8-1 Chapter 8: Image Restoration Image enhancement: Overlook degradation processes, deal with images intuitively Image restoration: Known degradation processes; model the processes and reconstruct images based on the inverse model ○ Degradations e.g., noise, error, distortion, blurring

8-2 ◎ Degradation Model g(x,y): degraded image, f(x,y): image, h(x,y): degradation process n(x,y): additive noise From the convolution theorem, Difficulties: (a) unknown N(u,v), (b) small H(u,v)

8-3 ◎ Noise (originating from image acquisition, digitization, or transmission) ○ White noise: the noise whose Fourier spectrum is constant ○ Periodic noise: Noisy image Original 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 -) e.g.,

8-5 。 Uniform noise: (a, b can be + or -)

8-6

8-7 。 Gaussian noise:

8-8 。 Rayleigh noise:

8-9 。 Erlang (gamma) noise:

8-10 。 Exponential noise:

8-11 ◎ 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

8-12 Examples: (a) Uniform noise: Given

8-13 (b) Rayleigh noise: Given

8-14 ○ Multiplicative noise: Each pixel is multiplied with a value (noise) chosen from a probability distribution e.g., Speckle noise

8-15 ◎ Noise removal ○ Salt-and-pepper noise – high frequency image component Low-pass filter median filter

8-16 。 Mean filter – tend to blur image (i) Arithmetic mean: 4 × 3 5 × 5

8-17 (ii) Geometric mean: (iii) Harmonic mean: (iv) Contraharmonic mean:

× 3 median filter 3 × 3 (twice) 5 × 5

8-19 。 Adaptive filter -- change characteristics according to the pixels under the window

8-20 3×3 5×5 7×7 9×9

8-21 Assume Gaussian noise n(x,y) is uncorrelated and has zero mean ○ Gaussian noise 。 Image averaging:

8-22 Example:

8-23 Band reject filter Notch filter ○ Periodic noise

8-24 In general case, Fourier spectrum noise Corresponding spatial noise

8-25 ○ Inverse filtering

8-26 Low-pass Filtering: Constrained Division d = 

8-27 ○ Wiener filtering -- Considers both degradation process and noise Idea: (Parametric Wiener filter)

8-28 When r = 1, If noise is zero,, ( Wiener filter) (Inverse filter) If noise is white noise, is constant

8-29 Input image k = k = k =

8-30 Image f(x,y) undergoes planar motion : the components of motion T: the duration of exposure Fourier transform, ○ Motion debluring

8-31

8-32 Suppose uniform linear motion: Note H vanishes at u = n / a (n: an integer) Restore image by the inverse or Wiener filter