Conclusions and Further Work

The qualitative results are given by the abundance images computed us-ing the endmember spectra found by both approaches. The abundance images obtained from the spectral unmixing with the GA-MI endmembers are shown in Figure 4.5. Figure 4.6 presents the abundance images obtained from the ES endmembers. We nd that the results are quite parallel: the abundance image of endmember #1 found by the GA-MI is similar to the one of endmember found by the Evolutionary Strategy, and the corresponding endmembers seem to be good detectors of vegetal cover including crops. Abundance image

#2 of GA-MI is similar to the abundance image #5 of the Evolution Strategy.

However, these images seem of little value. There are no interesting spatial structures discovered in these images. It seems that the corresponding end-member spectra are noise detectors. The abundance images and #4 of the GA-MI correspond to the abundance images #3 and #4 of the Evolutionary Strategy, respectively. The endmembers seem to correspond to detectors of articial constructs like buildings and roads.

An interesting result is that the GA-MI has obtained the same qualitative results with fewer endmembers than the ES. From a dimensionality reduction viewpoint, the GA-MI has been able to obtain a transformation of the im-age data from the original 14-dimensional space into a 4-dimensional space, preserving much of the qualitative information of the image and obtaining an optimized reconstruction of the original image. We assert that the qualitative information has been preserved because the main spatial features of the im-aged scene are detectable in the transformed images, and correspond to cover classes dened in the ground truth. We may say that GA-MI outperforms the ES in the sense of obtaining a more parsimonious representation of the image data.

We have proposed an Evolutionary Algorithm that uses the notion of morpho-logical independence and the Morphomorpho-logical Autoassociative Memory for test-ing it, for the task of hyperspectral image linear unmixtest-ing. Linear unmixtest-ing is both a detection process and a dimension reduction process. It is very efficient for the detection of small features, which can be blurred by other methods, like clustering-based unsupervised analysis of the images. The Evolutionary Algorithm ensures some degree of optimality of the endmembers extracted from the image, in the sense of minimum unmxing (and reconstruction) error.

We have found that our algorithm improves over an Evolutionary Strategy tailored to the problem, in the sense of the minimization of the proposed t-ness function. We have also found that the GA-MI algorithm spontaneously performs a selection of the appropriate number of endmembers. An added ap-peal of the approach proposed is that it uses the spectra found in the image.

Using the actually measured spectra may reduce the interpretation problems when trying to decide the actual physical materials present in the scene.

Acknowledgments

References

Further work must be addressed to the experimentation with other hyper-spectral images, and the extensive testing of the parameter sensitivities of the approach, keeping in mind that hyperspectral images are very large and need speedy analysis methods.

The authors received partial support from projects of the Ministerio de Cien-cia y Tecnologia MAT1999-1049-C03-03, 0739-C04-02, and TIC2000-0376-P4-04, and UPV/EHU 00140.226-TA-6872/1999 of the University of The Basque Country (UPV/EHU). The Center for Ecological Research and Forestry Applications (CREAF) of the UAB kindly provided the experimental multispectral image and corresponding ground truth.

1. Asano A., K. Matsumura, K. Itoh, Y. Ichioka, S. Yokozeki (1995) Optimization of morphological lters by learning, Optics Comm. 112 : 265-270

2. Bäck T., H.P. Schwefel (1993) An overview of Evolution Algorithms for parameter optimization. Evolutionary Computation, 1:1-24.

3. Bäck T., H.P. Schwefel (1996) Evolutionary computation: An overview. IEEE ICEC96, pp.20-29.

4. Th. Bäck (1996) Evolutionary Algorithms in Theory and Practice, Oxford Uni-versity Press, New York.

5. Carpenter G.A., S. Grossberg, D.B. Rosen (1991) Fuzzy ART: Fast stable learning of analog patterns by an adaptive resonance system, Neural Networks, 4:759-771, 6. Craig M., Minimum volume transformations for remotely sensed data, IEEE

Trans. Geos. Rem. Sensing, 32(3):542-552.

7. Gader P.D., M.A. Khabou, A. Kodobobsky (2000) Morphological regularization neural networks, Pattern Recognition, 33:935-944.

8. Goldberg D.F. (1989) Genetic Algorithms in Search, Optimization & Machine Learning, Addison-Wesley, Reading, MA.

9. Graña M., B. Raducanu (2001) On the application of morphological heteroasso-ciative neural networks. Proc. Intl. Conf. on Image Processing (ICIP), I. Pitas (ed.), pp. 501-504, Thessaloniki, Greece, October, IEEE Press.

10. Graña M., B. Raducanu, P. Sussner, G. Ritter (2002) On endmember detection in hyperspectral images with Morphological Associative Memories, Iberamia 2002, LNCS Springer-Verlag, in press.

11. Hopeld J.J. (1982) Neural networks and physical systems with emergent collec-tive computational abilities, Proc. Nat. Acad. Sciences, 79:2554-2558,

12. Ifarraguerri A., C.-I Chang (1999) Multispectral and hyperspectral image analysis with convex cones, IEEE Trans. Geos. Rem. Sensing, 37(2):756-770.

13. Keshava N., J.F. Mustard (2002) Spectral unimixing, IEEE Signal Proc. Mag.

19(1):44-57

14. Kohonen T., (1972) Correlation Matrix Memory, IEEE Trans. Computers, 21:353-359.

15. Pessoa L.F.C , P. Maragos, (1998) MRL-lters: a general class of nonlinear systems and their optimal design for image processing, IEEE Trans. on Image Processing, 7(7):966 -978,

16. Pessoa L.F.C , P. Maragos (2000) Neural networks with hybrid morphologi-cal/rank/linear nodes: A unifying framework with applications to handwritten character recognition, Patt. Rec. 33:945-960

17. Raducanu B., M. Graña, P. Sussner (2001) Morphological neural networks for vision based self-localization. Proc. of ICRA2001, Intl. Conf. on Robotics and Automation, pp. 2059-2064, Seoul, Korea, May, IEEE Press.

18. Rand R.S., D.M. Keenan (2001) A Spectral Mixture Process conditioned by Gibbs-based partitioning, IEEE Trans. Geos. Rem. Sensing, 39(7):1421-1434.

19. Ritter G.X., J.L. Diaz-de-Leon, P. Sussner. (1999) Morphological bidirectional associative memories. Neural Networks, 12:851-867.

20. Ritter G.X., P. Sussner, J.L. Diaz-de-Leon. (1998) Morphological associative memories. IEEE Trans. on Neural Networks, 9(2):281-292.

21. Ritter G.X., G. Urcid, L. Iancu (2002) Reconstruction of patterns from noisy in-puts using morphological associative memories, J. Math. Imag. Vision, submitted.

22. Ritter G.X., J.N. Wilson, Handbook of Computer Vision Algorithms in Image Algebra, CRC Press:Boca Raton, Fla.

23. Rizzi A.,M. ,F.M. Frattale Mascioli (2002) Adaptive resolution Min-Max classi-ers, IEEE Trans. Neural Networks 13(2):402-414.

24. Salembier P. (1992) Structuring element adaptation for morphological lters, J.

Visual Comm. Image Repres., 3:115-136.

25. Sussner P. (2001) Observations on Morphological Associative Memories and the Kernel Method, Proc. IJCNN2001, Washington, DC, July

26. Sussner P. (2002) , Generalizing operations of binary autoasso ciative morpholog-ical memories using fuzzy set theory, J. Math. Imag. Vision, submitted.

27. Won Y., P.D. Gader, P.C. Coffield (1997) Morphological shared-weight neural network with applications to automatic target detection, IEEE Trans. Neural Networks, 8(5):1195-1203.

28. Yang P.F., P. Maragos, (1995) Min-max classiers: Learnability, design and ap-plication, Patt. Rec., 28(6):879-899.

29. Zhang X.; C. Hang; S. Tan; PZ. Wang (1996) The min-max function differentiation and training of fuzzy neural networks, IEEE tTrans. Neural Networks 7(5):1139 -1150.

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On a Gradient-based Evolution Strategy