An python implementaion of

**"Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation"  
by Gillis. (2014), doi : 10.1137/130946782  

**"Using Separable Nonnegative Matrix Factorization Techniques for the Analysis of Time-Resolved Raman Spectra"  
by Luce et al. (2016), doi : 10.1177/0003702816662600**

This code is revised from the matlab code provided by the author.

(matlab code : https://sites.google.com/site/nicolasgillis/code)

For fundamental knowledge of time-resolved spectrum, refer :
https://en.wikipedia.org/wiki/Time-resolved_spectroscopy

Given a time-resolved sepctrum M ∈ Rm×n+, (m : time slices, n : pixels)
we aimed to factorized M = W * H, where
W ∈ Rm×r+ is the temporal profiles of the species involved in the reaction, (kinetics)
and H ∈ Rr×n+ is their hyperspectra. (ex : absorption spectra, emission spectrum, fluorescence spectra, etc.)
The decomposition can be done by the nonnegative matrix factorization techniques.

Since the general NMF problem is a difficult problem (ill-posed, NP hard),
it is more practical to solve the "separable NMF" problem, which gives an approximated solution.
For details of NMF and separable-NMF, refer the Luce's work linked above.

A matrix M is r-separable (near-separable) if there exist an index set K of cardinality r and a nonnegative matrix H ∈ Rr×n+ with M = M(:,K) H. That is, M can be spanned by a convex hull formed by M(:,K).  

If you are confused by the idea of convex hull, imaging a r-dimensional space D ∈ Rr, where each axes correspond to M(:,K). The nonnegative H indicates that all elements of M has to live in the subspace D ∈ Rr+, which results a convex hull. ex : for r = 3, the covex hull is the first quadrant.

A time-resolved sepctrum M is near-separable, if there exist a certain pixel that has only one hyperspectrum for all time slices.   Hence, the approximation of separable NMF highly depends on the overlapping between hyperspectra.
i.e. The less hyperspectra overlap, the better approximation you get.

(1) Install libraries : numpy, scipy, matplotlib

(2) Run time_resolved_spectra_NMF.py 

An artificial time-resolved spectra is given M = W * H:
where

from top to bottom:

(1) H (hyperspectra from SNPA).

(2) spectra at different time slices.

(3) W(kinetics) and comparison of real ideal(real)-trace.