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Spectrochemical analysis

We address the problem of spectral unmixing when positivity and additivity constraints are imposed on the mixing coefficients. A hierarchical Bayesian model is introduced to satisfy these two constraints. A Gibbs sampler is then proposed to generate samples distributed according to the posterior distribution of the unknown parameters associated to this Bayesian model. Simulation results conducted with synthetic data illustrate the performance of the proposed algorithm. The accuracy of this approach is also illustrated by unmixing spectra resulting from a multicomponent chemical mixture analysis by infrared spectroscopy.

The spectral source separation procedure and the main results are detailed in the papers presented at the IEEE Workshop SSP 2007 and published in Signal Processing (EURASIP), respectively:

The MATLAB codes are available online:

Synthetic data

Simulation results conducted with synthetic data illustrate the performance of the proposed algorithm. The spectral sources used in the mixtures are simulated to get observed signals similar to real spectrometric data. Each spectrum is obtained as a superposition of Gaussian and Lorentzian shapes with different and randomly chosen parameters (location, amplitude and width). The mixing coefficients are also chosen to get evolution profiles similar to kinetic reactions. Figure 1 shows an example of M=3 source signals of L=1000 spectral bands. The abundance evolution profiles are simulated for N = 15 observation times. An i.i.d. Gaussian sequence is added to each observation to obtain a signal to noise ratio (SNR) equal to 20dB in each mixture.
Simulated spectral sources
Fig. 1. Example of simulated spectral sources.

Figure 2 summarizes the result of 100 Monte Carlo runs for which the mixing matrix is kept unchanged while new sources and noise sequences are generated at each simulation. Figure 2-a shows a comparison between the true concentrations and their minimum mean square error (MMSE) estimates. The proposed Bayesian model has also been implemented with exponential priors for the spectral sources . The results obtained with such priors are shown in Figure 2-b. It can be firstly noticed that the estimated mixing coefficients satisfy positivity and additivity constraints, whatever the source prior. Moreover, an improvement in the abundance estimation accuracy is noted when using an exponential source prior.
Mixing coefficients
Fig. 2. Simulated (cross) and estimated (circles) mixing coefficients with positive Gaussian (a) and exponential (b) priors on the sources.

Real data

To validate the algorithm with real data, an experiment has been performed on mixture data obtained from near infrared (NIR) spectroscopy measurements. Three known chemical species (cyclopentane, cyclohexane and n-pentane) are mixed experimentally with monitored concentrations. These species have been chosen for two main reasons: their available spectra in the NIR frequency bands are highly overlapping. In addition, as these species are inert solvents, they do not interact when they are mixed, which ensures that no new component appears. Figure 3 shows the pure spectra of the chemical species.
Real spectral sources
Fig. 1. Measured pure spectra of the mixed alcanes.

The results of estimation are reported in Fig. 4. It can be noticed that the abundance fraction profiles are estimated correctly, while a small difference between the actual abundances and the estimated ones can be noted. This is explained by the baseline that can be observed in the actual spectra in the spectral domain ranging from 2.4μm to 3.5μm.
Mixing coefficients
Fig. 2. True (cross) and estimated (circles) mixing coefficients using exponential priors on the sources.

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