7 SCHMIDT et al. : IMPLEMENTATION STRATEGIES FOR HYPERSPECTRAL UNMIXING... Algorithm Without CHO With CHO Time ratio BPSS 166 400 (OMEGA-5) 1 468 (OMEGA-7) 113.28 BPSS2 332 680 (OMEGA-6) 3 176 (OMEGA-8) 104.71 Run Id Image Algo pixel H$_{2}$O CO$_{2}$ dust selection OMEGA-1 41 1.CUT BPSS no 0.883 0.955 0.542 OMEGA-2 41 1.CUT BPSS2 no 0.823 0.958 0.980 OMEGA-3 41 1.CUT BPSS yes 0.956 0.951 0.766 OMEGA-4 41 1.CUT BPSS2 yes 0.894 0.910 0.975 OMEGA-5 41 1 BPSS no 0.773 0.957 0.555 OMEGA-6 41 1 BPSS2 no - 0.953 0.512 OMEGA-7 41 1 BPSS yes 0.940 0.953 0.372 OMEGA-8 41 1 BPSS2 yes 0.450 0.954 0.982 V. DISCUSSION AND CONCLUSION Fig. 3. Reference spectra of the OMEGA hyperspectral image 41 1: (i) in blue: synthetic H$_{2}$O ice with grain size of 100 microns, (ii) in red: synthetic CO$_{2}$ ice with grain size of 10 centimeters, (iii) in black: OMEGA typical dust materials with atmosphere absorption. Fig. 2. Sources estimated by BPSS2 (blue lines) and their spectral matches (red dotted lines), for an artificial dataset with 3 endmembers and 100 -times amplified OMEGA-like noise (no cutoff). TABLE VI RESULTS ON ALGORITHMS BPSS AND BPSS2 ON A PORTION OF OMEGA IMAGE (41 1.CUT) AND ON THE ENTIRE IMAGE (41 1). FOR 41 1.CUT, THE PROPORTION OF PIXELS WITH DETECTED CO 2 IS 48 . 72% AND RESPECTIVELY 63 . 48% FOR H $_{2}$O [39]. FOR 41 1, THE PROPORTION OF PIXELS FOR CO 2 AND H $_{2}$O IS 16 . 76% AND 21 . 84% . THE COLUMNS H$_{2}$O, CO$_{2}$ AND DUST INDICATE THE CORRELATION COEFFICIENT BETWEEN THE ESTIMATED SOURCES AND THE REFERENCE SPECTRA. (-) INDICATES THAT NO IDENTIFICATION OF H $_{2}$O NEITHER FROM SPECTRAL NOR SPATIAL RESULTS. THIS SOURCE HAS BEEN DETECTED TO BE CO 2 ICE (CORRELATION 0 . 911 ). TABLE V COMPUTATION TIMES IN SECONDS, FOR OMEGA 41 1 IMAGE WITH 3 ENDMEMBERS, FOR BOTH BPSS AND BPSS2, WITH AND WITHOUT CHO. IN THIS EXAMPLE, 670 PIXELS HAVE BEEN SELECTED WITH THE CHO, AMONG A TOTAL OF 111488 .THE NAME OF THE RUN OF TABLE IV-B3C IS NOTED IN PARENTHESIS. [39] and unsupervised classification [47], 3 endmembers have been detected: dust, CO$_{2}$ and water ice. The number of sources has been tuned to 3 in our study. The proportion of pixels containing CO$_{2}$ice and H$_{2}$O ice on the 41 1 image is estimated to be 16 . 76% and 21 . 84% , respectively [39]. The first 300 lines of the 41 1 image (subset named 41 1.CUT) contain all spectra containing ices. For this subset, the proportion of pixels with detected CO$_{2}$ and H$_{2}$O is 48 . 72% and 63 . 48% , respecively. for dataset 41 1 (run OMEGA-5 to OMEGA-8) in comparison with 41 1.CUT (run OMEGA-1 to OMEGA-4). This result suggests that both BPSS and BPSS2 are less efficient in a case of an asymmetric distribution of the sources. b) BPSS vs. BPSS2: The algorithm BPSS gives signifi- cantly better results than BPSS2 (for instance run OMEGA-3 vs OMEGA-4). This is due to non-linearity in the radiative transfer and noise in the dataset in contradiction with the full additivity constraint. c) Effect of the pixel selection: When the convex hull se- lection has been used as a pre-processing step to BPSS/BPSS2, the estimation is significantly better (see fig. 4, for run OMEGA-5 and fig. 5 for run OMEGA-7). These results show that pixel selection is a way to better take into account the occurrence of rare endmembers and thus is an interesting method to provide better results. 2) Performance: Computation times are about 100 times shorter when pixel selection by convex hull (CHO) has been performed as a preprocessing step (see Table V). 3) Accuracy: Table IV-B3c reports the results from differ- ent tests, each run is defined by a number. To estimate the quality of the estimation, the correlation between the reference spectra and the estimated sources has been computed. The attribution of each source has been done ad hoc using both spectral source and spatial abundances. a) Asymmetric abundances of the sources: The quality of estimation with both BPSS and BPSS2 is significantly lower For the first time, a MCMC-based blind source separation strategy with positivity and sum-to-one constraints has been effectively applied on a complete hyperspectral image with a typical size frequently encountered in Earth and Planetary Science. The optimization of BPSS [17] and BPSS2 [18] presented in this article consists of two independent parts: (i) Technical Optimization (TO) reduces the memory footprint, lowers the average cost of algorithmic operations, and makes smart re-use of memory (ii) Convex Hull Optimization (CHO) reduces the number of spectra to process.