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An evaluation framework and a benchmark formulti/hyperspectral image compression

Jonathan Delcourt, Alamin Mansouri, Tadeusz Sliwa, Yvon Voisin

To cite this version:Jonathan Delcourt, Alamin Mansouri, Tadeusz Sliwa, Yvon Voisin. An evaluation framework and abenchmark for multi/hyperspectral image compression. Computer Vision Graphics and Image Pro-cessing, Elsevier, 2011, 1 (1), pp.55-71. �10.4018/ijcvip.2011010105�. �hal-00637853�

https://hal-univ-bourgogne.archives-ouvertes.fr/hal-00637853

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An Evaluation Framework and a Benchmark For Multi/HyperspectralImage Compression

Jonathan Delcourt, Alamin Mansouri, Tadeusz Sliwa and Yvon VoisinUniversity of Burgundy

Laboratoire Le2i, BP 16 Route des Plaines de l’Yonne,89010 Auxerre Cedex, France

Email: [email protected]

In this paper, we investigate different approaches for multi/hyperspectral image compression.In particular, we compare the classic Multi-2D compression approach and two different im-plementations of a 3D approach (Full 3D and Hybrid) with regards to variations in spatial andspectral dimensions. All approaches are combined with a spectral Principal Component Analy-sis (PCA) decorrelation stage to optimize performance. For consistent evaluation, we proposea larger comparison framework than the conventionally used PSNR, including eight metricsdivided into three families. We also discuss the time and memory consumption differencebetween the three compression approaches. The results show the weaknesses and strengths ofeach approach.

IntroductionA hyper/multispectral imaging system splits the light

spectrum into more than three frequency bands (dozens tohundreds) and records each of the images separately as a setof monochrome images. This type of technique increasesthe number of acquisition channels in the visible spectrumand extends channel acquisition to the light that is outsidethe sensitivity of the human eye. Such systems offer severaladvantages over conventional RGB imaging and have, there-fore, attracted increasing interest in the past few years. How-ever, multispectral uncompressed images, in which a singleimage-band may occupy hundreds of megabytes, often re-quire high capacity storage. Compression is thus necessaryto facilitate both the storage and the transmission of multi-spectral images.

Generally, a multispectral image is represented as a 3Dcube with one spectral and two spatial dimensions. The factthat a multispectral image consists of a series of narrow andcontiguously spectral bands of the same scene produces ahighly correlated sequence of images. This particularity dif-ferentiates multispectral images from volumetric ones withthree isotropic spatial dimensions, and also from videos withone temporal and two spatial dimensions. Conventional com-pression methods are not optimal for multispectral imagecompression, which is why compression algorithms need tobe adapted to this type of image.

One of the most efficient compression methods formonochrome images compression is the JPEG 20001

(Boliek, Christopoulos, & Majani, 2000 ; Boliek, Ma-jani, Houchin, Kasner, & Carlander, 2000 ; Christopou-los, Skodras, & Ebrahimi, 2000 ; Taubman, 2000 ; Taub-man, Marcellin, & Rabbani, 2002). Its extension tomulti/hyperspectral images yields to different approaches.These approaches depend on the manner of which one con-sider the multi/hyperspectral cube after the decorrelation

stage (figure 1):• in the Multi-2D approach, each image band of the

multi/hyperspectral image is considered separately (2Dwavelets + 2D SPIHT) (Du & Fowler, 2007 ; Kaarna, Toiva-nen, & Keranen, 2006 ; Mielikainen & Kaarna, 2002),• the whole cube is considered as input leading to two

main implementations: the Hybrid approach (3D wavelets+ 2D SPIHT), as used in (Penna, Tillo, Magli, & Olmo,2006), and the Full 3D approach (3D wavelets + 3D SPIHT).For this latter we used an anisotropic 3D wavelets decom-position. Many works in literature have explored the 3Dwavelet transform for multi/hyperspectral image compres-sion but they only use isotropic 3D wavelets (same type ofwavelets following all directions) (Kaarna, Zemcik, Kaelvi-ainen, & Parkkinen, 1998 ; Kaarna & Parkkinen, 2000b,2000a ; Kaarna, 2001 ; Kaarna et al., 2006 ; Kim, Xiong, &Pearlman, 2000 ; Lim, Sohn, & Lee, 2001 ; Mielikainen &Kaarna, 2002 ; Penna et al., 2006 ; Tang, Cho, & Pearlman,2003), but only Kaarna and Parkkinen describe a compres-sion method based on an anisotropic wavelets decompositionin (Kaarna & Parkkinen, 1998).

We tested the three compression approaches with the samelifting scheme wavelet transform and compared them. Toprovide a more objective benchmark, we propose a frame-work of evaluation composed of seven metrics in addition tothe classic PSNR. These metrics evaluate the quality of re-construction in terms of signal, spectral reflectance and per-ceptive aspects.

In the next section, we provide a short overview of howwe use the principal composant analysis algorithm (PCA)(Pearson, 1901) within the three compression approaches,before describing them into the second section. The thirdsection introduces the framework of comparison by splitting

1 http://www.jpeg.org

2 JONATHAN DELCOURT, ALAMIN MANSOURI, TADEUSZ SLIWA AND YVON VOISIN

Figure 1. Graphical representation of the three compression ap-proaches.

the metrics into three families and gives the explicit formulafor each metric. We discuss our experiments and their resultsin the fourth section and highlight the strengths and weak-nesses of the three approaches. In order to take the algorith-mic aspects of these strategies into account, we will also dis-cuss them in terms of time and memory consumption in thefourth section. Conclusions are presented in the last section.

Compression approachesFor the implementation of the three approaches we chose

to use the wavelets of JPEG 2000 standard because it is areference for 2D compression. The JPEG 2000 standardwavelets are ”Le Gall 5/3” for lossless compression and”Cohen-Daubechies-Feauveau 9/7” (or CDF 9/7) for lossycompression. In our case we perform lossy compression, sowe will use the CDF 9/7 wavelet.

As previously reported, multispectral images have a highcorrelation between image-bands. To achieve the best com-pression ratios it is necessary to take this correlation into ac-count.

PCA decorrelationIn order to optimize multi/hyperspectral image compres-

sion, a decorrelation step is often used. In this con-text, several methods have been developed. Classic algo-rithms are based on vector quantization (Gupta & Gersho,1992), wavelets or Hybrid methods, such as DPCM-DCT(Abousleman, Marcellin, & Hunt, 1995), KLT-DCT (Saghri& Tescher, 1991) and PCA (KLT). The PCA has been shownto be one of the most efficient spectral decorrelators (Ready& Wintz, 1973) and is used in many compression methods.

Epstein et al. propose in (Epstein, Hingorani, Shapiro, &Czigler, 1992) a method for landsat thematic mapper multi-

spectral imagery. The method first removes inter-band cor-relation by PCA to produce principle components of sevenlandsat bands. The principle components are then com-pressed using wavelet and lossless compression techniqueslike run length encoding. Harsanyi and Chang (Harsanyi &Chang, 1994) applied PCA to hyperspectral images to re-duce data dimensionality, suppress undesired or interferingspectral signatures, and classify the spectral signatures of in-terest. In (Mielikainen & Kaarna, 2002), Mielikainen andKaarna applied PCA to reduce correlation among spectralbands, but they only selected a small number of spectra fromthe image for the calculation of the eigenvectors. They thenapplied integer wavelet transform to the residual image toconcentrate energy and reduce entropy. Du and Fowler im-plemented PCA along with JPEG 2000 for hyperspectral im-age compression (Du & Fowler, 2007). They assumed thatPCA would help in spectral decorrelation and JPEG 2000would help in compression. They found that the method per-formed better than the combination scheme of wavelet forspectral decorrelation and JPEG 2000 for compression. Theytested both methods and found that, for rate-distortion and in-formation preservation, PCA with JPEG 2000 outperformedJPEG 2000 alone.

Other spectral decorrelators based on PCA may be used.In (Chang, Cheng, & Chen, 2000), adaptive KLT is usedfor decorrelation. The original image is divided into properregions, and transforms each image data-set region by thecorresponding transformation function. The results of theirsimulations show that the performance of adaptive KLT isbetter than KLT alone. In (Gu, Zhang, & Zhang, 2002) Gu etal. proposed a Kernel Based Nonlinear Subspace Projection(KNSP) method followed by kernel PCA. They partitionedthe full data space into different subspaces. Next, they usedKernel PCA for feature selection based on class separatibilitycriteria. The authors claim that the method is more suitablefor feature extraction than linear PCA and segmented linearcomponent transformation, particularly when hyperspectraldata have non-linear characteristics.

Some compression algorithms are optimised for specificapplications (classification, visualisation, dimension reduc-tion, etc.). For this they will use PCA variants. In our casewe simply seek to compress, without knowledge of the finalimage utilisation, which is why we use classical PCA, whichdoes not favor any particular use.

In our experiments, we applied PCA to the originalmulti/hyperspectral image in the spectral dimension. As aresult, we obtain a new multiband image in the transformdomain in which the spectral correlation is reduced. Theimage-bands in the transform domain were sorted with de-creasing variance (according to the values of the eigenval-ues). We finally applied the three compression approachesto all bands of the transformed image, unlike in dimensionreduction (Harsanyi & Chang, 1994 ; Mielikainen & Kaarna,2002 ; Du & Fowler, 2007 ; Gu et al., 2002) where only a fewbands were selected. This procedure allowed us to preservethe maximum amount of image information.

AN EVALUATION FRAMEWORK AND A BENCHMARK FOR MULTI/HYPERSPECTRAL IMAGE COMPRESSION 3

First approach – Multi-2DThis approach consists of applying the same 2D wavelets

transform on each band of the resulting PCA image. Becauseof PCA, the resulting image has decreasing energy bands. Inorder to take this fact into account, it is preferable to weigheach band. For weights, we define the energy E of each bandas in the formula:

E =

��

x,y

Iλ(x, y)2

XY(1)

where Iλ is the image band at the λ wavelength, X and Y areits dimensions, and x and y are the position of a pixel in theband.

Afterwards, we apply a 2D SPIHT (Said & Pearlman,1996) coding to each band of the wavelet transform resultsto achieve compression.

Second approach – Full 3DThe Full 3D approach consists of considering the whole

multi/hyperspectral image cube as an input for a 3D wavelettransform. In our case the input is the result of the PCA.Then a 3D extension of SPIHT encoder is applied. The 3DSPIHT encoder of Kim et al. (Kim et al., 2000) is appropriateto the 3D block shape of the decomposition. Dragotti et al.(Dragotti, Poggi, & Ragozini, 2000) also propose a 3D ex-tension of SPIHT for multispectral image compression, butthis encoder works like a temporal compensator and is moreappropriate for video coding.

However, since the spectral dimension of the multispectralimages is lower than the two other spatial dimensions andsince following this dimension the correlation is higher, it isappropriate to use a different type of wavelets for this dimen-sion. In (Kaarna & Parkkinen, 1998) Kaarna and Parkkinenrecommend a short wavelets basis as a good choice for spec-tral wavelets. This recommendation is confirmed by the re-sults obtained by Mansouri et al. in (Mansouri, Sliwa, Hard-eberg, & Voisin, 2008) in which the authors propose the Haarlifting scheme wavelets basis as an appropriate short supportbasis for reflectance representation and estimation from mul-tispectral images.

Technically speaking, it is possible to use two meth-ods to apply the Full 3D wavelets transform. The classicsquare wavelets transform method produces a multidimen-sional wavelets transform by applying one level of the one-dimensional (1D) transform separately in all dimensions andthen iterating this procedure on the approximation cube. Theother way of obtaining a multidimensional wavelets trans-form consists first of computing all the desired decomposi-tion iterations along one dimension, then all the desired it-erations on the next dimension, and so on. This method iscalled the hybrid rectangular/square wavelets transform. Wedepict these two principles in Fig.2.We use the classic squarewavelets transform in the Full 3D approach followed by 3D-SPIHT algorithm and the hybrid rectangular/square waveletstransform is used in the Hybrid approach.

(a)

(b)Figure 2. Graphical representation of the two ways of 3D waveletsdecomposition: (a) square decomposition by its first and secondsteps; (b) hybrid rectangular/square wavelets decomposition withtwo spatial decompositions followed by two spectral decomposi-tion.

This Full 3D implementation of the wavelets transformtakes into account the high spectral correlation of the imageand its anisotropy.

Third approach – Hybrid

The third approach consists of applying a Full 3Dwavelets transform on the PCA result as in the Full 3D ap-proach. But the square wavelets transform is replaced by ahybrid rectangular/square wavelets (Fig.2) as used by Pennaet al. in (Penna et al., 2006). This wavelets transform takesinto account the multispectral image properties. But thefact that this wavelets transform has two differentiated stages(spatial transform is followed by spectral transform) allowsits result to be considered as multiple 2D plans. For this rea-son we apply 2D SPIHT coding on each resulting band toachieve compression as in the Multi-2D approach. In orderto take the difference of energy bands into account we weigheach band with its energy E as in equation (1).

Compression evaluationframework

When lossy compression methods are used, quality mea-surements are necessary to evaluate performance. Accordingto Eskicioglu (Eskicioglu & Fisher, 1995), the main problemin evaluating lossy compression techniques is the difficulty


of describing the nature and importance of the degradationon the reconstructed image. Furthermore, in the case of ordi-nary 2D images, a metric often has to reflect the visual per-ception of a human observer. This is not the case for hyper-spectral images, which are first used through classification ordetection algorithms. Therefore, metrics must correspond toapplications. This is why instead of evaluating compressionperformances according to one metric or one type of metric,we propose the utilization of eight known metrics belongingto three categories to do so. We call this a framework forcompression evaluation. In (Christophe, Leger, & Mailhes,2005), Christophe et al. show that the use of a set of metricsis more relevant than using just one.

The metrics we propose can be divided into three fam-ilies: signal processing isotropic extended metrics (PSNR,RRMSE, MAE and MAD), spectral oriented metrics (Fλ,MSA and GFC), and an advanced statistical metric takingsome perceptive aspects into account (UIQI). We use thePSNR in order to facilitate comparison with other methods,since it is the metric most employed in image compression.

In the following sections this notation will be used: I is theoriginal multispectral image and I is the reconstructed mul-tispectral image. The multispectral images are representedin three-dimensional matrix form: I(x, y, λ), x is the pixelposition in a row, y the number of the row and λ the spectralband. nx, ny, nλ respectively the number of pixels in a row,the number of rows and the number of spectral bands.

We also introduce the notation I(x, y, ·) stands forI(x, y, ·) = {I(x, y, λ) | 1 ≤ λ ≤ nλ}. In this case I(x, y, ·)corresponds to a vector of nλ components.

For simplification, we note I(x, y, λ) and I(x, y, λ) by I and

I, and alsonx�

x=1

ny�

y=1

nλ�

λ=1

I by�

x,y,λ

I.

Signal processing isotropic extended metricsThese metrics come from classic statistical measures.

They do not take into account the difference between spatialand spectral dimensions. The structural aspect of errors doesnot appear.

Relative root mean square error (RRMSE). It is a classicstatistical measure based on MSE (Lp norm) with a normal-ization by the signal level.

RRMSE(I, I) =

��1

nxnynλ

�

x,y,λ

�I − I

I

�2(2)

Mean absolute error (MAE).

MAE(I, I) =1

nxnynλ

�

x,y,λ

��I − I�� (3)

Maximum absolute distortion (MAD). The MAD is usedto give a upper bound on the entire image.

MAD(I, I) = max��I − I

��

(4)

Spectral oriented metricsThese metrics are specially adapted for the multispectral

field.

Goodness of fit coefficient (GFC).

GFC(I, I) =

��

�

j

RI(λ j)RI(λ j)

��

�

j

�RI(λ j)

�2��

12��

�

j

�RI(λ j)

�2��

12

(5)

where RI(λ j) is the original spectrum at wavelength λ j andRI(λ j) is the reconstructed spectrum at the wavelength λ j.

The GFC is bounded, facilitating its understanding. Wehave 0 ≤ GFC ≤ 1. The reconstruction is very good forGFC > 0.999 and perfect for a GFC > 0.9999.

Spectral fidelity Fλ. This metric was developed by Eski-cioglu (Eskicioglu & Fisher, 1993). We define fidelity by :

F(I, I) = 1 −

�

x,y,λ

�I − I�2

�

x,y,λ

[I]2(6)

We will take into account the following adaptation focus onspectral dimension to obtain spectral fidelity:

Fλ(I, I) = minx,y

�F�I(x, y, ·), I(x, y, ·)

��(7)

Maximum spectral angle (MSA). The MSA is a metricused in (Keshava & Mustard, 2002). The spectral angle rep-resents the angle between two spectra viewed as vectors inan nλ-dimensional space.

SAx,y = cos−1

�

λ

I.I

��

λ

I2�

λ

I2

(8)

In our case we take the maximum of SA with:

MSA = maxx,y

�SAx,y

�(9)

Universal image quality index (UIQI)The UIQI was developed by Wang (Wang & Bovik, 2002)

for monochrome images. This metric uses structural distor-tion rather than error sensibility. It is an advanced statisticalmetric. The UIQI is based on three factors: loss of correla-tion, luminance distortion and contrast distortion.

Q(U,V) =4σUVµUµV�

σ2U + σ

2V

� �µ2

U + µ2V

� (10)


with σUV the cross correlation E�(U − µU) (V − µV )

�, µ is

the mean and σ2 the variance. The result is bounded by:−1 ≤ Q ≤ 1.

The UIQI can be applied in three different ways, on eachband, on each spectrum of the image or on both. We use iton each spectral band of the image as follows:

Qx,y = minλ

�Q�I(·, ·, λ), I(·, ·, λ)

��(11)

Experiments and results

We conducted our experiments on the largely usedAVIRIS2 images SanDiego, JasperRidge and MoffettField(Fig. 3). These images represent very different land-scapes, JasperRidge represent uniform spatial area whereasSanDiego represents an airport and MoffettFiel represents anurban landscape with many high frequencies.

Experiments

First experiment. The first experiment we conductedaimed to compare the performance of the three approachesregarding different compression bitrates when using differentspatial dimensions of images. We conducted the experimentson 32 bands of the SanDiego image with spatial dimensionsof 64 ∗ 64, 96 ∗ 96 and 128 ∗ 128 pixels, on 32 bands ofthe JasperRidge and MoffettField images with with spatialdimensions of 64 ∗ 64 and 128 ∗ 128 pixels. All images arecoded in 16 bit integer.

Second experiment. The second experiment sought toevaluate the performance of the three approaches regard-ing different compression bitrates when the number of bandschanges. So we used different spatial sizes of the SanDiegomultispectral image with a different number of bands (32, 64,96, 128, 160 and 192).

Results

Representing the results of the experiments within theframework of eight metrics is difficult. A good way to rep-resent the results is to use a star (radar) diagram (As in(Christophe, Leger, & Mailhes, 2008)) which gives a moreintuitive vision than a classical x-y representation in thiscase. The eight axes of the diagram correspond to the eightmetrics. All star diagrams have the same scale, minimum andmaximum are given on each axis for graphical interpretationand ease of comparaison. Axis of RRMSE, MAD, MAE andMSA are inverted, the extremity corresponds to minimumdegradation and the origin of the axeis corresponds to max-imum degradation. This representation permits good read-ability but does not allow us to show bitrate variation. Thatis why in Fig. 7 and 8 we only show results for a bitrate of1 bpp.

Results in terms of PSNR for the SanDiego image areshown in Fig. 5 and 6 and for other metrics in Fig. 7 and 8.Results for JasperRidge and MoffettField images are shownin Fig. 4.

(a)

(b)

(c)Figure 3. Multi/hyperspectral Aviris images we used in our exper-iments, (a) SanDiego , (b) JasperRidge and (c) MoffettField

2 http://aviris.jpl.nasa.gov


First experiment results. The results of the first experi-ment (regarding image spatial dimension variations) are rep-resented in figure 4 in terms of PSNR for the Cuprite image.

Results show that compression results for the three usedimages have the same trend. This trend is characterized bythe Full 3D approach which outperforms Multi-2D and Hy-brid approaches for high bitrate values. For small bitrate val-ues the Multi-2D approach gives the best results. The Hy-brid approach never has the best results. The results for allapproaches decreases when the spatial image dimensions in-crease.

Second experiment results. The results of the second ex-periment (regarding image spectral dimension variations) interms of PSNR are represented in figures 5, 6, 7 and 8 for theSanDiego image.

Graphics show that Multi-2D and Full 3D approacheshave the best results for a number of 96 spectral bands. Whenthe number of spectral bands deviates from this value, theresults proportionally decrease. For the Hybrid approachPSNR results increase proportionally to the number of bands.

The star diagrams show that all metrics don’t have thesame results. It’s particularly visible for the Hybrid approachwith PSNR, GFC, MAD, MAE and UIQI which have similarresults but for RRMSE, Fλ and MSA have inverted results.For the Multi-2D and the Full 3D approaches all the resultsare similar except regarding to UIQI metric.

Speed and memory consumptioncomparison

We can estimate speed and memory need for each com-pression approach by comparing it to the two others for eachpart of the compression.

First we applied spectral PCA for all approaches, takingthe same amount of time and memory. Secondly we appliedwavelet decompositions. For the Full 3D and Hybrid ap-proaches, decompositions are very similar and are performedon the entire image, taking similar computation time andcomputation memory. For the Multi-2D approach it dependson the implementation of the algorithm. If we consider eachband of the image separately, the decomposition of the entireimage takes a little more time than 3D decompositions, butless memory (a ratio equal to the number of bands). We canalso apply all 2D decompositions to the image at the sametime: the spectral wavelet decomposition time is less, butrequires as much memory as in 3D decompositions. Finallywe applied SPIHT and 3D SPIHT algorithms. These algo-rithms are identical, the only differences are the number ofpixels with children (three over four for SPIHT and sevenover eight for 3D SPIHT), the number of children (four forSPIHT and eight for 3D SPIHT) and their positions. The 3DSPIHT is slower than SPIHT and also takes more memory.

The speed and memory used by the three algorithms de-pend on image complexity but also on algorithm implemen-tations. The Multi-2D approach is the fastest, ahead of theHybrid approach; the Full 3D approach is the slowest. TheFull 3D approach also requires more memory than the twoothers. For large spatial dimension images the results show

(a)

(b)

(c)Figure 4. Compression results for (a) SanDiego, (b) JasperRidgeand (c) MoffettField images in terms of PSNR


Figure 5. Compression result in terms of PSNR for 32, 64 and 96bands of the SanDiego image.

Figure 6. Compression result in terms of PSNR for 128, 160 and192 bands of the SanDiego image.


Figure 7. Compression result for 32, 64 and 96 bands of theSanDiego image with a bitrate of 1 bpp.

Figure 8. Compression result for 128, 160 and 192 bands of theSanDiego image with a bitrate of 1 bpp.


that it is better to use high bitrate values to limit degradationsintroduced by the compression.

DiscussionThe two experiments we performed allow us to see the

effects of variations in spatial and spectral dimensions oncompression approaches. A general trend is observed: forsmall values of bitrate the Multi-2D approach gives the bestresults and for high values the Full 3D approach gives thebest results. Results of the Hybrid approach fall between thetwo others.

This trend could be explained by two major points:• For small values of bitrates, the Full 3D approach gives

bad results because the 3D SPIHT used in this approach uselists (list of significant and insignificant pixels, list of in-significant sets) which grow very fast compared to lists of2D SPIHT (each pixel has eight children for the 3D versionand only four in 2D). And for high values of bitrates fewercoefficients are added to the lists. This could explain the factthat the Multi-2D approach gives better results than the Full3D approach only for small values of bitrates.• The Hybrid approach gives inferior results because it

is a combination of 2D and 3D approaches. So using a 2DSPIHT after a 3D decomposition is not the best method.

Our results are contradictory to those of Penna et al.(Penna et al., 2006) who compare Full 3D and Hybrid ap-proaches. The authors found that the Hybrid method givesbetter results than the 3D method. In their article, they com-pared wavelet decompositions in various types of hybrid and3D approaches within. Then the results obtained showed thatthe square 3D wavelets decomposition gives results whichare not as good as the hybrid rectangular/square wavelets de-composition. We could probably explain this by the fact thatPenna et al. use the same filter (CDF 9/7) for each dimen-sion thus ignoring multispectral image anisotropy and highspectral correlation.

ConclusionIn this article, we have compared three approaches of mul-

tispectral image compression. These approaches are Multi-2D, Full 3D and Hybrid compression approaches, combinedwith a PCA decorrelation. The comparison of these ap-proaches is performed within a framework containing eightmetrics belonging to three different categories: signal pro-cessing isotropic extended metrics, spectral oriented metrics,and perceptive metrics. All metrics show the same trend:the Multi-2D approach is better than the Full 3D approachfor low bitrate values, but this trend is inverted for higherbitrate values. The Hybrid approach has intermediate resultsor results which are inferior to the two other approaches.

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an evaluation framework and a benchmark for multi ... · erations on the next dimension, and so on....

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