Image Degradation Model

  • Mathematical model for image degradation, i.e., the observed image g(x,y) = f(x,y)*h(x,y) + e(x,y) where * denotes convolution. f(x,y) is the noiseless image. h(x,y) is the degradation function (assumed to be linear). e(x,y) is the noisy perturbations of each pixel value.
  • Block diagram of the image degradation model with time domain and frequency domain interpretations.
  • Problems associated with image restoration, identifying the unknown degradation function h(x,y) and the unknown noise source e(x,y).

Some new probability distributions

  • Discussed new distributions which the noise e(x,y) may have.
  • Introduced the Rayleigh Distribution.
  • Introduced the Erlang (Gamma) Distribution.
  • Introduced the Exponential Distribution.
  • Introduced the Impulse Distribution.

Estimating Noise Parameters

  • Assume the degradation function is trivial, i.e., h(x,y) = delta(0,0).
  • Estimate the noise distribution e(x,y).
  • Approach by inputing a constant image (with pixel value k) to the system.
  • The histogram of the output image is an approximation of the convolution of the true noise distribution with the image histogram (a delta function at the constant k).
  • How to estimate the true noise distribution using this technique. An example for estimating the parameters of Rayleigh noise from a perturbed constant image.

Restoration in the presence of only noise

  • Arithmetic Mean
  • Geometric Mean
  • Harmonic Mean
  • Contraharmonic Mean

Order Statistic Filters

  • Revisit the median, min, and max filters.
  • Introduce the midpoint filter.
  • Introduce the alpha-trimmed mean filter.

Adaptive Filters

  • Defined adaptive filter
  • Introduced an adaptive local noise reduction filter.

Periodic Noise Reduction by Frequency Domain Filtering

  • Assume h(x,y) = delta(0,0) and e(x,y) is not random but a deterministic periodic function.
  • We wish to remove e(x,y) from the image g(x,y). This is done by filtering in the frequency domain.
  • Introduced bandreject filtering : Ideal, Butterworth, and Gaussian Bandreject filters.
  • Introduced notchreject filtering : Ideal, Butterworth, and Gaussian Notchreject filters.