laboratory of mathematical methods of image processing
DESCRIPTION
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 @ cs.msu.su ). Hong-Kong , November 2, 20 10. - PowerPoint PPT PresentationTRANSCRIPT
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Laboratory of Mathematical Methods of Image ProcessingFaculty of Computational Mathematics and Cybernetics
Moscow State University
Hong-Kong, November 2, 2010
Andrey S. Krylov( kryl @ cs.msu.su )
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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.
0
4 29
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nn
n i ˆ
n
xn
n
xn
n dx
ed
n
ex
)(
!2
)1()(
22 2/
B) They form a full orthonormal in system of functions.
A)),(2 L
sin 1
2
1 lim
cos 1
2
1 lim
124
24
xn
xn
xn
xn
nn
n
nn
n
The Hermite functions are defined as:
C)
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, )()(0
i
ii xcxF
dxxxfc ii )( )(
General form of expansion:
N
mmm
nN
mnx
m
N
mm
nn
xfxN
xHexfAc m
11
2/
1
)( )( 1
)( )( 1 2
)(
)()( 2
1
1
mN
mnm
nN
x
xx
where
and
mx are zeros of Hermite polynomial )(xH N
Fast implementation:
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nmmn
nm iF )(
m
ym
n
xn
mn
yxmn
nm dy
ed
dx
ed
mn
eyx
)()(
!!2
)1(),(
2222 2/2/
2D case
The graphs of the 2D Hermite functions:
),(0,0 yx ),(1,1 yx ),(2,2 yx
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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
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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
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Image filtering
Detail (increased)
Detail (increased)
Filtered image
Scanned image
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384512,3.1,4 xnrK
384512,3.1,8 xnrK
384512,5.1,4 xnrK
384512,2.1,16 xnrK
Original image
Hermitefoveation
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384512,3.1,4 xnrK
384512,3.1,8 xnrK
384512,5.1,4 xnrK
384512,2.1,16 xnrK
Original image
Hermitefoveation
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Texture sample
Standard coding
Hierarchical coding
Hierarchical coding without
subtractions A1
A2
A3
A4
B1
B2
B3
B4
C1
C2
C3
C4
Texture Texture ParameterizationParameterization
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Image segmentation taskImage segmentation task
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Information parameterization for image database retrieval
= +HF Hermite
component component LF Hermite
Informationused for identification
Normalized picture
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Image matching and identification results
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Iris biometry with hierarchical Hermite projection Iris biometry with hierarchical Hermite projection methodmethod
Iris normalization
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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 Iris biometry with hierarchical Hermite projection methodmethod
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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 Iris biometry with hierarchical Hermite projection method – Comparison stagemethod – Comparison stage
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Some ReferencesSome 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.