallelized but requiresthree coding passes for each bit-plane. Second, theoptimal JPEG2000 rate control has high implementationcomplexity whereas the sub-optimal rate control isinaccurate [16]. In [17], some contexts in the EBCOT coderhave been removed in order to reduce JPEG2000 complex-ity. Although a good compression performance wasmaintained, the complexity gain was not sufficient.

The paper is organized as follows: Section 2 presentsthe needs and constraints for on-board spacecraft com-pression. Section 3 introduces the general principles of thepost-transform for compression. Section 4 proposesdifferent post-transform dictionaries. Section 5 studiesthe associated compression performance obtained withthese bases. Section 6 concludes the paper.

2. On-board compression: needs and constraints

Table 1 gives the data rates at the input of thecompression system of three recent French Earth observa-tion missions: SPOT4, SPOT5 and PLEIADES. This tableillustrates the increasing on-board compression needs.The main on-board constraints are strip-based inputformat produced by push-broom acquisition mode, lim-ited down-link capacity and limited on-board computa-tional capacity. First, as the satellite travels Earth surfaceup and down, the optical sensors produce an image offixed width but with a virtually endless length. Therefore,this image has to be compressed and transmitted duringits acquisition. For this reason, images will be processedby blocks of 16 lines on the future PLEIADES satellites.Second, even when the satellite is visible from a receiverstation on-ground, the down-link rate is limited. Hence, abuffer is generally used on-board to store the data thatcannot be transmitted in real time. Nevertheless, thebuffer capacity remains limited. Consequently, the outputbit-rate of the compressor must be controlled. Coders thatproduce embedded bit-streams such as the BPE of theCCSDS [1] are thus recommended. Bit-rate regulation isthen possible by simple truncation of the bit-streamwhatever the bit-rate. On the contrary, as mentioned inthe Introduction, a reliable tuning of JPEG2000 bit rate hasa high computational cost [16]. Indeed, JPEG2000 encoderimposes specific truncation points.

On the SPOT5 satellite launched in 2002, the decorr-elator is the discrete cosine transform (DCT) [18]. OnPLEIADES satellites, the decorrelator will be the DWT.Fig. 1 displays a simulated PLEIADES panchromatic image

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Fig. 1. PLEIADES panchromatic simulated image and the associated DWT.

Table 2Statistical dependencies between pairs of wavelet coefficients.

Distance d 1 2 3

High pass direction

Correlation coefficient r �0.54 0.13 �0.01

Mutual information Ir (%) 15.3 4.4 2.8

Low pass direction

Correlation coefficient r 0.18 �0.01 0.01

Mutual information Ir (%) 8.4 4.8 3.8

X. Delaunay et al. / Signal Processing 90 (2010) 599–610 601

with a resolution of 70 cm and the associated DWT. ThePLEIADES compression scheme associates the DWT to alow complexity bit-plane coder which produces anembedded bit-stream. A challenging issue for future on-board compressor is to exploit or to reduce the remainingdependencies between DWT coefficients to improvecompression performance. The proposed post-transformcompression scheme is intended to reduce these depen-dencies while maintaining a low complexity.

3. Post-transform in the wavelet domain

The DWT of the images is performed on three levels ofdecomposition using the lossy 9

7 biorthogonal float filtersrecommended by the CCSDS [1]. In order to exploitremaining redundancies between DWT coefficients,blocks of coefficients in all subbands, except the low-frequency subband (LL3), are further transformed inde-pendently. No blocking artifacts are visible on thereconstructed image since the blocks are processed inthe wavelet domain. First, the choice of an appropriateblock size must be discussed.

3.1. Remaining redundancies

Joint-probabilities between DWT coefficients havebeen modeled in [4]. In [3], mutual information measure-ments between two or more DWT coefficients have shownthat most dependency is intra-band. The proposed post-transform precisely aims at exploiting intra-band redun-dancies. Table 2 displays the estimated statistical depen-dencies in terms of correlation coefficient r and mutualinformation Ir between pairs of DWT coefficients as afunction of their distance. The estimation has beenperformed from seven satellite images, i.e. on more than7.2 millions of DWT coefficients. The distance d betweenpairs of coefficients is 1, 2 or 3 pixels in vertical orhorizontal direction. High pass direction denotes pairs ofcoefficients horizontally (respectively, vertically) alignedin the subband HL1 (respectively, LH1). Low pass direction

denotes pairs of coefficients vertically (respectively,horizontally) aligned in the subband HL1 (respectively,LH1). The relative mutual information Ir is defined by

IrðX;YÞ ¼2ðH0ðYÞ � H0ðYjXÞÞ

H0ðXÞ þ H0ðYÞ, (1)

where H0ðXÞ is the zero order entropy of X. IrðX;YÞ can beinterpreted as the proportion of information that can besaved to encode Y if X is already known. For entropyderivation, the quantization operation must be taken intoaccount. A quantization step q lead to a particular bit-rateafter arithmetic coding. For Ir computation in Table 2, thequantization step has been set to q ¼ 32. This value leadsto a bit-rate near 2 bpp targeted for on-board compressionof 12-bit Earth observation images. Table 2 emphasizesthat, in the DWT of 12-bit high resolution satellite Earthobservation images, intra-band redundancies betweenDWT coefficients are confined to a small neighborhood.Thus, the post-transform is applied to small blocks. For anon-board satellite compression application, processingsmall blocks is less memory intensive and requires lesscomputational capabilities. Moreover, since the blocks areprocessed independently, the process can be parallelizedto achieve real-time compression. Finally such processingis suitable for a strip-based input format as produced bypush-broom type sensors. Indeed, the compression canbegin whereas the image acquisition is not completed. Inthe following, DWT coefficients are processed by blocks ofsize 4� 4. This block size is a good compromise for the

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Fig. 2. Post-transform compression scheme.

Fig. 3. Post-transform of one block of wavelet coefficients.

X. Delaunay et al. / Signal Processing 90 (2010) 599–610602

compression: the bigger the blocks, the more complex thecomputation. However when the block size decreases, thenumber of blocks and thus the side information increasesas explained in the following.

3.2. Post-transform general principle

For each block of 4� 4 DWT coefficients, the best post-transform decomposition basis is chosen among a dic-tionary of bases by minimization of a rate-distortioncriterion. This method leads to compression performanceimprovement in terms of rate distortion with respect tothe DWT alone. Indeed, the canonical basis, whichcorresponds to the absence of post-transform, belongs tothe dictionary. However, there are two main drawbacks:the need to signal to the decoder which post-transformbasis has been chosen on each block and the computa-tional cost of the basis selection. Indeed, the selected basisidentifier, or so-called side information in the following,must be sent for each block. Moreover, although only onebasis is retained for each block, the decomposition on allthe bases in competition must be computed to select thebest one. Robert et al. have designed a similar process in[19] to enhance the compression performance of the videocoder H.264. Blocks of size 4� 4 are circularly shiftedaccording to nine different orientations before DCTderivation. The best orientation is selected by optimiza-tion of a rate-distortion criterion. The optimizationrequires the derivation of the nine possible shifts andtheir associated DCT on each block. The selected shift

must be transmitted. Unfortunately, the slight decorrela-tion improvement is compensated by the bit-rate increasedue to shift signaling. On the contrary, the post-transformproposed in this paper globally enhances the compressionperformance.

The post-transform compression scheme is outlined onFig. 2. Blocks of DWT coefficients of all the subbandsexcept the LL3 (low-pass) subband are post-transformedand quantized. The compressed bit-stream is composed ofthe entropy coded post-transform coefficients and sideinformation. The entropy coder is the adaptive arithmeticcoder provided in [20].

The post-transform of one block is detailed in Fig. 3.Each block of DWT coefficients denoted by f is linearlytransformed in order to obtain a representation f b� moreeffectively compressible. The linear post-transforms aredefined in a dictionary D known by both the encoder andthe decoder. This dictionary contains NB different ortho-normal bases. Let Bb ¼ ff

bmg

Mm¼1 denote the basis number

b with b 2 f1; . . . ;NBg. The fbm are the associated basis

vectors and M ¼ 16 is the space dimension. Blockscan be viewed as vectors of R16. The post-transformedrepresentation f b of f simply results from a change ofbasis, from the canonical basis to one of the bases of thedictionary D:

f b¼XMm¼1

of ; fbm4fb

m.

The quantized representations are denoted by f bq where q

is the quantization step of the uniform scalar quantizer

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Fig. 4. Post-transform compression algorithm.

X. Delaunay et al. / Signal Processing 90 (2010) 599–610 603

with double dead-zone Qq:

QqðxÞ ¼0 if jxjoq;

signðxÞðkþ 1=2Þq if kq � jxjoðkþ 1Þq:

(

This is the commonly used quantizer for DWT compres-sion. It has been shown theoretically and numerically in[21] that it is close to the best choice to minimize thedistortion. On each block, f b�

q denotes the best representa-tion in terms of compression among all the quantizedrepresentations f b

q, b 2 f1; . . . ;NBg.The post-transform coding algorithm is summarized in

Fig. 4. The different steps of the algorithm as well as thechosen parameters are discussed in the following. Thedecoding process is very simple. The post-transformedcoefficients f b�

q are first inversely quantized. The DWTcoefficients can be reconstructed using the side informa-tion on each block. Finally, the inverse DWT gives thereconstructed image.

Table 3Rate-distortion trade-off for Fig. 1(a) with the quantized wavelet

coefficients W and an adaptive arithmetic coder.

3.3. Rate-distortion trade-off

The best representation f b�q of each block is selected

using a rate-distortion criterion after quantization of therepresentations f b; b 2 f1; . . . ;NBg.

Entropy H0ðWÞ (bpp) Bit-rate RðWÞ (bpp) PSNR(W) (dB)

1.675 1.714 50.04

1.746 1.794 50.48

Table 4Rate-distortion trade-off for Fig. 1(a) with the quantized post-trans-

formed coefficient PT and an adaptive arithmetic coder.

H0ðPTÞ H0ðPTÞ RðPTÞ RðPTÞ þ Rðside informationÞ PSNR(PT)

1.589 bpp 1.584 bpp 1.627 bpp 1.794 bpp 51.17 dB

3.3.1. Lagrangian criterion

The rate-distortion trade-off can be adjusted using theLagrangian Lðf b

qÞ defined by

Lðf bqÞ ¼ Dðf b

qÞ þ lRðf bqÞ, (2)

where the distortion Dðf bqÞ is a measure of the square error

due to the quantization and Rðf bqÞ is an estimation of the

bit-rate needed to encode the post-transformed block andthe side information. Finally, l is the Lagrangian multi-plier optimized for the compression. The distortion is

computed by

Dðf bqÞ ¼ kf

b� f b

qk2 ¼

XMm¼1

jab½m� � Qqðab½m�Þj2,

where the ab½m�, m 2 f1; . . . ;Mg are the coefficients of therepresentation f b. Since the post-transform is orthonor-mal, the square error can be computed in the transformdomain.

3.3.2. Bit-rate estimation

The bit-rate can be expressed by Rðf bqÞ ¼ Rcðf

bqÞ þ Rb,

where RcðfbqÞ is the bit-rate allocated to the quantized

representation f bq and Rb is the bit-rate allocated to the

side information. The first term RcðfbqÞ is estimated by the

entropy of the quantized post-transformed coefficients:

RcðfbqÞ ¼ �

XMm¼1

log2 PrðQqðab½m�ÞÞ.

The probabilities PrðQqðab½m�ÞÞ of the post-transformedcoefficients cannot be directly estimated at this stagesince they depend on the selection of the best representa-tion of each block. Therefore, these probabilities areestimated from the histogram of each DWT subband.However, for a given quantization step, the observedkurtosis (respectively, entropy) is higher (respectively,lower) after the post-transform. Indeed, the kurtosis(respectively, entropy) measures how peaky (respectively,smooth) is the probability distribution of a randomvariable. For example, on the HL1 subbands of the imagedisplayed on Fig. 1, the kurtosis of the post-transform is41.9 and the kurtosis of the DWT is 27.2. According toTables 3 and 4, the entropy of the DWT H0ðWÞ is 1.675 bppand the entropy of the post-transform H0ðPTÞ is 1.584 bppfor the same image and quantization step q ¼ 32. Never-theless, the total bit-rate of the post-transform RðPTÞ þ

Rðside informationÞ is increased up to 1.794 bpp becauseof the side information. This bit-rate increase is balancedby a quality increase according to the PSNR comparison inTables 3 and 4. Table 4 shows that the entropy of the post-transform coefficients estimated from the DWT histogramH0ðPTÞ is very close to the true entropy H0ðPTÞ. Moreover,the true bit-rate RðPTÞ obtained with the adaptive

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Fig. 5. (a) The 12 grouping configurations. (b) Basis vectors for direction #2.

Fig. 6. PELICAN airborne sensor image (30 cm resolution) and the

selected bandelet bases for LH1 subband.

X. Delaunay et al. / Signal Processing 90 (2010) 599–610604

arithmetic coder is well estimated by H0ðPTÞ with arelative error of only 2.4%. Finally, several tests on largesatellite images lead to the same observation. Note thatTables 3 and 4 are not intended to compare the DWT andpost-transform compression performance. They ratheraim at illustrating the sharing out between rate anddistortion for the post-transform. The DWT is used as areference. Compression performance will be compared inSection 5.

The second term Rb is computed by

Rb ¼ �log2 PrðbÞ with PrðbÞ ¼0:5 if b ¼ 0;

0:5=NB if b 2 f1; . . . ;NBg:

(

The post-transform denoted by b ¼ 0 is the identity, thatis, the block of DWT coefficients is not post-transformed.In most blocks, DWT coefficient decorrelation is sufficientfor an efficient compression. Therefore, to favor this ‘‘non-transformation’’, the prior probability Prð0Þ is fixed to ahigher value than the other probabilities PrðbÞ forb 2 f1; . . . ;NBg. This reduces the rate-distortion costLðf 0

qÞ compared to the other costs Lðf bqÞ.

3.3.3. Lagrangian multiplier

In [22], Le Pennec gives a theoretical expression of theoptimal Lagrangian multiplier: l ¼ ð3=4g0Þq

2 withg0 ¼ 6:5. This expression is based on a low bit-rateassumption. g0 is the ratio between the total bit-rateand the number of non-zero quantized coefficients. Thisratio has been found approximately constant at low andmedium bit-rates in [21]. This optimal Lagrangian multi-plier jointly minimizes the rate and the distortion for the

quantization step q. Another value of l would give ahigher distortion with the same bit-rate or higher bit-ratewith the same distortion, for this quantization step. In[23], we have derived a similar expression under the highbit-rate assumption: l ¼ ðlnð2Þ=6Þq2. This assumption mayhold at bit-rate greater than 2 bpp targeted for on-boardcompression. The numerical value of this expression isvery close to the previous one: l � 0:115q2. However, anoptimization process, similar to the Shoham and Gershoapproach [24], performed on several Earth observationimages, has shown that l � 0:15q2 leads to a slightperformance improvement [23].

4. Dictionaries of bases

This section studies the dictionary composition. Theaim is to obtain a low complexity yet efficient post-transform.

4.1. Bandelet transform by groupings

4.1.1. The banlet bases

In [15], Peyre has proposed the bandelet bases bygroupings and described the post-transform framework.Directional bases are built by linking coefficients along thesame direction as displayed in Fig. 5(a). Discrete Legendrepolynomials up to a degree n� 1 are assigned to thegroupings of n coefficients to form orthonormal bases. Thevectors fb;m;m ¼ 1; . . . ;M, of the directional basis #2 aredisplayed in Fig. 5(b). The DCT basis and two Haar baseswith different levels of resolution are added to these basesto form the dictionary of bandelet bases. Those additionalbases are intended to be selected on blocks wheredirectional basis are ineffective like blocks of textures.Fig. 6(b) presents the results of the bandelet analysis onthe LH1 subband of the Fig. 6(a). The bases have beenselected according to the rate-distortion criterion (2) andare depicted on each 4� 4 blocks. Blocks marked with awhite line are post-transformed in a directional basis,blocks marked with H0 or H1 in a Haar basis and blockmarked with a C in the DCT basis. Other blocks are notpost-transformed. The selected basis directions are rarelyrelated to the edge directions for two main reasons. First,directions can hardly be distinguished on small blocks of4� 4 coefficients. Second, geometrical information is notconsidered in the best basis selection criterion (2).

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X. Delaunay et al. / Signal Processing 90 (2010) 599–610 605

4.1.2. Exploited mutual information

The intra-block statistical redundancies exploited bythe bandelet bases is analyzed on the subbands HL1, LH1

and HH1 of seven Earth observation images of size1024� 1024. Estimations are performed on 7� 16 384blocks for each subband HL1, LH1 and HH1. The quantiza-tion step remains q ¼ 32. Blocks of DWT coefficients areclassified according to their best post-transform repre-sentation. DWT blocks that are best represented in thedirectional basis #b according to the rate-distortion cost(2) compose the bth class. Fig. 7 displays the relativemutual information in each pair of quantized DWTcoefficients from blocks of the subbands HL1, LH1 andHH1 that are best represented in the directional basis #2.This class contains 4335 blocks for the HL1 subbands,4340 blocks for the LH1 subbands and 1909 blocks for theHH1 subbands. The lines between two coefficientsrepresent the relative mutual information Ir defined inEq. (1). Plain lines are for Ir � 20%, dashed lines are for16% � Iro20% and dotted lines are for 10% � Iro16%.Mutual information of DWT coefficient pairs fit thedirectional groupings of the bandelet basis #2. Thedependency is thus well exploited by this basis in all thesubbands. Nevertheless, it can be observed that depen-

Fig. 7. Mutual information of pairs of wavelet coefficients from blocks of

class #2.

Fig. 8. Correlations in pairs of wavelet coefficients from blocks of class

#2.

Fig. 9. Bases built by PCA for each subband HL1, LH1 and HH1: (a) PC

dency in the direction of the high-pass wavelet filter is notalways exploited in the subbands HL1 and LH1.

4.1.3. Correlation analysis

Similarly, Fig. 8 displays the correlation coefficients r

for each pair of DWT coefficients from blocks in class #2.Plain lines are for jrj � 0:5, dashed lines are for 0:4 �jrjo0:5 and dotted lines are for 0:25 � jrjo0:4. Since fewcorrelations appear in the direction of the groupings, itcan be said that the directional basis #2 does not exploitmuch correlations. The correlations observed correspondto the direction of the high-pass wavelet filter. The sameholds for the other block classes. Based on these observa-tions, new bases that better decorrelate the DWTcoefficients are built in Section 4.2. In particular, thestatistical properties of each subband are taken intoaccount.

4.2. Dictionary of bases derived from image statistical

analysis

A better reduction of redundancies may be obtainedfrom a dictionary derived from a statistical image analysis.A set of relevant images is processed off-line to build a so-called exogenic dictionary which will be used for thecompression of any other images. The natural way toreduce correlations in a random process is to perform aprincipal component analysis (PCA). This is consistentwith the transform coding principles in which data arefirst decorrelated so as to reduce the zero order entropy[2]. One key point to improve the compression perfor-mance is to build one dictionary per subband. In this way,neither the complexity nor the side information areincreased.

4.2.1. Bases built by PCA on each subband

One PCA has been performed on each subband of theDWT of a learning set of images. The resulting bases aredisplayed in Fig. 9. The vectors are sorted column-wise bydecreasing eigenvalues. Note that the vectors of principalcomponent exhibit the low-pass and high-pass directionsof the wavelet filters. For compression, blocks of DWTcoefficients can either be post-transformed in the adaptedPCA basis or not post-transformed. The computationalcost is limited since only one post-transform is computed.

A basis on HL1, (b) PCA basis on LH1 and (c) PCA basis on HH1.

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Fig. 10. Bases built by PCA on the block classes #2 for each subband: (a) HL1, (b) LH1 and (c) HH1.

Fig. 11. Basis vectors by decreasing energy order for HL1 subband: (a) PCA basis on HL1, (b) DCT basis and (c) Hadamard basis.

Fig. 12. Approximation rates of HL1 wavelets coefficients.

X. Delaunay et al. / Signal Processing 90 (2010) 599–610606

One key advantage of the PCA bases over the bandeletbases is that vectors are sorted in decreasing energy order.Thus, high magnitude coefficients are expected on the firstvectors of those bases. An entropy coder could exploit thisprior knowledge.

4.2.2. Bases built by PCA on each class of blocks

For a fair comparison between PCA and the directionalbandelets with respect to the number of bases we havebuilt 12 PCA bases adapted to the 12 classes of blocks builtin Section 4.1 [25]. The PCA bases built on class #2 in thesubbands HL1, LH1 and HH1 are displayed in Fig. 10. Thegoal is to remove the correlations observed in thoseclasses in Fig. 8. These new PCA bases are successful inthis task. The maximum correlations observed on a set ofsix test images, outside the learning set, are less than 0.2in these PCA bases although they are more than 0.5 in thedirectional bandelet bases. Other dictionaries of basesmay be created based on other criteria to form thedifferent classes of blocks. For example, specialized basesmay be built for seas, forests, fields or city areas.

4.3. Standard bases

As shown in Fig. 11, when the vectors have beenreorganized, the DCT basis and the PCA basis are similar.Thus, the DCT can be used as a post-transform with themain advantage that it can be computed using well-knownfast algorithms. The same holds for the even simplerHadamard transform involving only sums and differencesof the DWT coefficients without multiplication. The required

normalization by 4 can be done by 2 bit shifts. Fig. 12shows the approximation rate of HL1 wavelet coefficientsfor the post-transform in the PCA, DCT and Hadamardbases. f M denotes the approximation of the block of DWTcoefficients f from the M largest post-transformed coeffi-cients. The approximation rate is similar in the threebases. The advantage of the PCA is thus limited since theother bases require less computational complexity.

5. Compression results

5.1. Post-transform with a multiple bases dictionary

If the computational power is sufficient to allow a largedictionary, PCA is a possible solution to build multiple

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Fig. 13. Gain of PSNR with respect to the wavelet transform of the post-transform with the 12 PCA bases, the 12 directional bandelet bases and the 15

bandelet bases.

Fig. 14. PSNR (a) and gain of PSNR with respect to the DWT (b) using a one basis dictionary.

X. Delaunay et al. / Signal Processing 90 (2010) 599–610 607

bases. One basis may be built for each relevant class ofblocks as described in Section 4.2. Fig. 13 presents themean compression results obtained on a set of six 1024�1024 Earth observation images from PLEIADES satelliteand PELICAN airborne sensor. The PELICAN sensor pro-vides images with the spatial resolution expected forfuture satellite images. These images are 12-bits depth.This figure compares the compression performanceobtained with the 12 directional bandelet bases, the 15bandelet bases (including the DCT and the two Haar basis)and the 12 bases built by PCA. According to Fig. 13, thePSNR with the 12 PCA bases is around 0.2 dB greater thanthe PSNR with the 12 directional bandelet bases and isclose to the PSNR with the 15 bandelet bases. Yet it hasbeen observed that the addition of the DCT and two Haarbases to the 12 PCA dictionary does not significantlyincrease the performance. In this case, at 2 bpp the PSNR is50.65 dB while it is 50.67 dB with the 15 bandelet bases.At other rates, PSNR are always slightly lower than thoseobtained with 15 bandelet bases. Indeed, the PCA aremore similar to the DCT basis than the directionalbandelet bases. Consequently, the DCT basis is moreefficient and more often selected when used in conjunc-tion with the directional bandelet bases. Another way toenlarge the PCA dictionary is to define more blocks classes

and thus more PCA bases. However, the complexityincreases almost linearly with the number of basescontrarily to the compression performance: the 12 PCApost-transform is far from being 12 times more efficientthan a one PCA post-transform. In terms of compressionperformance, 15 bandelet bases seems a better choicethan the 12 or even 15 PCA bases when the dictionary sizeis not limited by the complexity. Moreover, note that 70%of the bandelet coefficients are equal to zero. Hence, asmart implementation may lead to a smaller computa-tional complexity with the 15 bandelet bases than withthe 12 PCA bases, regarding the projection on the basiselements. This advantage must be balanced by theincreased complexity of the rate-distortion criterionderivation for the 15 bases with respect to 12 basesregarding the rate estimation. Nevertheless, if sufficientcomputational power is available, the bandelet transformwith 15 bases is a good choice.

5.2. Post-transforms using one basis

However, on-board satellites, the computational powermay allow only a one basis post-transform. Fig. 14 displaysthe compression performance obtained in this case. Fig.14(a) compares compression results obtained with the

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Fig. 15. Hadamard post-transform and bandelet transform compared to JPEG2000 and the CCSDS coder on six Earth observation images.

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Hadamard post-transform and the DWT coefficients with-out post-transform. Fig. 14(b) compares the performanceof the different post-transform dictionaries, i.e. the PCAbases (one per subband), or the DCT basis, or theHadamard basis relatively to the compression results ofthe DWT coefficients. Obviously, the post-transform usinga dictionary consisting of only one basis improves thecompression performance. For an approximate target rateof 2 bpp, a typical bit-rate in on-board applications, thePSNR increase is between 0.4 and 0.6 dB compared to theDWT alone. The results obtained with the PCA, the DCTand the Hadamard bases are similar. When complexityconstraints impose a one basis post-transform, the use ofa PCA basis on each subband is not recommended unlessthe learning set is sufficiently representative. The DCTpost-transform is less complex and gives good results. Yet,Hadamard basis could be advised since it is even lesscomplex with similar compression performance. More-over, the energy compaction property of the Hadamardbasis, emphasized on Fig. 12, could also be exploited bythe entropy coder.

2 Ref. 1 has been obtained with the QccPack [20], ref. 2 with an

implementation of the CCSDS 122 coder [1] from NASA and ref. 3 with

Kakadu software.

5.3. Comparison to the standards

Fig. 15 compares the performance of the post-trans-form with the Hadamard basis and bandelet transformwith 15 bases to the CCSDS and JPEG2000 coder at bit-rates close to the targeted bit-rate of 2.0 bpp. In the CCSDScoder, the BPE uses tables of variable length codes for avery low complexity entropy coding. On the contrary inJPEG2000, EBCOT uses the contextual adaptive arithmeticcoder called MQ-coder [9] with a rate-distortion optimi-zation process to sort the ‘‘code-blocks’’ in the best order.There exists no such optimization process in the BPE.Although a context modeling could take advantage ofparticular statistical distribution of the post-transformcoefficients, in the proposed work, an adaptive arithmeticcoder without context is applied directly on the post-transform coefficients.

The compression performance obtained with thebandelet transforms in 15 bases are still 0.5 dB underthe results obtained with JPEG2000. However, even theless complex Hadamard post-transform outperforms theCCSDS coder. Moreover, as post-transforms applied on

small blocks, they are suitable for the scan-based mode.Note that the performance in Fig. 15 are obtained in scan-based mode for the CCSDS but in frame-based mode forJPEG2000. Although JPEG2000 includes a scan-basedmode compatible with strip compression, this feature isnot provided in available softwares such as OpenJPEG orKakadu. According to [16,26], the scan-based JPEG2000performance is approximately reduced by 0.5 dB at2.0 bpp compared to the frame-based mode, due toadditional headers in each strip. In [27], a full lowcomplexity compression scheme using the post-transformin Hadamard basis and the BPE coder has been proposed.This compression scheme even reduces the computationalcomplexity of the best basis selection. The Lagrangianrate-distortion criterion is replaced by a l1 norm mini-mization: a sum of the post-transform coefficientsreplaces the rate and distortion estimations. This com-pression scheme has a complexity sufficiently small to beimplemented on-board satellite. Moreover the resultsobtained are 0.17 dB better than those of the CCSDS.Table 5 sums up the performance of the compressionschemes mentioned. Reference compression times2 havebeen obtained on a linux PC with an Intel CPU at 1.86 GHzand 1 Go memory for the compression of a simulatedPLEIADES image of size 2048� 2048 at 2 bpp. Fromsimulations, JPEG2000 frame-based results at 2 bpp arebetter than those of the CCSDS by 1.1 dB. According to[16,26], JPEG2000 scan-based results may be around0.6 dB better than those of the CCSDS. Thus, with a post-transform compression scheme having a low complexity,the results obtained may be only around 0.5 dB lower thanthose of JPEG2000 in scan-based mode.

6. Conclusion

This article has presented a new coding scheme basedon post-transforms in the wavelet domain and inspiredfrom the bandelet theory. The remaining linear depen-dencies between wavelet coefficients have been analyzed.The post-transform compression scheme has been de-

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Table 5Performance summary of different compression schemes.

Coder Rate control Transform On-board compliance Relative complexity PSNR gain at 2.0 bpp

Adaptive arithmetic coder Impossible DWT No Ref. 1 ¼ 0:52ms=sample 0 dB (reference)

15 bandelets ðNB ¼ 15Þ No Ref. 1þ ðNB � 2M2Þ=M þ0:97 dB

Hadamard PT No Ref. 1þ ðM log2 MÞ=M þ0:54 dB

BPE Full and easy DWT (CCSDS) Yes Ref. 2 ¼ 0:36ms=sample þ0:37 dB

Hadamard PT Yes Ref. 2þ ðM log2 MÞ=M þ0:54 dB [27]

EBCOT (JPEG2000) Limited and complex Scan-based Yes � þ1 dB [16,26]

Frame-based No Ref. 3 ¼ 0:56ms=sample þ1:49 dB

X. Delaunay et al. / Signal Processing 90 (2010) 599–610 609

signed to efficiently reduce these remaining correlations.This paper shows also that, in the case of satellite images,the bandelet directional bases are hardly related to theimage geometric features. The redundancy actuallyexploited in the grouping direction can be measured bythe mutual information but some correlations foundbetween neighboring wavelet coefficients are notexploited. This type of redundancy is better exploited bythe use of PCA bases. The post-transform process requiresthe decomposition on all the dictionary bases for rate-distortion optimization. If enough computational power isavailable, the choice of the dictionary of 15 bandelet basesis advised rather than the dictionary of PCA bases. Usingan arithmetic coder, the average gain over the wavelettransform would reach 1.0 dB at the targeted rate of2.0 bpp. In the case of on-board satellite compression, thelow complexity yet efficient Hadamard basis wouldgenerally be advised. The average gain at 2.0 bpp wouldthen be 0.5 dB in PSNR. The post-transform process can beadapted to make it suitable to an embedded bit-planecoder such as the BPE of the CCSDS recommendation,resulting in a sufficiently low-complexity coder for on-board compression. Note that the average gain obtainedwith the post-transform scheme results from a localdistortion decrease near the edges. Usual image post-processing such as segmentation should take advantage ofthis property.

Acknowledgments

This work has been carried out under the financialsupport of the French space agency CNES (www.cnes.fr)and NOVELTIS company (www.noveltis.fr). We also thankG. Peyre for the useful discussions and for the softwarecomponents provided.

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