Image Restoration: Noise Removal, Blur Correction & Artifact Recovery

TAKEAWAYS:

• Image restoration reverses degradation (noise, blur, or distortion) to recover the original image using mathematical models like inverse filtering or Wiener filtering.
• Noise models (Gaussian, salt-and-pepper, Poisson) dictate the restoration approach—e.g., median filters for salt-and-pepper noise.
• Blind deconvolution estimates both the blur kernel and original image when the degradation process is unknown (common in astronomy).
• Geometric corrections (e.g., affine transformations) fix distortions from camera motion or lens defects.
• Regularization (e.g., Tikhonov) prevents overfitting in ill-posed restoration problems.
• Real-world applications include medical imaging (MRI/CT artifact removal), satellite imagery (cloud/atmospheric correction), and facial recognition (low-light enhancement).

1. Why Image Restoration?

Digital images degrade due to:

• Noise: Random variations (e.g., sensor thermal noise, transmission errors).
• Blur: Motion, defocus, or atmospheric turbulence.
• Defects: Sensor malfunctions (e.g., "dead pixels") or compression artifacts.

Goal: Recover the original (latent) image $f(x,y)$ from the degraded observation $g(x,y)$, where: $$g(x,y)=H\{f(x,y)\}+n(x,y)$$

• $H$: Degeneration operator (e.g., blur kernel).
• $n(x,y)$: Noise.

2. Noise Models and Filters

Noise types and their restoration strategies:

Worked Example: Median Filter for Salt-and-Pepper Noise

Scenario: A corrupted image from a Nepalese traffic camera (e.g., NTC surveillance) has 20% salt-and-pepper noise. Restore it using a 3×3 median filter.

Steps:

1. Input: Degraded image $g(x,y)$ with noise.
2. Window: Slide a 3×3 window over $g(x,y)$.
3. Median: Replace center pixel with the median of the 9 values.
4. Output: Restored image $\hat{f}(x,y)$.

Visual:

Result: Sharpens edges (e.g., license plates) while removing noise spikes.

3. Blur Models and Deconvolution

Blur arises from:

• Motion blur: Camera/object movement (e.g., Pathao driver’s phone camera).
• Defocus blur: Out-of-focus lens (e.g., Nepalese smartphone photos).
• Atmospheric blur: Haze in outdoor images (e.g., mountain photos from Pokhara).

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

• $h(x,y)$: Point Spread Function (PSF) or blur kernel.
• $\ast$: Convolution operator.

Deconvolution Methods

Worked Example: Wiener Filter for Motion Blur Scenario: A Daraz delivery photo is blurred due to camera shake. The blur kernel $h(x,y)$ is a linear motion blur of length 5 pixels. Noise variance ${\sigma }_n^2=10$.

Steps:

1. Compute Fourier transforms: $$G(u,v)=\mathcal{F}\{g(x,y)\},H(u,v)=\mathcal{F}\{h(x,y)\}$$
2. Apply Wiener filter: $$\hat{F}(u,v)=\frac{H^{\ast }(u,v)G(u,v)}{|H(u,v){|}^2+{\sigma }_n^2}$$
3. Inverse Fourier transform to get $\hat{f}(x,y)$.

Visual:

graph TD
    A["Blurred Image\n(g(x,y))"] -->|"Fourier Transform"| B["Frequency Domain\nG(u,v), H(u,v)"]
    B -->|"Wiener Filter"| C["Restored Frequency\nŶ(u,v)"]
    C -->|"Inverse FT"| D["Deblurred Image\n(ŷ(x,y))"]

Output: Sharper edges (e.g., product details in the Daraz image).

4. Geometric Distortions and Corrections

Caused by:

• Lens distortion: Barrel/pincushion (e.g., GoPro cameras).
• Camera motion: Rotation/scaling (e.g., drone footage).
• Perspective: Vanishing points (e.g., Kathmandu street photos).

Correction Methods:

Worked Example: Affine Correction for Drone Imagery Scenario: A Ncell drone survey image of a Pokhara field is skewed due to tilt. Estimate the affine transform: $$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}e\\ f\end{array}\right]$$ Steps:

1. Identify 4+ control points (e.g., road intersections).
2. Solve for $A$ and $\mathbf{b}$ using least squares.
3. Apply inverse transform to warp the image.

Visual:

graph LR
    A["Distorted Drone\nImage"] -->|"Affine Transform"| B["Control Points\n(4+ corners)"]
    B -->|"Solve for A,b"| C["Corrected Ortho-\nImage"]

Output: Rectified image for accurate land measurement.

5. Regularization and Ill-Posed Problems

Problem: Restoration is ill-posed (no unique solution) due to:

• Noise amplification (inverse filtering).
• Missing high-frequency details.

Solution: Add a regularization term to constrain the solution: $$\hat{f}=\arg \underset{f}{\min }\Vert g-Hf{\Vert }^2+\lambda \Vert Lf{\Vert }^2$$

• $L$: Regularization operator (e.g., Laplacian for smoothness).
• $\lambda$: Trade-off parameter.

Example: Tikhonov Regularization (for blur + noise): $$\hat{F}(u,v)=\frac{H^{\ast }(u,v)G(u,v)}{|H(u,v){|}^2+\lambda }$$

6. Real-World Applications

In the Real World

1. Medical Imaging (Nepal Health Hospitals)

   • MRI/CT Artifact Removal: Wiener filtering removes streaking artifacts from Nepal’s Patan Hospital scans, improving diagnosis.
   • Noise Reduction: Median filters clean ultrasound images (e.g., fetal scans) by removing speckle noise.

2. Satellite and Remote Sensing (Nepal’s Department of Survey)

   • Cloud Correction: Blind deconvolution removes atmospheric blur from Nepal’s satellite images (e.g., landslide monitoring).
   • Geometric Correction: Affine transforms align drone maps for accurate disaster response (e.g., floods in Chitwan).

3. Facial Recognition (eSewa/Khalti Biometrics)

   • Low-Light Enhancement: Wiener filters sharpen nighttime ID photos in Khalti’s facial recognition, reducing false rejects.

4. Autonomous Vehicles (Pathao/Nepal’s EV Startups)

   • Motion Deblurring: Inverse filtering sharpens dashboard camera feeds for real-time obstacle detection.

Worked Example: Restoring a Blurry NEPSE Stock Chart

Scenario: A NEPSE daily closing price chart is blurred due to poor resolution. The blur kernel is a Gaussian with σ=2.

Steps:

1. Model: $g=h\ast f+n$, where $h$ is Gaussian.
2. Wiener Filter: Compute $\hat{F}=\frac{H^{\ast }G}{|H{|}^2+K}$, with $K=0.01$.
3. Result: Smoother price trends (e.g., clearer support/resistance levels).

Visual:

7. Exam Tip

What Examiners Look For:

1. Mathematical Rigor:
   • Derive the Wiener filter formula from scratch (show steps).
   • Explain why inverse filtering fails for noisy images (amplification of high frequencies).
2. Practical Scenarios:
   • Link restoration to real applications (e.g., "How would you restore a blurry Ncell drone image?").
   • Compare filters (e.g., "When to use median vs. Gaussian filter?").
3. Visuals:
   • Draw before/after restoration (e.g., noisy → denoised image).
   • Sketch frequency domain plots (e.g., $H(u,v)$ for blur).
4. Common Pitfalls:
   • Forgetting to normalize the blur kernel in deconvolution.
   • Ignoring noise variance $K$ in Wiener filtering.
5. Shortcut for Full Marks:
   • For theory questions, use the degradation model $g=Hf+n$ as your framework.
   • For numerical questions, always show:
     • Fourier transform steps (if frequency domain).
     • Median/mode calculations (if spatial domain).
     • Control point selection (for geometric corrections).

Sample Exam Question: "A 256×256 image is corrupted by salt-and-pepper noise with density 0.1. Design a restoration pipeline using spatial filters. Justify your choice."

Model Answer Structure:

1. Noise Type: Salt-and-pepper → nonlinear filter needed.
2. Filter Choice: Median filter (3×3 or adaptive).
3. Steps:
   • Apply median filter iteratively (2–3 passes).
   • Compare with mean filter (show why median preserves edges).
4. Visual: Before/after images with PSNR/SSIM metrics.
5. Real Tie-In: "This is used in NTC traffic cameras to clean license plate images for automated toll collection."

8. Summary Table

9. Key Formulas to Memorize

1. Degradation Model: $$g(x,y)=H\{f(x,y)\}+n(x,y)$$
2. Inverse Filtering (Frequency Domain): $$\hat{F}(u,v)=\frac{G(u,v)}{H(u,v)}$$
3. Wiener Filter: $$\hat{F}(u,v)=\frac{H^{\ast }(u,v)G(u,v)}{|H(u,v){|}^2+{\sigma }_n^2}$$
4. Median Filter (Spatial Domain): $$\hat{f}(x,y)=\text{median}\{g(x+i,y+j){\}}_{i,j=-1}^1$$
5. Affine Transform: $$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}e\\ f\end{array}\right]$$