A fast classification based method for fractal image encoding

$: A fast classification based method for fractal image encoding$
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Image and Vision Computing 26 (2008) 1129–1136

A fast classification based method for fractal image encoding

Tamas Kovacs *

Institute of Informatics, Kecskemet College, Faculty of Mechanical Engineering and Automation, Izsaki 10, 6000 Kecskemet, Hungary

Received 6 February 2007; received in revised form 5 October 2007; accepted 23 December 2007

Abstract

In the present paper a fast and efficient fractal image encoding method based on classification of image blocks is presented. Twoparameters are used to sort image blocks into disjoint classes: the direction of the approximate first derivative and a normalized rootmean square error of the fitting plane in the given block. With the help of these parameters the number of domain blocks examinedfor a range block is reduced dramatically, and thus, the classification results in a considerable acceleration of the encoding process, with-out loss of the reconstruction fidelity. The proposed method is compared to recently developed fast classification algorithms and a ‘Nosearch algorithm’, and its rate-distortion performance under the same encoding time limit is proved to be better than that of the others.� 2008 Published by Elsevier B.V.

Keywords: Fractal image compression; Classification

1. Introduction

Since the basic idea of fractal image compression tech-nique was introduced by Barnsley et al. [1,2] in late1980s, the various applications and developments of thismethod have become an area of high interest. The mainreason for this was that it seemed to be a new alternativein compression theory promising a decent rate-distortionperformance.

The idea of this alternative method originates in thefact that our natural environment generally shows self-similarity on a wide scale of some physical parameter (likespace-coordinates or time), and this property is reflectedon all (or almost all) photographed pictures. Therefore,the considerable amount of redundancy implied by thisself-similarity can be exploited. By the means of the Iter-ated Function System proposed by Barnsley, there is acontractive transformation for each image that has thefixed-point identical to the image itself. In other words,applying that transform (function) iteratively on an arbi-

0262-8856/$ - see front matter � 2008 Published by Elsevier B.V.

doi:10.1016/j.imavis.2007.12.008

* Tel.: +36 76 516 401; fax: +36 76 516 399.E-mail address: [email protected]

trary starting image, the result converges to the originalimage. Thus, the image is encoded by the transformation.

Unfortunately, the form of this transformation is farfrom trivial. A relatively simple method to establish sucha transform was invented by Jacquin [3,4], who partitionedthe range and the domain image into square blocks, andthen established a simple affine transformation for eachsingle domain block mapping it into a specific range block,as introduced in the next section. This scheme proved to beuseful, however, its search process is extremely time-con-suming. Since then, most of the research works have beenconcentrated on the remedy of this serious drawback.The restriction of the domain pool was the main point ofnumerous later efforts [5–8], which will be discussed inmore detail in the next section.

Other researchers focused on improvements of thesearch process to make it faster by tree structure searchmethods [9,10], parallel search methods [11,12] or quad-tree partitioning of range blocks [13,14,27].

The organization of this paper is as follows. In Section2 the basic fractal image encoding method is describedtogether with some later developments. Section 3 intro-duces the proposed classification algorithm, in Section 4the tests of the proposed method are presented and

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1130 T. Kovacs / Image and Vision Computing 26 (2008) 1129–1136

compared to that of other fast classification schemes. Inthe last section conclusions of the comparison are drawn.

2. The basic fractal encoding algorithm and its variations

In this chapter the theoretical background of fractalencoding will be treated only briefly in order to devotemore space to the various methods developed to speed upthe basic algorithm.

2.1. The basic algorithm

The contractive transformation mentioned in the intro-duction is constructed locally, that is, the range picture isdivided into disjoint square blocks (‘‘range blocks”) andthe parameters of the transformation are established foreach range block.

The range blocks are non-overlapping of size B � B,while the domain blocks are overlapping and of size2B � 2B. The pixel values of a given range block areobtained by transforming a certain domain block into therange block in three consecutive steps:

� reduction of the domain block into a B � B size blockby replacing its 2 � 2 sub-blocks by a single pixel, thevalue of which is the average of the sub-blockpixels,� shift of the reduced B � B size domain block by a vector

v into the position of the range block,� an affine transformation of the pixel values of the form

p 7! ap þ t; ð1Þwhere p denotes the pixel values, while a and t are param-eters of the transformation.

By introducing the notation R and D for the B � B

matrices of pixel values of the range block and the reduceddomain block, respectively, the transformation can be sum-marized as

R ¼ aDþ t: ð2ÞSince the transformation is local the parameter set (v,a, t)should be established for each range block. Thus, thecode-book consists of the stored parameter set{(vi,ai, ti)ji = 1, . . .N}, where N is the number of rangeblocks.

Given this parameter set, the encoded image can bereconstructed by applying the transformation several timessuccessively on an arbitrary starting image, until the resultconverges to a fixed-point.

It is possible to avoid the search process for theparameter ti, if the following level-shift transformation[15,16] is performed on the range and domain blocksRi and Di:eRi ¼ Ri � �ri � I ;eDi ¼ Di � �di � I ;

ð3Þ

where eRi and eDi are the level-shifted matrices, �ri and �di arethe means of the pixels in Ri and Di, respectively, and I is amatrix with all elements equal to 1. The means of the pixelvalues of both eRi and eDiare obviously zero, therefore, thevalue of ti must be also zero for all eRi. Thus, Eq. (2) is re-duced toeRi ¼ ai

eDj; ð4Þ

and instead of ti the mean value �ri is to be stored.The encoding procedure, that is, finding the parameters

vi and ai, can be described by the following algorithm:

� Calculate �ri and �di

� Calculate eRi and eDi

� For each eRi do:{

– find eDj and ai, that minimize the root mean square (rms)

error defined as dði; jÞ :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikeRi�aieDjk

B2

r,

– store vi, ai and �ri, where vi is the vector giving the posi-tion of eDj relative to eRi.}

In the above algorithm k�k denotes the matrix normdefined as kaijk ¼

Pi;ja

2ij.

Note that the search of the minimizing eDj and ai impliesa double loop embedded in each other and in the mainloop. That is why a full-search algorithm (searching thewhole domain pool) is very time-consuming. eDj is usuallycalled the ‘‘best matching” domain block for eRi, becausethe best result can be achieved by finding this block. How-ever, this does not mean that there are no other suitableblocks in the domain pool, and partly this is the basis ofthe various methods that try to speed up the encodingprocess.

2.2. Quad-tree method

The choice of the block size (B) is essential because thebigger a range block is, the harder it is to find properdomain block that can be efficiently mapped into it, andthis reduces the quality of the reconstructed image. Onthe other hand, with bigger blocks, a much better compres-sion ratio can be obtained. (It is easy to verify that the com-pression ratio is proportional to the value of B2.)

The quad-tree method proposed by Fisher [14] uses thebenefits of bigger blocks without much loss of the recon-struction fidelity. In this scheme different block sizes (Bn)are used in more consecutive steps. In the first step therange image is divided into blocks of size B1 � B1, then,the search for the best matching domain block takes place.The found domain block and its ai value are accepted orrejected by a so called split decision function that comparesthe rms error d(i, j) to a given threshold T1, and accepts theblock only if d(i, j) is less than T1. Otherwise the search isconsidered to be unsuccessful, and the range block at hand

T. Kovacs / Image and Vision Computing 26 (2008) 1129–1136 1131

is divided into four equal parts of size B2 � B2, whereB2 = B1/2. When the first step is completed we have a num-ber of range blocks of size B2 that are uncovered i.e. havenot been encoded yet. In the next step the same method isrepeated, but with starting size and threshold B2 and T2,respectively. At the final level, when the smallest blocksare treated, the best matching domain blocks are acceptedunconditionally. A typical choice for the sizes used in thismethod are Bn={16,8,4,2}.

Distasi et al. [20] and later Furao et al. [8] establishedthat a better result is obtained by the application of varyingthreshold values in the consecutive steps as follows

T nþ1 ¼ 2T n ð5Þ

Distasi et al. examined more possible split decisionmethods, and it turned out that the above rms error basedsplit decision function outperforms the others.

2.3. Local search and No search methods

The local search methods are based on the assumptionthat the best matching domain block is situated spatiallynear the range block at hand, therefore, the search forthe proper domain block can be restricted to the vicinityof the range block [5,6]. This, of course, results in a consid-erable speedup of encoding and some damage of the recon-struction fidelity.

The so called ‘No search’ algorithms [7,8] do not searchthe domain pool, but a specific domain block is appointedto be the ‘‘best matching” one. These methods producepoorer reconstruction fidelity than other partial searchschemes, however, they are extremely fast, and give riseto the possibility of real time applications of fractal imageencoding. Another advantage of the ‘No search’ scheme isthat there is no need to store the relative position vector, vi.Therefore, better compression ratios can be achieved.

2.4. Classification search methods

The exhausting search process can be reduced also bydividing both sets of the domain and the range blocks intosubsets with the help of a hashing parameter. By using thisclassification, one assumes that the hashing parameter val-ues of a given range block and its best matching domainblock pair are equal, so the range block and its pair aresupposed to be in the same class (subset). The classificationis a part of the pre-process executed before the searchphase, and thus, the searched domain pool is reduced tothe actual class the range block belongs to.

Wang et al. [17] and Duh et al. [18] used the edge prop-erties of the blocks to group them into three or four classes,and this resulted in a speedup ratio of 3–4.

Fisher [14] divided the domain pool into 72 classesaccording to certain combinations of the four quadrantsof the block in question. His work proved the efficiencyof the classification schemes: the searching time is reduced

to the order of magnitude of seconds without a great loss ofreconstruction fidelity.

Tong et al. [16] and later Wu et al. [19] used the standarddeviation, given as

STDi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikDi � �diIk=B2

qð6Þ

in order to classify blocks. Wu enhanced Tong’s methodand achieved further speedup of the encoding with com-pression ratio close to 1 bit per pixel.

2.5. Hybrid and other fast methods

The most promising rate-distortion results so far wereproduced by the hybrid methods that apply a fractal cod-ing algorithm for the blocks of the multi-resolution waveletcoefficients of the image and an efficient arithmetic coder[21–25]. The most essential part of the new JPEG 2000compression standard is also such a combination of amulti-resolution wavelet transform and an arithmeticcoder. However, it does not apply fractal encoding, sinceit would detain the independent procession of the differentregions of the image.

Endo et al. [23] and later Bellouata [24] and Iano et al.[25] achieved good rate-distortion results with the help ofhybrid wavelet-fractal methods all three employing Fisher’sclassification scheme [14], although the speeds of thesemethods were limited by the still time-consuming fractalencoding part.

Another recently developed fast technique is to vectorquantize image blocks with the help of a quad-tree parti-tioning scheme and then replace the full-search by a nearestneighbor search algorithm [26,27]. The approximated near-est neighbor search method proposed by Tong and Wong[27] gained a speedup of about 4 compared to the basicnearest neighbor search algorithm. At the end of theirpaper, referring to Fisher’s work [14], they emphasize thatit is possible to incorporate a classification scheme intotheir algorithm.

Lastly, it is worth to mention other speedup techniquesas well. The algorithm proposed by Hartenstein and Saupe[28] takes advantage of the fact that the domain blocks areoverlapping and this sort of code-book coherence can beexploited by a FFT based cross correlation analysis. Veryrecently Chung and Hsu [29] published the idea of parti-tioning the original image into a set of approximatelyhomogeneous blocks before the application of the searchalgorithm resulting in a considerable speedup comparedto the full-search method.

3. The proposed classification method

The main purpose of the present work is to find a fastclassification based encoding algorithm to reduce theencoding time to the order of magnitude of the ‘No search’scheme, while preserving the rate-distortion performanceof the earlier classification methods. The basic idea of this


development is the choice of suitable hashing parameters,with the help of which an effective classification can beachieved. Nevertheless, here no great effort is made to opti-mize the developed code by other speedup techniques.

Similarly to Fisher’s method [14], two different quanti-ties are used to characterize image blocks.

One of these parameters refers to the basic symmetry ofimage blocks, and it will be quoted here as the Approxi-mated First Derivative (AFD) parameter. Consider theimage (with size of 512 � 512) as a function of the form

f ðxk; ykÞ : ½0; . . . ; 511; 0; . . . ; 511�7!½0; . . . ; 511�: ð7Þ

Though it is a discrete function, it is possible to calculatethe first derivative (gradient vector) $f = (oxf,oyf) at a cer-tain point numerically. In order to do so, we need anumeric formula by means of which the whole image blockcan be characterized independently from the block’s size. Avery simple and yet appropriate solution is to approximatethe function values {f(xk,yk)jk = 1, . . . ,B2} of the block athand by a linear function of the form

zðx; yÞ ¼ qxxþ qyy þ c ð8Þ

and use the first derivative of z(x,y) as the numeric firstderivative of f(xk,yk), that is $f = (qx,qy). Regarding thesymmetry of the block, only the direction of the gradientvector is of interest. Therefore, its polar angle, given as

u ¼ arctanqy

qx

; u 2 � p2;p2

h ið9Þ

is used to sort the blocks into classes. It can be said that theimage block at hand roughly shows axial symmetry with re-spect to the gradient vector, and the basic point of the clas-sification is that a given range block and its best matchingdomain block pair have the same axial symmetry.

The AFD based sorting of blocks alone is not enough toperform a real fast algorithm. The other parameter pro-posed for classification is a normalized root mean squareerror (NRMS) of the fitting linear function Eq. (8), definedhere as

NRMSðRÞ ¼ RMSðRÞqðRÞ ð10Þ

where RMS(R) and q(R) are the root mean square error ofthe fitting linear function and the magnitude of the approx-imated gradient in the image block R, respectively. Theyare calculated as

RMSðRÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

B2

XB2

k¼1

½pk � ðqxxk þ qyyk þ cÞ�2vuut ð11aÞ

qðRÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

x þ q2y

qð11bÞ

where pk is the k-th pixel value in the block R. It is easy toshow that a range block eRi and its best matching domainblock eDj should have the same NRMS parameter value,and therefore, this parameter is capable for classificationof blocks. Since the RMS(R) and the q(R) functions are

homogeneous, by the means of Eq. (4) the following rela-tions hold

RMSðeRiÞ ¼ aiRMSðeDjÞ ð12aÞqðeRiÞ ¼ aiqðeDjÞ ð12bÞ

whence

NRMSðeRiÞ ¼ NRMSðeDjÞ ð13Þ

what was to be proven.Similarly to the STD parameter, which measures the

deviation from the block’s mean, the NRMS parametercharacterizes the deviation of the pixels from the fittedplane given by Eq. (8).

After the u and NRMS parameters are determined forthe block at hand, they are quantized to the integer num-bers uk=[0, . . . , (K�1)] and NRMSl=[0, . . . , (L�1)] accord-ing to the following equations

uk ¼uþ p=2

p� K

� �ð14Þ

NRMSl ¼(

NRMS2<NRMS>

� L� �

if NRMS2<NRMS>

� L < L

ðL� 1Þ otherwiseð15Þ

where bxc denotes the highest integer not greater than xand <NRMS> denotes the average NRMS of the imageblocks, which was measured in the pre-process phase.

By means of the classification method, the set of rangeblocks and the domain pool is divided into separate setsidentified by the pairs of the quantized parameter values(uk,NRMSl), and thus, there are altogether K � L disjointsets for the range blocks and the same for the domainblocks. If a given range block in question turns out to bein the specific set identified by (uk,NRMSl), only thedomain block set belonging to the same parameter-pairshould be searched.

However, in many cases the best matching domain blockcan be found in one of the eight neighboring lists{(uk+i,NRMSl+j)ji = �1, 0, 1; j = �1, 0, 1} or even in afurther neighboring list of (uk,NRMSl). For the sake ofsimplicity a control parameter Nex is introduced anddefined as the number of the domain blocks to be examinedfor a range block. First the domain block list (uk,NRMSl)is searched, and then its neighbors, and then further neigh-bors, until the number of the examined domain blocksexceeds Nex. When applying this procedure, the choice ofK and L has little effect, since the speed of the algorithmand the reconstructed image quality depend on Nex in thefirst place.

Besides, Fisher’s quad-tree scheme [14] is used in thepresent work with block sizes Bn={16, 8, 4, 2} and withthe rms based adaptive tolerance method proposed byDistasi et al. [20] (see Eq. (5)).

The spirit of the adaptive method is also applied for Nex

as follows:

ðN exÞnþ1 ¼ 2ðN exÞn ð16Þ

Fig. 1. Visual comparison of the original (a) and reconstructed 512 � 512 test images of Lena at 0.8 bpp compression ratio and with 5.0 s encoding time inall cases: (b) Encoded by Fisher’s method (PSNR = 29.32), (c) encoded by the STD classification method (PSNR = 31.62), (d) encoded by the proposedAFD–NRMS classification method (PSNR = 33.08).


since using greater Nex value for the much more populateddomain pool of smaller blocks is reasonable. The optimalvalue of the coefficient in relation (16) was establishedexperimentally.

Furthermore, the De-Redundancy Method (DRM)introduced by Wu et al. [19] is also adopted in anunchanged form.

The search for an ai value that minimizes d(i, j) is carriedout by pre-quantizing ai to the values {±1, ±0.75, ±0.5,±0.25}, and then substituting the eight values into d(i, j)one by one (for details of this method see Ref. [8]). Thealternative of this solution, on the other hand, is the leastsquare method that solves the equation

odði; jÞoai

¼ 0: ð17Þ

This latter one gives ai directly as the solution of the Eq.(17), therefore, it is somewhat faster, but producespoorer image quality. However, here we employ the leastsquare method as a pre-test before each comparison ofeRi and eDj, and if the mean square error of the least

square method is higher than a certain threshold (givenin terms of Tn) the eDj block is omitted. If not, the valueof ai obtained by the pre-test is used for a further reduc-tion: if it is positive then only the positive quantized ai

values will be substituted into d(i, j), otherwise only thenegative ones.

4. Experiments and results

The recently published STD and the frequently citedFisher’s classification schemes [14,19] and, in addition tothese, Furao’s ‘No search’ algorithm [8] were chosen forcomparison with the present method. All four methodswere implemented in NET developing environment (VisualStudio C#) and the codes were run on a Windows XP, Pen-tium IV, 3 GHz platform.

The compression ratio and the encoding time were set tothe desired values with the help of the tolerance threshold T

and the control parameter Nex.The centers of the overlapping domain blocks are situ-

ated on a square lattice, the lattice constant of which is

Fig. 2. Visual comparison of the original (a) and reconstructed 512 � 512 test images of the baboon at 0.8 bpp compression ratio and with 8.0 s encodingtime in all cases: (b) encoded by Fisher’s method (PSNR = 22.02), (c) encoded by the STD classification method (PSNR = 19.96) and (d) encoded by theproposed AFD–NRMS classification method (PSNR = 23.25).


B/2 pixels. Thus, the total domain pool consists of 4(512/B)2 blocks.

The reconstruction fidelity was measured by the peaksignal to noise ratio (PSNR), defined as

PSNR ¼ 10log10

5122NpPNp

i¼1ðp0i � p1iÞ2

ð18Þ

where p0i and p1i are the pixels in the original and in thereconstructed image at the same position, respectively.The parameter Np denotes the number of pixels in theimage.

The compression ratio is expressed in bit per pixel(bpp). In the proposed scheme, encoding a B � B sizeimage block in a 512 � 512 size picture requires altogether8þ 2log2

512B þ 3þ 1 bits: 8 bits for storing the mean value

ð�riÞ of the range block, 2log2512B bits for the location infor-

mation of the domain block pair, 3 bits for ai, and finallyan identity bit to sign if the block is further divided in the

quad-tree scheme or not. (In he case of 2 � 2 range blocksthis last bit is unnecessary.)

The implemented algorithms were compared with thehelp of the frequently used grayscale test images of Lena,and the baboon (see Figs. 1 and 2) of size 512 � 512.

Fig. 3. gives the comparison of rate-distortion curves ofthe various methods measured on the Lena image underthe same encoding time limit (5.0 s), except the ‘No search’algorithm, which used 0.55 s. (In the case of this latter onethe encoding time cannot be varied by control parametersand depends only on the performance of the computing.)It can be seen that for all compression ratios the proposedscheme gives better image quality. For visual comparisonFigs. 1 and 2 show the two test images processed (encodedand decoded) by the three classification methods with sim-ilar compression ratios and encoding times. The advantageof the proposed method is seen more clearly on the baboonpicture.

The image quality (measured by PSNR) versus theencoding time curves of the Lena image are shown in

Table 1Encoding results of the proposed model and the three reference models

Test image compressionratio

Lena 512 � 512 0.80 bpp Lena 512 � 512 1

Method Encoding time(s)

PSNR Encoding time(s)

Fisher’s 22.63 31.26 36.95STD classification 5.59 31.88 17.81‘No search’ 0.55 25.84 0.55AFD–NRMS (no pre-test

of ai)2.69 32.01 3.98

AFD–NRMS (with pre-testof ai)

1.77 32.00 2.53

The last two rows show the results of the proposed model with and without t

25

27

29

31

33

35

0 10 15 20 25Encoding time(s)

PS

NR

Fisher'sSTD (Wu etal.)'No Search'Proposed

5

Fig. 4. The encoding time versus the PSNR of the proposed model andthe other reference models measured on the 512 � 512 Lena image underthe same encoding time limit (5.0 s) in the cases of the three classificationmethods and with the encoding time 0.55 s in the case of the ‘No search’method.

22

24

26

28

30

32

34

36

38

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2bpp

PS

NR

Fisher'sSTD (Wu et al.)'No Search'Proposed

Fig. 3. The reconstructed image quality (measured by PSNR) versus thecompression ratio (given in bit per pixel) of the proposed AFD–NRMSclassification method and the other referenced fast methods measured onthe 512 � 512 Lena image with encoding time fixed at 5.0 s in all cases.


Fig. 4. All of the curves were measured at bit rate 0.8 bpp.Due to the aforementioned property of the ‘No Search’model, its performance is represented by only a point at0.55 s in this figure. These curves show a saturation behav-ior, that is, there is no remarkable improvement of theimage quality beyond a certain value of the encoding time.The proposed model has a much smaller such saturationtime-value than the other two classification models, andit produces definitely better image quality than the ‘NoSearch’ model with similar order of magnitude of theencoding time.

Table 1 gives the encoding times and image qualitiesmeasured on the two test images at compression ratios0.8 and 1.44 bpp by the four models.

The last two rows of the table show the results of theproposed model with and without the pre-test of the coef-ficient ai introduced at the end of the previous section. Itcan be seen that this simple pre-test yields a speedup ofabout 1.5.

5. Conclusions

The applications of fractal image compression meth-ods are limited due to the exhaustive encoding time,although they could be useful components in recentlydeveloped promising hybrid methods. In this paper anew classification method is proposed in order toreduce the encoding time and so make the fractalmethod more acceptable for further applications. Thetests have shown that classification by the introducedAFP and NRMS parameters reduces the encoding timeconsiderably compared to earlier classification methodswithout loss of the rate-distortion performance. Thus,it seems that the above two parameters characterizewell the similarity of image blocks, that is, two imageblocks having the same AFP and NRMS parametervalues turn out to be similar to each other from fractalcoding point of view.

In addition to this, the proposed pre-test of the pre-quantized scale parameter a by a least square fittingmethod yielded a further speedup of about 1.5.

.44 bpp Baboon 512 � 512 0.80 bpp Baboon 512 � 512 1.44 bpp



PSNR

32.60 14.30 22.36 22.88 23.9333.27 12.00 23.47 8.55 22.6928.33 0.55 20.46 0.55 21.4834.89 2.80 22.37 3.86 24.13

34.90 1.89 22.41 2.63 24.18

he pre-test of the coefficient ai.


Surely, it is possible to find even more characteristicparameters, with the help of which the classification canachieve further speedup and even reach the theoretical timelimit represented by the ‘No search’ method.

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A fast classification based method for fractal image encoding

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