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

Designing Perceptually-Based Image Filters in the Modulation Domain Joseph P. Havlicek School of Electrical and Computer Engineering University of Oklahoma 3 May 2011

Eigenfunctions of LTI System ◮ For any ω 0 ∈ R , the signal x ( t ) = e jω 0 t is an Eigenfunction of any 1-D continuous-time LTI system. h ( t ) ∗ x ( t ) = h ( t ) ∗ e jω 0 t y ( t ) = � LTI h ( τ ) e jω 0 ( t − τ ) dτ = x(t) y(t) R H � e jω 0 t h ( τ ) e jω 0 τ dτ = R � �� � a number x ( t ) = e jω 0 t H ( ω 0 ) e jω 0 t = | H ( ω 0 ) | e { jω 0 t +arg H ( ω 0 ) } =

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 X ( ω ) e jωt dt. 2 π R ◮ 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 H ( ω ) X ( ω ) e jωt dt. 2 π R • Each term is scaled by | H ( ω ) | and shifted by arg H ( ω ) .

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? G ( ej ω ) • attenuate additive noise 1 + δ p _ with a stationary spec- 1 δ p trum. Passband Stopband • in music: boost the δ s bass and attenuate the ω 0 ω p ω s 0 π midrange – related to Transition how human hearing per- band ceives the signal.

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

What About Human Vision? ◮ Are these Eigenfunctions ◮ Closely related to human visual perception of this?

167 167 167 167 Gabor Aspects of Mammalian Biological Vision (a) (a) ◮ Complex Gabor filter: ( ) ( ) ◮ Biologically motivated (a) (a) ( ) ( ) Gabor filter bank: (b) (b) (d) (d) (b) (b) (d) (d) Figure 4.6: Spa e domain represen tation of Filter 10. (a) Real part plotted as Figure 4.6: Spa e domain represen tation of Filter 10. (a) Real part plotted as Figure 4.6: Spa e Figure domain 4.6: represen Spa e domain tation represen of Filter tation 10. (a) of Real Filter part 10. plotted (a) Real as part plotted as a surfa e. (b) Real part depi ted as a gra y s ale image. ( ) Imaginary part a surfa e. (b) Real part depi ted as a gra y s ale image. ( ) Imaginary part a surfa e. (b) a Real surfa e. plotted part as a depi ted (b) surfa e. Real (d) as part Imaginary a gra depi ted y s ale part depi ted as image. a gra as y a ( ) s ale gra Imaginary y s ale image. image. ( ) part Imaginary part plotted as a surfa e. (d) Imaginary part depi ted as a gra y s ale image. plotted as a surfa e. plotted (d) as Imaginary a surfa e. (d) part Imaginary depi ted as part a gra depi ted y s ale as image. a gra y s ale image. of the individual parameters app ear in T able 4.3. The real and imaginary of the individual parameters app ear in T able 4.3. The real and imaginary omp onen ts of the unit-pulse resp onse of �lter 10 are depi ted in Figure 4.6, omp onen ts of the unit-pulse resp onse of �lter 10 are depi ted in Figure 4.6, of the individual of the parameters individual app parameters ear in T able app 4.3. ear in The T able real 4.3. and imaginary The real and imaginary b oth as surfa es and as grey s ale images. b oth as surfa es and as grey s ale images. omp onen ts of omp the unit-pulse onen ts of the resp unit-pulse onse of �lter resp 10 onse are of depi ted �lter 10 in are Figure depi ted 4.6, in Figure 4.6, b oth as surfa es b oth and as as surfa es grey s ale and images. as grey s ale images. 4.5 The Dominan t Comp onen t P aradigm 4.5 The Dominan t Comp onen t P aradigm The ob je tiv e of dominan t omp onen t analysis is to estimate, at ea h p oin t in The ob je tiv e of dominan t omp onen t analysis is to estimate, at ea h p oin t in a m ulti- omp onen t image, the v alues of the mo dulating fun tions of the om- a m ulti- omp onen t image, the v alues of the mo dulating fun tions of the om- 4.5 The Dominan 4.5 The t Dominan Comp onen t Comp t P aradigm onen t P aradigm p onen t that dominates the lo al image sp e trum at that p oin t. The dominan t p onen t that dominates the lo al image sp e trum at that p oin t. The dominan t The ob je tiv e of The dominan ob je tiv t omp e of dominan onen t analysis t omp onen is to t estimate, analysis is at to ea estimate, h p oin t in at ea h p oin t in omp onen t amplitude estimates ma y b e in terpreted as ontr ast . Lik e the fre- omp onen t amplitude estimates ma y b e in terpreted as ontr ast . Lik e the fre- a m ulti- omp onen a m t ulti- omp image, the onen v alues t image, of the the mo v alues dulating of the fun tions mo dulating of the fun tions om- of the om- p onen t that dominates p onen t that the dominates lo al image the sp e trum lo al image at that sp e trum p oin t. The at that dominan p oin t. t The dominan t omp onen t amplitude omp onen estimates t amplitude ma y estimates b e in terpreted ma y b as e in ontr terpreted ast . Lik as e the ontr fre- ast . Lik e the fre-

Modulation Domain Signal Representation ◮ A nonstationary image component: t k ( x ) = a k ( x ) exp[ jϕ k ( x )] . ◮ Modulation domain signal model: K K � � t ( x ) = t k ( x ) = a k ( x ) exp[ jϕ k ( x )] . k =1 k =1 ◮ AM: a k ( x ) • local texture contrast. ◮ FM: ∇ ϕ k • local texture orientation and granuliarity.

AM-FM Signal Components ◮ Image: ◮ Steerable Pyramid: ◮ AM-FM Components: (d)

Nonlinear Demodulation Algorithm ◮ AM-FM image component: t k ( x ) = a k ( x ) exp[ jϕ k ( x )] ◮ Interpolate t k ( x ) with a cubic tensor product spline. ◮ FM: local texture orientation and granuliarity: � ∇ t k ( x ) � ∇ ϕ k ( x ) = Re jt k ( x ) ◮ AM: local texture contrast: a k ( x ) = | t k ( x ) |

✩✪ ✏✑ ✘✙ ✣✤ ✥✦ ✧★ ✫✬ ✖✗ ✔✕ ✒✓ ✍✎ ✜✢ ☞✌ ✡☛ ✟✠ ✝✞ ☎✆ ✂✄ �✁ ✭✮ ✚✛ Modulation Domain Signal Processing a 1 G DEMOD 1 ϕ 1 SIGNAL PROCESSING RECONSTRUCTION a 2 G DEMOD 2 ϕ 2 t y a 3 G DEMOD 3 ϕ 3 a K G DEMOD K ϕ K

Orientation Selective Attenuation • Best LTI filtering result: • Modulation domain filtering result:

FM Processing Examples • Least squares phase reconstruction: • Spline-based phase reconstruction:

Lena Example

Barbara Example