Are these two methods identical to get the noise only image

Under the same circumstances, I get 15 X ray images I1, I2, I3, …,I15.
Then two methods were used to get the noise only image.

First method:
Noise image=(I1-I2)/sqrt(2). 
They said that subvided by sqrt(2) can maintain the variance of the Noise
image identical to I1 and I2. 

Second method:
Noise image=I1-(I1+I2+I3+…+I15)/15. Can this method maintain the variance
of noise image identical to original images?

If I don't consider the coefficient of sqrt(2), I think first method can
also be explained as
Noise image=I1-(I1+I2)/2.
The above equation is similar to second method.

Please give me some suggestion, Thank you.

On 11/19/2010 07:26 PM, tjucruiser wrote:
> Under the same circumstances, I get 15 X ray images I1, I2, I3, …,I15.
> Then two methods were used to get the noise only image.
>
> First method:
> Noise image=(I1-I2)/sqrt(2).
> They said that subvided by sqrt(2) can maintain the variance of the Noise
> image identical to I1 and I2.
>
> Second method:
> Noise image=I1-(I1+I2+I3+…+I15)/15. Can this method maintain the variance
> of noise image identical to original images?
>
> If I don't consider the coefficient of sqrt(2), I think first method can
> also be explained as
> Noise image=I1-(I1+I2)/2.
> The above equation is similar to second method.
>
> Please give me some suggestion, Thank you.

Assuming that the images are perfectly registered, of the same scale, 
has the same contrast & brightness, etc., etc.:

The first method gives you noise that's a combination of the noise in 
image 1 and the noise in image 2.  Assuming that the noise in image 1 
and image 2 have the same variance and is independent, then the variance 
of the noise you compute will be the same.  If the images start out with 
Gaussian noise then the noise you compute will have the same statistics 
-- but I doubt that the noise in an x-ray image will be Gaussian, given 
that you're looking at intensity.

The second method gives you a much closer approximation to the noise in 
I1, although it is still not exact.  When you calculate the average 
image you'll be reducing the variance significantly, but there will 
still be some, and the noise statistics will tend toward Gaussian. 
You'll also have a general whitening of the image, as I suspect that the 
noise is not zero-mean.  So you'll need to divide by something like 
sqrt(1 + 1/15), or some similar factor (I would have to do some math to 
know).  _And_, you won't have noise statistics that end up the same, 
unless you started with Gaussian noise.

Note that image registration, contrast, brightness, etc., problems, will 
sink this technique deeper than deep.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

On 11/19/2010 7:26 PM, tjucruiser wrote:
> Under the same circumstances, I get 15 X ray images I1, I2, I3, …,I15.
> Then two methods were used to get the noise only image.
>
> First method:
> Noise image=(I1-I2)/sqrt(2).
> They said that subvided by sqrt(2) can maintain the variance of the Noise
> image identical to I1 and I2.
>
> Second method:
> Noise image=I1-(I1+I2+I3+…+I15)/15. Can this method maintain the variance
> of noise image identical to original images?
>
> If I don't consider the coefficient of sqrt(2), I think first method can
> also be explained as
> Noise image=I1-(I1+I2)/2.
> The above equation is similar to second method.
>
> Please give me some suggestion, Thank you.
>
>

If you superimpose two equal random noises together then the resulting 
standard deviation (the root mean square value) increases as the sqrt of 
2.  So, maybe that's what "they" mean.

Note that these are average measurements.
Also note that since the noises are random and since we're using 
averages of one sort or another then "subtracting" makes little sense. 
That's why is said "superimpose" above.

But you can subtract the constant stuff - so subtraction makes sense - 
although I'd be careful of any systemic differences in scale from one 
image to another.  I know you said "under the same circumstances" but 
this would be a key point in having "the same circumstances".  Also, 
that the underlying images are equal spatially as well of course.

I don't know what a "noise image" is in the context of random noise - 
other than measuring some parameters of it.  Calling it an "image" 
connotes a degree of stability which eludes me.  But, I suppose it could 
be common language that I'm not familiar with.

Your second method was garbled in the text here.
Did you mean:
            N
I1 - (1/N)sum I(n)
           i=1
?

If all you're interested in is an "image" of noise then I guess the 
scale factor might not matter.  It depends a lot on what you're trying 
to do.

One finds: http://ip.com/patent/US7834918


Anyway, it appears that it's useful to do noise averaging in images to 
reduce the random noise.

So, I can imagine doing something like this:

       N
(1/N)sum I(n)
      i=1

Which is the average of a number of images.  If they are of the same 
subject then the noise would be averaged down.  If the noise is zero 
mean then I suppose eventually or theoretically, with enough images, one 
would see the noise mean at each pixel and that would be zero.  Somehow 
I have to think there would be a practical point of diminishing returns.

I've tried to figure out what your objective is here.

If you define a noise image by subtracting the average of a number of 
images (with the subject being constant and with presumed zero or 
reduced noise) from a single image then the result would be an estimate 
of the noise in that single image.
Then you could repeat the process for each image and get a statistic of 
the image noises.
Is that what you're trying to measure?
In other words, this would be a noise enhancement / subject suppression 
scheme, eh?

Surely one would not calculate a "noise image" and then subtract it from 
any image with the idea of reducing the noise.  That would only make 
things worse - even if the "noise image" were taken from the same set of 
images.

Fred