1
Lossy Compression of Images Corrupted by
Mixed Poisson and Additive Gaussian Noise
Lossy Compression of Images
Corrupted by Mixed Poisson and
Additive Gaussian Noise
Vladimir V. Lukina, Sergey S. Krivenkoa, Mikhail S. Zriakhova,
Nikolay N. Ponomarenkoa, Sergey K. Abramova,
Arto Kaarnab, Karen Egiazarianc
a National
Aerospace University, 61070, Kharkov, Ukraine;
b Lappeenranta University of Technology, Institute of Signal Processing,
P.O. Box-20, FIN-53851, Lappeenranta , Finland;
c Tampere University of Technology, Institute of Signal Processing,
P.O. Box-553, FIN-33101, Tampere, Finland
Vladimir Lukin
19/08/2009
2
Contents
Contents
• Introduction
• Signal and Noise Models
• Peculiarities of Lossy Compression of Noisy Images
• Similarities and Differences Between Transform Based Filtering and
Compression
• Quantitative Criteria
• Optimal Operation Point
• Problems and Ways of Reaching OOP in Practice
• Noise Removal Properties of Lossy Compression for Artificial Test Image
• Noise Removal Properties of Lossy Compression for Real-life Test Images
• Proposed Modified Procedure for Compressing Images Corrupted by
Signal-dependent Noise
• Conclusions
Vladimir Lukin
19/08/2009
3
Introduction
Applications: CCD color imaging systems, CCD multi- and hyper-spectral imaging systems
Goal: Analyzing main approaches to lossy compression with filtering effect of raw image data
corrupted by mixed Poisson and additive Gaussian noise
Reason: photon-counting image registration
principle
Poisson noise
CCD
matrix
Additive noise
Requirements (alternative):
1) essential compression ratios;
2) sufficient noise removal;
3) useful information preservation
Color
(multichannel)
image
Reason: instrumentation and ambient influences
Vladimir Lukin
Lossy compression techniques
Achievement of the Optimal
Operation Point (OOP)
19/08/2009
4
Signal and Noise Models
Gijn  GijP  nij , i  1,..., I , j  1,..., J
GijP defines an ij-th image pixel corrupted by Poisson noise with the true value
equal to Gij ;
Gij defines zero-mean additive Gaussian noise with variance  2 ;
I and J denote an image size.
This model simulates real life situation of noise in R, G, and B components of
color images under assumption that variance of fluctuations induced by
Poisson noise for majority of image pixels is larger than variance of additive
noise considered constant.
The model also relates to other optical and infrared sensors like those ones
applied in multi- and hyperspectral remote sensing imaging.
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Peculiarities of Lossy Compression
of Noisy Images
Why lossy (not lossless) compression?
1) Lossy compression is able to provide considerably larger CRs (compared to lossless coding)
without degrading image resolution and introducing disturbing artefacts;
2) A positive effect of image filtering can be observed due to lossy compression if introduced
losses mainly relate to noise removal and useful image content is preserved.
The RS (Helsinki region) image corrupted by
additive Gaussian noise with σ2 = 100
The decoded lossy compressed image
(bpp = 0.75)
Vladimir Lukin
19/08/2009
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Similarities and Differences Between Transform
Based Filtering and Compression
Similarity:
In both orthogonal based filtering and compression, an image is subject to
orthogonal transform applied either to entire image or locally, in blocks. Then,
orthogonal transform coefficients are quantized in the case of image compression or
thresholded if an image is denoised.
Difference I:
If hard thresholding is used, then for small amplitude coefficients that are assigned
zero values there is no difference between quantization and denoising. But for large
amplitude coefficients quantization used in lossy compression introduces losses in
information content. Due to this, filtering observed in lossy compression of noisy
images is always less efficient than denoising.
Difference II:
For improving performance of transform based denoising, a spatially invariant
approach is used. Such approach is not and cannot be employed in compression.
This is the second reason why filtering observed in lossy compression is less
efficient than denoising.
Vladimir Lukin
19/08/2009
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Quantitative Criteria
The standard measures to characterize a compressed image quality
I
J
- MSEor   (Gijd  Gijn )2 /( IJ  1), where Gijd is the decompressed image;
i 1 j 1
- PSNRor  10 log10 (2552 / MSEor ) - for 8 bits image representation.
Alternative measures to characterize a compressed image quality
I
J
- MSEnf   (Gijd  Gij )2 /( IJ  1), where Gij is the noise free image;
i 1 j 1
- PSNRnf  10log10 (2552 / MSEnf ).
It is more reasonable to characterize a compressed image quality by
quantitative measures calculated with respect to the corresponding
noise-free image (MSEnf, PSNRnf) rather than to the original noisy
one (MSEor, PSNRor).
Vladimir Lukin
19/08/2009
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Optimal Operation Point
Optimal operation point (OOP):
The argument of the curves MSEnf (CR), MSEnf (bpp) or MSEnf (QS) for which these
curves reach theirs minima have been called optimal operation point (OOP): CROOP ,
bppOOP or QSOOP..
σ2 MSE400
= 400
Main idea:
It is worth compressing a noisy image
in the neighborhood of OOP.
Main problem:
In practice, noise-free image is not at
disposal.
MSE50
σ2 = 50
250
200
MSEnf
OOP is observed and commonly occurs
to be more “obvious” for less complex
content images and/or for rather
intensive noise.
2 = 100
σMSE100
150
100
50
0
2
3
4
5
QSn OOP
6
7
QSn
8
9
10
11
QSn  QS 
Dependences MSEnf (QSn) for the noisy test
gray-scale image Lena for different additive
noise levels
Vladimir Lukin
19/08/2009
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Problems and Ways of Reaching OOP
in Practice
Case I: pure additive noise
Proposed procedure I: iteratively compressing/decompressing an image several times
JPEG2000, PSNRor
JPEG2000, PSNRnf
AGU, PSNRor
AGU, PSNRnf
33
31
PSNR, dB
with calculating of standard MSE between
original (noisy) and decompressed images.
Using the interpolation of the obtained
curve MSE(CR) (or MSE(bpp)) to
determine an estimate of CROOP or bppOOP
as such CR or bpp for which MSE was
equal to variance of noise in original
(noisy) image (a priori known or preestimated).*
29
27
25
23
* N.N. Ponomarenko, V.V. Lukin, M.S. Zriakhov,
and K. Egiazarian, “Lossy compression of
images with additive noise”, in Proc. Intern.
Conf. on Advanced Concepts for Intelligent
Vision Systems, Belgium, 2005, pp. 381-386.
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
bpp
Dependences of PSNRnf and PSNRor on bpp
for the test gray-scale image Lena in
conventional 8-bit representation for σ2=200
(PSNRor=25)
Vladimir Lukin
19/08/2009
10
Problems and Ways of Reaching OOP
in Practice
Case I: pure additive noise
Proposed procedure II: for coders with CR controlled by quantization step QS (standard
*http://www.ponomarenko.info/agu.htm and
http://www.ponomarenko.info/adct.htm
**N. Ponomarenko, V. Lukin, M. Zriakhov,
K. Egiazarian, and J. Astola, “Estimation of
accesible quality in noisy image
compression”,
in
CD-ROM
Proc.
EUSIPCO, Italy, 2006, 4 p.
σ2 MSE400
= 400
σ2MSE100
= 100
2 = 50
σMSE50
250
200
MSEnf
JPEG, AGU and ADCTC*, etc.).
For such coders non-iterative
procedure can be used. One has to
set QSOOP approximately equal to
4.5σ where σ is a standard deviation
of additive noise**.
150
100
50
0
2
3
4
5
QSn OOP
6
7
QSn
8
9
10
QSn  QS 
11
Dependences MSEnf (QSn) for the noisy test
gray-scale image Barbara for different
additive noise levels
Vladimir Lukin
19/08/2009
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Problems and Ways of Reaching OOP
in Practice
Case II: mixed additive and signal-dependent (multiplicative or Poisson) noise
Possible strategies:
1) To apply lossy compression directly to an original image.
Problem I: it is difficult to recommend a way of setting parameters of a coder to provide
compression in OOP neighbourhood.
Problem II: for multiplicative noise case more essential filtering effect of lossy
compression was mainly observed for image regions with relatively small local means
whilst for image regions with rather large local means noise was mainly not suppressed.
2) To apply a three-state compression.
At the first stage, the corresponding homomorphic transform is used, namely, of
logarithmic type for pure multiplicative noise or Anscombe transform for compressing
images corrupted by Poisson noise.
At the second stage, it becomes possible to apply known methods of compression.
At the third stage, decompressed images are subject to the corresponding inverse
homomorphic transform.
The question is what strategy is better?
Vladimir Lukin
19/08/2009
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Noise Removal Properties of Lossy
Compression for Artificial Test Image
Test image: Artificial image of size 512x512 pixels has 16 vertical strips of width 32
pixels. For each strip, the values are the same,
i.e. constant and equal to 20 (for the leftmost
strip), 30, 40,…, 170.
Peculiarities:
After simulating noise ( 2  10 ) the following
conditions have been satisfied:
Gij  3  2   P2 ij  0 and Gij  3  2   P2 ij  255
for any Gij  20,30,...,170 .
The strip width suits well to operation
principle of AGU coder that exploits just
32x32 pixel size of blocks. This allows
minimizing blocking artifacts.
2
For all strips  2  0.5 Pij
(prevailing influence
of signal-dependent noise for all strips and
entire image).
Vladimir Lukin
Artificial noisy test image
19/08/2009
Noise Removal Properties of Lossy
Compression for Artificial Test Image
(Strategy I: direct approach)
13
To analyze noise suppression, we have determined residual variance
for each l-th strip

2
res l

32( l 1)  30 512
  (G
i  32( l 1)  3 j 1
d
ij
 Gl ) 2 /(512  28),
where Gl is the is the true value for the l-th strip (equal to 10+10 l);
2
2
It is also possible to analyze ratios l   res l /(  Gl ) to study noise
suppression due to lossy compression quantitatively ( 1/  l shows
how many times noise variance has been reduced).
For the coders AGU and SPIHT, we have obtained dependences of  l
on l for several QS. The minimal QS was equal to 4.5  2  G1 whilst
the maximal QS was about 4.5  2  G16 .
Vladimir Lukin
19/08/2009
Noise Removal Properties of Lossy
Compression for Artificial Test Image
(Strategy I: direct approach)
Dependences of  l on l for different QS for
the coder AGU
28.46
42.69
Dependences of  l on l for different bpp for
the coder SPIHT
60.37
0.89
1.00
1.00
0.90
0.90
0.80
0.80
Suppression Ratios
Suppression Ratios
24.65
0.70
0.60
0.50
0.40
0.67
0.11
0.01
0.70
0.60
0.50
0.40
0.30
0.30
0.20
0.20
0.10
0.10
0.00
14
0.00
1
3
5
7
9
11
13
15
1
Strip Index
3
5
7
9
11
13
15
Strip Index
Preliminary conclusion: For rather large bpp (quite small CR), small variance of
residual noise is observed only for the leftmost strips (small l). If bpp becomes
smaller, noise suppression increases (  l reduces for all strips).
Vladimir Lukin
19/08/2009
Noise Removal Properties of Lossy
Compression for Artificial Test Image
(Strategy II: three-stage approach)
Direct Anscombe-like Transform
 255
GijA n  
 DBI
15
Inverse Anscombe-like Transform

Gijn 

Gijinv A
2
 DBI


 
GijAd  
 
 255
 

where DBI is the maximal value for a given image representation (e.g., 255 for 8 bits);
G Ad denotes the decompressed image;  defines rounding to the nearest integer.
Note that compression is applied to the image GijAn. Inverse transform is
carried out for an image GijAn after decompression. Small bias
introduced by the pair of Anscombe-like transforms is neglected.
For Poisson noise case, after applying the direct transform one gets
an image corrupted by pure additive noise with practically constant
variance  a2  0.25  2552 / DBI   64 .
The presence of additive noise component in the considered model,
although it is not predominant, changes the situation.
Vladimir Lukin
19/08/2009
Noise Removal Properties of Lossy
Compression for Artificial Test Image
(Strategy II: three-stage approach)
255
For an l-th strip GijAn 
Then if
GijAn 
16
Gij  yij , where yij  GijP  Gij  nij .
DBI
Gij    Gij one has
255
DBI
2
Gij  yij 
Variance is defined as 
one obtains
255
DBI
2
Anij
2
 Anij
Gij (1  yij / Gij ) 
255
DBI
Gij (1  yij / 2Gij ).
2552

 2y ; since  2y  Gij (1   2 / Gij ) ,
4DBI Gij
2552  2
2552


.
4DBI 4DBI Gij
This means that the image GijAn is corrupted by Gaussian noise with
zero mean and variance which is equal for all pixels of the same
strip but with variance slightly larger for strips with smaller l.
Vladimir Lukin
19/08/2009
Noise Removal Properties of Lossy
Compression for Artificial Test Image
(Strategy II: three-stage approach)
Dependences of  l on l for different QSA for
24
28
42
60
the coder AGU
Dependences of  l on l for different bpp for
0.66
0.34
0.009
0.007
the coder SPIHT
1.00
24
42
1.00
60
1.00
0.80
1.00
0.80
0.90
0.70
0.90
0.70
0.80
0.60
0.70
0.50
0.60
0.40
0.50
0.30
0.40
0.20
0.50
0.30
0.40
0.20
0.20
0.00
5
7
9
11
13
15
0.007
0.60
0.40
0.20
0.00
3
0.009
0.70
0.50
0.30
0.10
1
0.34
0.80
0.60
0.30
0.10
0.10
0.66
0.90
Suppression Ratios
Suppression Ratios
Suppression Ratios
Suppression Ratios
0.90
28
17
0.10
1
Strip Index
3
5
7
9
11
13
15
Strip Index
0.00
3
5
7
9
11
 res13 l of 15about
We can recommendStrip
toIndex
use QSA about 32…40 that 1produces
almost
constant
Strip Index
1…3 which is practically not seen in decompressed image (for the coder AGU).
It is possible to provide very efficient noise suppression in image homogeneous regions if
quantization step is set large enough or bpp is set small enough. If QS increases,
residual noise from signal-dependent transforms to almost additive.
0.00
1
3
5
7
9
11
13
15
Vladimir Lukin
2
19/08/2009
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Noise Removal Properties of Lossy
Compression for Real-life Test Images
Let us denote direct application of lossy compression, i.e., without the pair of
Anscombe-like transforms as DC (direct compression).
On the contrary, the compression procedure that exploits the Anscombe-like
transforms will be denoted as HBC (homomorphic based compression).
Real-life test image Airfield
Real-life test image Frisco
Vladimir Lukin
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Noise Removal Properties of Lossy
Compression for Real-life Test Images
PSNRnf (bpp ) for both strategies, the coders
PSNRnf (bpp ) for both strategies, the coders
AGU and SPIHT for the image Airfield
PSNRnf
SPIHT DC
AGU DC
SPIHT HBC
AGU HBC
AGU and SPIHT for the image Frisco
PSNRnf
32
SPIHT DC
AGU DC
SPIHT HBC
AGU HBC
27
31
26
30
25
29
24
28
23
27
22
21
26
0
1
2
3
4
5
bpp
6
0
1
2
3
4
5
bpp
6
All obtained curves have maxima. For the image Airfield these maxima appear themselves
less clearly than for the image Frisco.
Maximal values for the image Frisco are larger than for the image Airfield. This is
explained by less complex structure of information content for the image Frisco and
the presence of rather large quasi-homogeneous regions in it.
For more complex images curves maxima take place for larger bppOOP.
Vladimir Lukin
19/08/2009
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Noise Removal Properties of Lossy
Compression for Real-life Test Images
It is possible to recommend using the HBC procedure for both coders. For the
HBC procedure it is recommended to set fixed QSA about 35.
The noisy real-life test image Frisco
The compressed image (HBC strategy,
AGU coder with QSA=35)
Vladimir Lukin
19/08/2009
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Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise
Conclusion resulting from previous analysis: for efficient suppression of
noise it is enough to have a lossy coder quantization step approximately
equal to 4.5 standard deviations of noise in a given region.
Main idea: instead of performing homomorphic transformations, it seems
possible to set an appropriate individual QS for each particular image block if
noise standard deviation for this block is a priori known or can be preestimated.
Difficulties:
It might seem that the use of specific (not equal) quantization steps for each
block leads to necessity to save their values as side information at image
coding stage. But this problem can be avoided.
One thing we need before compressing an image is a priori known or pre2
estimated dependence of local variance  loc on local mean Gloc .
Vladimir Lukin
19/08/2009
Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise:
Coding Stage
22
The sequence of operations (for the AGU coder) performed for a given block:
1) Calculate DCT in a block and obtain DCT coefficients Dn,m , n  0,...,31; m  0,...,31 ;
2) Determine the block mean Gbl using D0,0 ; for example, for DCT of size 32x32 pixels
Gbl  D0,0 32;
3) Quantize Gbl using quantization step QSD0 =10: Gblq  round [Gbl /10] (the value 10 is de-
fined empirically in experiments);
rec
q
4) Reconstruct the block mean Gbl by multiplying Gbl by 10;
5) Calculate quantization step QSDCT for other DCT coefficients (other than D0,0 ) using
rec
2
known dependence  loc  Gloc  as QS DCT  k loc (Gbl ) where k is a parameter to be analyzed later (e.g., for the model of noise considered in this study QSDCT  k 10  Gblrec );
6) Quantize all DCT coefficients of the given block and pass them to further coding.
Vladimir Lukin
19/08/2009
Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise:
Decoding Stage
23
The sequence of operations (for the AGU coder) performed for a given block:
1) Reconstruct a given block mean by multiplying Gblq by 10: Gblrec  10Gblq ;
2) Reconstruct D0,0 ; for example, for 32x32 blocks D0,0  32Gblrec ;
3) Calculate quantization step QSDCT for other DCT coefficients taking into account that
QS DCT  k loc (Gblrec );
4) Reconstruct other than D0,0 DCT coefficients of the given block using the decoded
values and QSDCT for this block;
5) Carry out inverse DCT in the block.
Note: at the coding stage there is no need to code the values QSD0 and QSDCT for image
blocks. At decoding stage, they are calculated using decoded values Gblrec and known
dependence of local variance on local mean.
Vladimir Lukin
19/08/2009
Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise:
Post-filtering
24
Background:
The coder AGU can use post-processing of decompressed images.
Similar post-processing can be carried out for the proposed modification of
the AGU coder (further denoted as AGU-M).
Obtained results:
Test image
Airfield (bpp=0.70)
Frisco (bpp=0.28)
Without post-filtering
With post-filtering
PSNRnf  25.05 dB PSNRnf  27.42 dB
PSNRnf  28.92 dB PSNRnf  31.45 dB
Preliminary conclusion: The values of PSNRnf with post-filtering are better
(larger) than the corresponding maximal values for the coding procedures
considered earlier.
Vladimir Lukin
19/08/2009
Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise:
Pre-filtering
25
Background:
The quality of compressed images can be additionally improved if one uses
image lossy compression with k considerably smaller than 4.5 with further
post-filtering (for this strategy it was reasonable to set the parameter k ≈1.3).
Obtained results:
Image
Airfield
Frisco
bpp
2.80
2.41
k=1.0
with NPF
26.10 dB
26.70 dB
with PF
29.77 dB
33.51 dB
bpp
2.45
2.06
k=1.3
with NPF
25.88 dB
26.48 dB
with PF
29.68 dB
33.38 dB
Preliminary conclusions: As it is seen, PSNRnf for the case of post-filtering has
been improved. But this is reached by the expense of larger bpp, i.e., smaller
CR provided.
There is almost no difference in PSNRnf for k=1.0 and k=1.3. Then, it is
reasonable to use k=1.3 since in this case larger CR values are provided.
In practice, one has to decide what is of prime importance, larger PSNRnf or
larger CR.
Vladimir Lukin
19/08/2009
26
Conclusions
1. The task of compressing images corrupted by mixed Poisson and additive
Gaussian noise is considered. It is shown that different approaches to
compression are possible.
2. All approaches result in some noise suppression due to lossy compression,
i.e., to noise filtering. However, statistics of residual noise considerably
depends upon a compression procedure used.
3. It is demonstrated that more efficient ways are either to exploit root-square
transforms (the use of Anscombe transform and its modifications will be
considered in future) or to adjust coder parameters to statistical
characteristics of mixed noise.
4. It is possible to perform “careful” compression with small CR and then to
carry out post-filtering.
5. Image pre-filtering and lossy compression are possible as well.
6. Recommendations concerning parameter selection for the considered
approaches are presented.
Vladimir Lukin
19/08/2009