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.