Lecture 8

Applying noise using noise models to images

In digital image processing, degradation is typically modeled as an additive process:

g (x, y) = f (x, y) + η (x, y)

Where $f (x, y)$ is the original clean image, $η (x, y)$ is the noise component generated from a specific Probability Density Function (PDF), and $g (x, y)$ is the resulting corrupted image.

Let's look at a simple $3 \times 3$ patch of an image. Assume this patch is a solid, flat mid-gray area where every pixel has an intensity value of 100 (on a standard 0–255 8-bit scale).

f (x, y) = [\begin{matrix} 100 & 100 & 100 \\ 100 & 100 & 100 \\ 100 & 100 & 100 \end{matrix}]

Here is how two different noise models would numerically attack this matrix:

Example 1: Adding Impulse (Salt-and-Pepper) Noise

Salt-and-Pepper noise doesn't add a continuous value; it acts as a probability-based replacement.

1. Define the PDF Rules:

Probability of Salt ( $P_{s}$ ): 10% (0.1). The pixel becomes 255 (pure white).
Probability of Pepper ( $P_{p}$ ): 20% (0.2). The pixel becomes 0 (pure black).
Probability of remaining unchanged: 70% (0.7). The pixel stays 100.

2. The Random Process:

The computer evaluates every single pixel and generates a random probability number between 0.00 and 1.00 for each one. Let's assume the computer rolls the following random numbers for our 9 pixels:

Random Rolls = [\begin{matrix} 0.45 & 0.05 & 0.88 \\ 0.12 & 0.76 & 0.25 \\ 0.91 & 0.55 & 0.08 \end{matrix}]

3. Applying the Thresholds:

If roll < 0.10 $\to$ Pixel becomes 255 (Salt).
If 0.10 $\leq$ roll < 0.30 $\to$ Pixel becomes 0 (Pepper).
If roll $\geq$ 0.30 $\to$ Pixel stays 100.

4. The Final Corrupted Matrix, $g (x, y)$ :

g (x, y) = [\begin{matrix} 100 & 255 & 100 \\ 0 & 100 & 0 \\ 100 & 100 & 255 \end{matrix}]

Notice how the noise completely obliterated the original data in the affected pixels, creating extreme black and white spikes.

Example 2: Adding Gaussian Noise

Gaussian noise is strictly additive. It generates random numerical offsets based on a bell curve. Most offsets will be small (near the mean), but a few will be large.

1. Define the PDF Rules:

Mean ( $\bar{z}$ ): 0. (The noise is centered around zero, meaning it will brighten and darken pixels equally).
Standard Deviation ( $σ$ ): 15. (Controls how wide the bell curve is. A higher $σ$ means more severe noise).

2. The Random Process:

The computer samples 9 values from a Gaussian distribution with $\bar{z} = 0$ and $σ = 15$ . Here are the generated noise values, $η (x, y)$ :

η (x, y) = [\begin{matrix} + 12 & - 5 & + 3 \\ - 22 & + 31 & - 8 \\ + 1 & - 15 & + 6 \end{matrix}]

3. The Additive Math ( $g = f + η$ ):

We simply add the noise matrix to the original image matrix. (Note: If a value drops below 0 or exceeds 255, we clip it).

g (x, y) = [\begin{matrix} 100 & 100 & 100 \\ 100 & 100 & 100 \\ 100 & 100 & 100 \end{matrix}] + [\begin{matrix} + 12 & - 5 & + 3 \\ - 22 & + 31 & - 8 \\ + 1 & - 15 & + 6 \end{matrix}]

4. The Final Corrupted Matrix, $g (x, y)$ :

g (x, y) = [\begin{matrix} 112 & 95 & 103 \\ 78 & 131 & 92 \\ 101 & 85 & 106 \end{matrix}]

Notice how unlike Salt-and-Pepper, Gaussian noise affects absolutely every pixel, but the original 100 intensity is still "hidden" within the noisy values.