Chapter 8 Image Restoration Types of image degradations

Chapter 8: Image Restoration ○ Types of image degradations: Noise, error, distortion, blurring, etc. Degradation model: where g(x, y): degraded image, f(x, y): true image, h(x, y): degradation process n(x, y): additive noise 8 -1

○ Two ways to recover image degradations: 1) Image enhancement: Overlook degradation processes, deal with images intuitively 2) Image restoration: Known degradation processes; model the processes and reconstruct images based on the inverse model 8 -2

○ Start with the degradation model Fourier transform From the convolution theorem, Difficulties: (a) Unknown N(u, v), (b) Small H(u, v) 8 -3

◎ Types of Noises: ○ White noise: the noise whose Fourier spectrum is constant ○ Periodic noise: Original image Noisy image ○ Additive noise: each pixel is added a value (noise) chosen from a probability distribution 8 -4

• Salt-and-pepper (impulse) noise Let x : noise value (a, b can be + or -) 8 -5

• Uniform noise: (a, b can be + or -) 8 -6

8 -7

Given histogram h 8 -8

• Gaussian noise: 8 -9

Method 1: 2 -10

Method 2: 2 -11

• Rayleigh noise: 8 -12

• Erlang (gamma) noise: 8 -13

• Exponential noise: 8 -14

◎ Estimation of noise parameters Steps: 1. Choose a uniform image region 2. Compute histogram 3. Compute mean and variance 4. Determine the probability distribution from the shape of 5. Estimate the parameters of the probability distribution using 8 -15

Examples: (a) Uniform noise: Given 8 -16

(b) Rayleigh noise: Given 8 -17

○ Multiplicative noise: Each pixel is multiplied with a value (noise) chosen from a probability distribution, e. g. , speckle noise 8 -18

◎ Noise removal ○ Salt-and-pepper noise – high frequency image component low-pass filter median filter 8 -19

。 Mean filter (i) Arithmetic mean: 4× 3 5× 5 8 -20

(ii) Geometric mean: (iii) Harmonic mean: (iv) Contraharmonic mean: 8 -21

3 × 3 median filter 3 × 3 (twice) 5× 5 8 -22

。Adaptive filter -- change characteristics according to the pixels under the window 8 -23

3× 3 5× 5 7× 7 9× 9 8 -24

○ Gaussian noise Assume Gaussian noise n(x, y) is uncorrelated and has zero mean Image averaging: 8 -25

Example: 8 -26

Periodic noise Band reject filter Notch filter 8 -27

In general case, Fourier spectrum noise Corresponding spatial noise 8 -28

○ Inverse filtering 8 -29

Low-pass Filtering: Constrained Division d = 40 60 80 100 8 -30

○ Wiener filtering -- Considers both degradation process and noise Idea: The derivation leads to (Parametric Wiener filter) 8 -31

(Wiener filter) When r = 1, If noise is zero, , (Inverse filter) If noise is white noise, is constant 8 -32

Input image k = 0. 001 k = 0. 00001 8 -33

○ Motion debluring Image f(x, y) undergoes planar motion : the components of motion T: the duration of exposure Fourier transform, 8 -34

Shifting property: 8 -35

Suppose uniform linear motion: Restore image by the inverse or Wiener filter 8 -36