A Novel Method for Burst
Error Recovery of Images
First Author:
Second Author:
Affiliations:
E-mail:
S. Talebi
F. Marvasti
King’s College London
farokh.marvasti@kcl.ac.uk
• I. The Proposed Technique
• II. Simulation
• III. Noise Sensitivity
• IV. Conclusion
• V. Future Work
I. The Proposed Technique
• We use a new transform such that the kernel
of the transform is equal to
2
2
exp(  j  m  i  q1  j  n  k  q2 )
N
N
• 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
j 2im q1 
j 2kn q2
H ( si , pk )   ( si  exp(
))  ( pk  exp(
))
N
N
m 1
n 1



  ht , f  si t  pk  f
(1)
H ( sim , pn )  0 m, n 1,,
(2)
t 0 f 0
The inner summation is the SDFT of the missing
samples e(im,kn), hence


(3)
h

E
(


r

t
,


d

f
)

0
 t , f
t 0 f 0
The remaining values of E(r,d) can be found from (4) by the
following recursion


1
E ( r, d )  
  ht , f  E ( r  t , d  f )
h0,0 t 0 f 0
(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
aSDFTm,n  aDFTmod(q m, N ), mod(q n , N )
1
2
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.
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
1.0910-15.
b
a
c
e
d
f
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
ye1
x1
a
c
e
y
ye
y
b
d
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.