A Novel Method for Burst Error Recovery of Images

First Author: S. Talebi Second Author: F. Marvasti Affiliations: King’s College London

E-mail: farokh.marvasti@kcl.ac.uk

• I. The Proposed Technique

• II. Simulation

• III. Noise Sensitivity

• IV. Conclusion

• V. Future Work

I. The Proposed Technique

)22

exp( 21 qknN

jqimN

j

• We use a new transform such that the kernel of the transform is equal to

• where q1 and q2 are positive prime integers with respect to N. It can be shown that this kernel is a sorted kernel of DFT.

The polynomial error locator is

1 1

21 ))2

exp(())2

exp((),(m n

nk

miki N

qkjp

N

qijspsH

0 0,

t f

fk

tift psh (1)

0),( npsHim ,,1, nm (2)

The inner summation is the SDFT of the missing samples e(im,kn), hence

0 0

, 0),(t f

ft fdtrEh (3)

The remaining values of E(r,d) can be found from (4) by the following recursion

0 0,

0,0

),(1

),(t f

ft fdtrEhh

drE (4)

The SDFT is actually derived from DFT and the fast algorithm can still be used. Because the SDFT transform can be handled by DFT and sorting; the sorting of the elements is as follows

for m,n =1,..,N that aSDFT is the element of the SDFT transform matrix and aDFT is the element of the DFT transform matrix.

),2mod(),,1mod(, NnqNmqnm DFTSDFT aa

Figure 1. The zeros are inserted around the original matrix Xorg

(m,n) to get the Xover (m,n)

( “ ” are the padded zeros and “ * “ are the elements of the Xorg

(m,n) ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

* * * * * * * * * * * * * * *

Rows of zeros inserted; 2 {

Columns of zeros inserted; 2

II. Simulation

• A (256 256) image is used for the simulation of the algorithm. A number of zeros are inserted in the rows and columns of the SDFT matrix and a new matrix of size of (512 512) is derived. The Mean Squared Error for this simulation is equal to 1.0910-15.

ab

c d

ef

III. Noise Sensitivity• III(a)- Quantization Noise:

• For our proposed method, the sensitivity of the algorithm is simulated as follows. The over-sampled image is quantized to 8 bits before the transmission. The corresponding SNRs before the transmission and after reconstruction of this simulation are equal to 38.2 dB and 35.02 dB, respectively.

III(b) Additive Noise

A white random noise of uniform distribution is added to the quantize transmitted image (SNR=28.6dB). The SNR after the recovery of the original image is equal to 28.193dB. The result for this case is shown in the next Fig. The SNR values show that the new method is robust against additive and quantization noise. This can be attributed to the low dynamic range of the coefficients ht,f ,which results in small amount of accumulated round off error in the solution of the difference equation (4).

x y

ye1 ye

x1 y

a b

c d

e f

IV. Conclusion

– We have shown that the new algorithm has three advantages:

– 1- It is ideal to recover the missing pixels for large blocks of bursty errors.

– 2- In terms of complexity, it is simpler than other techniques.

– 3- It is very robust in correcting bursts of errors with respect to additive and quantization noise.

– Disadvantage of this method is that the ratio between the recovery pixels and the added zeros is 1/3.

V. Future Work

– Application of this method in the case of compressed image will be consider next.

– The extension of the method to recover randomly distributed pixel losses is currently under investigation.