Negative amplitude of cross-correlation function in ICCS?

[Kavorka]

I am an undergrad working in biophysics research, and I was hoping you guys may have input on a simple but strange problem I'm facing. I don't know if this is appropriate for this forum, but it is not an especially specialized or complex problem and I am desperate.

I am currently conducting analysis on fluorescent images using image cross-correlation spectroscopy. I am trying to quantify the colocalization of two fluorescent tags in images of cells. Simply put, I have 2 pictures per image of the cell, one for each wavelength of light that each the fluorescent tag emits. They are simple black and white images 500x400, with pixels having certain intensities. In order to quantify how much the two fluorescent tags overlap, I am using this process which compares the two images and comes up with coefficients that quantify this overlap. The process uses a spatial correlation function:

r(ε,η) =
<(I₁(x,y) - <I₁>)>*<(I₂(x+ε,y+η) - <I₂>)>
<I₁>*<I₂>

I₁(x,y) is the intensity at each pixel (x,y) in image 1
<I₁> is the average intensity of all pixels in image 1
The brackets <> denote an average over all pixels, or ensemble average
(ε,η) are the spatial lag variables

This may seem complicated but its simple: When cross-correlating, you place image 2 directly on top of image 1 so there is no spatial lag and (ε,η) = (0,0), and run this function which gives you a negative or positive number indicating the correlation (positive if the two images correlate strongly negative if they don't). Image 2 is then shifted over by one pixel on top of image 1 and this function is run again, and again for every position (ε,η) image 2 can take on top of image 1.

When you use the correlation function on two images which are the same, this is auto-correlation and it gives a 2-D function which is ideally Gaussian, with a peak at the center point (0,0) (because the images are perfectly correlated at the center).

If you think about it and plug values into the function, an autocorrelation function can only take on positive values and a crosscorrelation function can take on either negative or positive values. If none of the fluorescent tag in image 1 colocalized with the tag in image 2, the crosscorrelation amplitude would be negative.

However, the theory of ICCS ultimately leads to this equation as well:

<N>_1/2 = r_1/2(0,0) / [r₁(0,0) * r₂(0,0)]

where <N>_1/2 is the average number of co-localized fluorescent particles per area
r_1/2(0,0) is the amplitude of the cross-correlation function between images 1 and 2
[r₁(0,0) is the amplitude of the auto-correlation function of image 1
[r₂(0,0) is the amplitude of the auto-correlation function of image 2

So this begs the question: if the cross-correlation amplitude can be positive or negative and the autocorrelation amplitudes have to be positive, how can this equation hold? <N>_1/2 must be positive.

This means that ICCS theory states that the cross-correlation peak must positive. But, in my analysis, I get negative cross-correlation peaks, which can be proven to be possible from the correlation function. The only conclusion I can reach is that the assumptions ICCS makes were violated in my analysis. I looked long and hard through papers but found that the number of pixels I was using was ideal for ICCS, and I can't find any assumptions of ICCS theory that leads to the seconds equation that I've broken.

I'm just wondering if there is something I haven't realized mathematically about this problem, and the positive and negative values, or if anyone here knows anything about this type of analysis. You can also see a succinct and exact explanation of what I'm doing in this paper: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1779921/
If you go to the Theory section, then the ICCS subsection.

 

[jambaugh]

From my cursory understanding, $r_{1,2}(0,0)$ is the intensity covariance between the two images divided by the mean intensities. This of course can be both positive or negative. then $r_i(0,0)$ is the variance of the i'th image divided by its mean intensity squared. So your $\langle N \rangle_{1,2}$ is the correlation coefficient (times each images mean intensity?) and so can be both positive or negative. I presume the intensities themselves are magnitudes so that's not an issue. In particular if image 1 is the negative image of image 2 you should get -1 (times the product of mean intensities?).

Your thinking seems correct to me and the claim that this colocalization average must be positive must then be invoking a physical constraint instead of a mathematical one. This is based on what I see here. I have no direct experience with this topic, only general statistics.

 

[Stephen Tashi]

What definition of "amplitude of a function" is being used? Perhaps that definition requires "amplitude" to be a non-negative number.

 

[Kavorka]

Using the absolute value of the amplitude (defined as the zero-lag (0,0) value which is theoretically the peak, but a Gaussian fit allows for a shift from zero-lag due to experimental error) was the first thing that occurred to me, but from studying the literature I've learned that they assume the peak is positive, they don't take the absolute value. If you took the absolute value it would like saying that a peak of 0.77 and a peak of -0.77 represent the same image correlation, but one is actually positively correlated and one is negatively.

Jambaugh, I believe you are correct in that there is a physical constraint, because much of this theory is making assumptions in terms of confocal microscopy and laser focal area overlap and such things, which leads me to think that maybe the only option is that the experiment itself didn't meet the assumptions of ICCS - perhaps the time between taking image 1 and image 2 at different wavelengths (20 seconds about) was too long and caused pixel decorrelation due to cellular movement and diffusion. I am working to improve the process, but still looking for ideas because I'm still not sure what the problem is.

 

[Andy Resnick]

The process uses a spatial correlation function:

r(ε,η) =
<(I₁(x,y) - <I₁>)>*<(I₂(x+ε,y+η) - <I₂>)>
<I₁>*<I₂>

<snip>

I'm not an expert in this method of analysis, but I wonder if your equation above should instead be:

r(ε,η) =
<(I₁(x,y)>*<(I₂(x+ε,y+η)>
<I₁>*<I₂>

That would seem to eliminate negative values and reproduce your other equation.

 

[Kavorka]

I'm not an expert in this method of analysis, but I wonder if your equation above should instead be:

r(ε,η) =
<(I₁(x,y)>*<(I₂(x+ε,y+η)>
<I₁>*<I₂>

That would seem to eliminate negative values and reproduce your other equation.

Unfortunately the terms in the numerator of the spatial correlation function are not just average intensities across all pixels, they are the average intensity fluctuation across all pixels which can be positive or negative, that I'm sure of.

 

[atyy]

The article you linked http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1779921/ says "If there is complete spatial overlap of the foci of the two laser beams and no quenching or fluorescence enhancement upon interaction of the two fluorophores ..." just before Eq (5).

Do you have a source with the derivation?