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 )

Outline• Motivation• Hermite Projection Method• Fast Hermite Projection Method• Applications

•Image enhancement and analysis •Iris recognition

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.

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

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)

, )()(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:

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

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

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

Image filtering

Detail (increased)

Detail (increased)

Filtered image

Scanned image

384512,3.1,4 xnrK

384512,3.1,8 xnrK

384512,5.1,4 xnrK

384512,2.1,16 xnrK

Original image

Hermitefoveation

384512,3.1,4 xnrK

384512,3.1,8 xnrK

384512,5.1,4 xnrK

384512,2.1,16 xnrK

Original image

Hermitefoveation

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

Image segmentation taskImage segmentation task

 

Information parameterization for image database retrieval

= +HF Hermite

component component LF Hermite

Informationused for identification

Normalized picture

Image matching and identification results

Iris biometry with hierarchical Hermite projection Iris biometry with hierarchical Hermite projection methodmethod

Iris normalization

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

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

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.