Channel error recovery for transform image coding - CiteSeerX

1754

TRANSACTIONS IEEE

ON COMMUNICATIONS, VOL. COM-29, NO. 12, DECEMBER 1981

Channel Error Recovery for Transform Image Coding 0 . ROBERT MITCHELL,

MEMBER, IEEE,

AND

ALI J . TABATABAI,

STUDENTMEMBER, IEEE

transmitted consistsof the coefficient valuesthemselves. If channel error protection is required, the quality of the reconstruction must be sacrificed to achieve the same data rate. Modestino and Daut [8] have considered the tradeoffs involved in assigning channel error protection bits and quantization bits for DPCM transmission in noisy channels. The problem issimilarwhen transform codingis considered: as the channel becomes noisy, the dominant errors are due to bit errors and not to quantization noise, and thus more channel I. INTRODUCTION error protection is appropriate. However, it may be that the WO-DIMENSIONAL block transform coding methods pro- channel protection is not provided due to the increase in bit vide very effective image compression. At image data rates rate required or due to the fact that the channel errors are of 0.5 to 1.5 bits/pixel, we have found these methods to be unexpected. judged subjectively superior to other methods at low channel An example of a received reconstructed image in the error rates [l]-[3]. The adaptive two-dimensional block presence of channel errors (random binary symmetric channel) transform coding method described by Chen and Smith [4] is shown in Fig. 6. The channel error probability was lo-'. We have noticed that a human observer can spot many major has produced someof the best results. In this method, the two-dimensional fast discrete cosine transform [5], [6] is errors due to channel errors quite readily. Errors appear as performed over subblocks (typically size 16 X 16) of the incorrect 16 X 16 blocks. A human observes a discontinuity originalimage. The blocks are sorted into classesbased on across the block boundaries inconsistent with typical image one or more features. (Chen and Smith use the total ac energy data. A single coefficient error in a block manifests itself in in the block as the feature.) Then bits areassigned to each the reconstructed imageasan additive basis function of the two-dimensional cosine transform. The most predominant class based on coefficient sample varianceswithin each class. A typical adaptive multiclass zone method is diagrammed errors areusually due to dc and low frequency basis funcin Fig. 1. An original image is shown in Fig. 2. The classifica- tions because they represent the largest energy content in the tion and bit allocation maps used for a compression to 1.0 blocks and thus have the most bits assigned to them. This paper presents a method to automatically inspect bitslpixel are shown in Figs. 3 and 4, respectively. The reconthe block boundaries of a reconstructed two-dimensional structed image is shownin Fig. 5 . When uncorrected channel errors are present, the effects transform coded image, to locate blocks which are most onthe reconstructed picture are much different than for likely to contain errors, to approximate the size and type of PCM image transmission. Some bits in the compressed data are error in theblock, and to eliminate this estimated error critical and require that channel errors be detected and cor- from the picture. This method uses redundancy in the source rected to prevent total loss of the image. This critical informa- data (assumed continuity across block boundaries) to provide tion is indicated in Fig. 1 and includes the classification map, channel error correction without any additional channel protection bits or changes to the transmitter. It can be used bit assignment tables, and scale factor. As an example, a 5 12 X 5 12 pixel image coded using 16 X when channel errors are unexpected prior to reception. 16 blocks and four classes would require 2048 bits for the 11. CHANNEL ERROR RECOVERY ALGORITHM classification map, 3072bitsforthebit assignment tables, The basic approach is to check the four boundaries around and 16 bits for the scale factor. These 5136 bits can be protected using a (127, 71) BCH code [7] which will reduce an each reconstructed image block. If a discontinuity exists unprotected bit error rate of lo-' to one error every 21 000 along these boundaries that is consistent with a single domiimages. This requires 9187 bits or 0.0350 bitslpixel for the nant transform coefficient error, the coefficient location and critical overhead information. The bulk of the data that .is amplitude are estimated and a basis picture corresponding to the estimated error is subtracted from the block. A block diagram indicating the order of calculations performed is Manuscript received December 8, 1980;revised July 16, 1981. 0. R. Mitchell is with the School of Electrical Engineering, Purdue shown in Fig. 7. University, West Lafayette, IN 47907. Assume that one error has occurred at the kth row and Zth A. J. Tabatabai was with the School of Electrical Engineering, Purcolumn of a transmitted block of transform coefficients. due University, West Lafayette, IN 47907. He is now with Bell Laboratories, Holmdel, NJ 07733. Hence, the decoded block at the receiver, denoted by matrix Absrrucr-A method is presented to automatically inspect the block boundariesof areconstructedtwo-dimensional'transformcoded image, to locate blockswhich are most likely to contain errors, to approximate the size and type of error in the block, and -to eliminate this estimated error from thepicture.This method uses redundancy in the source data toprovidechannelerror correction. No additional or changes to thetransmitterare channelerror protectionbits required. It can be used when channel errors are unexpected prior to reception.

T

0090-6778/81/1200-1754$00.75 0 1981 IEEE

1755

MITCHELL AND TABATABAI: CHANNEL ERROR RECOVERY

Classification ~ a p

B i t assignment

bit

input

-

Calculate

Channel

over sub-blocks

picture

Fig. 1.

Adaptive multiclass zone block transform coding transmitter.

8 6 5 5 4 4 3 3 3 2 2 2 1 1 1 1

class 1 6 5 4 : 4 4 3 3 2 2 2 2 1 1 1 0 5 . 5 . , 4 4 3 3 2 2 2 2 1 1 1 1 1 5 ~ ~ 4 ~ 4 4 3 3 2 2 2 1 1 1 1 0 0 w 4 4 4 3 3 2 2 2 1 1 1 1 0 0 4 4 4 3 3 3 2 2 2 1 1 1 1 0 0 4 4 3 3 3 3 2 2 1 1 1 1 0 0 0 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 3 3 3 2 2 2 2 2 1 1 1 1 0 0 0 2 3 2 2 2 2 2 1 1 1 1 0 0 0 0 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 2 2 2 2 2 1 1 1 1 0 0 0 0 0 0 2 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 3 0 0 0 1 1 1 1 1 0 0 0 ~ 0 0 0 0 0 0 0

class 2

8 5 5 4 4 4 3 3 2 2 2 2 1 1 1 1

Airportoriginal, 512 X 512 pixels, 256 gray levels (8 bits).Fig.

Fig. 2.

5 5 4 4 4 3 3 3 2 2 2 2 1 1 1 1

4 4 4 4 4 3 3 3 2 2 2 1 1 1 1 1

4 4 4 4 3 3 3 3 2 2 2 1 1 1 1 1

class 3

8 4 4 3 3 3 3 2 2 2 2 1 1 1 0 0

4 4 3 3 3 2 2 2 1 1 2 1 1 0 0 0

3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0

3 3 3 2 2 2 2 2 1 1 1 1 1 0 0 0

3 3 2 2 2 2 1 1 1 1 1 0 0 0 0 0

2 2 2 2 2 1 1 1 1 1 1 0 0 0 0 0

2 2 2 2 1 1 1 1 1 1 1 0 0 0 0 0

2 2 1 1 1 1 1 0 1 1 0 0 0 0 0 0

2 1 1 1 1 1 1 0 u 0 0 u 0 0 0 0

1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 0 u 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 u 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 u 0 0 0 0 0 0 0

class 4

4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 1

3 3 3 2 2 2 1 1 1 0 1 3 3 2 2 2 2 1 1 1 1 1 3 3 2 2 2 2 1 1 1 0 0 3 3 2 2 2 1 1 1 0 0 0 3 3 2 2 2 1 1 1 1 0 0 3 2 2 2 1 1 1 1 1 0 0 2 2 2 1 1 1 1 1 1 0 0 2 2 2 1 1 1 1 1 0 0 0 2 2 2 2 1 1 1 0 0 0 0 2 2 2 1 1 1 0 0 0 0 0 2 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

n 3 2 1 1 0 O 0 0 0 0 0

3 2 1 0 0 0 O 0 0 0 0 0

2 ~ o o o ~ o o o o o o o o 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . O O O O O O O O O d O O O O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

o o o o o o o o o o o o u o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4. Bit allocationmapsforthefourclassesshownin Fig. 3. Bits are assigned proportional to the sample variance of each coefficient of each class and so that the average data rate including overhead is 1.0 bits/pixel.

2 2 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 1 1 2 2 2 1 2 1 1 1 1 1 2 2 1 1 2 3 2 1 3 3 1 3 2 1 2 2 2 2 2 2 3

1 2 2 1 1 2 2 1 ? 2 1 2 2 2 1 1 1 2 1 2 2 1 1 2 3 3 3 3 3 2 3 2 2 3 2 1 1 1 2 2 2 3 2 3 2 3 1 2 2 2 2 1

1 1 1 1 1 2 3 2 3 3 3 3 3 3 1 2 2 2 1 1

1 1 2 3 2 1 1 3 1 2 2 2 1 1 2 3 2 1 2 2 3 3 2 1 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 1 1 1 1 2 2 2 2 2 2 3 2 2 1 2 1 2 2 1 1 2 2 2 1 1 2 2 1 2 1 2 2 2 2 1 1 2 1 1 1 1 1 3 3 1 2 2 2 2 3 1 2 2 2 3 2 2 1 1 2 2 2 2 2 2 1 2 2 1 1 2 1 2 2 2 2 2 2 3 2 1 2 1 1 1 1 1 1 2 2 J 2 J 2 1 3 3 3 3 3 1 2 2 2 2 3 1 1 2 2 2 2 1 1 1 3 1 1 2 3 2 3 2 3 3 3 3 4 3 5 3 2 3 2 2 3 1 1 3 2 1 1 1 2 2 3 1 1 1 1 2 3 4 4 3 4 3 3 3 3 4 3 2 2 2 1 1 1 1 1 2 1 2 3 1 2 2 1 3 3 4 4 4 3 3 3 3 3 3 3 4 4 3 3 2 2 1 1 1 1 3 2 2 1 1 3 3 1 4 3 3 3 3 5 3 5 3 3 4 3 3 3 2 2 3 1 1 1 2 2 1 3 2 3 1 2 1 1 4 3 3 3 3 3 3 3 3 3 2 3 3 3 2 2 2 1 1 1 2 2 3 1 1 1 1 2 3 1 4 3 3 3 3 3 3 3 3 3 3 3 4 4 3 2 2 1 2 1 2 2 2 2 3 2 1 2 1 1 1 1 3 3 3 3 4 3 4 4 2 3 3 1 1 1 3 2 1 1 1 1 2 3 3 1 3 1 1 2 3 1 3 3 3 3 4 4 4 4 1 3 3 2 1 2 2 2 3 3 2 3 2 2 3 2 3 1 2 3 3 3 3 3 3 3 3 4 4 4 2 3 3 1 2 2 1 3 3 2 1 1 1 2 3 2 3 1 2 2 2 1 2 3 4 4 4 4 4 4 1 2 2 1 1 2 1 1 1 2 1 1 3 2 3 1 3 3 3 3 3 4 3 4 3 4 4 4 4 4 2 2 1 1 2 2 3 4 1 2 1 1 2 1 3 2 3 3 4 4 4 4 4 4 4 4 4 4 4 4 2 2 1 1 2 1 3 4 4 1 1 1 1 1 2 1 3 4 3 4 4 4 4 4 4 4 4 4 4 4 1 2 1 2 2 2 1 2 3 3 3 3 3 3 1 1 3 4 4 4 4 4 4 4 4 4 4 4 4 4 1 1 1 1 1 2 2 1 1 2 3 3 4 3 3 3 4 4 4 4 4 4 4 4 ~ 4 4 4 4 4

1 3 2 1 1 1 1 1 1 2 3 3 4 3 4 3 4 4 4 4 4 4 4 4 ~ 4 4 4 4 4 4 4 4 2 3 2 2 1

2 3 1 3 1

2 1 1 1 1

1 1 1 3 2

1 2 1 2 2

2 3 3 1 2

3 3 3 1 2

2 3 3 3 1

3 3 3 3 3

3 3 3 3 3

3 3 3 3 3

3 3 3 3 3

3 4 4 3 3

4 4 4 3 4

4 3 3 4 4

4 4 4 3 4

4 4 4 3 4

4 4 4 4 4

4 4 4 4 4

4 3 4 4 4

3 4 4 4 4

4 4 4 4 4

4 4 4 4 4

4 4 4 3 4

4 4 4 4 4

4 4 4 4 4

4 4 4 4 4

4 4 4 4 4

4 4 4 3 4

4 4 3 4 4

4 4 4 4 4

4 3 4 4 4

1 1 1 1 1 1 1 1 3 4 4 3 3 4 4 3 4 4 4 4 4 4 4 3 4 4 4 ~ 4 4 4 4 2 1 1 1 2 3 2 1 3 4 4 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 3 1 3 1 1 3 3 3 4 4 4 4 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 3 4

Fig. 3. Classification map for each 16 X 16 block in Fig. 2 using four energy classes with an equal number of blocks in each class. Class 1 represents the highest ac energy and Class4 represents the lowest.

Fig. 5. Reconstructedimageryusingtheclassesandbitassignments shown in Figs. 3 and 4, respectively. This represents Fig. 2 compressed t o 1.0 bits/pixel with no channel errors. The mean-square error is 27.1.

1756

IEEE TRANSACTIONS ON CO"UNICATIONS,

VOL. COM-29, NO. 12, DECEMBER 1981

Fig. 6. Effects of random channel errors with PE = 10-2 on the compression system operating at 1.0 bitlpixel. The mean-square error is

346.8.

A

C a l c u l a t et h e 4 boundary d i f f e r e n c ev e c t o r s - Eq. ( 2 ) . Take DCT o f e a c hv e c t o r Eq. ( 3 ) .

T F i n dt h e most l i k e l y row,column l o c a t i o nf o r a c o e f f i c i e n t e r r o r B t o h a v eo c c u r r e dT. e stth e c o e f f i c i e n t si nt h el o c a t i o n s f o r amp1 i t u d e s i m i l a r i t y E q s . (7) - (IO).

/

- No c o r r e c t a \bCl eh e c k e r rdoert e c t ecdo.n s i s t e n soicgfyn s L

E coefficients

L

Fig. 7.

4

I

dominance). and rowandcolumn coefficients

,//

t e sBt os t h pos i.t i ve

s t l m a t em a q n l t u a e and s i g n o f c o e f l . i ne r r o ru s i n gt h e selected

Onlyone test positive

B o t ht e s t s positive

Ne i t h e r row n o rc o l u m nt e s t is, p o s i t i v e

-

Eas.

( l l .) - (.~l 3 ) .

\

\

\

[Check f o r 3 oo uf t 4 b o u n d a r i e sc o n s i s t e n t wlthsingleerror

of location

I

/

Neqative .

/

- /

I

Eas. ( l 6 .) - ( 1 8.) .

\

\

\

AII t e s t s positive

I

Block diagram of error detection and correction process.

I

1757

MITCHELL AND TABATABAI: CHANNEL ERROR RECOVERY

B= [bi,i]NXN,i,j=O,l,...,N-l,canbe~ttenas B=A+&+S

(1)

where A = [ a i j ]N X

= block to be transmitted

Q=

= error due to transform coefficient

[ 4 i , i ] NX N

quantization

S = two-dimensional inverse discrete cosine transform of E

E=

[ei,j]NxN

e1 .9 1. = e

i=k,j=

e.I 1. = 0

otherwise

= error matrix due to the noisy channel. Fig. 8.

Notation for neighboring block pixel values.

The block size N X N is 16 X 16 for the examples we have shown. If the transmitted block is not on the boundary ofthe m = 0, N - 1 picture, then it is surrounded by four blocks represented, respectively, by matrices U,D,L , and R , as shown in Fig. 8. n = 0, - , N - 1. Form the four difference vectors denoted by Xi, i = 0, 1, 2 , 3 of sizeN as follows. Let us denote the first row of the above matrix by e-,

A

Xi = [xi,O

Y

2N

XO,k=aO,k-UN-l,k

(2)

x1 ,k = a N - 1 ,k - dO,k X2,k=ak,0 - Zk,N-l

This represents block A in Fig. 7. Denote the discrete cosine transform [5] of vectors Xi by Yi. [Xi]

i= 0,1,2,3.

s= [Sm,n]NXN

(4)

where 2e

N

cos

(2m + 1)kn 2N

cos

where

(3)

We claim that, due to the inherent continuity of most image data, the two vectors Y o , Y , will approximate the first and the last rowsof the matrix H that is obtained by taking a one-dimensional inverse discrete cosine transform along each column of the error matrix E . The basis picture (2-D inverse cosine transform) associated with the error matrix E has the form

Sm,n = -

where we have implicitly assumed the case k , 1 # 0. The onedimensional DCT of So will yield

k=O,.**,N- 1.

X g , k = a k , N--l lk , o

Yi = DCT

(2n + 1)Zn

2N

1

k,IfO

k,l=0

t,,j = 0

i f 1.

The vector To is the first row of matrix H discussed earlier. The vector Yo is our estimate of vector To. It also includes variations due to original image row differences and quantization noise. We shall assume that the channel error component dominates theother sources of variation in vector Yo. (If it does not dominate, the error cannot be detected and corrected.) Thus, a channel error will be manifested as a single large element in vector Y o . The location of this element is 1 which is the column of the nonzero element in matrix E. The magnitude of the large element in Yo is related to e as shown in (5). Similarly, vector T 1 is the last rowof matrix H , and error location and sizecanbe estimated from vector Y , which is our estimate of vector T , . Similar to ( 9 , we find

k=O,l#O

-N

e cos

1

( 2 N - 1)ln

x i , 2 , ’*’, x i , N - 1 1

(2m + 1)kn 2.N

k#O,l=O

=O

i f1.

1758

TRANSACTIONS IEEE

ON COMMUNICATIONS, VOL. COM-29, NO. 12, DECEMBER 1981

Note from ( 5 ) and (6) that the magnitude of is the same as that of t l I . This fact will be used in the algorithm described later [see (rij$o)] . Similarly,two the vectors Y 2 and Y , will approximate the first and last columns of the matrix G, which is obtained by taking a one-dimensional inversecosine transform along used to find each row of the error rowthe location matrix E. of Vectors the coefficient Y 2 andinYerror , canand be

w

Y

i = 0 , 1,2,3 k = 0 , l;*.,N-

1

no

I

y2

its magnitude. Once the transform of the difference vectors has been obtained as in (3), the most likely row and column locations for an error to have occurred can be determined. Since a single channel error will be manifested as a singlelarge magnitude component in each of the four Y{s, the dominant magnitude coefficient in each Yi is investigated. This dominant coefficient is labeled ai. Let

(7)

pj-1 ‘0,

+---1---

I +

yi,

y3

a3

n,

!I

Fig. 9.

Example locations of coefficients used in (7) and (8).

That is, the row and column locations having the best amplitude similarity are chosen as the potential error location. In addition to the above,an “outlier” test [9] -[11] is also performed to check whether either of the values (Yk, a, can be regarded as a dominant coefficient in their respective vectors Y k , Y , . The test performed is the following.

where n j is the value of k at which the maximum occurs.

6, = sample standard deviation of y i , i i = 1,3

i=O, l;-,N- 1. zi = min (vZi, v2i+l)

i= 0 , l .

(9)

Equations (7)-(9) locate the maximum component in each Yi and compare the magnitude of this with the magnitude of the component having the same location but on the opposite sideof thematrix (seeFig. 9). Equations (5) and (6) indicate that these magnitudes should be identical if they are caused by a single error. The smaller the values of zo and z 1 in (9), the closer the amplitude match. This corresponds to block B in Fig. 7. A threshold value X = 1.6 has been empirically chosen as a measure Of Of values’ zi)s, i = 0,1 satisfy the following condition: ziQX

i = O , 1,

‘f

neither

Of

the

(lo)

then we conclude that no correctable error has occurred in the block. This corresponds to block C in Fig. 7. On the other hand,if both s2; satisfy the condition of (lo), then we choose the values of (Yk, a, such that

Finally, fromthetheory of discrete cosine transforms, the relationships between the signsof (Yk and can be established, in case an error has occurred in the block. Hence,

CY,

(Yk(Ym

.

>0

k=0,m=2

CYkym - ,n,

>0

Yk-l,nkam

>O

Y k - ,n kY; - ,n,

k=O,m=3

,

(1 3)

k=l,m=2 = 1, =

in case of a detected error. Therefore, if (lo), (12), and (13) are satisfied, we conclude that an error has occurred at the n,th row and nkth column of the block of received coefficients and we proceed to block E of Fig. 7. The error matrix E can be estimated by using the theory of the discrete cosine transform: E=

[ij,j]NXN

i,j=O,l,-,N-

1

1759

MITCHELL AND TABATABAI: CHANNEL ERROR RECOVERY

+

I+ 4&

IYS-m,n,

I sgn (pm)

n,, nk #

o

and either (Yk or a2 forma dominant coefficient (12) in their respective vectors Y k , Y 2 , we conclude that an error been has

COS%

n, nk

detected, and the error matrixbe can

estimated as

0 #0

where Po = a0 01 = Yo,fl J 02

= a2

P3 = ~ z , n 3 .

The last case to be considered is when only one of the zi)s satisfies (10) (see block F in Fig. 7). This case occurs when errors are present in adjacent blocks, causing one of the error vectors of (2) to be unreliable. For the sake of simplicity let us assume that

x.

zo

+- a22

-

n2 f O nk

=0

4 i

On theotherhand, if (17), (18), and dominance tests (12) are true for a3 instead of a2,then we simply replace a2 by a3(-l)"k in (19). In case (15) is satisfied for z1 instead of zo, steps similar to the process discussed above can be taken.

Therefore, we can find nk for k = 0 or 1 such that 111. RESULTS

Now if

Shown in Fig. 10 are the results of applying the preceding algorithm to the image in Fig. 6 . Note the improvement in image quality and in mean-square error. At this high error rate there isanaverageof 2.5 channel errors per block. The algorithm assumes that only a single error is present per block. However, when multiple errors are present, the dominant error is still often detected and corrected. When the

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IEEE TRANSACTIONS COMMUNICATIONS,'VOL. ON COM-29,

NO. 12, DECEMBER 1981

Fig. 10. Results of applyingthe error correction algorithm to the image in Fig. 6. The mean-square error is reduced to 92.5.

Fig. 12. Results of applying the error correction algorithm tothe image in Fig. 11. The mean-square error is reduced to 38.0.

Fig. 11. Effects of random channel errors witli PE = 10-3 on the coding system operating at 1.0 bit/pixel. The mean-square error is 59.1.

Fig. 13.

block to be tested occurs ontheboundary of the image, zerosareused forthenonexistent vectors in (2).Even in t h i s case, the algorithm often corrects errors since only three of the four boundary difference vectors must be consistent with a single transform error. A lower error rate used is to produce the image in Fig. 11. The result ofapplying the error correctionalgorithm to this image is shown in Fig. 12. In order to make the errors and corrections morevisible, the difference inthe pictures between the original (Fig. 2) andFigs. 11 and 12 are shown in,Figs. 13 and 14, respectively. Note that manyof theerrors visible in Fig. 13 are not visible in Fig. 11 due to image variation. Many of these errors were corrected, as can be seen in Fig. 14. In a few situations

the processhas introduced new errors. On the average, the improvement in subjective quality andmean-square error is remarkable: This improvement .has been achieved with no additional information required from 'the transmitter such as error correction or overhead bits. Figs. 15-17 show,the effects of this processing when coding at 0.5 bits/pixel and with a channel error probability of At this higher compression, most of the energy is concentrated in the lower frequencies. This is indicated by the bit assignment tables in Fig. 15. Thus, the errors aremainlylow frequency errors. Thealgorithmoperates onthe reconstructed image of Fig. 16 without needing to know the bit assignments or compression rate involved. Table I shows a comparison of mean-square error perform-

Differencepicturebetweenthe original (Fig. 2) and Fig. 11. The effectsof channel errors are more visible.

1761

MITCHELL AND TABATABAI: CHANNEL ERROR RECOVERY

F‘ig. 16. Effectsofrandomchannelerrorswith

PE = 10-2 on the compression system operating at 0.5 bits/pixel. The mean-square: error is 341.4.

Fig:. 14. Difference picture between the original (Fig. 2) and Fig. 12. The corrections made by the algorithm are more noticeable.

class 1

class 3

a 5 4 3 3 2 2 2 1 1 1 1 0 0 0 0 5 4 3 3 3 2 2 2 1 1 1 0 0 0 0

4 4 3 3 3 2 2 1 1 1 0 0 0 0 0

4 3 3 3 3 2 2 1 1 1 0 0 0 0 0

3 3 3 3 2 2 1 1 1 1 0 0 0 0 0

3 3 3 2 2 2 1 1 1 1 0 0 0 0 0

2 2 1 1 1 0 0 0 0 0 0 2 2 1 13 12 02 01 01 01 01 00 2 2 1 1 0 0 0 0 0 0 0 2 2 1 12 01 01 01 01 01 00 00 2 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 i J 0 0 0 0 0 0 0 1 0 0 01 00 00 00 00 00 00 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

. 0 0 0 0 0 0

8 3 0 0 2 0 0 2 2 1 1 0 0 1 0 0 0 0 0 0

0 0 0 0

3 3 3 3 3 2 2 2 1 1 1 0 0 0 0 0

3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 0

2 2 0 1 0 1 1 1 0 0 0 0 0 0 0 0

2 2 0 1 0 1 1 1 0 0 0 0 0 0 0 0

1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0

u 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 u 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 d ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

class 4

:Lass 2

4 4 3 3 3 2 2 2 1 1 1 1 0 0 0 0

3 3 0 2 0 1 1 1 0 0 1 0 0 0 0 0

2 2 2 2 2 2 2 1 1 1 1 0 0 0 0 0

2 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0

2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 0

1 1 1 1 0 l l l 0 0 ! 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Fig5. 15. Bit allocationmaps

a ~ ~ o o o o o o o o o o o o o 2 1 0 0 0 0 0 0 0 0 0 0 o 0

1 0 0 0 0 0 0 0 0 0 0 0 o 0

0 0 0 0 0 0 0 0 0 0 0 0 o 0

0 0 0 0 0 0 0 0 0 0 0 0 o 0

0 0 0 0 0 0 0 0 0 0 0 0 o 0

0 0 0 0 0 0 0 0 0 0 0 0 o 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 o 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o u o o o o 0 0 0 0 0 0 0 0

F’ig. 17. Resultsofapplyingtheerrorcorrectionalgorithm

to the image in Fig. 17. The mean-square error is reduced to 115.4.

TABLE I MEAN-SQUARE ERROR BETWEEN ORIGINAL (FIG. 2) WHICH HAS A VARIANCE O F 1551 AND THE VARIOUS RECONSTRUCTED IMAGES

-

o o o o o o o o o o o o u o o o

t o achieveanaveragedatarateof bitslpixel.

Compression Rate

0.5

2 bits/pixel No e r r o r s 27.

PE = 0.01 Corrected

PE

= 0.001 59.

Corrected

1.0 b i t s / p i x e l

0.5 b i t s / p i x e l

,7.3

I

64.2

365.5

346.8 92.5

341.4

I

82.3 70.4

81.6 26.2 12.9

38.0

115.4

1762

IEEE TRANSACTIONS COMMUNICATIONS, ON

ance for the examplesshown aswellas other data rates and error probabilities. We have experimented with applying the algorithm several timesrecursively onthe samereceived picture. Repeated applications result in marginalimprovement in mean-square error. IV.CONCLUSIONS

The algorithm described has been shown to be an effective tool in combatingchannel errors in blocktransformcoded imagery, especially when no a priori information is available about the nature of the channel noise. The receiversimply usesimage continutity assumptionsandknowledgeof the type of errors introduced by incorrect transfop coefficients.

been made of the time required to perform these calculations, the numberofoperationsrequiredis onthe order ofthat required for the inverse transform imagereconstruction. REFERENCES [I]

0. R.Mitchell, S. C.Bass, E. J . Delp, and T. W . Goeddel, “Coding of aerialreconnaissanceimagesfortransmissionover noisy channels,”Rome AirDevelopment Center,Griffiss Air Force Base, NY, RADC-TR-28-210, Final Tech. Rep., Sept. 1978, NTIS Accession Number ADA 061539. S . C . Bass, E. J . Delp, T. W . Goeddel,A. [21 0. R.Mitchell, Tabatabai, and P. Stiling,“Improvements in someimagecompression techniques for aerial reconnaissance analysis,” Rome Air Development Center, Griffiss Air Force Base, NY, RADC-TR-7999, Final Tech. Rep., May 1979,NTISAccessionNumber ADA 071088. [3] 0. R. Mitchell, S . C . Bass, E. J. Delp, T . W . Goeddel, and T. S . Huang, ‘!Image coding for photo analysis,” SOC. Inform. Display, vol. 21, pp. 293-304,1980. [4] W. H. Chen and C. H. Smith, “Adaptive coding ofmonochrome and color images,” IEEE Trans.Commun., vol.COM-25,pp. 1285-1292, Nov. 1977. [5] N. Ahmed, T. Natarjan, and K. R . Rao, “Discrete cosine transform,” fEEETrans.Comput., vol.C-23, pp. 90-93, Jan. 1974.

VOL. COM-29, NO. 12, DECEMBER 1981

[6] W . H.Chen. C . H.Smith,and S . Fralick,“Afast computational algorithm for the discrete cosine transform,” f E E h Trans. Commun., vol.COM-25,pp. 1004-1009, Sept.1977. [7] S . Lin, An IntroductiontoError-CorrectingCoding. Englewood Cliffs, NJ: Prentice-Hall, 1970. G . Daut,“Combined source-channel [8] J. W. ModestinoandD. fEEETrans. Commun., vol.COM-27,pp. coding ofimages,” 1644-1659,Nov.1979. [9] A. Papoulis, Probability,Random . Variables and Stochastic Processes. New York:McGraw-Hill,1965. Technometrics. vol. 2, [IO] F. J. Anscombe,“Rejectionofoutliers,” no.. 2, pp. 123-147, 1960. [ I I ] F. E. Grubbs,“Detectingoutlyingobervations in samples,” Technometrics, vol. 1 1 , no. I . pp. 1-21, 1969.

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Engineering.Hespentthe s;mmers-of 1977and1979atWhiteSands Missile Range, where he familiarized himself with the problems of realtimevideotrackingandpatternrecognition.While atWSMR, he developed several texture and shape description algorithms. He is active in teaching, research, and consulting in the areas of image coding, texture analysis, shape description, pattern recognition, and signal processing. Dr. Mitchell is a member of Eta Kappa Nu, Tau Beta andPi, Sigma Xi.

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