Q19Digital Image Processing

Image Restoration is a highly rigorous mathematical discipline. Unlike subjective enhancement, restoration attempts to computationally invert known physical degradation. The absolute most common degradation is Noise—chaotic mathematical variations in pixel intensity caused by sensor heat, electronic interference, or atmospheric radiation. To violently eliminate this noise, an engineer must first mathematically model its exact Probability Density Function (PDF).

A. Gaussian (Normal) Noise

The absolute most common mathematical model, representing chaotic thermal electronic noise in sensors. Its PDF is a perfect Bell Curve: $p(z)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-(z-\mu {)}^2/2{\sigma }^2}$. The noise is violently distributed across ALL pixels in the image, continuously.

B. Rayleigh Noise

A heavily skewed mathematical PDF. It is not symmetric like Gaussian. It is violently utilized in complex range-imaging (radar and sonar) to model skewed signal scatter.

C. Uniform Noise

The PDF is a perfectly flat, rigid rectangle. Every exact noise value between a minimum $a$ and maximum $b$ has the absolute exact same mathematical probability of occurring. Found in hardware quantization errors.

D. Salt-and-Pepper (Impulse) Noise

Catastrophic hardware failure. The PDF consists of two violently massive spikes (bipolar impulses). Random pixels are forced to absolute maximum (255, Salt) or absolute minimum (0, Pepper). The rest of the image is completely uncorrupted.

Used primarily for continuous noise like Gaussian.

• Arithmetic Mean Filter: Blindly averages all pixels in a $3\times 3$ mask. It reduces Gaussian noise but causes horrific blurring of sharp edges.
• Geometric Mean Filter: Mathematically multiplies the pixels and takes the $N$-th root. It achieves smoothing comparable to the arithmetic mean, but it violently preserves edge detail significantly better.
• Harmonic Mean Filter: Specifically mathematically engineered to aggressively destroy "Salt" noise, but fails catastrophically on "Pepper" noise.

These filters completely abandon arithmetic and violently sort the pixels based on absolute numerical ranking.

• Median Filter: The absolute apex weapon against Salt-and-Pepper noise. It replaces the center pixel with the 50th percentile value. It completely deletes impulse noise spikes while mathematically preserving perfect, razor-sharp physical edges.
• Max Filter: Selects the 100th percentile (absolute brightest pixel). Used to violently destroy "Pepper" (black) noise.
• Min Filter: Selects the 0th percentile (absolute darkest pixel). Used to violently destroy "Salt" (white) noise.