Image Restoration & Noise Models: Degradation, Filters & Cleanup

TAKEAWAYS:

• Noise is random variation in pixel values that degrades image quality, modeled as Gaussian, salt-and-pepper, or Poisson noise.
• Image degradation follows the degradation model $g(x,y)=h(x,y)\ast f(x,y)+n(x,y)$, where $h$ is the blur kernel, $\ast$ is convolution, and $n$ is noise.
• Restoration techniques include inverse filtering (for known blur), Wiener filter (for noise reduction), and morphological operations (erosion/dilation for salt-and-pepper noise).
• Histogram equalization and adaptive filtering are used to enhance degraded images before restoration.
• Real-world examples: Nepal’s NTC satellite images (restored for flood monitoring) and Daraz product photos (cleaned of noise for better e-commerce displays).
• Exam questions often test noise models, filter design, and restoration steps with numerical examples.

1. Why Image Restoration Matters

• Noise: Random fluctuations in pixel values (e.g., sensor noise, transmission errors).
• Degradation: Blur from motion, defocus, or atmospheric turbulence.
• Artifacts: Striations in satellite images, compression artifacts in JPEGs.

Example: A Daraz product photo with noise (e.g., grainy pixels) reduces buyer trust. Restoration smooths it for better sales.

2. Noise Models: Types and Effects

Noise is classified by its probability distribution and spatial pattern. Key models:

A. Additive Noise Models

Noise is added to the original image $f(x,y)$: $$g(x,y)=f(x,y)+n(x,y)$$ where $g$ is the degraded image.

Visual:

   Gaussian Noise:       Salt-and-Pepper Noise:
   [•••••••••]            [■■■■■■■■]
   [•••••••••]            [■■■■■■■■]
   [•••••••••]            [■■■■■■■■]

• = noisy pixel, ■ = black/white pixel

B. Multiplicative Noise

Noise scales pixel values (common in low-light photography): $$g(x,y)=f(x,y)\cdot n(x,y)$$ Example: Nepal’s NEPSE stock charts may have multiplicative noise from market volatility.

3. Image Degradation Model

1. Blurring (e.g., motion, defocus): Convolution with a kernel $h(x,y)$.
2. Noise addition: $n(x,y)$ (additive or multiplicative).
3. Atmospheric distortion (e.g., haze in satellite images).

Mathematical Model: $$g(x,y)=h(x,y)\ast f(x,y)+n(x,y)$$ where:

• $\ast$ = convolution,
• $h$ = Point Spread Function (PSF) kernel,
• $n$ = noise.

Example: A blurry Ncell selfie (motion blur) can be modeled as: $$g(x,y)=h(x,y)\ast f(x,y)$$ where $h$ is a Gaussian kernel.

4. Image Restoration Techniques

Goal: Recover $f(x,y)$ from $g(x,y)$.

A. Inverse Filtering

Assumes noise is negligible and blur is known: $$f(x,y)=H^{-1}(x,y)\ast g(x,y)$$ where $H^{-1}$ is the inverse of the blur kernel $h$.

Limitation: Amplifies high-frequency noise (e.g., ringing artifacts).

Worked Example: Restore a blurred Daraz product image with kernel: $$h=\left[\begin{array}{ccc}0.1& 0.2& 0.1\\ 0.2& 0.4& 0.2\\ 0.1& 0.2& 0.1\end{array}\right]$$ Steps:

1. Compute $H^{-1}$ (inverse of $h$).
2. Convolve $g(x,y)$ with $H^{-1}$.

Output:

Blurred: [•••••••••] (low contrast)
Restored: [□□□□□□] (sharp edges)

B. Wiener Filter

Reduces noise by balancing restoration and smoothing: $$H_{wiener}(u,v)=\frac{H^{\ast }(u,v)}{|H(u,v){|}^2+\frac{N(u,v)}{S(u,v)}}$$ where:

• $H^{\ast }$ = complex conjugate of blur kernel in frequency domain,
• $N(u,v)$ = noise power spectrum,
• $S(u,v)$ = signal power spectrum.

Advantage: Less sensitive to noise than inverse filtering.

Example: Restoring NTC flood monitoring images with Gaussian noise.

C. Morphological Restoration

For salt-and-pepper noise:

1. Erosion: Shrinks noisy pixels.
2. Dilation: Expands clean regions.
3. Opening/Closing: Combines erosion + dilation.

Worked Example: Clean a noisy Pathao driver’s license photo:

sequenceDiagram
  participant Image as Input Image
  participant Noise as Salt-and-Pepper
  participant Erosion as Erosion
  participant Dilation as Dilation

  Image->>Noise: Adds noise
  Noise-->>Image: Output: [■■■■■■■■]

  Image->>Erosion: Erosion Kernel (3x3)
  Erosion-->>Image: Removes isolated noise

  Image->>Dilation: Dilation Kernel (3x3)
  Dilation-->>Image: Restores edges

  Image->>Image: Cleaned Output: [Cleaned Image]

Output:

Before: [■■■■■■■■]
After:  [□□□□□□]

5. Real-World Applications

In the Real World

1. NTC Satellite Imagery:

   • Idea: Wiener filtering removes atmospheric haze from Nepal’s weather satellite images.
   • Impact: Accurate flood/landslide monitoring for disaster response.

2. Daraz Product Photos:

   • Idea: Morphological operations clean up noise in product images to improve e-commerce conversions.
   • Impact: Higher buyer trust and reduced returns.

3. Medical Imaging (X-rays, MRIs):

   • Idea: Inverse filtering sharpens blurry CT scans for precise diagnoses.
   • Impact: Early detection of diseases like tumors.

6. Worked Example: Restoring a Noisy Image

Given:

• Original image $f(x,y)$:

  [10  20  30]
  [40  50  60]
  [70  80  90]
  

• Blur kernel $h$:

  [0.1 0.2 0.1]
  [0.2 0.4 0.2]
  [0.1 0.2 0.1]
  

• Additive Gaussian noise $n$ (mean=0, variance=10).

Steps:

1. Convolve $f$ with $h$:

   g = h * f =
   [12.3 14.5 12.3]
   [24.6 29.0 24.6]
   [36.9 43.5 36.9]
   

2. Add noise $n$:

   g_noisy =
   [12.3+2.1  14.5-1.8  12.3+3.0]
   [24.6-0.5  29.0+1.2  24.6+2.7]
   [36.9+1.5  43.5-2.3  36.9-1.8]
   

3. Apply Wiener filter in frequency domain to restore $f$.

Output:

Original: [10 20 30]
         [40 50 60]
         [70 80 90]
Noisy:   [14.4 12.7 15.3]
         [24.1 30.2 27.3]
         [38.4 41.2 35.1]
Restored: [9.8 19.5 29.2]
          [39.7 51.0 59.8]
          [69.3 81.5 88.7]

7. Comparison of Restoration Techniques

8. Exam Tip

• Focus on:
  • Defining noise models (Gaussian, salt-and-pepper) and their probability distributions.
  • Explaining the degradation model $g=h\ast f+n$.
  • Steps for inverse filtering and Wiener filter (with frequency domain math).
  • Morphological operations for salt-and-pepper noise (erosion/dilation).
• Common Questions:
  • "Given a noisy image, apply Wiener filtering to restore it."
  • "Compare inverse filtering and Wiener filter."
  • "How does morphological restoration work for salt-and-pepper noise?"
• Worked Examples:
  • Always show convolution steps for blur kernels.
  • For Wiener filter, compute $H_{wiener}(u,v)$ explicitly.
  • Use small matrices (3x3) for clarity.

9. Key Formulas to Memorize

1. Degradation Model: $$g(x,y)=h(x,y)\ast f(x,y)+n(x,y)$$

2. Wiener Filter (Frequency Domain): $$H_{wiener}(u,v)=\frac{H^{\ast }(u,v)}{|H(u,v){|}^2+\frac{N(u,v)}{S(u,v)}}$$

3. Morphological Operations:

   • Erosion: $(f\ominus b)(x,y)={\min }_{b}f(x+b)$
   • Dilation: $(f\oplus b)(x,y)={\max }_{b}f(x+b)$

10. Summary Flowchart

flowchart TD
    A["Degraded Image"] -->|"Noise + Blur"| B["Degradation Model"]
    B --> C{"Noise Type?"}
    C -->|"Gaussian"| D["Wiener Filter"]
    C -->|"Salt-and-Pepper"| E["Morphological Restoration"]
    C -->|"Known Blur"| F["Inverse Filtering"]
    D --> G["Restored Image"]
    E --> G
    F --> G