Lossy Compression of Images Corrupted by Mixed Poisson and Additive Gaussian Noise

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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
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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

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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

• Requirements (alternative)
• essential compression ratios
• sufficient noise removal
• useful information preservation

CCD matrix
Color (multichannel) image
Poisson noise
Lossy compression techniques
Additive noise
Reason instrumentation and ambient influences
Achievement of the Optimal Operation Point (OOP)
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Signal and Noise Models
defines an ij-th image pixel corrupted by Poisson
noise with the true value equal to
defines zero-mean additive Gaussian noise with
variance
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 Compressionof 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 s2 100
The decoded lossy compressed image (bpp 0.75)
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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.
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Quantitative Criteria
The standard measures to characterize a
compressed image quality
-
, where is the decompressed image
-
- for 8 bits image representation.
Alternative measures to characterize a compressed
image quality
-
, where is the noise free image
-
.
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).
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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..
OOP is observed and commonly occurs to be more
obvious for less complex content images and/or
for rather intensive noise.
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.
Dependences MSEnf (QSn) for the noisy test
gray-scale image Lena for different additive
noise levels
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Problems and Ways of Reaching OOPin Practice
Case I pure additive noise
Proposed procedure I iteratively
compressing/decompressing an image several times
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 pre-estimated).
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.
Dependences of PSNRnf and PSNRor on bpp for the
test gray-scale image Lena in conventional 8-bit
representation for s2200 (PSNRor25)
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Problems and Ways of Reaching OOPin Practice
Case I pure additive noise
Proposed procedure II for coders with CR
controlled by quantization step QS (standard
JPEG, AGU and ADCTC, etc.). For such coders
non-iterative procedure can be used. One has to
set QSOOP approximately equal to 4.5s where s is
a standard deviation of additive noise.
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.
Dependences MSEnf (QSn) for the noisy test
gray-scale image Barbara for different additive
noise levels
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Problems and Ways of Reaching OOPin Practice
Case II mixed additive and signal-dependent
(multiplicative or Poisson) noise
Possible strategies

• 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.
• 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?
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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 (
) the following conditions have been satisfied
and
for any
. The strip width suits well to operation
principle of AGU coder that exploits just 32x32
pixel size of blocks. This allows minimizing
blocking artifacts. For all strips
(prevailing influence of signal-dependent noise
for all strips and entire image).
Artificial noisy test image
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Noise Removal Properties of Lossy Compression for
Artificial Test Image(Strategy I direct
approach)
To analyze noise suppression, we have determined
residual variance for each l-th strip
where is the is the true value for the l-th
strip (equal to 1010 l)
It is also possible to analyze ratios
to study noise suppression due to lossy
compression quantitatively ( shows how many
times noise variance has been reduced).
For the coders AGU and SPIHT, we have obtained
dependences of on l for several QS. The minimal
QS was equal to whilst the
maximal QS was about .
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Noise Removal Properties of Lossy Compression for
Artificial Test Image(Strategy I direct
approach)
Dependences of on l for different QS for
the coder AGU
Dependences of on l for different bpp for
the coder SPIHT
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 ( reduces for all
strips).
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Noise Removal Properties of Lossy Compression for
Artificial Test Image(Strategy II three-stage
approach)
Direct Anscombe-like Transform
Inverse Anscombe-like Transform
where DBI is the maximal value for a given image
representation (e.g., 255 for 8 bits)
denotes the decompressed image defines
rounding to the nearest integer.
Note that compression is applied to the image
. Inverse transform is carried out for an image
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
. The
presence of additive noise component in the
considered model, although it is not predominant,
changes the situation.
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Noise Removal Properties of Lossy Compression for
Artificial Test Image(Strategy II three-stage
approach)
For an l-th strip
, where .
Then if one has
Variance is defined as
since , one
obtains
This means that the image 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.
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Noise Removal Properties of Lossy Compression for
Artificial Test Image(Strategy II three-stage
approach)
Dependences of on l for different QSA for
the coder AGU
Dependences of on l for different bpp for
the coder SPIHT
We can recommend to use QSA about 3240 that
produces almost constant of about 13 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.
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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
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Noise Removal Properties of Lossy Compression for
Real-life Test Images
for both strategies, the
coders AGU and SPIHT for the image Airfield
for both strategies, the
coders AGU and SPIHT for the image Frisco
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.
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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 compressed image (HBC strategy, AGU coder
with QSA35)
The noisy real-life test image Frisco
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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 pre-estimated.
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
pre-estimated dependence of local variance
on local mean .
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Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise
Coding Stage
The sequence of operations (for the AGU coder)
performed for a given block

• Calculate DCT in a block and obtain DCT
  coefficients
  
• Determine the block mean using for
  example, for DCT of size 32x32 pixels
• Quantize using quantization step QSD0 10
  (the value 10 is de-
• fined empirically in experiments)
• Reconstruct the block mean by multiplying
  by 10
• Calculate quantization step QSDCT for other DCT
  coefficients (other than ) using known
  dependence as
  where k is a parameter to be ana-lyzed later
  (e.g., for the model of noise considered in this
  study )
• Quantize all DCT coefficients of the given block
  and pass them to further coding.

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Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise
Decoding Stage
The sequence of operations (for the AGU coder)
performed for a given block

1. Reconstruct a given block mean by multiplying
   by 10
2. Reconstruct for example, for 32x32
   blocks
3. Calculate quantization step QSDCT for other DCT
   coefficients taking into account that
4. Reconstruct other than 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 and known dependence of local
variance on local mean.
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Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise
Post-filtering
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 Without post-filtering With post-filtering
Airfield (bpp0.70)
Frisco (bpp0.28)
Preliminary conclusion The values of PSNRnf with
post-filtering are better (larger) than the
corresponding maximal values for the coding
procedures considered earlier.
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Proposed Modified Procedure for Compressing
Images Corrupted by Signal-dependent Noise
Pre-filtering
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 k1.0 k1.0 k1.0 k1.3 k1.3 k1.3
Image bpp with NPF with PF bpp with NPF with PF
Airfield 2.80 26.10 dB 29.77 dB 2.45 25.88 dB 29.68 dB
Frisco 2.41 26.70 dB 33.51 dB 2.06 26.48 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 k1.0 and k1.3. Then,
it is reasonable to use k1.3 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.
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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.