1. Laboratory of Mathematical Methods of Image Processing Faculty of Computational Mathematics and Cybernetics Moscow State University Hong-Kong, November 2, 2010 Andrey S. Krylov ( kryl @ cs.msu.su )

2. Outline Motivation Hermite Projection Method Fast Hermite Projection Method Applications Image enhancement and analysis Iris recognition

3. Fourier transform is widely used in different areas off theoretical and applied science. The “frequency” concept is the basic tool for signal processing. Nevertheless the data is always given on a finite interval so we can not really process the data for a continuous Fourier transform based model. Reduction of the problem using DFT (and FFT) is not correct. The suggested Hermite projection method to reduce the problem enables to enhance the results using rough estimation of the data localization both in time and frequency domains.

4. The proposed methods is based on the features of eigenfunctions of the Fourier Transform - Hermite functions. An expansion of signal information into a series of these computationally localized functions enables to perform information analysis of the signal and its Fourier transform at the same time.

5. B) They form a full orthonormal in system of functions. A) The Hermite functions are defined as: C)

6. General form of expansion: where and are zeros of Hermite polynomial Fast implementation:

7. 2D case The graphs of the 2D Hermite functions:

8. Original image 2D decoded image by 45 Hermite functions at the first pass and 30 Hermite functions at the second pass Difference image (+50% intensity) Image filtering

9. Original image 2D decoded image by 90 Hermite functions at the first pass and 60 Hermite functions at the second pass Difference image (+50% intensity) Image filtering

10. Detail (increased) Filtered image Scanned image

11. Original image Hermite foveation

12. Original image Hermite foveation

13. TextureParameterization

14. Image segmentation task

15. Information parameterization for image database retrieval = + HF Hermite component component LF Hermite Information used for identification Normalized picture

16. Image matching and identification results

17. Iris biometry with hierarchical Hermite projection method Iris normalization

18. First level of the hierarchy: vertical OY mean value for all OX points is expanded into series of Hermite functions Second level of the hierarchy Forth level of the hierarchy Iris biometry with hierarchical Hermite projection method

19. l2 metrics for expansion coefficients vectors. Database image sorting is performed for all hierarchical levels. Cyclic shift of the normalized image to 3, 6, 9, 12, 15 pixels to the left and to the right to treat [ ‑ 10º, 10º] rotations. ~91% right results for CASIA-IrisV3 database ( the rest 9% were automatically omitted at the initial iris image quality check stage) Iris biometry with hierarchical Hermite projection method – Comparison stage

20. Some References A.S.Krylov, A.V.Vvedenskii “Software Package for Radial Distribution Function Calculation”//Journal of Non-Crystalline Solids, v. 192-193, 1995, p. 683-687. A.S.Krylov, A.V.Liakishev "Numerical Projection Method for Inverse Fourier type Transforms and its Application" // Numerical Functional Analysis and Optimization, v.21, 2000, No 1-2, p.205-216. D.N.Kortchagine, A.S.Krylov, “Projection Filtering in image processing,” //Proceedings of the International conference on the Computer Graphics and Vision (Graphicon 2000), pp. 42–45. L.A.Blagonravov, S.N.Skovorod’ko, A.S.Krylov A.S. et al. “Phase transition in liquid cesium near 590K”// Journal of Non-Crystalline Solids, v. 277, № 2/3, 2000, p. 182-187. A.S.Krylov, J.F.Poliakoff, M. Stockenhuber “An Hermite expansion method for EXAFS data treatment and its application to Fe K-edge spectra”//Phys. Chem. Chem. Phys., v.2, N 24, 2000, p. 5743-5749. A.S.Krylov, A.V.Kutovoi, Wee Kheng Leow "Texture Parameterization With Hermite Functions" // 12th Int. Conference Graphicon'2002, Conference proceedings, Russia, Nizhny Novgorod, 2002, p. 190-194. A.Krylov, D.Kortchagine "Hermite Foveation" // Proceedings of 14-th International Conference on Computer Graphics GraphiCon'2004, Moscow, Russia, September 2004., p. 166-169. A.Krylov, D.Korchagin "Fast Hermite Projection Method" // Lecture Notes in Computer Science, 2006, vol. 4141, p. 329-338. E.A.Pavelyeva, A.S.Krylov "An Adaptive Algorithm of Iris Image Key Points Detection" // Proceedings of GraphiCon'2010, Moscow, Russia, October 2010, pp. 320-323. S.Stankovic, I.Orovic, A.Krylov "Video Frames Reconstruction based on Time-Frequency Analysis and Hermite projection method" // EURASIP J. on Adv. in Signal Proc., Vol. 2010, ID 970105, 11 p., 2010. S.Stankovic, I.Orovic, A.Krylov "The Two-Dimensional Hermite S-method for High Resolution ISAR Imaging Applications" // IET Signal Processing, Vol. 4, No. 4, August 2010, pp.352-362.