Image Restoration & Compression: Noise Removal, Deconvolution, JPEG, DCT, Huffman

What is Image Restoration?

Types of Degradation

1. Noise: Random variations in pixel intensity (e.g., salt-and-pepper, Gaussian).
2. Blur: Loss of sharpness due to motion, defocus, or camera shake.
3. Sensor Defects: Dead pixels or stuck bits in the camera sensor.

Restoration Techniques

1. Noise Removal

Goal: Reduce random noise while preserving edges. Methods:

• Linear Filtering: Average/smoothing filters (e.g., mean filter).
• Nonlinear Filtering: Median filter (better for salt-and-pepper noise).
• Wiener Filter: Optimal for Gaussian noise (uses statistical properties).

2. Deblurring (Inverse Filtering)

Goal: Reverse blur caused by a known point-spread function (PSF). Steps:

1. Model blur as a convolution: $g=f\ast h+n$, where:
   • $g$: degraded image,
   • $f$: original image,
   • $h$: PSF (blur kernel),
   • $n$: noise.
2. Apply inverse filtering in frequency domain: $$F=\frac{G}{H}\text{(where G,H are Fourier transforms of g,h)}$$
3. Problem: Noise amplification if $H$ has near-zero values. Solution: Wiener deconvolution (regularization).

3. Geometric Distortion Correction

Goal: Fix distortions from lens or sensor misalignment. Methods:

• Affine Transformations: Correct perspective or skew.
• Homography: Map distorted image to a reference plane (used in satellite imagery).

graph LR
    A["Degraded Image"] -->|"Noise"| B["Noise Removal"]
    A -->|"Blur"| C["Deblurring"]
    A -->|"Distortion"| D["Geometric Correction"]
    B -->|"Filtering"| E["Restored Image"]
    C -->|"Inverse Filter"| E
    D -->|"Transformation"| E

Example: Restoring a blurry photo from a shaky phone camera. Assume a 3×3 blur kernel:

[0.11 0.11 0.11]
[0.11 0.11 0.11]
[0.11 0.11 0.11]

Apply inverse filtering in the frequency domain to recover sharp edges.

Image Compression

Compression reduces file size while minimizing quality loss. Two main types:

1. Lossless: No data loss (e.g., ZIP, PNG). Used for medical/scientific images.
2. Lossy: Sacrifices quality for higher compression (e.g., JPEG, MP3). Used for photos/videos.

Key Techniques

1. Transform Coding (DCT)

Goal: Convert spatial redundancy into frequency coefficients. Steps:

1. Divide image into 8×8 blocks.
2. Apply Discrete Cosine Transform (DCT): $$F(u,v)=\frac{1}{4}C(u)C(v)\sum _{x=0}^7\sum _{y=0}^{7}f(x,y)\cos \left[\frac{(2x+1)u\pi }{16}\right]\cos \left[\frac{(2y+1)v\pi }{16}\right]$$ where $C(u)=\frac{1}{\sqrt{2}}$ if $u=0$, else 1.
3. Quantize high-frequency coefficients (small values → zero).
4. Encode remaining coefficients (e.g., zigzag scan).

Visual: Note: Low-frequency coefficients (top-left) carry most energy.

2. Entropy Coding (Huffman)

Goal: Assign shorter codes to frequent symbols. Steps:

1. Calculate symbol probabilities (e.g., DCT coefficient frequencies).
2. Build Huffman tree:
   • Combine least frequent symbols iteratively.
   • Assign binary codes (e.g., 0 for frequent, 10 for rare).
3. Encode image data using the Huffman table.

Example:

Compression Ratio: $$\text{Ratio}=\frac{\text{Uncompressed Size}}{\text{Compressed Size}}=\frac{8\text{ bits/symbol}}{2.5\text{ bits/symbol}}=3.2$$

JPEG Compression Pipeline

flowchart LR
    A["Original Image"] --> B["Divide into 8×8 Blocks"]
    B --> C["DCT Transform"]
    C --> D["Quantization"]
    D --> E["Zigzag Scan"]
    E --> F["Huffman Encoding"]
    F --> G["Compressed JPEG"]

Real-World Example:

• Daraz/Flipkart Product Images: Compressed with JPEG to load faster on mobile networks.
• WhatsApp Media: Uses JPEG for photos and Huffman-like encoding for metadata.

In the Real World

1. eSewa/NTT Data: Compress transaction receipts (JPEG) to reduce server storage.
2. Pathao Driver App: Uses image compression to send real-time traffic snapshots to the backend.
3. NEPSE Stock Charts: Apply noise removal to clean up low-light stock photos before analysis.

Worked Example: Compressing a 512×512 grayscale image (262,144 pixels × 8 bits = 2.1 MB).

1. Divide into 8×8 blocks → 4,096 blocks.
2. Apply DCT → Quantize high frequencies (e.g., discard coefficients < 5).
3. Huffman encode → Assume 50% reduction → Final size: ~1.05 MB (50% compression).

Exam Tip

1. Restoration:
   • Know the inverse filtering equation and its limitations (noise amplification).
   • Compare Wiener filter vs. median filter (Wiener for Gaussian noise, median for salt-and-pepper).
2. Compression:
   • Memorize DCT steps and the zigzag scan pattern.
   • Calculate compression ratio for Huffman tables.
3. Applications:
   • Link JPEG to web images, PNG to logos, and lossless to medical imaging.
4. Math Shortcuts:
   • For DCT, focus on the energy compaction property (most energy in low frequencies).
   • For Huffman, practice building trees from probability tables.