Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

Reference

Gabor Filter and Its Use in Fingerprint Technology Xu Dong Email: [email protected]

March 14, 2008

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Contents 1

Gabor Formula Concept Formula Mathematical Formula Engineering Formula

Application Examples

Reference

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Contents 1

2

Gabor Formula Concept Formula Mathematical Formula Engineering Formula Gabor Filter Features Space/Frequency Feature Biological Explanation

Application Examples

Reference

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Contents 1

2

3

Gabor Formula Concept Formula Mathematical Formula Engineering Formula Gabor Filter Features Space/Frequency Feature Biological Explanation Gabor Filter Design

Application Examples

Reference

Thank you

Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

Contents 1

2

3 4

Gabor Formula Concept Formula Mathematical Formula Engineering Formula Gabor Filter Features Space/Frequency Feature Biological Explanation Gabor Filter Design Application Examples Palmprint recognition Fingerprint orientation field computing Fingerprint enhancement Fingerprint prime line detection

Reference

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

Contents 1

2

3 4

5

Gabor Formula Concept Formula Mathematical Formula Engineering Formula Gabor Filter Features Space/Frequency Feature Biological Explanation Gabor Filter Design Application Examples Palmprint recognition Fingerprint orientation field computing Fingerprint enhancement Fingerprint prime line detection Reference

Reference

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

Contents 1

2

3 4

5 6

Gabor Formula Concept Formula Mathematical Formula Engineering Formula Gabor Filter Features Space/Frequency Feature Biological Explanation Gabor Filter Design Application Examples Palmprint recognition Fingerprint orientation field computing Fingerprint enhancement Fingerprint prime line detection Reference Thank you

Reference

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

Reference

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Concept Formula

Concept Formula

Concept Formula Definition (Gabor Filter) Gabor filter is a sinusoidal carrier multiplies a Gaussian envelop.

−u ) = s(→ −u ) × w (→ − g(→ r u)

(1.1)

Gabor Formula

Gabor Filter Features

Gabor Filter Design

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Mathematical Formula

Mathematical Formula Mathematical Formula (2D) Definition (Gabor Filter)

g(x, y) = exp{−π[(x − x0 )2 a2 + (y − y0 )2 b2 ]} × exp{−2πi[u0 (x − x0 ) + v0 (y − y0 )]}

(1.2)

Fourier Transform (2D) Definition (Gabor Filter Frequency Domain)

G(x, y) = exp{−π[(u − u0 )2 /a2 + (v − v0 )2 /b2 ]} × exp{−2πi[x0 (u − u0 ) + y0 (v − v0 )]}

(1.3)

Gabor Formula

Gabor Filter Features

Gabor Filter Design

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Engineering Formula

Engineering Formula

Engineering Formula Definition (Gabor Filter)

gx0 ,y0 ,f ,θ,ϕ (x, y) = exp[−(x02 + γ2 y02 )/(2δ)2 ] cos(2πfx0 + ϕ), x0 = (x − x0 ) cos θ − (y − y0 ) sin θ, y = (x − x0 ) sin θ + (y − y0 ) cos θ 0

(x0 , y0 ) is a orthogonal transform of (x, y).

(1.4)

Gabor Formula

Gabor Filter Features

Gabor Filter Design

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Space/Frequency Feature

Space/Frequency Feature Frequency/Orientation selective feature The orientation of the Gaussian is roughly equivalent to the orientation of the carrier. f , θ be frequency and orientation selected value.

Perfect resolution feature both in spacial and frequency domain Uncertainty principle. A nonzero function and its Fourier transform cannot both be sharply localized.

∆t∆f ≥

1 4π

(2.1)

Spacial/frequency bandwidth is defined to be the second moment of its energy distribution.

R∞

ff ∗ x2 dx R∞ (∆x)2 = −∞ ff ∗ dx −∞

(2.2)

Gabor Formula

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Gabor Filter Design

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Space/Frequency Feature

Space/Frequency Feature Perfect resolution feature both in spacial and frequency domain Uncertainty principle’s minimum value can be achieved if and only if the function is Gabor function. Proof : from the mathematical form of Gabor function, for any function in Gabor function bank, there is orthogonal transform that make x0 = y0 = 0, with no modify to its space/frequency bandwidth. Then we have,

1 √ , 2a π a ∆u = √ , 2 π

∆x =

1 √ 2b π b ∆v = √ 2 π

∆y =

(2.3)

then

∆x∆y∆u∆v = 1/(16π2 )

(2.4)

Thus Gabor filter is the perfect linear filter in both spacial and frequency domain performance.

Gabor Formula

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Gabor Filter Design

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Biological Explanation

Biological Explanation

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Gabor filter fit the transform function of striate cortex( , V1) single cell of mammal. List below is the experiment result on three simple cells in cat striate cortex.

ú™

Gabor Formula

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Biological Explanation

Biological Explanation

Simple-cell constraints on degrees of freedom in Gabor filter



Phase: adjacent pairs of simple cells are in quadrature( ) phase relation. This means simple cells have no preferred phase ϕ (cos or sin). Orientation bandwidth: about 40◦ . Spatial-frequency bandwidth: 1.2 − 1.5 octaves. ∆ω −1 Correlation on orientation-frequency bandwidth:∆θ ≈ 2λ 22∆ω +1

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

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Gabor Filter Design Form of Gabor function: Engineering view Take only the real part.

G0 (u, v) = Re[G(u, v)]

(3.1)

Add with conjugate amplitude.

G0 (u, v) = 1/2[G(u, v) + G(−u, −v)]

(3.2)

The two equations are consistent when u0 = v0 = 0, in which case the Gabor filter is degenerated to a Gaussian filter.

Gabor Formula

Gabor Filter Features

Gabor Filter Design

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Gabor Filter Design Values of freedom degrees. General idea. x0 , y0 , u0 , v0 can be 0 in 2D image processing. a and b are local scale rate, and a/b determines the Gaussian filter’s curvature of half-magnitude profile. f and θ determine the very shape of Gabor filter, see list below.

Featured in biological simulation: A non-tutorial method Seen application in texture segment, visual object sensor, etc.

Featured in enhancement of textures that its frequency and orientation is pre-defined: A tutorial method Palmprint/fingerprint identification/enhancement, texture segment, etc.

Gabor Formula

Gabor Filter Features

Gabor Filter Design

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Palmprint recognition

Palmprint recognition (Kong&Zhang03Palmprint, PR) Form of Gabor filter.

g(x, y) =

1 x 2 + y2 exp{− } exp{2πi(ux cos θ + uy sin θ)} (4.1) 2πδ2 2δ2

Degrees of freedom. Filter Type Lv1 (9*9) Lv2 (17*17) Lv3 (35*35) Simple cell

u 0.3666 0.1833 0.0916 -

δ 1.4045 2.8090 5.6179 -

∆F 0.5387 0.1332 0.0333 1.2-1.5

∆θ 1.0069 0.5035 0.2517 0.69

1/u 2.7278 5.4555 10.9170 -

Gabor Formula

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Fingerprint orientation field computing

Fingerprint orientation field computing (LinHong98Fingerprint, PAMI) Perform Gabor bank filter to get 8 directional filtered image. orientation bandwidth: 2.5 octaves. central frequency: 60 cycles/width.

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Gabor Formula

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Gabor Filter Design

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Fingerprint orientation field computing

Fingerprint orientation field computing (LinHong98Fingerprint, PAMI)

Find ridge in each filtered image.

Fusion ridges and compute genuine orientaiton field.

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Gabor Formula

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Fingerprint enhancement

Fingerprint orientation field computing (our algorithm)

A Gabor filter based enhancement algorithm. For each block of feature, calculate its ridge frequency f and orientation θ. Perform Gabor filter to each block with selected f and θ. A wide used algorithm.

Gabor Formula

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Gabor Filter Design

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Fingerprint prime line detection

Fingerprint prime line detection (My ongoing work) Problem overview. A new problem in our camera based capture system, comparing to the traditional sensor based devices.

Information (minutiae) beneath the prime line is useless, for most of fingers. The prime line must been detected to ensure the feature extraction result as well as speed up the extraction algorithm.

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

Reference

Fingerprint prime line detection

Fingerprint prime line detection (My ongoing work) Gabor filter is performed. size:50*50. σ:a = 10, b = 30. θ:90◦ . f :1/23.

Algorithm procedure. Normalize image→Frequency domain Gabor filter→ Normalize image→Binary filtered image→ Calculate column sum→Find sum peak.

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

Reference

Fingerprint prime line detection

Fingerprint prime line detection (My ongoing work) Some result.

Future work. Orientation problems. Extract genuine prime lines.

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

Reference

Reference

Denis Gabor, Theory of communication, J.Inst.Electr.Eng., 1946 John G. Daugman, Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters, J.Opt.Soc.Am.A, 1985 Peter Kruizinga and Nikolay Petkov, Nonlinear operator for oriented texture, IP, 1999 Javier R. Movellan, Tutorial on Gabor filters, (Network) Ferald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J.Four.App., 1997 And some palmprint and fingerprint papers.

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Gabor Formula

Gabor Filter Features

Gabor Filter Design

Application Examples

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You can’t predict the future, but you can invent it. –Denis Gabor.

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