Designing Perceptually-Based Image Filters in the Modulation Domain - - PowerPoint PPT Presentation

SLIDE 1

Designing Perceptually-Based Image Filters in the Modulation Domain

Joseph P. Havlicek School of Electrical and Computer Engineering University of Oklahoma 3 May 2011

SLIDE 2

Eigenfunctions of LTI System

◮ For any ω0 ∈ R, the signal x(t) = ejω0t is an Eigenfunction of any 1-D continuous-time LTI system.

H y(t) LTI x(t)

x(t) = ejω0t y(t) = h(t) ∗ x(t) = h(t) ∗ ejω0t =

• R

h(τ)ejω0(t−τ)dτ = ejω0t

• R

h(τ)ejω0τdτ

• a number

= H(ω0)ejω0t = |H(ω0)|e{jω0t+arg H(ω0)}

SLIDE 3

Representing Signals as Sums of Eigenfunctions

◮ We use the Fourier transform to write an arbitrary input x(t) as a sum of Eigenfunctions: x(t) = 1 2π

• R

X(ω)ejωtdt. ◮ The action of the LTI system H is that each term in the sum gets multiplied by a corresponding Eigenvalue H(ω): y(t) = 1 2π

• R

H(ω)X(ω)ejωtdt.

• Each term is scaled by |H(ω)| and shifted by arg H(ω).

SLIDE 4

The LTI Filter Design Problem

◮ Design the Eigenvalues H(ω), e.g., the frequency response, to achieve some desired signal processing goal. ◮ What kinds of problems is this approach good for?

• attenuate additive noise

with a stationary spec- trum.

• in

music: boost the bass and attenuate the midrange – related to how human hearing per- ceives the signal.

1 + δp 1 δp G(ejω) Transition band δs ωs ω π ωp Passband Stopband _

SLIDE 5

Human Auditory Perception

◮ But is human auditory perception really very closely related to the Eigenfunction representation?

SLIDE 6

What About Human Vision?

◮ Are these Eigenfunctions ◮ Closely related to human visual perception of this?

SLIDE 7

Gabor Aspects of Mammalian Biological Vision

◮ Complex Gabor filter:

167 (a) ( ) (b) (d) Figure 4.6: Spa e domain represen tation

• f

Filter 10. (a) Real part plotted as a surfa e. (b) Real part depi ted as a gra y s ale image. ( ) Imaginary part plotted as a surfa e. (d) Imaginary part depi ted as a gra y s ale image.

• f

the individual parameters app ear in T able 4.3. The real and imaginary

• mp
• nen

ts

• f

the unit-pulse resp

• nse
• f

lter 10 are depi ted in Figure 4.6, b

• th

as surfa es and as grey s ale images. 4.5 The Dominan t Comp

• nen

t P aradigm The

• b

je tiv e

• f

dominan t

• mp
• nen

t analysis is to estimate, at ea h p

• in

t in a m ulti- omp

• nen

t image, the v alues

• f

the mo dulating fun tions

• f

the

• m-

p

• nen

t that dominates the lo al image sp e trum at that p

• in

t. The dominan t

• mp
• nen

t amplitude estimates ma y b e in terpreted as

• ntr

ast. Lik e the fre- 167 (a) ( ) (b) (d) Figure 4.6: Spa e domain represen tation

• f

Filter 10. (a) Real part plotted as a surfa e. (b) Real part depi ted as a gra y s ale image. ( ) Imaginary part plotted as a surfa e. (d) Imaginary part depi ted as a gra y s ale image.

• f

the individual parameters app ear in T able 4.3. The real and imaginary

• mp
• nen

ts

• f

the unit-pulse resp

• nse
• f

lter 10 are depi ted in Figure 4.6, b

• th

as surfa es and as grey s ale images. 4.5 The Dominan t Comp

• nen

t P aradigm The

• b

je tiv e

• f

dominan t

• mp
• nen

t analysis is to estimate, at ea h p

• in

t in a m ulti- omp

• nen

t image, the v alues

• f

the mo dulating fun tions

• f

the

• m-

p

• nen

t that dominates the lo al image sp e trum at that p

• in

t. The dominan t

• mp
• nen

t amplitude estimates ma y b e in terpreted as

• ntr

ast. Lik e the fre- 167 (a) ( ) (b) (d) Figure 4.6: Spa e domain represen tation

• f

Filter 10. (a) Real part plotted as a surfa e. (b) Real part depi ted as a gra y s ale image. ( ) Imaginary part plotted as a surfa e. (d) Imaginary part depi ted as a gra y s ale image.

• f

the individual parameters app ear in T able 4.3. The real and imaginary

• mp
• nen

ts

• f

the unit-pulse resp

• nse
• f

lter 10 are depi ted in Figure 4.6, b

• th

as surfa es and as grey s ale images. 4.5 The Dominan t Comp

• nen

t P aradigm The

• b

je tiv e

• f

dominan t

• mp
• nen

t analysis is to estimate, at ea h p

• in

t in a m ulti- omp

• nen

t image, the v alues

• f

the mo dulating fun tions

• f

the

• m-

p

• nen

t that dominates the lo al image sp e trum at that p

• in

t. The dominan t

• mp
• nen

t amplitude estimates ma y b e in terpreted as

• ntr

ast. Lik e the fre- 167 (a) ( ) (b) (d) Figure 4.6: Spa e domain represen tation

• f

Filter 10. (a) Real part plotted as a surfa e. (b) Real part depi ted as a gra y s ale image. ( ) Imaginary part plotted as a surfa e. (d) Imaginary part depi ted as a gra y s ale image.

• f

the individual parameters app ear in T able 4.3. The real and imaginary

• mp
• nen

ts

• f

the unit-pulse resp

• nse
• f

lter 10 are depi ted in Figure 4.6, b

• th

as surfa es and as grey s ale images. 4.5 The Dominan t Comp

• nen

t P aradigm The

• b

je tiv e

• f

dominan t

• mp
• nen

t analysis is to estimate, at ea h p

• in

t in a m ulti- omp

• nen

t image, the v alues

• f

the mo dulating fun tions

• f

the

• m-

p

• nen

t that dominates the lo al image sp e trum at that p

• in

t. The dominan t

• mp
• nen

t amplitude estimates ma y b e in terpreted as

• ntr

ast. Lik e the fre-

◮ Biologically motivated Gabor filter bank:

SLIDE 8

Modulation Domain Signal Representation

◮ A nonstationary image component: tk(x) = ak(x) exp[jϕk(x)]. ◮ Modulation domain signal model: t(x) =

K

• k=1

tk(x) =

K

• k=1

ak(x) exp[jϕk(x)]. ◮ AM: ak(x)

• local texture contrast.

◮ FM: ∇ϕk

• local texture orientation and granuliarity.

SLIDE 9

AM-FM Signal Components

◮ Steerable Pyramid: ◮ Image: ◮ AM-FM Components:

(d)

SLIDE 10

Nonlinear Demodulation Algorithm

◮ AM-FM image component: tk(x) = ak(x) exp[jϕk(x)] ◮ Interpolate tk(x) with a cubic tensor product spline. ◮ FM: local texture orientation and granuliarity: ∇ϕk(x) = Re ∇tk(x) jtk(x)

• ◮ AM: local texture contrast:

ak(x) = |tk(x)|

SLIDE 11

Modulation Domain Signal Processing

ϕ1 a1

✁ ✂✄ ☎✆ ✝✞ ✟✠ ✡☛ ☞✌ ✍✎ ✏✑ ✒✓ ✔✕ ✖✗

DEMOD DEMOD DEMOD DEMOD

✘✙ ✚✛ ✜✢ ✣✤ ✥✦ ✧★

SIGNAL PROCESSING

✩✪ ✫✬ ✭✮

a ϕ a ϕ a ϕ

2 2 3 3 K K

G G G G

1 2 3 K

RECONSTRUCTION

t y

SLIDE 12

Orientation Selective Attenuation

• Best LTI filtering result:
• Modulation domain filtering result:

SLIDE 13

FM Processing Examples

• Least squares phase reconstruction:
• Spline-based phase reconstruction:

SLIDE 14

Lena Example

SLIDE 15

Barbara Example