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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 5, MAY 2012 2641

Hierarchical Oriented Predictions for ResolutionScalable Lossless and Near-Lossless Compression

of CT and MRI Biomedical ImagesJonathan Taquet and Claude Labit

AbstractWe propose a new hierarchical approach to resolution

scalable lossless and near-lossless (NLS) compression. It combines

the adaptability of DPCM schemes with new hierarchical orientedpredictors to provide resolution scalability with better compressionperformances than the usual hierarchical interpolation predictoror the wavelet transform. Because the proposed hierarchical ori-ented prediction (HOP) is not really efficient on smooth images, wealso introduce new predictors, which are dynamically optimized

using a least-square criterion. Lossless compression results, whichare obtained on a large-scale medical image database, are more

than 4% better on CTs and 9% better on MRIs than resolutionscalable JPEG-2000 (J2K) and close to nonscalable CALIC. The

HOP algorithm is also well suited for NLS compression, providingan interesting ratedistortion tradeoff compared with JPEG-LS

and equivalent or a better PSNR than J2K for a high bit rate onnoisy (native) medical images.

Index TermsHierarchical prediction, image coding, losslesscompression, medical imaging, near-lossless (NLS) compression.

I. INTRODUCTION

E VEN if numerous studies have been developed in this fieldof interest, compression of biomedical images [1] remainsan important issue. Since the emergence of digital acquisition inmedical imaging, the data production is continuously growing.In recent years, it has been subject to a quasi-exponential in-

crease, in particular, because of an extensive use of MRI imagesand, even more, computed tomography(CT). These are both vo-lumic modalities that can be viewedas a sequence of 2-Dimages

(slices). The successive improvements of acquisition equipmenttends to amplify the resolution of those images, which intensi-fies the mass of data to archive [2]. All this makes them reallymuch more cumbersome than other imaging modalities. This iswhy we focused on CT and MRI.

Medical images may require to be saved for periods of over30 years. They are stored in picture archiving and communi-

cation systems for which efficient compression algorithms areof great interest. They could also benefit from functionalities

Manuscript received May 25, 2011; revised September 21, 2011 and De-cember 16, 2011; accepted January 04, 2012. Date of publication January 26,2012; date of current version April 18, 2012. This work was supported in partby the Brittany Council under Grant 4591. The associate editor coordinatingthe review of this manuscript and approving it for publication was Prof. HitoshiKiya.

The authors are with INRIA, Centre Inria Rennes Bretagne Atlan-tique, IRISA, 35042 Rennes, France (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2012.2186147

allowing an intelligent navigation through a network that may

reduce the data transmissions.Legally, the diagnostic information must be kept in the same

state as during the initial diagnosis stage to allow their reconsid-

eration in case of judicial proceedings. Therefore, if some lossesappear as compression consequences, the radiologists would

have to study the degraded images when doing their diagnosis.Although recent research efforts are focused on lossy compres-sion of biomedical images, such as in [3] or [4] (where lossy

compression is obtained using vector quantization), and even ifavant-garde radiologists advocate using lossy compression [5]

and recommend compression ratio that ranges with JPEG and

JPEG-2000 (J2K) standards for various imaging modalities [6](emphasizing that these are only guidelines), it is difficult to

convince all medical specialists who really do not want to lose

any important information. Therefore, for ethical reasons, med-ical images are most often stored without any loss, even if they

always contain unnecessary noisy information that could be re-

moved by using a less drastic lossy compression that can ensurea control on the losses, such as near-lossless (NLS) algorithms,

to preserve a visually lossless quality.Focusing on 2-D algorithms, the best lossless compression re-

sults are usually obtained with efficient DPCM schemes. They

follow a row-scan-ordered prediction and use adaptive methodsexploiting causal information. JPEG-LS (JLS) standard [7] and

CALIC [8] are often used as references [9]. Such coders lacka progressive model, which is important for distant access ofbiomedical images. In lossless context, resolution or rate scala-

bility coding is allowed by HIP approaches such as hierarchicalinterpolation (HINT) [10] and derivative works such as inter-

leaved HINT (IHINT) [11], as well as by an integer wavelettransform (IWT) used for example in the J2K standard [12].

HIPs use subsampling to extract a low-resolution image that

is employed for an interpolated prediction of the missing sam-

ples with finite support filters. To provide resolution or rate scal-ability, the residual data can be compressed by a subband bit-

plane coder such as EBCOT [13] or SPIHT [14], or by anyspecific -ary dictionary specialized entropy coder that will be

more optimal for resolution only scalability.

Lifting schemes commonly used for IWT act similar to an in-terpolated prediction but also concentrate energy in low bands.

Such integer wavelet bases provide antialiased multiresolution

representations and have better PSNRthan HIPs when used withrate scalable coders, but it is necessary to perform an additional

optimization scheme to exploit them efficiently for NLS com-

pression (see Section VI).In [15], the authors introduced a new hierarchical oriented

prediction (HOP) that combines DPCM with HIP. It provides

1057-7149/$31.00 2012 IEEE


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2642 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 5, MAY 2012

TABLE IDATABASES COMPOSITION

only resolution scalability but exploits already coded pixels ofthe same subband to improve decorrelation, compared withIWT

or HIP. In this paper, the algorithm is more detailed and in-cludes some minors changes with no significant impact on theresults. These modifications concern the previously used

threshold that is removed from the definition of the static pre-dictor, and the simplification of the context selection for biascancelation. New experiments are added to validate three main

contributions: 1) the HOP approach; 2) the sequential context-based error correction; and 3) the entropy coding technique.To improve the lack of efficiency of the predictors on smooth

images, two new extensions, which exploit dynamically con-structed predictors using a least square optimization, are alsoproposed. The hierarchical approach of IHINT, which performs

better than HINT, was retained for lossless and NLS compar-isons with the proposed methods. Then, the used implementa-

tion of IHINT will also be detailed throughout this paper.In Section II, the data sets used for the experiments are

introduced. The hierarchical decomposition of IHINT andthe proposed one for HOP are described in Section III. TheHOP prediction approaches are then presented in Section IV,

which details the static predictors (see Section IV-A) and thedynamically least-square-estimated ones (see Section IV-B).

The proposed sequential prediction bias cancelation is ex-plained in Section V. The NLS extensions of IHINT and HOPare exposed in Section VI; then, the proposed coding scheme

that is used with both algorithms is described in Section VII.Finally, lossless and NLS results on CT and MRI images arecompared with usual 2-D reference algorithms in Section VIII.

II. DATA SETS USED FOR EXPERIMENTS

A very large amount of data has been used for this paper.

Experiments were carried out on more than 6000 CT and 3000

MRI slices. The database used contains 12-bit images: native

CT and MRI slices of the National Library of Medicines Vis-

ible Human Project,1 resulting from the acquisitions of two full

bodies (VHP-Male/Female) and a head (VHP-Harvard); Med-

ical Database for the Evaluation of Image and Signal Processing

Algorithms (MeDEISA)2; and 8-bit images from Mallinckrodt

Institute of Radiology, Image Proc. Lab., available at CIPR.3

The structure of the whole data set is further detailed in Table I.

From the VHP-Harvard data set, because it has given

atypical results compared with the other MRI (see results in

Section VIII), a special 3-D MRI volume has been removed

1http://www.nlm.nih.gov/research/visible/visible_human.html

2http://www.medeisa.net

3http://www.cipr.rpi.edu/resource/sequences/sequence01.html

and is considered separately (Harvard-3D). It contains textured

noise, which has correlated information, and well-pronounced

Gibbs reconstruction artefacts. The CIPR data set contains

postprocessed images for which values are remapped to 8 bits;

therefore, most of the noise is removed. However, it is often

used as a reference in the literature of volumic medical images

compression, such as in [16] and [17]. MeDEISA is composed

of postfiltered CT slices, native and postprocessed MRI slices,and 3-D MRI slices. Postprocessed MRI slices have parts of

the background noise (regions of no interest) set to 0. The

VHP data sets are native acquisitions coming directly from the

acquisition material; therefore, the images are the most noisy

of the whole database.

For validation experiments, three subsets show the different

kinds of results that may be obtained. Smooth images results are

showed by those obtained on the CT slices of MeDEISA, and

noisy native CT/MRI slices by those obtained on VHP-Male

CT/MRI data sets. Three slice examples, shown in Fig. 1, are

also extracted for rate/distortion comparisons.

III. HIERARCHICAL DECOMPOSITIONS

A. IHINT Decomposition

To hierarchically decompose an image, a prediction level

of IHINT [11] can be summarized in the two prediction steps

showed in Fig. 2. Let be the set of horizontally even indexed

pixel values, and let be the set of horizontally odd indexed

pixel values; the first step (HStep) consists of predicting the

pixels of using an interpolative finite impulse response filter

on . then contains the residual values of the prediction. The

second step (VStep) is the mathematical transposition of HStep

applied independently on to obtain two sets and , and

on to obtain and .The set then contains a subsampled lower resolu-

tion image that can be recursively decomposed. The pixel

values/residual of each set can be reorganized to obtain a

dyadic representation.

For comparisons to be held in the following, the IHINT pre-

diction of an odd indexed value is the average of its two (even

indexed) neighboring values. This high-pass filter was retained

because of its similarity with predictors used by HOP [see (3)

in Section IV-A] and has been well known to be an efficient

predictor.

B. Proposed Decomposition

Similar to IHINT, a prediction level of the HOP scheme [15]is performed in two steps (see Fig. 3). The first one (HStep) con-

sists of predicting horizontally odd indexed pixels values with

the aid of already known pixels: the proposed approach uses

not only the even indexed ones but it can also take advantage of

any previously predicted ones (which are now causal values).

Thus, we can obtain a horizontally subsampled image and the

residual of the odd predicted pixels. The second step (VStep) is

the mathematical transposition of HStep and acts on the lower

resolution image.

With the HOPs prediction approach (see next section), the

subset is sufficiently decorrelated and does not require to be

decomposed once again.

Similar to IHINT, HOP can be computed in the same

memory space as the image, but it requires that the current row


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TAQUET AND LABIT: HOPs FOR LOSSLESS AND NLS COMPRESSION OF CT AND MRI BIOMEDICAL IMAGES 2643

Fig. 1. 12-bit-slice examples: contrast LUT has been adjusted for 8-bit displayand printing. (a) Not native CT of MeDEISA [CT_data_1(91)]. (b) Native CTof VHP-Male [normalCT(248)]. (c) Native MRI of VHP-Male [pd-1(21)].

Fig. 2. One prediction level of the IHINT algorithm.

Fig. 3. One prediction level of the HOP algorithm.

(or column) and the previous ones that are necessary for the

prediction are buffered (it represents here a total of three rows).

IV. PROPOSED HOPS

A. Static Predictor

HOP is mostly designed for noisy images containing struc-

tured objects with sharp edges (contrasted data), which is the

case for most of native medical images.

Inspired by the efficient gradient-adjusted predictor of

CALIC, an orientation estimation is done using the pattern

presented in Fig. 4. First, the absolute value of the local dif-ferences is computed for each orientation belonging to set

, i.e.,

Card(1)

where is the image, , and is defined by the

set of linked pixels in Fig. 4. Then, using a noise threshold

(defined later in Section IV-C), the diagonal gra-

dient , and the horizontal/vertical

gradient , the most favorable orienta-

tion is selected for the

prediction

if

else

withif

else.(2)

It allows a choice between five predictors: non-oriented ,

vertical , horizontal , diagonal, and diag-

onal. The prediction value is then given as follows:

if

if

if

if

(3)


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Fig. 4. (left) Contextual prediction pattern and (right) linked pixels used forgradient estimation.

Fig.5. Sets ofcausal pixelsusedfor(left)HOP-LSE, and for (leftand right)

HOP-LSE .

to perform an oriented prediction along edges, and

if (4)

for estimating the value by a more robust filter in homogeneous

areas that mainly contain noise.

The local activity for the predictors orientation

if

else(5)

will be used in the next sections (Sections IV-B and V).

B. Least-Square-Estimated Dynamic Predictors

The HOPs prediction is not really effective on smooth

images mainly because the small prediction support size (3) is

not adequate for the decorrelation of diffuse information. Be-

cause higher order predictors would result in better estimation

on such images (using a wider prediction support size), we

also introduce two new predictive approaches: HOP-LSE and

HOP-LSE . To build , they both exploit the extended sets of

causal pixels compared with HOP (see Fig. 5). For an efficient

TABLE IIIMPACT OF THE THRESHOLD ON THE COMPRESSION (bpp)

use of these extended sets, the predictors are dynamically built

using least square estimations, giving a better adaptation to the

specific characteristics of each image.

For HOP-LSE , the following optimization is always done,

whereas for HOP-LSE, if , the nonoriented static pre-

dictor (4) of the previous section is used. It reduces the com-

plexity of the algorithm by computing the least square optimiza-

tion only in the nonhomogeneous areas, which represent about

30% of the pixels on the whole data sets and then allows the di-

vision of the computation time, for a nonoptimized implemen-

tation, by around 3 compared with HOP-LSE .

To compute different predictors depending on the local ac-

tivity, (5) is logarithmically quantized to

(6)

where gives the index of the highest bit of the integer

part of . A prediction context that combines both

the local activity and the local orientation is then selected for

the prediction.

Let be the value of the current pixel to be predicted,

where is the context selected for the prediction and is the

number of times that has been previously selected as a pre-diction context. Let be the column vector containing the

values of the causal pixels belonging to the set (see Fig. 5).

The prediction will be

(7)

where is the vector containing the least-square-optimized

linear prediction coefficients. It is classically computed by

(8)

where and.

Because this least square optimization needs to be non-

singular and because it requires a certain number of learning

and to be sufficiently effective, as long as ( was fixed

to 64 for experiments), and whenever is not far from sin-

gularity, the static predictors associated to (2) and depicted in

the previous section are used.

To avoid keeping incrementally large matrix and

vector for each context, the implementation is done by

iteratively updating the positive definite symmetric matrix

AC (noncentered autocorrelation of the

prediction pixels) as follows:

AC AC (9)


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TABLE IIICOMPRESSION RESULTS FOR DIFFERENT LENGTH OF THE SEQUENTIAL CONTEXT ERROR CORRECTION

Best results are in bold, whereas worst are underlined.

and the vector CC (noncentered cross

correlation between prediction pixels and predicted values) asfollows:

CC CC (10)

with in lossless mode and being its

reconstructed value if any quantization is used (see Section VI).

C. Threshold

In order not to spend too much time optimizing the

threshold, it is set to a noise estimation (using the Donohos

thresholding formula [18]) computed on the highest frequencies

of an orthonormal Haar transform of the image to compress.For HOP and HOP-LSE , even if it is not always optimal in

terms of coding efficiency, it allows a slight improvement of

the compression, up to about 1% for HOP on noisy data sets

and for HOP-LSE on smooth data sets. However, on noisy

CTs, HOP-LSE compression is not improved because the

noise in CTs has strong location dependence and then cannot

be captured by the used least square optimization process.

Table II shows results obtained with and without the threshold.

Results of HOP-LSE are not presented because HOP-LSE has

no sense without the threshold: It would then be equivalent

to HOP-LSE , the predictors optimization would be done for

almost every pixel, and then the statically defined nonoriented

predictor characterizing HOP-LSE would not be used.

For practical uses, depending on the material and acquisition

parameters, the threshold could be optimized offline.

V. PREDICTION BIAS CANCELATION

A. Overview

To avoid systematic errors, also called biases, generally oc-

curring with context-based static predictors, a common tech-

nique is to correct prediction in a context with the av-

erage of the previous errors that occurred in the same

context

(11)

can bedefined from any causal computation. For example,

JLS is quantizing the gradient on the causal neighborhood ofthe predicted value, and CALIC is also using the prediction

value itself to define a binary texture pattern. In [19], the au-

thors extended this approach by using a cumulated prediction

error correction corresponding to the average of weighted errors

from different contexts selections as follows:

(12)

and showed some compression improvements.

B. SCEC

We propose to extend (12) by using a sequential context-

based error correction (SCEC). This is done by sequentially ap-plying the following correction scheme for to :

(13)

with . The final prediction value is , with

as the rounding to the nearest integer operator. The prediction

error is then used to update each .

When the context selection does not depend on , (13) is

equivalent to (12), but when it depends on it, the prediction

is successively refined and allows the selection of more and

more accurate contexts. For experiments, the coefficients are

simply fixed to be .

The context selection used with HOP is now described.Since oriented approaches are not always effective on textured

areas, the , depending of , are defined by retaining the

texture capture process of CALIC. A local texture information

nn ww is binarized

to define a 5-bit number , with if

, or else, . is combined with the local

activity quantization (6) and with the employed orientation

numbers as follows: 0 for noise, 1 for horizontal, 2 for vertical,

3 for diagonal, and 4 for diagonal.

C. SCEC Validation

Some SCEC results, obtained when varies, are reported

in Table III. The analysis is done for HOP, HOP-LSE, and

HOP-LSE in lossless mode PAE and NLS mode


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(PAE , see next section). The predictors performances will

be discussed later in Section VIII.

With static predictors (HOP), SCEC always allows the

asymptotic improvement of the compression, with further ame-

liorations for smooth images. As an example, on the smooth

MeDEISA CT slices, the lossless compression rate is reduced

(compared with no correction: ) by 3.1% using only

one correction context , whereas it is reduced by3.55% using a sequence of five corrections . Then,

on noisy native VHP-Male MRI images, the improvement is

of 0.64% when and 0.91% when , and on noisy

native VHP-Male CT images, where the kind of noise is much

more location dependent, the rate is only reduced from 0.51%

to 0.76% . In NLS mode, the compression

amelioration is better again, with a rate reduced by up to

6.25% for MeDEISA CT images, 1.35% for CT

images, and 1.87% for MRI images of VHP-Male. In terms of

distortions in NLS mode, only the first context correction seems

to improve the PSNR; then, both the PSNR and the rate slightly

decrease, keeping the ratedistortion tradeoff approximatelysimilar.

With dynamic predictors (HOP-LSE and HOP-LSE ), the

same observations can be made on smooth images and native

CT images. However, on native MRI images, when ,

the bias cancelation seems to fail. This is due to the smaller

size of the images (256 256) and the highly contrasted na-

ture of MRI. For the smallest resolutions, the images are aliased

due to the subsampling: Pixels are less correlated, and the least

square estimation is more difficult. The prediction is therefore

less efficient, and the bias cancelation is not effective enough in

small resolutions due to a lack of data for correctly estimating

correction statistics. This could be improved by disabling the

contextual correction for a given context while the number ofprevious occurrences in the same context is smaller than a cer-

tain threshold. However, since allows the bypass of this

drawback (by averaging the errors of different contexts) and im-

proves results, no threshold was used.

Results show that SCEC allows the reduction of compressed

image size in both compression modes. In NLS, it has a small

impact on the ratedistortion tradeoff with static predictors, but

with dynamically optimized ones, would be better being null

or greater than 1. For the experiments that follow, is always

fixed to 3.

VI. NLS EXTENSIONS

A. Brief Review

When introducing slight compression losses in medical

imaging, a frequently used error control is the peak of absolute

error (PAE). Its value is the maximum absolute difference

between the pixels of the original and the ones of the recon-

structed images. A PAE-constrained lossy compression is often

referred to as NLS compression.

With predictive coding, NLS compression is performed by

predicting pixels from the NLS causal reconstructed values. The

residual error is then generally quantized by a uniform quanti-

zation as follows:

(14)

which ensures a PAE less or equal to when reconstructing

(15)

Note that it is verified that

(16)

This is useful for NLS hierarchical predictions (see

Sections VI-B and VI-C).

JLS has adopted a slightly different approach, which pro-

motes run length encoding (RLE) whenever the RLE value sat-

isfies the PAE and uses the quantized residual if otherwise.

For high PAE, it generates line artefacts but allows an effi

cientcompression without employing arithmetic coding.

When using transform coding for NLS compression, one of

the best known way is to use a lossy compression, followed by a

residual coding of the quantized error image [20], [21]. This

kind of approach requires an optimization of the coding cost

of joint lossy residual representation. Some research efforts

were also carried out on wavelet-based scalable -oriented

compression [22].

In terms of PSNR, the transform approach is more effective

than predictive coding for high PAE because the lossy compres-

sion tends to minimize the MSE and thus preserves the most

important energy information. It also results in good compres-

sion properties of the residual probability distribution function(PDF). However, for small PAE, when really good quality must

be preserved, as in medical imaging, the predictive approach is

currently recognized as performing an equivalent or a better job.

B. IHINT NLS Extension

Let be the residual of when predicted from .

For NLS extension of IHINT (see Section III-A and Fig. 2),

the approach starts, with the pyramidal dyadic reorganization

of the pixels, by approximating the lowest resolution image

. Then, a pyramidal descent is done, starting

from this lowest resolution until the full-resolution NLS approx-

imation of the image is obtained: is used to predict andto get their approximated residuals and . The

prediction of is reverted to build and allows the com-

putation of . By reverting

the prediction steps, and are also built. Then, , ,

, and data are reorganized to obtain an upper resolution

NLS image that can be used as the band of the next resolu-

tion level.

When the full-resolution NLS image is obtained, the hierar-

chical prediction (which will be the exactly the same as during

the pyramidal descent) is done in inverse order (from highest

to lowest resolution, as during a lossless compression). The

residual values and the lowest resolution image are quan-

tized again to obtain exactly the same quantized residual values,

as during the pyramidal descent.


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C. HOP NLS Extension

The NLS extension of HOP follows the merged guidelines

of HIP and DPCM: it starts by approximating the lowest

resolution image used to predict higher resolution ones, and al-

ways performs its successive predictions from previously ap-

proximated values. For each increasing resolution, it starts by

a VStep (see Fig. 3) to reconstruct a completely NLS band,which is then used for the NLS prediction of the band.

Similar to IHINT, when the full-resolution NLS image is ob-

tained, the decomposition is done once again, in the same order

as during lossless compression, and the residual values and the

lowest resolution image are quantized again.

For both IHINT and HOP, the NLS decomposition requires

the NLS image estimation (pyramidal descent), followed by

the decomposition (pyramidal ascent). The time complexity is

then around two times the one of the lossless decomposition,

which only requires the pyramidal ascent decomposition, but the

memory consumption stays the same. Another implementation

using temporary storage of the residual data obtained during the

pyramidal descent would allow it to perform with the same timecomplexity as lossless but with a loss of memory of the size of

the image.

VII. CODING SCHEME

A. Residual Remapping

The residual remapping is often used by predictive coders to

reduce the alphabet size (by a factor of 2 compared with the

full-residual-range values) to make the entropy coding easier.

Knowing and (with

being the image), the prediction value can be forced to lie

in . If and, respectively, , it is

forced to be resp. . The residual prediction

is then bounded by

and . Now, it can be remapped using a

bijective function as follows:

(17)

To obtain a PDF that tends to be decreasing, positive and nega-

tive residual values are interlaced using the following:

if andif and

else if

else if

(18)

with , , and

(see also Fig. 6).

The remapping is employed with both IHINT and HOP.

B. Resolution Scalable Coding

For the following results, the image decomposition is fully

done for both IHINT and HOP. The value of the lowest resolu-

tion image, which is then a single pixel, is first transmitted.

To allow a resolution-progressive transmission, the decom-

position is compressed starting from the lowest resolution.

Fig. 6. Residual remapping.

For IHINT, each remapped quantized residual band ,

, and then ; for HOP, each remapped quantized

residual band and then are sequentially coded

row by row and column by column. A coding context CC

is determined from quantized local error energy, modeled by

causal neighbors, and already coded lower band hierarchical

neighbors of the current symbol (remapped quantized residual

value). CC allows an entropy coding of , using a PDF that isiteratively computed each time a symbol has previously been

coded in the same context.

From the minimal resolution of 32 32 (width 32 and

height 32), these statistics and the associated coders are reset

to their initial states before the coding of the next resolution.

This allows the independent decoding of a small image and of

each successive expansion.

This multiresolution scalable coding method can be em-

ployed, e.g., in quick view for a fast navigation approach: In a

huge set of images, it can be used to quickly decode/transmit

subresolution images; a selection can then be easily made, and

only the information required to reconstruct upper resolutionsneed to be decoded/transmitted.

To be effective with the resolution progressive constraints,

the entropy coder needs to estimate quickly and efficiently the

PDF of the data that must be compressed. In the literature, there

are fast adaptation entropy coders exploiting usual PDF mod-

eling framework (generalized Gaussian, Laplacian, geometric,

etc.), and we need only adjust a small number of parameters

for the PDF estimation. However, obviously, they are not as ef-

ficient as if a finer PDF estimation was used. The problem of

a finer PDF estimation is that it requires more parameters and,

therefore, more input data to correctly adjust to it.

The approach proposed for HOP is a combination of the two

previously mentioned ones. It uses a small set of parameters toquickly estimate the data distribution and to have a rapid coding

efficiency, and a wider set of parameters for a slower but better

fit to the true distribution, which allows a more powerful com-

pression. This is done with the aid of two distinct coders used

in a two-stage coding shown in Fig. 7. A reduced-parameter en-

tropy coder (stage-0 coder S0C) allows us to quickly approxi-

mate the data distribution and to compress the first coefficients,

whereas another coder (stage-1 coder S1C) is learning finer dis-

tribution statistics. When the S1C coder is estimated to have suf-

ficiently learned the statistics, it is used instead of S0C for the

next coefficients compression. S1C is continuing to update the

data distribution. When a coefficient value is unknown by S1C,

an escape symbol is output, and S0C is employed to compress

this value. The algorithm details are shown in the Appendix.


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TABLE IVRESOLUTION SCALABILITY RESULTS

Uncompressed images have size with 12 bits per pixel. The bpap notation means bits per added pixels and correspondsto the bit rate used for coding the pixels required to get this resolution from the previous one.Best results are in bold, whereas the worst results are underlined.

Fig. 7. Two-stage coder.

C. Resolution Scalability Validation

Table IV shows some coding results obtained for various res-

olution decoding of the three images in Fig. 1. The efficiency of

HOP, HOP-LSE, and HOP-LSE is compared with IHINT and

J2K. Comparing all results to J2K, which is a multiscale refer-ence coder, it shows that the proposed coder is also well suited

for scalability.

For lossless mode, even if it is difficult to compare J2K with

IHINT because of the energy compaction of wavelets, the pre-

sented entropy coder seems to have equivalent or better perfor-

mance in terms of scalable compression. For the small resolu-

tions, in both lossless and NLS modes, oriented predictive ap-

proaches (HOP, HOP-LSE, and HOP-LSE ) are more efficient

than the nonoriented IHINT. HOP also gives better results than

HOP-LSE and HOP-LSE , which can be explained by a lack of

data for least square estimations of aliased (subsampled only)

subresolution images. This remark could be taken into account

to improve the compression by disabling the optimization on

lowest subresolutions.

VIII. EXPERIMENTAL RESULTS

A. Lossless Compression

HOP, HOP-LSE, and HOP-LSE are compared with the de-

scribed implementation of the IHINT algorithm, with the stan-

dards JLS4 and J2K5 and with the reference software of SPIHT6

(quality scalability) and CALIC.7 Table V reports the lossless

results averaged by the data set for the whole database.

On CTs, CALIC always gives the best compression perfor-

mances, except for smooth images data set (MeDEISA), where

least square dynamically optimized predictors perform better.

On MRI, except on the smooth data set (Harvard-3D), it often

performs equivalent compression to HOP.

For scalable coders only, J2K is most often the worst or not far

from the worst coding algorithm, leaving out MeDEISA CT im-

ages for which HOP is not efficient. However, except on smooth

data sets, HOP is always better than SPIHT, J2K, and IHINT.

IHINT obtains results similar to J2K on CT images but is com-

petitive with CALIC and HOP on MRI.

The proposed least square optimization of the predictors al-

lows us to bypass the inefficiency of HOP on smooth images.

Both HOP-LSE and HOP-LSE give the best compression re-

sults on the smooth data sets. They also outperform HOP most

of the time.

4http://www.hpl.hp.com/loco/

5http://www.kakadusoftware.com/

6http://www.cipr.rpi.edu/research/SPIHT/spiht3.html (linux version)

7ftp://ftp.csd.uwo.ca/pub/from_wu/v.arith (sun version)


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TABLE VLOSSLESS RATES AVERAGES

Best results are in bold, whereas worst are underlined. Cells of the best results by category (scalable/nonscalable)are grayed.

TABLE VI

NLS RATES AND PSNR AVERAGES FOR A PAE OF 4

Best rate averages and best PSNR averages for NLS coders are in bold. For each data set, the PSNR obtained by the best ratealgorithm (BRA) is compared with the average of PSNR obtained by J2K-9/7: The bit rate obtained by the BRA on each slice isused to constrain the J2K-9/7 compression. The best PSNR average between the BRA and J2K are underlined.

On MeDEISA CT images, HOP is 3% worse than J2K or

IHINT, but with HOP-LSE and HOP-LSE , 3.3% and 6.8%

of the space required by J2K and IHINT is saved. Moreover,

HOP, HOP-LSE, and HOP-LSE improve the compression by

more than 3% on the others CT images. On 8-bit CT images,

HOP-LSE also reduces space by more than 10% compared

with J2K.

On common MRI, the least square optimization of the pre-

dictors improves HOP compression by up to around 2%, which

leads to improvements of 5% to more than 7% compared withJ2K, and around 3% compared with IHINT. On 8-bit MRI im-

ages, HOP, HOP-LSE, and HOP-LSE improve compression

by 8.4%, 11.1%, and 12.5% compared with J2K; and 2.9%,

5.7%, and 7.2% compared with IHINT. On smooth MRIs (Har-

vard-3D) HOP-LSE saves 17.8% of the space used by J2K and

13.7% of the IHINT requirements.

Averaging the results for all the database, HOP, HOP-LSE,

and HOP-LSE are 2.24%, 3.87% and 4.28% better than J2K

on CT images; and 4.22%, 7.45% and 9.13% better on MRI

images, respectively.

B. NLS Compression

For NLS comparisons, only IHINT and JLS are kept because

the other previously used software do not provide PAE-con-

strained compression. However, PSNR performances of the

PAE constrained algorithms are compared with the lossy ver-

sion of J2K (rate constrained), using real valued biorthogonal

wavelets (9/7 kernel). Averaged rates and PSNR obtained by

JLS, IHINT, HOP, HOP-LSE, and HOP-LSE for a slight

PAE of 4 are given in Table VI. The J2K PSNR results are

obtained with the Kakadu v6.0 software, using a quantization

step (Qstep) set to , with being the bit depth of the

original image (8 or 12).

The images presented in Fig. 1 are used in Fig. 8 to showratedistortion curves that may be obtained with JLS, IHINT,

and the best HOP HOP-LSE . For better readability, HOP and

HOP-LSE results are not plotted. On the CT of the MeDEISA

data set, which shows results on smooth images and is one of

the worst losslessly coded by HOP (compared with the other

algorithms), HOP would be about 0.21 bpp less efficient than

IHINT when PAE (the PSNR is estimated to be 1.92 dB

worst) and 0.06 bpp less efficient when PAE (PSNR is

estimated to be 0.96 dB worst). The HOP-LSE results would

be between IHINT and HOP-LSE . On the two others images,

which give examples for the compression of noisy native CT

images and MRI images, HOP and HOP-LSE results would be

similar to HOP-LSE .


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Fig. 8. NLS rate/distortions results.

A PAE of 4 allows a compression improved by a factor

greater than 2 compared with lossless mode and has no vi-

sual impact. HOP, HOP-LSE, and HOP-LSE results show

that, in terms of PSNR, they allows us to obtain equivalent or

better performances than J2K on noisy native images. PSNR

is slightly worse on 8-bit data sets and on smooth images.

However, the NLS approaches enjoy the PAE quality control,

which is not the case of J2K.

Similar in lossless mode, the three HOP approaches

have relatively equivalent results on most noisy images.

The HOP oriented approach is, again, better than the

nonoriented IHINT one, except on the smoothest data sets

where HOP-LSE and HOP-LSE make the difference with

the least square optimization, which improves HOP results

by more than 10% and up to 20% on Harvard-3D set.

Compared with the JLS standard, HOPs improves both


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TABLE VIIEXECUTION TIME (SECS CODER/SECS DECODER) FOR IMAGES IN FIG. 1

rates and PSNR and, moreover, provides the resolution

scalability.

C. Indicative Execution Time

Just for information, we give some execution timesin Table VII obtained with an Intel(R) Core(TM)2 DuoE6750 CPU at 2.66 GHz. IHINT and HOP applications aremonothreaded and are not optimized. The main optimizationsthat could be performed are related to the entropy coder and to

the least square optimization.

IX. CONCLUSION AND PERSPECTIVES

A new HOP approach, and two variants using least squareoptimization (HOP-LSE and HOP-LSE ) have been presentedin the context of resolution scalable lossless and NLS biomed-ical image compression. A new sequential context-based biascancelation method was proposed and analyzed to improve theprediction efficiency. The last original contribution was an en-tropy coding technique based on a two-stage coder designed toimprove the compression in the resolution scalable context.

This paper has shown that, even if providing resolution scal-ability, some compression improvements could be obtained on

noisy native medical images both in lossless and NLS modescompared with the reference algorithms. The least square opti-mization has allowed us to boost the prediction on smooth im-ages, where HOP was not really efficient.

For small distortions, HOP obtains equivalent or better PSNRthan J2K on noisy native images and moreover enjoys the PAEquality control, which is not the case of J2K. An NLS compres-sion using a PAE of 4 allows a division by 2 of the archivalrequirements, compared with lossless compression, with no vi-sual impact.

As an extension of this paper, complementary experimentshave been carried out, particularly on virtual microscopy slides.These are really huge anatomical pathology images that require

from hundreds of megabytes to tens of gigabytes. Some pre-liminary tests on those images have given promising results.HOP obtained 10% lossless compression improvements com-pared with CALIC. The uses and needs for this emerging imagemodality are considerable, which is why we will focus our fu-ture work on it.

APPENDIX

DETAILS ON THE TWO-STAGE ENTROPY CODING

The first-stage coder (S0C) performs compression with a lim-

ited-parameter infinite alphabet adaptive coder. Each symbol

is decomposed in

(19)

(20)

with being the number of bits required to represent

. This Golomb-Rice-like decomposition is then

coded by adaptive binary arithmetic coders: one for each

bit of (mainly noise), and one for the unary representation

of (for IHINT, is fixed to 0. Since no predictors

adaptation is done and is not used during prediction,

better compression results were obtained like this).

The binary arithmetic coders, used for , independentlymaintain the frequencies of the symbols 0 and 1 .

When reaches a threshold , and are di-

vided by 2. It allows a better adaptation to local statistics. The

binary arithmetic coder used for the unary representation of

compresses a sequence of ones followed by one zero, and

also continuously maintains the frequencies of the symbols 0

and 1. From experiments, since it is less dependent on local sta-

tistics, the threshold only serves to ensure that the integer

capacities are not reached. Some details on this unary arithmetic

coding approach can be found in [23], where it is originally de-

signed for Laplacian distributions sources.

During thefi

rst-stage coding, as described before, the statis-tics of a size-adaptive reduced alphabet, which includes an es-

cape symbol, are learned. Starting with an alphabet that only

contains the escape symbol (ESC) with frequency ESC

, the size-adaptive reduced alphabet is designed as follows.

When a symbol is encountered, ESC is incre-

mented by one, is set to , and .

When a symbol is encountered, is incre-

mented by one.

When SF ESC is greater than

threshold , frequencies are reset: The frequency

of each symbol ESC is set to , and

ESC is set to ESC . Every symbol

with null frequency is removed from .The switching to the second stage occurs when the most

frequent symbols have been taken (referring to escape symbol

probability). This is estimated by simply comparing ESC

to threshold SF . If ESC is lower, it means that, most

of the time, the encountered symbols are present in . Then, be-

cause frequent symbols are in , ESC will be required less, and

the ESC will probably decrease after each new frequency

reset. Instead of waiting for it (and because ESC will

induce a higher coding cost of other symbols in ),

is empirically reduced to .

Once the second-stage coding (S1C) is started, the next sym-

bols are compressed by an adaptive arithmetic coder guided by

the reduced alphabet , and the same rules used during its de-

sign are used to continue to update thestatistics. When an escape

symbol is encountered, S0C is retained to code escaped symbol

value by Card . Because this remapping

trick (reducing coding cost) principally induces modifications

in the binary coder statistics used for the unary representation,

when the coder switching occurs, the frequencies of 0 and 1

symbols are reinitialized to 1.

In Table VIII, some results obtained with the proposed en-

tropy coding approach are compared with those obtained using

the stage-0-only entropy coding and the stage-1 only entropy

coding. It shows that S0C alone can give interesting results.

Since S1C alone gives slightly less efficient results but the pro-posed combination of the two coders gives the best results, it


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TABLE VIIIHOP COMPRESSION RATES (bpp) USING THE STAGE-0-ONLY CODER, THE

STAGE-1-ONLY CODER, AND THE PROPOSED COMBINATION

can be deduced that the dynamic size alphabet allows a good

PDF approximation once frequency tables are fed enough, and

that a two-stage approach is relevant, particularly for resolu-

tion progressive coding: S0C is more suited for small dimen-

sions/resolutions images, whereas S1C boosts the compression

of the wider residual bands.

For all experiments of this paper, the thresholds/constants

have been empirically fixed to , , and

.

ACKNOWLEDGMENT

The authors would like to thank the reviewers and C. Hay-

ward who helped to improve this paper.

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Jonathan Taquet was born in France in 1983. Hereceived the S.M. degree in computer science from

the University of La Rochelle, La Rochelle, France,in 2007 and the Ph.D. degree in signal proces singand telecommunications from the University ofRennes 1, Rennes, France, in 2011, after a studyon the compression of biomedical images with theINRIA-Rennes research center.

His major research interests include signal pro-cessing, analysis and synthesis, image compressionand communication for multimedia or medical

applications, and machine learning.

Claude Labit was born in France in 1956. He re-ceived the S.M. degree in computer science/telecom-munications or tfrom Ecole Nationale Suprieure desTlcommunications, Paris, France, in 1979 and thePh.D. degree in computer science and the DocteurdEtat degree from University of Rennes 1, Rennes,France, in 1982 and 1988, respectively.

Since 1982, he has been with INRIA. In 1985, hecreated and managed the TEMIS research team foradvanced image sequence processing. From 1999 to2007, he was a Director of the INRIA-Rennes/IRISA

Research Center. He is currently the Vice President of the scientific board atUniversity of Rennes 1 and the President of RennesAtalante Science Park. Hismajor research interests include motion analysis; image compression; image se-quence coding, both for digital video and multimedia service; and 3-D TV orlarge-scale image databases for medical or satellite applications.

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