Image Restoration. Enhancement v.s. Restoration. Typical Degradation Sources. Enhancement vs. Restoration. Image Enhancement: Image Restoration:

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Ginger Curtis
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Image Retoration Retoration v.. Enhancement Image Denoiing Image Retoration Enhancement v.. Retoration Image Enhancement: A roce which aim to imrove bad image o they will look better.

1 Image Retoration Retoration v.. Enhancement Image Denoiing Image Retoration Enhancement v.. Retoration Image Enhancement: A roce which aim to imrove bad image o they will look better. Image Retoration: A roce which aim to invert known degradation oeration alied to image. Enhancement v. Retoration Tyical Degradation Source Better viual rereentation Subjective Remove effect of ening environment Objective Otical ditortion (geometric, blurring) Low Illumination No quantitative meaure Mathematical, model deendent quantitative meaure Senor ditortion (quantization, amling, enor noie, ectral Atmoheric attenuation enitivity, de-moaicing) (haze, turbulence, ) 1

Image Preroceing Degradation Model Enhancement Retoration f(x,y) h(x,y) Σ g(x,y) Satial Domain Sectral Domain Denoiing Invere filtering Wiener filtering n(x,y) Degradation Model: g = hf + n Point

2 Image Preroceing Degradation Model Enhancement Retoration f(x,y) h(x,y) Σ g(x,y) Satial Domain Sectral Domain Denoiing Invere filtering Wiener filtering n(x,y) Degradation Model: g = hf + n Point oeration Satial oeration Filtering So what i the roblem? Examle g f + n = H 1 f = H ( g n) The noie n i unknown: only it tatitical roertie can be learnt. Hazing The oeration H i tyically ingular or illoed.

Echo image Motion Blur Blurred image Blurred image +

Examle of Indeendent and identically ditributed (i.i.d) Noie Gauian white noie (i.

8 P 1 ( f g ) ( ) σ g f = e πσ The noie value i not known

3 Echo image Motion Blur Blurred image Blurred image + additive white noie Additive noie: Image Denoiing g = f + n Examle of Indeendent and identically ditributed (i.i.d) Noie Gauian white noie (i.i.d.):.8 P 1 ( f g ) ( ) σ g f = e πσ The noie value i not known but it characteritic are known: Parametric model Parameter (mean, variance,...) f σ= 3

Imule noie (S & P): Uniform noie: P ( g f ) 1 = ( b a) a ( g f ) otherwie b Pa for g = a P( g f ) = Pb for g = b 1 Pa Pb for g = f Note: thi noie i not additive!.1.8.5.4.6.4. -1-5 5 1 f b-a=.3.

4 Imule noie (S & P): Uniform noie: P ( g f ) 1 = ( b a) a ( g f ) otherwie b Pa for g = a P( g f ) = Pb for g = b 1 Pa Pb for g = f Note: thi noie i not additive! f b-a= Bayeian denoiing Aume the noie comonent i Gauian white : g=f + n where n i ditributed ~N(,σ) A MAP etimate for the original f: P ( ) ( g f ) P( f ) f ˆ = arg max P f g = arg max f f P( g) Uing Baye rule and taking the log likelihood : fˆ = arg min f {( g f ) + λr( f )} data term rior term where R(f) i a enalty for non robable f Examle 1: Prior Term Similarity of neighboring ixel ( f ) = W ( ) ( f f ) N W(-) i a Gauian rofile giving le weight to ditant ixel. R (f -f ) 4

5 R ( f ) = W ( ) ( f f ) N Thi lead to Gauian moothing: fˆ = ( 1 α ) g + α N W N ( ) W g ( ) Noiy Image Reduce noie but blur out edge. The arameter α deend on the noie variance. Filtered Image Examle : Prior Term Edge enitive imilarity: R ( f ) = W ( ) log( 1+ ( f f ) ) N log(1+(f -f ) ) 5

Middle: weight Right: filtered image W 1 i a monotonically decending atial

6 R ( f ) = W ( ) log( 1+ ( f f ) ) N Thi lead to edge-reerving moothing: fˆ = ( 1 α) g + α ( ) W ( g g ) W 1 N ( ) W ( g g ) W 1 N g Left: noiy image Middle: weight Right: filtered image W 1 i a monotonically decending atial weight W i a monotonically decending hotometric weight Noiy Image Filtered Image 6

7 Invere Filtering Degradation model: g(x,y) = h(x,y)f(x,y) Edge reerving moothing Gauian moothing G(u,v)=H(u,v) F(u,v) F(u,v)=G(u,v)/H(u,v) Invere Filtering (Cont.) Two roblem with the above formulation: 1. H(u,v) might be zero for ome (u,v).. In the reence of noie the noie might be amlified: F(u,v)=G(u,v)/H(u,v) + N(u,v)/H(u,v) Solution: Ue rior information Fˆ = arg min F ( HF G) + λr( F ) data term rior term Otion 1: Prior Term Ue enalty term that retrain high F value: where Solution: Fˆ Fˆ = arg min E E F ( F ) ( F ) = ( HF G) + λf ( F ) E F ( HF G) + λf = H = H G H H + λ = H >> 1 Fˆ = G H H << 1 Fˆ = 7

8 Degraded Image (echo) F=G/H Fˆ H G H H + λ = Degraded Image (echo+noie) 8

Fˆ H G H H + λ = The invere filter i C(H)= H /(H H+ λ) At ome range of (u,v): S(u,v)/N(u,v) < 1 noie amlification. 18 16 14 1 C(H) 1 8 6 λ=1-3 4-1 -.5.5 1 1.5 H Otion : Prior Term 1.

9 Fˆ H G H H + λ = The invere filter i C(H)= H /(H H+ λ) At ome range of (u,v): S(u,v)/N(u,v) < 1 noie amlification C(H) λ= H Otion : Prior Term 1. Natural image tend to have low energy at high frequencie. White noie tend to have contant energy along freq. where Fˆ = arg min E F ( F ) ( F ) = ( HF G) + λ( u v ) F E Solution: ( F ) P F = H ˆ H = H H + λ ( HF G) + λ( u + v ) F = Thi olution i known a the Wienner Filter Here we aume N(u,v) i contant. F ( u v ) G

10 Degraded Image (echo+noie) Wienner Filtering Wienner Previou Advanced Technique 1

11 Image inainting Image Denoiing (alt & eer) 11

12 Image Demoaicing Color Denoiing Image deblurring Zoom in 1

13 Hazing image Dehazed image 13

11.5 MAP Estimator MAP avoids this Computational Problem!

11.5 MAP Estimator MAP avoids this Computational Problem! .5 MAP timator ecall that the hit-or-mi cot function gave the MAP etimator it maimize the a oteriori PDF Q: Given that the MMS etimator i the mot natural one why would we conider the MAP etimator? A: If