Image Sampling & Quantization: Pixels, Resolution & Color Depth

TAKEAWAYS:

• Sampling converts spatial coordinates into discrete pixels via the Nyquist theorem (sampling rate ≥ 2× highest frequency).
• Quantization maps continuous intensity values to discrete levels (e.g., 8-bit = 256 grayscale shades).
• Pixel relationships define resolution: width × height × bits per pixel = total storage (e.g., 1024×768×24 = 2.25 MB).
• Color models (RGB, CMYK, grayscale) determine quantization depth and file size trade-offs.
• Aliasing occurs when undersampling distorts high-frequency details (e.g., jagged edges in low-res images).
• Real-world impact: eSewa’s ID photos use 300 DPI (sampling) and 24-bit RGB (quantization) for clarity; Daraz thumbnails use 72 DPI to balance speed and storage.

1. Digital Image Formation: Sampling Spatial Coordinates

How Analog Images Become Pixels

An analog image (continuous light intensity) is converted to a digital image by:

1. Sampling: Measuring light intensity at discrete points (pixels) across x and y axes.
2. Quantization: Assigning each sampled value to a finite set of discrete levels (e.g., 0–255 for 8-bit grayscale).

Key Definitions

• Pixel (Picture Element): Smallest addressable unit in a digital image, defined by its spatial coordinates (row, column) and intensity value.
• Spatial Resolution: Number of pixels along width/height (e.g., 1920×1080 = Full HD).
• Sampling Rate: Pixels per unit length (e.g., 72 pixels/inch = PPI/DPI for printers).

Nyquist Sampling Theorem

To avoid aliasing (false high-frequency artifacts), the sampling rate must be at least twice the highest frequency in the image: $$f_s\ge 2\times f_{\text{max}}$$ Example: A 300 DPI scanner samples at 300 pixels/inch. If the highest spatial frequency in the image is 150 cycles/inch, $f_s=300\ge 2\times 150$ → no aliasing.

graph LR
    A["Analog Image\n(Continuous Light)"] -->|"Sampling"| B["Discrete Pixels\n(Spatial Grid)"]
    B -->|"Quantization"| C["Digital Image\n(Discritized Intensity)"

Real-World Example: eSewa ID Photos

• Sampling: 300 DPI (1 pixel = 1/300 inch) to capture fine facial details.
• Why? Undersampling (e.g., 72 DPI) would cause aliasing (jagged edges around hair/eyes), making verification harder.
• Quantization: 24-bit RGB (8 bits per channel) for color accuracy.

2. Quantization: Mapping Intensity to Discrete Levels

How Many Levels?

Quantization converts continuous intensity $I$ to discrete levels $L$: $$L=\text{round}\left(\frac{I}{\Delta }\right),\Delta =\frac{I_{\text{max}}}{2^b-1}$$

• $b$ = bits per pixel (e.g., $b=8$ → 256 levels).
• Grayscale: 0 (black) to 255 (white).
• RGB: 3 bytes (8 bits each for R, G, B).

Color Depth vs. File Size

Example Calculation: For a 1028×1028 image with 512 colors (9-bit quantization): $$\text{Size}=1028\times 1028\times \frac{9}{8}\text{ bytes}=1.15\text{ MB}$$ (9 bits = 1.125 bytes per pixel; 512 colors = $2^9$)

Real-World Example: Daraz Product Images

• Thumbnails: 72 DPI × 8-bit grayscale (0.25 MB) for fast loading.
• High-res: 300 DPI × 24-bit RGB (2.25 MB) for zoomable details.
• Trade-off: Lower quantization (e.g., 16-bit) reduces file size but loses color fidelity.

3. Pixel Relationships and Image Storage

Neighborhood Operations

Pixels are related via 8-neighborhood (including diagonals) or 4-neighborhood (only horizontal/vertical):

8-neighborhood:
A B C
D E F
G H I

• E is the center pixel; A–I are neighbors.
• Used in edge detection (e.g., Sobel operator) and morphological operations.

Storage Formula

Total image size (bytes) = width × height × bits per pixel / 8. Example: A 2048×1536 grayscale image (8 bits/pixel): $$2048\times 1536\times 1=3.14\text{ MB}$$

Real-World Example: Medical Imaging (X-rays)

• Sampling: 100–200 DPI to detect fine bone structures.
• Quantization: 12-bit grayscale (4096 levels) for high contrast.
• Why? Low quantization (e.g., 8-bit) would lose subtle tissue differences.

4. Aliasing and Its Effects

graph LR
    A[High-Res Image
(300 DPI)] -->|Downsampled
(72 DPI)| B[Aliased
(Jagged Edges)]
    B -->|Anti-Aliasing
(Gaussian Blur)| C[Smooth
(No Artifacts)]

Cause: Undersampling

When $f_s<2f_{\text{max}}$, high frequencies appear as false low frequencies (e.g., jagged edges). Example: Zooming a low-res image reveals "staircase" artifacts.

graph TD
    A["High-Frequency\nSignal"] -->|"Undersampled"| B["Aliased\nLow-Frequency\nArtifact"]

Real-World Example: Kathmandu Traffic Signs

• Problem: Low-resolution CCTV cameras (e.g., 640×480) undersample license plates, causing aliasing in number recognition.
• Solution: Use 1280×720+ resolution and anti-aliasing filters (e.g., Gaussian blur before downsampling).

5. Color Models and Quantization

Common Models

Real-World Example: WhatsApp Profile Pictures

• Quantization: 24-bit RGB (16.7M colors) for vibrant displays.
• Sampling: 480×480 pixels (scaled from higher-res uploads).
• Trade-off: Smaller thumbnails (e.g., 96×96) use dithering to simulate colors with fewer bits.

6. Worked Example: Calculating Image Size

Problem: A 512-color image has dimensions 1028×1028. Calculate its size in MB. Solution:

1. Determine bits per pixel: 512 colors = $2^9$ → 9 bits/pixel.
2. Calculate total bits: $1028\times 1028\times 9=9,437,184$ bits.
3. Convert to bytes/MB: $9,437,184/8=1,179,648$ bytes ≈ 1.13 MB.

7. Exam Tip: Common Pitfalls and Focus Areas

• Sampling vs. Quantization:

  • Sampling = spatial resolution (pixels/inch).
  • Quantization = color depth (bits/pixel).
  • Exam trick: Questions often mix these—always specify which is being asked!

• Nyquist Theorem:

  • Memorize: $f_s\ge 2f_{\text{max}}$.
  • Example: If $f_{\text{max}}=150$ cycles/inch, $f_s\ge 300$ DPI.

• File Size Calculations:

  • Use the formula: width × height × (bits/pixel) / 8.
  • Shortcut: For 24-bit RGB, divide by 8 first (3 bytes/pixel).

• Aliasing:

  • Always mention jagged edges or moiré patterns as visual clues.
  • Example: "A 72 DPI scan of a 300 DPI photo will show aliasing."

• Color Models:

  • RGB is for screens; CMYK for print.
  • Exam tip: "Indexed color" uses a palette (e.g., GIFs).

Labelled illustration of an analog-to-digital conversion pipeline showing a continuous image sampled into a pixel grid, then quantized to discrete intensity levels.

Side-by-side comparison of a properly sampled image vs. an aliased version with jagged edges (e.g., a circle appearing square).

In the real world

• eSewa ID Photos: Uses 300 DPI sampling (Nyquist-compliant for facial details) and 24-bit RGB quantization to ensure clear biometric verification, avoiding aliasing in edge regions (eyes, hairlines).
• Daraz Product Images: Thumbnails use 72 DPI × 8-bit grayscale (0.25 MB) for fast loading, while high-res versions use 300 DPI × 24-bit RGB (2.25 MB) to balance storage and zoomable detail.
• NTC Traffic Cameras: Low-resolution CCTV (640×480) causes aliasing in license plates, requiring anti-aliasing filters (Gaussian blur) before downsampling for OCR accuracy.