detection and correction of transmission errors in pyramid images

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 8, NO. 1, FEBRUARY 1998 25

Detection and Correction of Transmission Errors in Pyramid ImagesLi-Jae Wang and Jin-Jang Leou

Abstract—In this study, two detection and correction approach-es to protection from transmission errors in pyramid images areproposed. For entropy-coded pyramid images, a single transmis-sion error in a codeword will not only affect the underlyingcodeword, but also may affect subsequent codewords. Conse-quently, a single error in an entropy-coded system may result insignificant degradation. In this paper, two types of variable-lengthcodes with the synchronization capability are proposed to encodetwo types of pyramid images, namely, the difference pyramid andthe reduced-difference pyramid, to overcome the synchronizationproblem. The proposed approaches detect transmission errorsby examining error checking conditions. When a transmissionerror is detected, a sequence of bit inversions and redecodingprocesses is performed on the corrupted part of the compressedimage bitstream until a feasible redecoding solution is found. Thesimulation results show significant quality improvement.

Index Terms—Detection and correction approach, differencepyramid, progressive image transmission, reduced-differencepyramid, transmission error, variable-length code.

I. INTRODUCTION

T RANSMISSION of high resolution images over a low-speed channel has been a challenging problem for many

applications such as medical image archives and communi-cations and picture browsing via the Internet. Progressiveimage transmission [1]–[3] is a technique to alleviate the longtransmission time by first transmitting a low-resolution approx-imation image and progressively refining the image by sendingmore detail information. One popular technique of progressiveimage transmission is to use pyramids. Pyramid is a typeof hierarchical representations of images with ordered spatialresolution. There are several variants of pyramids accordingto the type of data represented and the decomposition process.Given an image containing pixels, a pyramid canbe defined as a sequence of matrices such that is areduced-resolution version of and the top-level imagebecomes a single pixel. An image containing pixelsis first decomposed into spatially contiguous, nonoverlappingblocks of nodes (pixels). A pyramid data structurecan be formed by successively operating over neighboringnodes, i.e., the value of a node in level is obtainedfrom the values of corresponding neighboring nodesin level . Typical pyramid data structures include the meanpyramid, the truncated mean pyramid, the difference pyramid,

Manuscript received August 23, 1997. This paper was recommended byAssociate Editor K.-H. Tzou. This work was supported in part by the NationalScience Council, Republic of China, under Grant NSC 85-2213-E-194-004and Grant NSC 87-2213-E-194-016.

L.-J. Wang is with the Department of Information Management, TranswordJunior College of Commerce, Touliu, Yunlin, Taiwan 640, R.O.C.

J.-J. Leou is with the Institute of Computer Science and InformationEngineering, National Chung Cheng University Chiayi, Taiwan 621, R.O.C.

Publisher Item Identifier S 1051-8215(98)01513-4.

the reduced-difference pyramid, the reduced-sum pyramid, theGaussian–Laplacian pyramid, and the S-transform pyramid[1]–[3]. Two types of pyramids, namely, the difference pyra-mid and the reduced-difference pyramid, with aretreated in this study.

In the coding system, a variable-length code was assumed.For entropy-coded pyramid images, a single transmission errorin a codeword will not only affect the underlying codeword,but may also affect subsequent codewords. To cope withthe synchronization problem, two types of variable-lengthcodes with self-synchronizing codewords and the universalsynchronizing sequence are employed. A binary prefix code issaid to be self-synchronizing if it contains some synchronizingcodewords [4], [5]. After the decoder receives any synchro-nizing codeword, it will be resynchronized regardless of thepreceding slippage. Similarly, the universal synchronizingsequence can be inserted into the compressed image bitstream.After the decoder receives a universal synchronizing sequencerecognized by its special codeword pattern, it will also beresynchronized regardless of the preceding slippage [6], [7].A simple universal synchronizing sequence is the end-of-line (EOL) codeword used in the digital facsimile codingstandard. Using the two types of variable-length codes, lossof synchronization due to transmission errors is thus confinedto the next synchronizing codeword or the next universalsynchronizing sequence. Both self-synchronizing codewordsand the universal synchronizing sequence can help to regaincodeword synchronization when transmission errors occur.

To protect image quality against transmission errors, thereare several proposed approaches, including: 1) the channelcoding approach [8], [9]; 2) the detection and concealment ap-proach [10]–[13]; and 3) the detection and correction approach[14]–[17]. For the channel coding approach, some error correc-tion codes are used to encode images such that transmissionerrors can be detected and corrected by adding redundancyto the transmitted information. However, the channel codingapproach will moderately increase the transmission bit rate.Error concealment techniques, on the other hand, alleviatevisual degradation by using digital signal processing. For thedetection and correction approach, redundancy informationinherent in neighboring pixels is transmitted, which enablesthe receiver to detect and correct transmission errors.

In this study, two detection and correction approaches totransmission errors in pyramid images are proposed. Pyra-mid images are encoded and decoded block-by-block with

. Within the two proposed approaches, transmissionerrors are detected by two error checking conditions. Whena transmission error is detected, a sequence of bit inversionsand redecoding processes is performed on the corrupted partof the compressed image bitstream until a feasible redecoding

1051–8215/98$10.00 1998 IEEE

26 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 8, NO. 1, FEBRUARY 1998

solution is found. For simplicity, two assumptions are made:1) the compressed data of each block contains at most onetransmission error and 2) the top level (not necessarily a uniquenode) of each pyramid is correctly received.

The paper is organized as follows. The two types of pyra-mids and two types of variable-length codes studied aredescribed in Section II. The proposed detection and correctionapproaches to transmission errors in pyramid images areaddressed in Section III. Simulation results are included inSection IV, followed by concluding remarks.

II. PYRAMID IMAGES AND VARIABLE-LENGTH CODE DESIGN

A. Pyramid Image Construction

For an image containing pixels, the truncatedmean pyramid is formed by successively averaging withtruncation over 2 2 neighboring nodes. For level ,

and for each spatially contiguous, nonoverlappingblock of 2 2 in level , where , the truncatedmean in level is given by

(1)

where , denotes the truncation of, and [ ] denotes the truncation of . It is noted that the

bottom level is the original image and the top level isthe truncated mean value of the image. For a given truncatedmean pyramid , the difference pyramid is formedby simply taking the differences between successive levels inthe truncated mean pyramid, i.e.,

(2)

where and . Thecorresponding reduced-difference pyramid is formed bytaking the differences between the neighboring nodes for eachspatially contiguous, nonoverlapping block of 22 nodes inlevel , i.e.,

(3)

where and . It is notedthat any three of the four differences in (3) are sufficient torecover the truncated mean pyramid.

To easily implement the two proposed approaches, thetruncated mean value of in (1) is redefined as the truncationof (instead of ) and the difference pyramid andthe reduced-difference pyramid are generated by (2) and (3),respectively.

B. Variable-Length Code Design

In this study, two types of variable-length codes, namely,the variable-length code with self-synchronizing codewordsand the variable-length code with the universal synchronizingsequence, will be generated and used to encode the pyramidimages. That is, we intend to perform near lossless coding.

Montgomery and Abrahams [4] developed a method forgenerating variable-length codes with self-synchronizing code-words through the use of self-synchronizing binary prefix-condition codes. Compared with optimal codes, their codeshave the minimum redundancy, which can be as little asone additional bit introduced into the least likely codewordfor a large class of sources. This code may have a muchbetter statistical synchronizing performance than that of anyoptimal code in many cases. In this study, their method isslightly modified to generate a variable-length code with self-synchronizing codewords. The details can be found in thethesis by Wang [18].

On the other hand, for a given pyramid, Lei and Sun [7]developed a systematic method for generating an optimalcode with the universal synchronizing sequence (the EOLcodeword). Using their method, two successive corruptedcodewords in two successive blocks may be misinterpretedas the universal synchronizing sequence, which may makethe recognition of the universal synchronizing sequence am-biguous. To cope with this problem, their method is slightlymodified to generate a variable-length code with the universalsynchronizing sequence. The details can also be found in [18].

III. D ETECTION AND CORRECTION OF

TRANSMISSION ERRORS IN PYRAMID IMAGES

A. Error Detection

Two error detection approaches are proposed for the twotypes of pyramids: the difference pyramid and the reduced-difference pyramid. For the difference pyramid, let denotethe sum of the four node values within a block in level,which is equal to the difference between the sum of the fourreconstructed block node values in levelof the associatedtruncated mean pyramid and four times the corresponding nodevalue in level of the associated truncated mean pyramid. Thepossible values of include 0, 1, 2, and 3 only. A singletransmission error within a block of a difference pyramid canbe detected by checking whether ornot, which is independent of the node values in lower levels.On the other hand, for the reduced-difference pyramid, the sum

of the four node values within each block is equal to zero. Asingle transmission error within a block of a reduced-differencepyramid can be detected by checking whether or not,which is also independent of the node values in lower levels.

However, for the reduced-difference pyramid, any threeof the four node values within each block are sufficient tospecify the pixel values (required to encode and transmit).Using the error checking condition , the four (insteadof three out of four) node values within each block ofthe reduced-difference pyramid are required to encode andtransmit, which leads to a 25% redundancy. To reduce such a


Fig. 1. A bitstream segment of a reduced-difference pyramid using coding scheme 2.

heavy redundancy, the proposed error detection approach usesa variable-length code with the universal synchronizing se-quence (coding scheme 2). Because the total number of blocksbetween two successive universal synchronizing sequences isknown in advance, a bitstream segment of a reduced-differencepyramid using coding scheme 2 is illustrated in Fig. 1, wherethe number of data blocks between theth and ( )thuniversal synchronizing sequences is. The th data blockconsists of three out of the four node values within thethblock, where . The th bit of the paritycheck bits is the parity check bit of theth data block. A singletransmission error within theth data block can be detected bychecking the th parity check bit. Hence, transmission errorsin a reduced-difference pyramid using coding scheme 2 canbe detected and corrected without the need of transmittingredundant node values. The proposed error detection approachrequires one bit per block as overhead. Compared with the25% redundancy, the overhead is very moderate. On the otherhand, because a variable-length code with self-synchronizingcodewords (coding scheme 1) achieves synchronization dy-namically, the corresponding error detection approach for thereduced-difference pyramid using coding scheme 1 (withoutthe 25% redundancy) is not available.

The two proposed error detection approaches are performedin a block-by-block manner, which have the level-independentproperty, i.e., transmission errors in higher levels will notaffect the error detection and correction in lower levels.

B. Error Correction

In this study, we assumed that the compressed data ofeach block contains at most one transmission error, which isreasonable for the case of low bit error rate (BER). When a“corrupted” block is detected, it may be one of the followingcases.

1) The compressed data of the current block contains atransmission error.

2) The compressed data of the current block is error-free,but the compressed data of one of its previous blocks iserroneously redecoded.

3) The compressed data of the current block is error-free,but the compressed data of one of its previous blockscontaining a transmission error is not detected.

To deal with the above three cases, three processes are adopted,respectively, as follows.

1) For the first case, a sequence of bit inversions andredecoding operations is performed on the compressed

data of the current block until a feasible redecodingsolution is found.

2) For the second case, we backtrack to the compresseddata of the nearest previous neighbor block of the currentblock, which has been bit-inverted and redecoded. Thesame process as case 1) is then performed, but bitinversions and redecoding operations are performed onlyon the redecoding solutions that have not been examined.

3) For the third case, no feasible redecoding solution can befound after all possible bit inversions for the compresseddata of the current block (block k) have been performed.Then we backtrack to the compressed data of the preced-ing block (block ) and perform the same process ascase 1). If no feasible redecoding solution can be found,we backtrack to the compressed data of block .The process will be repeated until a feasible redecodingsolution is found.

As a summary, the proposed error correction procedure for a“corrupted” block can be listed as follows.

Proposed Error Correction Procedure:

Step 1) A sequence of bit inversions and redecoding pro-cesses is performed on the compressed data of thecurrent block (block k). If a feasible redecodingsolution is found, exit.

Step 2) Backtrack to the compressed data of the nearestprevious neighbor block of block, which has beenbit inverted and redecoded, and perform the sameprocess as Step 1). If a feasible redecoding solutionis found, exit.

Step 3) Backtrack to the compressed data of block andperform the same process as Step 1). If a feasibleredecoding solution is found, exit. Otherwise, set

and repeat Step 3).

IV. SIMULATION RESULTS

The proposed detection and correction techniques to copewith transmission errors in the two types of pyramids exam-ined in this study have been tested on two images “Lenna”and “Peppers” with different BER’s. The two pyramid im-ages “Lenna” and “Peppers” are encoded by two kinds ofvariable-length codes with self-synchronizing codewords (cod-ing scheme 1) and the universal synchronizing sequence(coding scheme 2). The two pyramid images (512512pixels) consist of ten levels (levels 0–9) and . Thepeak signal-to-noise ratio (PSNR) in each level of an imagepyramid, defined as PSNR MSE dB, is used


Fig. 2. The image “Lenna” transmitted as a difference pyramid atBER = 0:05%: (a) the corrupted pyramid image “Lenna” in levels 5–7and (b) the processed pyramid image “Lenna” in levels 5–7 by the proposedapproach using coding scheme 1.

as the objective performance measure. Here mean square error(MSE) in level containing nodes is given by

MSE

where and denote the value of node ( ) in levelwithin the original and processed (reconstructed) truncated

mean pyramids, respectively.The simulation results are illustrated in Tables I and II,

where “-” indicates no transmission error in the correspondinglevel. The simulation results for the image “Peppers” aresimilar to that of the image “Lenna” [18], and thus omittedhere. The processed images “Lenna” using coding scheme 1and coding scheme 2 at BER % are illustrated in Figs. 2and 3. The total bit rates (b/node) within an image pyramidfor Huffman coding, coding scheme 1, and coding scheme 2are listed in Table III, where “N/A” indicates the correspond-ing proposed approach (without the 25% redundancy) is notavailable.

Based on the simulation results in Tables I and II, theperformance (PSNR) of the proposed approach for codingscheme 1 is comparable to that of coding scheme 2. Ac-cording to Table III, although variable-length codes with self-synchronizing codewords (coding scheme 1) are suboptimal,the total bit rates (b/node) of coding scheme 1 are ap-proximately equal to that of Huffman coding. On the otherhand, since the universal synchronizing sequence is purelyoverhead information, the total bit rates of coding scheme 2are higher than that of coding scheme 1 and Huffman coding(about 5.466% 6.944%), respectively. Therefore, from theviewpoint of the total bit rates, coding scheme 1 is slightlysuperior to coding scheme 2.

As the simulation results illustrated in Tables I and II, theproposed detection and correction approaches may completely

Fig. 3. The image “Lenna” transmitted as a reduced-difference pyramid atBER= 0:05%: (a) the corrupted pyramid image “Lenna” in levels 5–7 and (b)the processed pyramid image “Lenna” in levels 5–7 by the proposed approachusing coding scheme 2.

TABLE ISIMULATION RESULTS OF THEPROPOSEDAPPROACHESUSING CODING SCHEMES

1 AND 2 FOR THE IMAGE “L ENNA” T RANSMITTED AS A DIFFERENCEPYRAMID

recover all the corrupted node values. An erroneously rede-coded block may occur when it has the same total number of


TABLE IISIMULATION RESULTS OF THEPROPOSEDAPPROACHUSING CODING SCHEME 2FOR THE IMAGE “L ENNA” T RANSMITTED AS A REDUCED-DIFFERENCEPYRAMID

TABLE IIITOTAL BIT RATES (B/NODE) OF THE TWO

PYRAMID IMAGES “L ENNA” AND “PEPPERS”

bits as that of the original block and satisfies the correspondingerror checking condition. However, it will not affect thedetection and correction (or decoding) process of subsequentblocks. As an example (transmitted as a difference pyramid)shown in Fig. 4, the erroneously redecoded block has thesame total number of bits (14 b) as that of the original blockand satisfies the corresponding error checking condition, i.e.,

. However, it iserroneously redecoded.

Fig. 4. An erroneously redecoded block that has the same total number ofbits as that of the original block and satisfies the corresponding error checkingcondition.

Fig. 5. An illustrative processed image “Lenna” containing the block shiftingeffect at BER= 0:1% (coding scheme 1).

Using coding scheme 1, the number of decoded blocks (orcodewords) of a processed bitstream segment may be erro-neously different from that of the original bitstream segment,resulting in a left or right shifting of decoded blocks. Anexample of this effect is shown in Fig. 5 at BER %.Because the total number of blocks (or codewords) betweentwo successive universal synchronizing sequences is knownin advance, the block shifting effect can be eliminated if theuniversal synchronizing sequence is correctly received, i.e.,the block shifting effect may be confined to some extent ifcoding scheme 2 is used. Therefore, as compared with codingscheme 2, the block shifting effect is the weakness of codingscheme 1.

V. CONCLUDING REMARKS

The two types of pyramids (the difference pyramid and thereduced-difference pyramid) examined in this study have thelevel-independent property, i.e., transmission errors in higher


levels will not affect the error detection and correction in lowerlevels. Because the objective performance measure PSNRis measured by the original and processed (reconstructed)truncated mean pyramids, transmission errors (not completelycorrected) in higher levels will affect the “PSNR” values inlower levels, even if the two types of pyramids have thelevel-independent property.

In this study, complete ten levels of each pyramid imagecontaining 512 512 pixels ( ) are encoded andtransmitted. This may result in too many levels to processand hence degrade the coding performance. In practice, thetotal number of levels required to transmit and process fora pyramid image containing 512 512 pixels is usuallyless than ten. Therefore, the second assumption made in thisstudy is: the top level (not necessarily a unique node) ofa pyramid is correctly received. Additionally, we assumedthat the compressed data of each block contains at most onetransmission error, which is reasonable for the case of lowBER. However, the proposed approaches can be modified toprocess noisy pyramid images, in which the compressed dataof each block may contain two or more transmission errors.

In this study, the detection and correction approaches detecttransmission errors in pyramid images by examining errorchecking conditions. When a transmission error is detected,a sequence of bit inversions and redecoding processes isperformed on the corrupted part of the compressed imagebitstream until a feasible redecoding is found. The simulationresults show significant quality improvement.

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detection and correction of transmission errors in pyramid images

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