Linear spectral unmixing of Fluorescence spectra

 Hey,

I am trying to write a program for linear spectral unmixing with known endmembers. I have a fluorescence spectrum and the subspectra of the endmembers (all separately measured). What I know want to do is numerically estimate the intensities of the subspectra. According to a least square approximation, with a linear spectral unmixinf model (S = A*S1 + B*S2 + C*S3....). I have 9 endmembers in the spectrum. As I have never done any numerical analysis, could someone point me in the right direction to start with?

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Let me wade in here. From your question, you have a measured aggregate spectrum, and on the side, measured components that you will assume of which the aggregate is composed. Since they are measured, they are NOT Gaussian components, which is often only a poor approximation to the shape of those components. (Gaussians are symmetric and they have a very specific shape.) And since you have them measured, there seems no reason to approximate them with Gaussians anyway.

 

So, given a "function", F, sampled at a set of discrete set of wavelengths. Thus you have the measured spectrum at a set of n wavelengths. At those same wavelengths, you have 9 separate components, I'll call them S_i. Actually, F is a discretely sampled function of wavelength, lambda, as are the components.

 

You now pose the mixture model for F,

 

F = a_1*S_1 + a_2*S_2 + a_3*S_3 + ... + a_9*S_9

Thus at any wavelength, the measured spectrum is presumed to be some (unknown) linear combination of the measured component sub-spectra. You wish to estimate the component fractions perhaps as a vector

A = [a_1; a_2; ... ; a_9]

Logically, the a_i will be constrained to be non-negative, an issue I'll discuss at some length below. I've defined A as a column vector because that is how most code would return it in MATLAB.

 

The simple approach to estimation of the mixture coefficients in A is to use a basic linear regression. Here we would minimize the sum of squares of the residuals for the mixture model. The simple solution to that is:

 

Assuming columns vectors for F and the S_i that are all the same lengths, then define the n by 9 matrix S where the columns of S are the 9 component subspectra.

 

S = [S_1,S_2,S_3,S_4,S_5,S_6,S_7,S_8,S_9];

Then if F is also a column vector of length n,

A = S\F;

This is a simple linear regression (not unlike that which regress would return), and it will work acceptably SOME of the time, but it will fail terribly on occasion, because it employs no non-negativity constraints on the coefficients in A.

 

The point is, a negative component makes no physical sense. You cannot have a negative amount of some sub-spectra in the mixture, yet the simple linear regression will probably yield exactly that. It will happen because you have some noise in the measurement process, because your measured spectra were not perfectly measured, or because you might have some contribution from something you have not actually measured (often described as lack-of-fit), or for a few other problems I'm forgetting to mention. The point is, it WILL happen.

 

A negative component here might indicate a serious problem in your data, or it might be just trash. So it is always a good thing to look at the coefficients you would generate, to look at the resulting fit. Plot the residuals. Is there significant lack of fit?

 

Anyway, a more logical and better solution is to use a non-negative least squares solution. MATLAB offers such a solver in the form of lsqnonneg.

 

 

A = lsqnonneg(S,F);

A will now be a vector with non-negative components, that yield the best possible solution, subject to non-negativity constraints. In fact, sometimes some of the components of A may have some TINY negative numbers in them, on the order of

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