Lecture 01 - Discretization, Sampling & Color Spaces

Discretization refers to the process of transforming a continuous domain into a discrete one by representing it with a finite set of elements.
In the context of images, this typically involves converting a continuous spatial domain Ω ⊂ ℝ² (where an image is defined as a continuous function over space) into a finite grid of samples (pixels).

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Sampling (Spatial Discretization)

Sampling refers to discretizing the continuous spatial domain Ω.
We choose specific points (xᵢ, yⱼ), turning f(x, y) into a grid of values:

fᵢⱼ = f(xᵢ, yⱼ)

Example:

• Cameraman image at resolution 256×256 → 256 samples along each axis → 65,536 pixels.
• Reducing to 64×64 or 32×32 leads to fewer pixels and loss of detail.

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Tonal Domain and Quantization (Intensity Discretization)

The tonal domain (also known as the intensity domain or range) refers to the set of all possible intensity or color values a pixel in an image can take.

Quantization is the process of discretizing the range of grey levels, mapping the continuous intensity f(x, y) to a finite set of values, typically integers.

Common case:

• 8-bit quantization → values in { 0, 1, ..., 255 }
  • 0: black, 255: white

Binary images:

• Co-domain: { 0, 1 }, often rescaled to { 0, 255 }
• Result from thresholding operations, useful in document scanning, object detection, etc.

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Digital Image Formation Summary

To convert a continuous image into digital format:

• Use Sampling (for space)
• Use Quantization (for intensity)

Example pipeline:

• Original grayscale image → 256×256, 8-bit depth
• Downsampling:
  • Sample to 64×64 → coarse detail
  • Sample to 32×32 → fine detail lost; image appears blocky
• Quantization/Binarization:
  • Apply threshold (e.g., 128)
    • Pixel ≥ 128 → white (255)
    • Pixel < 128 → black (0)
• Result: binary image highlighting structure or edges

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Color Images

A color image is defined as a function:

f(x, y): Ω → ℝ³

RGB Color Model

Standard model for display systems (monitors, cameras).
Each channel is typically 8 bits → values in [0, 255] per channel.

Examples:

• [255, 0, 0] → red
• [0, 255, 0] → green
• [255, 255, 255] → white

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HSV Color Space

HSV stands for:

• Hue (H): Color type (e.g., red, green)
• Saturation (S): Color purity/intensity
• Value (V): Brightness

Advantages:
Perceptually uniform – better aligns with how humans perceive color, making it useful in:

• Object tracking
• Image segmentation
• Thresholding in color space

RGB (R,G,B)	HSV (H,S,V)
[255, 0, 0]	[0°, 1, 1] (pure red)
[0, 255, 0]	[120°, 1, 1] (pure green)
[0, 0, 255]	[240°, 1, 1] (pure blue)
[128, 128, 128]	[0°, 0, 0.5] (grey)

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