Image Restoration: The Problem and Basic Approaches

Transcription

1 1 Image Restoration: The Problem and Basic Approaches What is Restoration (vs. Enhancement): Enhancement Making pleasing images Oten no speciic model o the degradation Ad hoc procedures Restoration Undoing (inverting) and unwanted eect Model-based approach Optimality criteria 1

2 3 The Image Restoration Problem: ( h( g c ( + g( Ideal image Blur n( Measured image Noise g( ( h( n( PROBLEM: Given the degraded image g ( ind the ideal image ( 4 The Forward Model Original image - Blur h ,g,n o size MM Noise - n Measured image - g

3 5 The Forward Model: Matrix Formulation CS CS = (An M bym matrix) = (An M by 1 vector) CS =n CS =g g=+n 6 A section through the image g

4 7 The importance o the problem The image restoration problem is common in imaging systems. Other important problems can be solved using the same tools developed here. The problem and its solutions orm a very interesting mathematical ield, called Inverse Problems. 8 Sources o Noise Three major sources: During acquisition (oten random) During transmission Eects o coding/decoding Image noise as a random variable: For every (, n( can be considered as a random variable. In general, we assume that the noise n( is not dependent on the underlying signal (. In general, we assume that the value o n( is not correlated with n(x,y ). (Spatially uncorrelated noise) 4

5 9 Examples o Noise 10 Examples o Noise 5

6 11 (Eas Examples o Degradation:Periodic Noise Really enhancement rather than restoration 1 Examples o Degradation: Atmospheric Turbulence (, ) e x y k 5 / 6 x y Example: Look out the window o an airplane, behind the engine, at the ground. 6

7 13 Examples o Degradation:Motion blur 1/ 3 h( / / 3 Example: Photography with slow shutter speed in the presence o relative motion between the camera and the scene being imaged. 14 PROBLEM: Linear Image Restoration: ( h( g c ( + g( Ideal image Blur n( Measured image A SOLUTION: Noise g ( ( ˆ ( Measured image h r Restoration ilter Restored/ Reconstructed Image ˆ ( g( h ( r 7

8 15 Image Restoration Is a ard Problem: ( h( g c ( + g( Inverse ilter pair? n( ˆ( h r ( Oten h( is a low-pass type ilter. (e.g. Motion blur) ˆ( g( h ( r r ( * h( n( * h ( ( * h( * h ( n( * h ( NOISE AMPLIFICATION r r 16 Image Restoration: Basics o the Frequency Domain Approaches 8

9 17 Frequency Domain Formulation: FT = = FT = = FT = = FT = = x, ) ( y F x, ) ( y N x, ) ( y G x, ) ( y Low-pass ilter G Point-by-point multiply F N 18 Basic Inversion Idea G F N Restored Neglect the noise G F 1 1 Element-by-Element inverse N Inverse ilter 9

10 19 Avoiding Singularities Note: I has zero elements, then at those requencies, the inverse ilter does not exist. Thus use the ollowing inverting equation instead: * * C 1 C 1 * or or C C Scales image by 1/C at high requencies 0 A Simple Example Normalized Exact inverse ilter Gaussian blur Increasing C Freq. 10

11 1 Restoration Example Restored : Inverse Filter Used: Fˆ F C= Error=164 C=0.063 Error=47 C=0.63 Error=046 Larger C suppresses high requency response in the inverse ilter WAT IS TE BEST COICE FOR C? Choice o the Parameter C Fˆ F Restoration error C This is not a practical way o inding the best C!! 11

12 3 Drawbacks o this approach Finding the best C is not simple. In the proposed strategy, C is constant or all requencies!! Maybe we can gain something by using a requency dependent value. For high requencies where the noise is dominant (over the signal), the image is ampliied by 1/C>>1. This is silly, since we ampliy mostly noise! 4 A Dierent Strategy Previous Strategy - Based on : Where is high, apply exact inverse, and where low, apply /C New Startegy - Based on [Signal/Noise]* : SNR>>1 - inverse SNR<<1-0 Signal/Noi se SNR F(, ) x y 1

13 5 The Wiener Filter SNR is used in the ollowing manner: * * C SNR SNR C 1, *, when when SNR SNR C C 6 Simple Example Exact inverse ilter 10 0 Old Strategy Gaussian blur New strategy

14 7 Choice o SNR SNR is not known! Assume white noise with unit variance & radialy decaying signal: SNR(r) Freq. N F r, 1 ( r, ) r Polar Coordinates in Freq. Domain 8 Restoration Example = Found empirically Error=106 (compared to the 47 in the previous approach) 14

15 9 Is this the best one can do? NO!