Design of an Error-Tolerance Schemefor Discrete Wavelet Transform

in JPEG 2000 EncoderChun-Lung Hsu, Member, IEEE, Yu-Sheng Huang, Ming-Da Chang, and Hung-Yen Huang

Abstract—The JPEG 2000 image compression standard is designed for a broad range of data compression applications. The discrete

wavelet transformation (DWT), central to the signal analysis and important in the JPEG 2000, is quite susceptible to computer-induced

errors. The errors can be spread to many output transform coefficients if the DWT is implemented by using lifting scheme. This paper

proposes an efficient error tolerance scheme (ETS) to detect errors occurring in DWT. A pipeline-based DWT structure is also

developed in this paper to speed up the error detection process. The proposed ETS design uses weighting sums of the DWT

coefficients at the output compared with an equivalent check value derived from the input-data. With the proposed ETS design, the

errors introduced at DWT can be effectively detected. Additionally, the results of error detection can be further analyzed and evaluated

to show the capability of error tolerance. Some standard images are used as test samples to verify the feasibility of the proposed ETS

design. Experimental results and comparisons show that the proposed ETS has good performance in error detection time and error

tolerance capability.

Index Terms—JPEG 2000, DWT, error detection, error tolerance.

Ç

1 INTRODUCTION

JPEG 2000 has recently been approved as an internationalstandard for the compression of still digital pictures [1].

Unlike JPEG which uses the discrete cosine transform(DCT), the new standard is entirely wavelet-based toprovide better compression performance and some richfeatures. For example, JPEG 2000 allows efficient lossy andlossless compressions within a single unified codingframework, provides superior image quality at low bitrates, supports a more flexible file format, and avoidsexcessive computational and memory complexity.

The block diagram of JPEG 2000 encoder is illustrated inFig. 1 [2]. The discrete wavelet transformation (DWT) is firstapplied on the source image data. The transform coeffi-cients are then quantized and entropy encoding is applied,before forming the output codestream (bitstream). Thedecoder is the reverse of the encoder. The codestream isfirst entropy decoding, inverse quantized, and inverse DWT(IDWT), thus, resulting in the reconstructed image data. TheDWT is central to the JPEG 2000 image compressionstandard which includes lifting configurations for imple-menting the forward and inverse transforms. The mainproperties of DWT are the space-frequency localization andinherent multiresolution structure. In other words, waveletsallow efficient representation of a signal with a small

number of nonzero coefficients. Also, wavelets takeadvantage of data correlation in space and frequency.DWT is implemented with computer hardware ultimately,the processing operations are susceptible to transientfailures, primarily single-event upsets, alternately termedas soft errors. These factors will increase the influences asVLSI feature sizes shrink [3], [4]. Although the implementa-tion of DWT is susceptible by computer-induced errors, theimage quality is also kept within the application-specific orrange of acceptability if an error tolerance structure can beeffectively developed. Thus, design an effective structurefor error tolerance evaluation becomes an important issue ofDWT error detection in JPEG 2000 applications.

Error tolerance is a new design and test paradigm, whichtakes into consideration whether erroneous outputs ofdefective circuits still produce acceptable results [5], [6].Error tolerance classifies a system as being acceptable/unacceptable by estimating the performance degradationdue to errors, rather than relying solely on the conventionalperfect/imperfect classification. Error tolerance analyzes thesystem-level effects of errors, and accepts circuits if theperformance degradation can meet the application-specificor range of acceptability. A common characteristic of allcompression standards for images is that they rely on lossycompression, that is, the decoded image is not an exact copyof the original. Thus, in this paper, we view the effect oferrors as potential additional distortion suffered by thedecoded image. This added distortion will in some cases stilllead to an acceptable output. Therefore, image compressionsystems are good applications for error tolerance techniques.

This paper proposes an ETS design that targeted fordetecting errors of the DWT subsystem in JPEG 2000, as DWTis one of the most important subsystems in terms of bothcomputation and memory requirements. The remainder of

628 IEEE TRANSACTIONS ON COMPUTERS, VOL. 60, NO. 5, MAY 2011

. The authors are with the Department of Electrical Engineering, NationalDong Hwa University, 1, Sec. 2, Da Hsueh Rd., Shou-Feng, Hualien 974,Taiwan, ROC. E-mail: cch@mail.ndhu.edu.tw, {d9523006, m9523030,m9723029}@ems.ndhu.edu.tw.

Manuscript received 6 June 2009; revised 3 Nov. 2009; accepted 16 Dec. 2009;published online 6 Dec. 2010.Recommended for acceptance by D. Gizopoulos.For information on obtaining reprints of this article, please send e-mail to:tc@computer.org, and reference IEEECS Log Number TC-2009-06-0260.Digital Object Identifier no. 10.1109/TC.2010.239.

0018-9340/11/$26.00 � 2011 IEEE Published by the IEEE Computer Society

this paper is organized as follows: Section 2 reviews the basicprinciples and key features of DWT in JPEG 2000. Theproposed ETS structure, pipeline-based DWT design, errormodel definition, error detection, and tolerance strategy arepresented in Section 3. Section 4 shows the experimentalresults and comparisons for performance evaluation anddiscussion. Finally, Section 5 provides the final conclusions.

2 BACKGROUND

DWT is usually computed through convolution andsubsampling with a couple of filters to produce anapproximation low-pass filter result and a detail signalhigh-pass filter result. The multiresolution decomposition isobtained by iterating the convolution and subsampling ofthese two filters over the approximation components. Fortwo-dimension (2D) signals, there exist separable waveletsfor which the computation can decompose into horizontalprocessing (on the rows) followed by vertical processing (onthe columns), using the same one-dimension (1D) filter. Ateach level of the wavelet decomposition, each row of animage is first transformed using a 1D horizontal analysisfilter-bank. The same filter-bank is then applied vertical toeach column of the filtered and subsampled images,referred to as subbands. Fig. 2 illustrates a 2D DWToperation flow. The four subbands of Fig. 2 are denotedas horizontally and vertically low-pass (LL), horizontallyhigh-pass and vertically low-pass (HL), horizontally low-pass and vertically high-pass (LH), and horizontally andvertically high-pass (HH) [7].

In order to construct filter banks in an efficient way, thelifting scheme for both designing fast wavelets andperforming the discrete wavelet transform was presented[7], [8], [9], [10]. The lifting scheme can decompose DWTfilter into several lifting steps. If ehðzÞ and egðzÞ indicate thelow-pass and high-pass analysis filters, the polyphasematrix eP ðzÞ is defined as

eP ðzÞ ¼ eheðzÞ ehoðzÞegeðzÞ egoðzÞ� �

: ð1Þ

Additionally, the polyphase matrix eP ðzÞ can be factor-ized into a sequence of alternating upper and lowertriangular matrices multiplied by a constant diagonalmatrix, as shown in (2).

ePzÞ ¼ k 00 k�1

� �Ymi¼1

1 esiðzÞ0 1

� �1 0etiðzÞ 1

� �; ð2Þ

where k is a constant to scale the coefficient, esiðzÞ and etiðzÞindicate the predict and update operators, respectively[11], [12], [13]. JPEG 2000 adopts lifting technique as the

wavelet transform method and, generally, uses biorthogo-nal (9,7)/(5,3) filter for lossy/lossless compression [11]. Thegeneral block diagram of the lifting technique is illustratedin Fig. 3, which consists of four steps:

1. Split step. The original samples are separated intotwo disjoint sets, named even part and odd part.

2. Predict step. The even samples are multiplied by thetime domain equivalent of esðzÞ and are added to theodd samples.

3. Update step. The updated odd samples are multipliedby the time domain equivalent ofetðzÞ and are added tothe even samples.

4. Scaling step. The even and odd samples are multi-plied by k�1 and k, respectively.

3 PROPOSED ERROR TOLERANCE SCHEME

Fig. 4 shows the proposed ETS design, which consists of aninput parity procedure (IPP), an output parity procedure(OPP), and a parity analyzer. The main objective of theproposed ETS is to compare the differences between Cin andCout values to find the errors that occurred in the DWT.Then, the parity analyzer will further analyze whether theerrors can be tolerated or not. From Fig. 4, each row pixels ofan n� n image will be divided into the even and oddnumber of data samples input for DWT operation. Thus, thesize of input-data vector X shown in Fig. 4 is 1� n. A is an� n matrix of wavelet transform via lifting. Additionally,a 1� n tolerance weighting matrix W, has to be developedto assist establish the IPP and OPP structures for errordetection. The error models, tolerance weighting matrix W,pipeline-based DWT design for IPP and OPP structures,and parity analyzer are explicitly described in the following.

3.1 Modeling Errors

DWT is one of the most important subsystems in the JPEG2000 encoder. Generally, DWT is surrounded by sets ofmultipliers and adders that determine how data flowsthrough them. Thus, the realistic error assumption has tobe addressed for error detection [14]. However, theproposed ETS design mainly focuses on the effects ofcomputer-induced errors, which will be modeled throughtransfer matrices related to the lifting sections. For a

HSU ET AL.: DESIGN OF AN ERROR-TOLERANCE SCHEME FOR DISCRETE WAVELET TRANSFORM IN JPEG 2000 ENCODER 629

Fig. 2. Operation flow of a forward 2D DWT.

Fig. 1. Block diagram of JPEG 2000 compression standard.

Fig. 3. General block diagram of the lifting technique.

forward 2D DWT, the numerical errors, caused by anunderlying computer-induced errors, will propagate theircorrupting influence to the output. The computer-inducederrors can be modeled in a way at the numerical processing[15], [16], [17]. Especially, the data flow structure works inthe forward wavelet transform. Redinbo and Nguyen [17]clearly indicate how even a single numerical error, causedby underlying computer-induced errors, propagates itscorrupting influence to many outputs. Thus, any numericalerror effects can be spread by the data processing flow.Fig. 5 shows a general error model in the numericalprocessing. The error model in Fig. 5 does not imply thaterrors are additive, but rather, whenever an error occurs, itmay be described by adding a numerical value to the actualcalculated output. Thus, the whole row data of the actualoutputs will be influenced if the multinumerical errors arepresented. Additionally, visibility of DWT basis functionswill depend upon display visual resolution [18]. In terms ofsignal that reaches the eye, the visibility will be indistinct ifthe display resolution is decreased by a factor of error.

Based on the above-mentioned description, the error

mechanisms can be defined by two types such as intensive

error model (IEM) and distributed error model (DEM). If

the numerical errors will influence the pixels in contig-

uous rows of an image then the error model, IEM is

defined (see Fig. 6a). On the other hand, the DEM (see

Fig. 6b) is presented if many single row pixels of an image

will be influenced by the numerical errors. According to

the error models, the proposed ETS design will be

demonstrated as an effective method to explore the error

impact in the DWT and further analyze the tolerance of

errors for JPEG 2000 encoder applications.

3.2 Tolerance Weighting

In order to meet the human visual system and increase theflexibility of the proposed ETS design, a tolerance weight-ing matrix, W, has to be developed. The weighting factorsof W are very important for error tolerance analysis, sincethe most significant data are generally centralized in thecentral parts of an image. The error influence in the centralparts is more serious than that in the boundaries of animage [19]. Thus, a weighting matrix has to be built forsupporting the evaluation of error influence when theproposed ETS is active. The schematic representations ofparity weighting of the benchmark image “Lena” is shownin Fig. 7, which divides an image into some blocks to set thedifferent parity weighting factors. Generally, good parityweighting factors should have gain factors whose rangesspan three to four orders of pixel magnitude [17]. Based onthe parity weighting factors description in [17], this paperbuilt a 1� n tolerance weighting matrix for error detectionand error tolerance evaluation purpose.

3.3 Pipeline-Based DWT Design

The main components of the proposed ETS design are theIPP and OPP structures, whose objectives are generatingthe input/output checking values Cin and Cout and thendeliver them to parity analyzer for error detection. FromFig. 4, the checking values Cin and Cout are a weighted sumof the DWT inputs and outputs. Thus, in order to speed upthe DWT operations and error detection processes, apipeline-based DWT structure is proposed for supportingthe IPP and OPP designs.

630 IEEE TRANSACTIONS ON COMPUTERS, VOL. 60, NO. 5, MAY 2011

Fig. 4. Block diagram of the proposed ETS design.

Fig. 5. Errors in numerical processing. Fig. 6. Error models. (a) IEM. (b) DEM.

Based on the conventional structure for one level 9/7DWT via lifting [20], Fig. 8 shows the pipeline-baseddesign for a DWT via lifting. Input-data samples are splitinto two subsequences at the beginning of the pipeline-based DWT structure. For an n� n image, the input-datavector is represented by

X ¼ ½x0 x1 x2 x3 . . . xn�2 xn�1�;s0 ¼ ½x0 x2 . . . xn�2� ¼

�s0

0 s01 . . . s0

ðn=2Þ�1

�;

d0 ¼ ½x1 x3 . . . xn�1� ¼�d0

0 d01 . . . d0

ðn=2Þ�1

�;

ð3Þ

where sli and dli represent the ith even and odd samples inthe lth stage of lifting step.

Since the conventional DWT structure uses the paralleltechnique to deal with the input samples, the operation timeof the proposed pipeline-based DWT is less than that of theconventional one. For simplicity, the steps of the ith evenand odd samples shown in Fig. 8 are described as anexample and listed in (4-11).

1. Split step. The input sequences xi are split into evenand odd parts, s0

i and d0i .

d0i ¼ x2iþ1; ð4Þs0i ¼ x2i: ð5Þ

2. Lifting step. The two splitting sequences (s0i and d0

i )are performed by two lifting steps.

d1i ¼ d0

i þ ���s0i þ s0

iþ1

�;

s1i ¼ s0

i þ 2� ��d1i

�;

)Lifting step 1 ð6Þ

ð7Þd2i ¼ d1

i þ � ��s1i þ s1

iþ1

�;

s2i ¼ s1

i þ 2� ��d2i

�:

)Lifting step 1 ð8Þ

ð9Þ

3. Scaling step. Through the normalization factors k�1

and k, the low-pass and high-pass wavelet coeffi-cients si and di can be obtained.

di ¼ k� d2i ; ð10Þ

si ¼ k�1 � s2i : ð11Þ

Fig. 8 clearly indicates that the final output of the ith oddand even samples with two lifting steps can be obtainedafter four and five pipeline operations, respectively. Thefollowing samples (siþ1; diþ1; siþ2; . . . ) will be captured inturn based on the pipeline technique. Fig. 8 also shows thatthe proposed pipeline-based DWT design has a goodperformance in reducing the operation time. For example,the operations of lifting step 1 of the ith even sample, s1

i andthe (i+1)th odd sample, d1

iþ1 are working simultaneously(see Fig. 8b). The similar case can also be found in Fig. 8c,which shows that the lifting step 1 of the (i+1)th evensample, s1

iþ1 and the ðiþ 2Þth odd sample, d1iþ2 are working

at the same time. Figs. 8d and 8e illustrate the same caseswith the pipeline technique. According to the pipelineapproach, the final output of DWT can be consolidated inthe following matrix form:

d

s

� �¼

kI 0

0 k�1I

� �d2

s2

" #

¼kI 0

0 k�1I

� �I �Ma

�Md ��Mc

� �d1

s1

" #

¼kI 0

0 k�1I

� �I �Ma

�Md ��Mc

� �I �Ma

�Md ��Mb

� �d0

s0

" #;

ð12Þ

where Ma, Mb, Mc, and Md are the n� n matrices and theparameters �; �; �; �, and k represent floating point arith-metic for a 9/7 wavelet transform via lifting [20].

Based on the pipeline-based DWT design, the IPP, andOPP structures of the proposed ETS design shown in Fig. 4can be realized by using multipliers and adders. The twochecking values Cin and Cout from IPP and OPP structureshave to be designed to equal initially for detecting theerrors. From Fig. 4, the checking values Cin and Cout can bewritten as (13) and (14), respectively,

Cin ¼ XATWT; ð13ÞCout ¼WAXT; ð14Þ

where

W ¼ ½w0 w1 w2 w3 . . . wn�2 wn�1�:

Figs. 9a and 9b show the corresponding timing charts of Cinand Cout procedures to evaluate the timing consumption.Assume that an image has 512� 512 pixels using here foran example. The values Cin d0

and Cin s0of the first odd and

even samples of the input-data are obtained at clocks fiveand six, respectively, (see Fig. 9a). Based on the input-datavector shows in (3), the last odd and even input samples(Cin d255

and Cin s255) will be found at the 260th clock and the

261st clock, respectively. The same timing consumption isalso needed by determining the checking value Cout(see Fig. 9b). For the timing illustration in Fig. 9, the

HSU ET AL.: DESIGN OF AN ERROR-TOLERANCE SCHEME FOR DISCRETE WAVELET TRANSFORM IN JPEG 2000 ENCODER 631

Fig. 7. The schematic representations of parity weighting of an image.

632 IEEE TRANSACTIONS ON COMPUTERS, VOL. 60, NO. 5, MAY 2011

Fig. 8. The processes of pipeline-based DWT design.

difference between the values Cin and Cout can be capturedafter 261 clocks if the errors occurred in one row of animage.

3.4 Parity Analyzer

The parity analyzer shown in Fig. 4 plays the role of acomparator to check a syndrome (the difference betweenCin and Cout) and determine whether a syndrome is tolerantwith a chosen threshold or not. According to the errormodels in Section 3.1, the thresholds of IEM and DEM aredefined as THIEM and THDEM , respectively, to detect theintensive and distributed errors in an image. Equation (15)indicates that the errors of IEM can be found if thedifference between the checking values Cin and Cout aregreater than a threshold THIEM within the contiguous rowsfrom jth to kth. However, the definition of IEM detection

in (15) cannot be sure to detect the DEM if some of theerrors are occurred in only one row or some distributedrows. Thus, a threshold THDEM shown in (16) is used todetect the errors of DEM further:

Xk�jþ1

i¼jCin i � Cout ij j � THIEM; ð15Þ

Xni¼1

Cin i � Cout ij j � THDEM: ð16Þ

Although the errors of IEM and DEM can be captured bythe mathematical formulas shown in (15) and (16), thevalues of thresholds THIEM and THDEM are hardly used todetermine the capability of error tolerance in a real image.Generally, the human visual system is more sensitive about

HSU ET AL.: DESIGN OF AN ERROR-TOLERANCE SCHEME FOR DISCRETE WAVELET TRANSFORM IN JPEG 2000 ENCODER 633

Fig. 8. Continued.

the brightness variations than the changes in chrominance.Thus, the brightness variation detection method [21], whichplays an important role for error detection in an image, isadopted here to redefine the thresholds of error detection.For the computational efficiency consideration, the criterionof brightness variation is determined by using the histo-gram-based method, which usually shows sensitivity to theimage changes within a similar brightness condition [22].The histogram difference is defined by

DHIS ¼X

Hj �Hi

�� ��; ð17Þ

where Hi and Hj signify the histograms in the ith and

jth rows of an image. By setting threshold of the histogram

difference DHIS, the image holding brightness variations can

be detected. Equations (15) and (16) match up the concept

of histogram difference. Thus, two normalization of bright-

ness variation rates (BVRs) derived from (15) and (16) can

be defined and shown in (18) and (19), respectively.

634 IEEE TRANSACTIONS ON COMPUTERS, VOL. 60, NO. 5, MAY 2011

Fig. 9. Timing chart. (a) Cin procedure. (b) Cout procedure.

Significantly, in order to accurately evaluate the qualityimpact of an image by brightness variations, the BVRs arenormalized as fractional variations. The overall workingflow of the proposed ETS design is shown in Fig. 10:

BVRIEM ¼Pk�jþ1

i¼j Cin i � Cout ij jPk�jþ1i¼j Cin i

� 100%; ð18Þ

BVRDEM ¼Pn

i¼1 Cin i � Cout ij jPni¼1 Cin i

� 100%: ð19Þ

4 EXPERIMENT RESULTS

The 9/7 lifting DWT of the JPEG 2000 image compressionstandard is used as a circuit under test (CUT) todemonstrate the good performance in error detection timeand error tolerance capability of the proposed ETS design.Consider the lifting DWT operating with error injectionvalues modeled by a Gaussian noise source [23]. The errordetection performance is strongly dependent on the noisevariation and the selected detection threshold. Based onthe description of detection thresholds shown in [20],Fig. 11 illustrates the detection performance curves of the9/7 lifting DWT corresponding to the errors with twodifferent variations: 10�4�2

0 and 10�2�20, where �2

0 repre-sents the variation of input-data. Experimental results

show that the value, jCin � Coutj, from the parity analyzerof the proposed ETS is on the order of 10�10 with round-off errors. Thus, the necessary thresholds have to bechosen well above this level for error detection. Fig. 11clearly indicates that the system has high detectionperformance when the thresholds of IEM and DEMdetection are less than 10�4 and 10�7, respectively. Toexamine the effectiveness of the proposed ETS design indifferent experimental conditions, six benchmark se-quences (Lena, Peppers, Baboon, Barbara, Goldhill, andCameraman) are selected in the experiments. The compar-isons between the proposed ETS design and the work in[17] are presented in this section to demonstrate that theproposed ETS design has good performance in errordetection time. Additionally, quality observation andobjective analysis are discussed to evaluate the capabilityof error tolerance.

4.1 Performance Evaluation

The error detection time significantly affects the circuitperformance when the circuit is under testing. However,good performance in speeding up the error detectiongenerally depends on the quality of circuit design. Apipeline-based DWT design is used of the proposed ETS toaccelerate the speed of error detection.

The computational complexity is the critical factor toevaluate the performance of error detection time. Based onFigs. 4 and 13, since the sizes of matrices X and W are both1� n and A is a n� n matrix, the computational complexityof IPP structure is about 6n2 � n� 1 (see Fig. 12a) with onerow input-data if the conventional DWT is used. However, agreat reduction of computational complexity can be obtainedby using the proposed pipeline-based DWT design. Fig. 9aclearly indicates the actuality and shows that the first evendata will be exported after six clocks with 13 times ofmultiplication and six times of addition. Thus, for one rowinput-data, the computational complexity of the IPP struc-ture is only 19n. For quickly approximating the operationspeed of a procedure, the notation Big-O is usually used toestimate the running time of a procedure [24]. Thus, for ann� n image, the computational complexity of IPP structurewith conventional DWT and with pipeline-based DWT areOðn3Þ and Oðn2Þ, respectively. Additionally, the computa-tional complexity of OPP structure shown in Fig. 4 is

HSU ET AL.: DESIGN OF AN ERROR-TOLERANCE SCHEME FOR DISCRETE WAVELET TRANSFORM IN JPEG 2000 ENCODER 635

Fig. 10. The overall working flow of the proposed ETS design.

Fig. 11. Detection threshold of the proposed ETS.

evaluated by the data after two lifting steps and the

weighting matrix W. Thus, the computational complexity

of OPP structure is about 2n� 1 for one row input samples

(see Fig. 12b). Fig. 12b shows that the operations of DWT are

independent of OPP execution, thus the computational

complexity of OPP is about Oðn2Þ for an n� n image even

if different types of DWT are applied. According to the

above-mentioned discussion, the proposed ETS with pipe-

line-based DWT has better performance in error detection

time than the previous work in [17].

4.2 Error Tolerance Discussion

The quality observation and objective evaluation of bench-

mark images are presented here to discuss the capability of

error tolerance when the proposed ETS design applied to the

JPEG 2000 compression standard. Six 512� 512 standard

images, “Lena,” “Peppers,” “Baboon,” “Barbara,” “Gold-

hill,” and “Cameraman,” are selected to experiment. The

experiments of the standard image “Lena” shown in Fig. 13

act as an example to demonstrate the results of quality

observation. Fig. 13a illustrates the error-free case. Fig. 13b

shows an error case with the value of BVRDEM is equal to

3.85 percent. Although the quality of image in Fig. 13b is

affected by the errors, the performance of quality observation

is still good. In other words, the errors occurred in Fig. 13b

can be acceptable by human visual system. However, the

image quality is getting worse when the value of BVRDEM

changes to 6.69 percent. Fig. 13c shows the error case.

Additionally, if theBVRDEM value increases to 16.51 percent

(see Fig. 13d), the distortion of the image is serious and hard

to observe. From the quality observation in Fig. 13, the values

of BVRIEM and BVRDEM of the standard image “Lena” are

defined to four percent for evaluating the capability of error

tolerance. In other words, if the value ofBVRDEMðBVRIEMÞis less than four percent then the errors occurred in the

standard image “Lena” can be acceptable, whereas the errors

are unacceptable when the BVRDEMðBVRIEMÞ value is

greater than four percent.

The objective evaluation of a specific image presented

here depends on the relation between BVR and acceptable

error rate (AER). Error rate (ER) is the percentage of vectors

for which values at a set of outputs deviate from error free

response, during normal operation [25]. Thus, the AER in

this paper is defined as (20). Since AER is a percentage of

acceptable errors for all injection errors, AER represents the

capability of error tolerance with injection errors in an image.

AER ¼ Acceptable Errors

Total Injection Errors� 100%: ð20Þ

636 IEEE TRANSACTIONS ON COMPUTERS, VOL. 60, NO. 5, MAY 2011

Fig. 12. Computation complexity estimation for one row data withtraditional DWT. (a) IPP design. (b) OPP design.

Fig. 13. Quality observation for the benchmark image “Lena.” (a) Errorfree. (b) BVRDEM ¼ 3:85%. (c) BVRDEM ¼ 6:69%. (d) BVRDEM ¼16:51%.

Two standard images “Baboon” and “Cameraman” arediscussed here (see Fig. 14) to describe the capability oferror tolerance since image “Baboon” is more complex andimage “Cameraman” is rather monotone than the otherbenchmark images. From the experimental results shownin Fig. 14, the AER for the images “Baboon” and “Camera-man” are about 12.6 and 5.8 percent, respectively, if thevalue of BVRDEM is set at 4 percent. Thus, if the number ofpixel errors are 262;144 ð512� 512Þ, which are caused bynumerical errors injecting in the lifting section, then about33,030 and 15,204 pixel errors in the images “Baboon” and“Cameraman” can be acceptable under four percentbrightness variations. Table 1 shows the experimentalresults of AERs for six benchmark images with differentbrightness variations. The value of BVRDEMðBVRIEMÞ hasto be appropriately defined in accordance with qualityobservation for accurately evaluating the AER of a specifictest image.

4.3 Comparisons

The proposed ETS design shown in Fig. 4 can be implemen-ted by using a comparator and some multipliers, adders,buffers, and registers. Based on the circuit design shown in[13], [26], [27], the number of logic gates of the proposed ETSdesign is about 6,972. However, about 180 k logic gates areneeded for a VLSI architecture design of JPEG 2000 encoder[28]. Thus, the area overhead of the proposed ETS design isonly about 3.87 percent, which is a reasonable design for

circuit testing. The comparisons between the proposed

method and previous work [17] are shown in Table 2, which

clearly indicates that the proposed ETS design has good

performance in computational complexity (error detection

time) and error tolerance capability with little area overhead.

5 CONCLUSIONS

An effectively ETS design for lifting DWT error detection

and error tolerance evaluation in JPEG 2000 encoder is

presented in this paper. The paper first developed a pipeline-

based DWT structure to support the proposed ETS design for

speeding up the error detection time. Then, an IPP, an OPP,

and a parity analyzer based on the weighting sum technique

are built of the proposed ETS design to detect the errors. The

error detection performance depends on the detection

thresholds, which are determined by the brightness varia-

tions. Experimental results show that the proposed ETS with

pipeline-based DWT design can significantly improve the

error detection time compared with the previous work with

conventional DWT structure. Additionally, according the

quality observation and objective evaluation for the test

images, the proposed ETS design also demonstrates the good

performance in error tolerance capability.

ACKNOWLEDGMENTS

The authors would like to thank the National Science Council

of the Republic of China, Taiwan, for financially supporting

this research under Contract No. NSC: 97-2221-E-259-032.

HSU ET AL.: DESIGN OF AN ERROR-TOLERANCE SCHEME FOR DISCRETE WAVELET TRANSFORM IN JPEG 2000 ENCODER 637

TABLE 1Acceptable Error Rate (AER) for Test Images

Fig. 14. Objective evaluation results.

TABLE 2Comparison Results

REFERENCES

[1] “ITU-T Recommend. T.800-ISO/IEC FCD15444-1: JPEG 2000Image Coding System,” Int’l Organization for Standardization,ISO/IEC JTC1 SC29/WG1, 2000.

[2] C. Christopoulos, A. Skodras, and T. Ebrahimi, “The JPEG2000Still Image Coding System: An Overview,” IEEE Trans. ConsumerElectronics Design, vol. 46, no. 4, pp. 1103-1127, Nov. 2000.

[3] C. Constantinescu, “Trends and Challenges in VLSI CircuitReliability,” IEEE Micro, vol. 23, no. 4, pp. 14-19, July/Aug. 2003.

[4] J. Seifert, Z. Xiaowei, and L.W. Massengill, “Impact of Scaling onSoft-Error Rates in Commercial Microprocessors,” IEEE Trans.Nuclear Science, vol. 49, no. 6, pp. 3100-3106, Dec. 2002.

[5] M. Breuer, S. Gupta, and T. Mark, “Defect and Error Tolerance inthe Presence of Massive Numbers of Defects,” IEEE Trans. DesignTest Computers, vol. 21, no. 3, pp. 216-227, May/June 2004.

[6] H. Chung and A. Ortega, “Analysis and Testing for Error TolerantMotion Estimation,” Proc. 20th IEEE Int’l Symp. Defect and FaultTolerance in VLSI Systems, pp. 514-522, 2005.

[7] M. Srikar and S. Srinivasan, “A Novel Architecture for Lifting-Based Discrete Wavelet Transform for JPEG 2000 StandardSuitable for VLSI Implementation,” Proc. Int’l Conf. VLSI Design,pp. 202-207, Jan. 2003.

[8] C.J. Lian, K.F. Chen, H.H. Chen, and L.G. Chen, “Analysis andArchitecture Design of Lifting Based DWT and EBCOT for JPEG2000,” Proc. Int’l Symp. VLSI Technology, Systems, and Applications,pp. 180-183, 2001.

[9] S.M. Aroutchelvamw and K. Raahemifar, “An Efficient Architec-ture for Lifting-Based Forward and Inverse Discrete WaveletTransform,” Proc. IEEE Int’l Conf. Multimedia and Expo, pp. 816-819, 2005.

[10] Y.H. Seo and D.W. Kim, “VLSI Architecture of Line-Based LiftingWavelet Transform for Motion JPEG 2000,” IEEE J. Solid-StateCircuits, vol. 42, no. 2, pp. 431-440, Feb. 2007.

[11] I. Daubechies and W. Sweldens, “Factoring Wavelet Transformsinto Lifting Steps,” J. Fourier Analysis and Application, vol. 4,pp. 247-269, 1998.

[12] T. Acharya and C. Chakrabarti, “A Survey on Lifting-BasedDiscrete Wavelet Transform Architectures,” J. VLSI Signal Proces-sing Systems, vol. 42, no. 3, pp. 321-339, Mar. 2006.

[13] B.F. Wu and C.F. Lin, “A High-Performance and Memory-EfficientPipeline Architecture for the 5/3 and 9/7 Discrete WaveletTransform of JPEG 2000 Codec,” IEEE Trans. Circuits and Systemsfor Video Technology, vol. 15, no. 12, pp. 1615-1628, Dec. 2005.

[14] D. Nowroth, I. Polian, and B. Becker, “A Study of CognitiveResilience in a JPEG Compressor,” Proc. Int’l Conf. DependableSystems and Networks, pp. 32-41, 2008.

[15] N. Seifert, Z. Xiaowei, and L.W. Massengill, “Impact of Scaling onSoft-Error Rates in Commercial Microprocessors,” IEEE Trans.Nuclear Science, vol. 49, no. 6, pp. 3100-3106, Dec. 2002.

[16] P. Shivakumar, M. Kister, S.W. Keckler, D. Burger, and L. Alvisi,“Malertodeling the Effect of Technology Trends on the Soft ErrorRate of Combinational Logic,” Proc. Int’l Conf. Dependable Systemsand Networks, pp. 389-398, 2002.

[17] G.R. Redinbo and C. Nguyen, “Concurrent Error Detection inWavelet Lifting Transforms,” IEEE Trans. Computers, vol. 53,no. 10, pp. 1291-1302, Oct. 2004.

[18] A.B. Watson, G.Y. Yang, J.A. Solomon, and J. Villasenor,“Visibility of Wavelet Quantization Noise,” IEEE Trans. ImageProcessing, vol. 6, no. 8, pp. 1164-1175, Aug. 1997.

[19] J.P. Hornak, The Encyclopedia of Imaging Science and Technology.John Wiley & Sons, Inc., 2002.

[20] C. Nguyen and G.R. Redinbo, “Fault Tolerance Design in JPEG2000 Image Compression System,” IEEE Trans. Dependable andSecure Computing, vol. 2, no. 1, pp. 57-75, Jan. 2005.

[21] H.W. Haussecker, “Simulation Estimation of Optical Flow andHeat Transport in Infrared Image Sequences,” Proc. IEEE WorkshopComputer Vision, pp. 85-93, June 2000.

[22] Y. Shen, D. Zhang, C. Huang, and J. Li, “Adaptive WeightedPrediction in Video Coding,” Proc. IEEE Int’l Conf. Multimedia andExpo, vol. 1, pp. 427-430, June 2004.

[23] A. Papoulis, Probability, Random Variable, and Stochastic Process.McGraw-Hill, 1991.

[24] M. Sipser, “Measuring Complexity,” Introduction to the Theory ofComputation, pp. 226-228, PWS Publishing, 1997.

[25] D. Shin and S.K. Gupta, “A Re-Design Technique for DatapathModules in Error Tolerant Applications,” Proc. Asian Test Symp.,pp. 431-437, Nov. 2008.

[26] K.J. Cho, K.C. Lee, J.G. Chung, and K.K. Parhi, “Design of Low-Error Fixed-Width Modified Booth Multiplier,” IEEE Trans. VeryLarge Scale Integration Systems, vol. 12, no. 5, pp. 522-531, May2004.

[27] S.W. Cheng, “A High-Speed Magnitude Comparator with SmallTransistor Count,” Proc. IEEE Int’l Conf. Electronics, Circuits andSystems, pp. 1168-1171, Dec. 2003.

[28] L. Liu, N. Chen, H. Meng, L. Zhang, Z. Wang, and H. Chen, “AVLSI Architecture of JPEG 2000 Encoder,” IEEE J. Solid-StateCircuits, vol. 39, no. 11, pp. 2032-2040, Nov. 2004.

Chun-Lung Hsu received the PhD degree inelectrical engineering from National TaiwanUniversity, Taipei, Taiwan, ROC, in 2000. He isinterested in the relationship of low-power circuitdesign, consumer electronics development, sys-tem on a chip design, and VLSI testing. He iscurrently working as an associate professor inthe Department of Electrical Engineering, Na-tional Dong Hwa University, Hualien, Taiwan,ROC. The major researches of his laboratory are

3D IC design, built-in self-detect/correct for 3D ICs, and tolerancescheme development for image system. He is now a member of theIEICE, IET, and the IEEE.

Yu-Sheng Huang received the bachelor ofengineering degree in computer science andinformation engineering from Chaoyang Univer-sity of Technology, Taichung, Taiwan, ROC., in2004, and the MS degree in engineering fromNational Dong Hwa University, Hualien, Taiwan,ROC., in 2006. He is currently working towardthe PhD degree at National Dong Hwa Univer-sity. His research is focused on video IC design,built-in self-test, and fault/error tolerance.

Ming-Da Chang received the bachelor ofengineering degree in electronic engineeringfrom Fu Jen Catholic University, Taipei, Taiwan,ROC., in 2006, and the MS degree in engineer-ing from National Dong Hwa University, Hua-lien, Taiwan, ROC., in 2008. He is currently ahardware engineer at Accton Technology Cor-poration. His research is focused on built-inself-test and still image fault/error tolerance.

Hung-Yen Huang received the bachelor ofengineering degree in electronic engineeringfrom Fu Jen Catholic University, Taipei, Taiwan,ROC., in 2008. He is currently working towardthe MS degree from National Dong Hwa Uni-versity. His research is focused on still imagefault/error tolerance.

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638 IEEE TRANSACTIONS ON COMPUTERS, VOL. 60, NO. 5, MAY 2011