Laboratory of Mathematical Methods of Image Processing

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Laboratory of Mathematical Methods of Image
Processing Faculty of Computational Mathematics
and Cybernetics Moscow State University
Numerical Hermite Projection Method in Fourier
Analysis and its Applications
Andrey S. Krylov ( kryl _at_ cs.msu.su )
Hong-Kong, November 2, 2010
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• Outline
• Motivation
• Hermite Projection Method
• Fast Hermite Projection Method
• Applications
• Image enhancement and analysis
• Iris recognition

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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.
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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.
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The Hermite functions are defined as
A)
B) They form a full orthonormal in
system of functions.
C)
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General form of expansion
Fast implementation
where
and
are zeros of Hermite polynomial
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2D case
The graphs of the 2D Hermite functions
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Image filtering
2D decoded image by 45 Hermite functions at the
first pass and 30 Hermite functions at the second
pass
Original image
Difference image (50 intensity)
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Image filtering
2D decoded image by 90 Hermite functions at the
first pass and 60 Hermite functions at the second
pass
Original image
Difference image (50 intensity)
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Image filtering
Scanned image
Detail (increased)
Detail (increased)
Filtered image
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Hermite foveation
Original image
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Hermite foveation
Original image
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Texture Parameterization
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Image segmentation task
 
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Information parameterization for image database
retrieval



HF Hermite component
component
LF Hermite
Normalized picture
Information used for identification
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Image matching and identification results
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Iris biometry with hierarchical Hermite
projection method

• Iris normalization

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Iris biometry with hierarchical Hermite
projection method

• 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
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Iris biometry with hierarchical Hermite
projection method Comparison stage

• 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)

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Some References

• A.S.Krylov, A.V.Vvedenskii Software Package for
  Radial Distribution Function Calculation// Journa
  l 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. 4245.
• L.A.Blagonravov, S.N.Skovorodko, 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.