SNR estimation of a single image

[Winnok H. De Vos]

Dear all

In order to test robustness of an image processing algorithm, I want to
estimate the signal to noise ratio of my images. It concerns a very rough
approximation since I merely want to take Poisson noise into account, hence
making abstraction of detector noise, dark noise etc. (at least for now)
I want to be able to compare with synthetic images (originally without
noise) in which I gradually raise the Poisson noise. When considering the
original noiseless image as O, the current image as C and i the voxel number
I approximate the signal-noise ratio in these synthetic images using
(Manders et al., 1993):

SNR=20log(sqrt(sum(Ci)²/sum(Ci-Oi)²))

Since there is no original image for real images, this formula doesn't
apply. I have found several definitions of defining SNR, but none of them
seem to give values that correspond with what I retrieve after using the
first formula on the synthetic images. The most common way seems to be using
the standard deviation of a region with 'no' signal to estimate noise and
using the dynamic range or maximum of the signal. But this seems rather
arbitrary and dependent on the chosen region. Is there a way of
approximating the noise and SNR of a single image - so not by recording the
image several times - without having to select a signal-free ROI and what
would be the most reliable definition?
Many thanks in advance.

Kind regards,


winnok




______
ir. Winnok De Vos
Research Assistant

dep. Molecular Biotechnology
Faculty of Bioscience Engineering
Ghent University
Coupure links 653
9000 Ghent
Belgium

tel [phone]
fax [phone]
www.molecularbiotechnology.ugent.be

[Frederic V. Hessman]

> In order to test robustness of an image processing algorithm, I  
> want to estimate the signal to noise ratio of my images. It  
> concerns a very rough approximation since I merely want to take  
> Poisson noise into account, hence making abstraction of detector  
> noise, dark noise etc. (at least for now)
> I want to be able to compare with synthetic images (originally  
> without noise) in which I gradually raise the Poisson noise. When  
> considering the original noiseless image as O, the current image as  
> C and i the voxel number I approximate the signal-noise ratio in  
> these synthetic images using (Manders et al., 1993):
>
> SNR=20log(sqrt(sum(Ci)²/sum(Ci-Oi)²))
>
> Since there is no original image for real images, this formula  
> doesn't apply. I have found several definitions of defining SNR,  
> but none of them seem to give values that correspond with what I  
> retrieve after using the first formula on the synthetic images. The  
> most common way seems to be using the standard deviation of a  
> region with 'no' signal to estimate noise and using the dynamic  
> range or maximum of the signal. But this seems rather arbitrary and  
> dependent on the chosen region. Is there a way of approximating the  
> noise and SNR of a single image - so not by recording the image  
> several times - without having to select a signal-free ROI and what  
> would be the most reliable definition?
> Many thanks in advance.

Without knowing the gain g of the current image (the number of  
photoelectrons per camera intensity unit) there will be no point in  
doing Poisson statistics.  Ignoring detector read-out noise, bias,  
and dark current (as you said), the signal-to-noise is

        S/N = S/sqrt(S/g) = sqrt(gS)

(the conversion to db's is trivial but unnecessary, like expressing  
the exposure times in fortnights).  If you don't have the gain, you  
have no idea about the noise and will make a mistake of a factor of  
sqrt(g): for typical gains in the range of 2-5, this is not a small  
effect.  The gain is easy to measure: take a series of images with  
high intensities and measure the slope of the variance versus signal  
relation, which is the gain.

Calculating S/N from comparisons with synthetic images seems like a  
very dangerous route - lots of possibilities for systematic errors -  
especially since you should be able to compute the TRUE S/N from  
knowing what your images are about.  With no gain, I'd simply use the  
local standard deviations in regions with no image structure - at  
least they are measured from the data.

Rick

------------------------------------------------------------------------
------------------------
Dr. Frederic V. Hessman     [hidden email]
Institut für Astrophysik          Tel.  [phone]
Friedrich-Hund-Platz 1         Fax [phone]
37077 Goettingen                 Room F04-133
http://www.Astro.physik.Uni-Goettingen.de/~hessman
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MONET: a MOnitoring NEtwork of Telescopes
http://monet.Uni-Goettingen.de
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-------------------------

[Winnok H. De Vos]

Dear all

Thank you for your replies.
I had also thought of using deconvolved images as noise free images
(estimation of original), but decided not to since the deconvolution
'erodes' objects by the PSF, hence causing objects to be smaller and have
different intensities. It might be possible to normalize the latter, but I
fear the shape adjustment will cause a bias in noise calculation.
It seems the best way must be to use the relation of the detector gain (CCD
or PMT) with the photon flux.
But as I want to compare with the synthetic images which are independent of
detector characteristics, I would prefer a way to directly compare the
images purely from image variables.
A variance estimation over small neighbourhoods throughout the image might
roughly give an idea of noise, but then again might be skewed by border
regions?

Winnok


______
ir. Winnok De Vos
Research Assistant

dep. Molecular Biotechnology
Faculty of Bioscience Engineering
Ghent University
Coupure links 653
9000 Ghent
Belgium

tel [phone]
fax [phone]
www.molecularbiotechnology.ugent.be
----- Original Message -----
From: "Alessandro Esposito" <[hidden email]>
To: <[hidden email]>
Sent: Friday, July 13, 2007 11:05 AM
Subject: Re: SNR estimation of a single image


Search the CONFOCAL archive at
http://listserv.acsu.buffalo.edu/cgi-bin/wa?S1=confocal

It could be useful the following:

"The Role of Photon Statistics in Fluorescence Anisotropy Imaging" by Lidke
et al.

They use multiple images of the same object to compute a variance from which
you may infer the number of collected photons in the assumption that only
poissonian noise is present.

Cheers,

Alessandro
--
Dr. Alessandro Esposito
Laser Analytics Group - Department of Chemical Engineering
University Cambridge
CV: home.quantitative-microscopy.org
Web: wikiscope.org

[Frederic V. Hessman]

Just be careful : Lidke et al. suggest (page 1238)

"A defocused object is an easy and suitable test image. A background  
dark level is measured and subtracted from the series. To rule out  
fluctuations of the light source, one normalizes every image by its  
integrated intensity. The mean intensity and variance is computed per  
pixel over the series. A plot of the variance versus intensity yields  
a line with slope equal to the gain..."

This is basically what I suggested, but DON'T normalize the images  
before calculating the mean intensity and variance: if you do, you'll  
mess up your result (I'm not sure what Lidke et al. originally meant  
by this).  You don't have to worry about "fluctuations of the light  
source" anyway, since if the level changes, so will the Poisson noise.

Rick


On 13 Jul 2007, at 4:07 pm, winnok ugent wrote:

> Dear all
>
> Thank you for your replies.
> I had also thought of using deconvolved images as noise free images  
> (estimation of original), but decided not to since the  
> deconvolution 'erodes' objects by the PSF, hence causing objects to  
> be smaller and have different intensities. It might be possible to  
> normalize the latter, but I fear the shape adjustment will cause a  
> bias in noise calculation.
> It seems the best way must be to use the relation of the detector  
> gain (CCD or PMT) with the photon flux.
> But as I want to compare with the synthetic images which are  
> independent of detector characteristics, I would prefer a way to  
> directly compare the images purely from image variables.
> A variance estimation over small neighbourhoods throughout the  
> image might roughly give an idea of noise, but then again might be  
> skewed by border regions?
>
> Winnok
>
>
> ______
> ir. Winnok De Vos
> Research Assistant
>
> dep. Molecular Biotechnology
> Faculty of Bioscience Engineering
> Ghent University
> Coupure links 653
> 9000 Ghent
> Belgium
>
> tel [phone]
> fax [phone]
> www.molecularbiotechnology.ugent.be
> ----- Original Message ----- From: "Alessandro Esposito"  
> <[hidden email]>
> To: <[hidden email]>
> Sent: Friday, July 13, 2007 11:05 AM
> Subject: Re: SNR estimation of a single image
>
>
> Search the CONFOCAL archive at
> http://listserv.acsu.buffalo.edu/cgi-bin/wa?S1=confocal
>
> It could be useful the following:
>
> "The Role of Photon Statistics in Fluorescence Anisotropy Imaging"  
> by Lidke
> et al.
>
> They use multiple images of the same object to compute a variance  
> from which
> you may infer the number of collected photons in the assumption  
> that only
> poissonian noise is present.
>
> Cheers,
>
> Alessandro
> --
> Dr. Alessandro Esposito
> Laser Analytics Group - Department of Chemical Engineering
> University Cambridge
> CV: home.quantitative-microscopy.org
> Web: wikiscope.org

------------------------------------------------------------------------
------------------------
Dr. Frederic V. Hessman     [hidden email]
Institut für Astrophysik          Tel.  [phone]
Friedrich-Hund-Platz 1         Fax [phone]
37077 Goettingen                 Room F04-133
http://www.Astro.physik.Uni-Goettingen.de/~hessman
------------------------------------------------------------------------
-------------------------
MONET: a MOnitoring NEtwork of Telescopes
http://monet.Uni-Goettingen.de
------------------------------------------------------------------------
-------------------------