🧠 Bilateral Filtering in Image Processing – Tutorial
🟡 1. Motivation
In image denoising, we often want to remove noise but preserve edges. Classical filters like Gaussian blur are fast but blur across edges, causing loss of detail.
Bilateral filtering solves this by combining:
- Spatial closeness (like Gaussian blur)
- Intensity similarity (to avoid averaging across edges)
🟩 2. Intuition
Imagine averaging only those pixels that:
- Are close in distance
- Have similar brightness/color
So, a pixel is influenced only by neighboring pixels that look similar.
It’s like a smart blur that doesn’t mix objects of different intensities.
🔢 3. Mathematical Formulation
Let $I(x)$ be the intensity at pixel $x$. Then the bilateral filter output at $x$ is:
\[I_{\text{out}}(x) = \frac{1}{W(x)} \sum_{x_i \in \Omega} I(x_i) \cdot f_s(\|x - x_i\|) \cdot f_r(|I(x) - I(x_i)|)\]Where:
- $\Omega$ = local neighborhood of $x$
- $f_s$ = spatial Gaussian → penalizes far-away pixels
- $f_r$ = range Gaussian → penalizes intensity differences
-
$W(x)$ = normalization factor:
\[W(x) = \sum_{x_i \in \Omega} f_s(\|x - x_i\|) \cdot f_r(|I(x) - I(x_i)|)\]
🧮 4. Kernel Details
- $f_s(d) = \exp\left(-\frac{d^2}{2\sigma_s^2}\right)$
- $f_r(\Delta I) = \exp\left(-\frac{(\Delta I)^2}{2\sigma_r^2}\right)$
| Term | Meaning |
|---|---|
| $\sigma_s$ | Controls spatial influence radius |
| $\sigma_r$ | Controls range sensitivity (how different intensities are tolerated) |
📊 5. Visual Flowchart
For each pixel x:
└─ For each neighbor x_i in window:
├─ Compute spatial weight: f_s(||x - x_i||)
├─ Compute range weight: f_r(|I(x) - I(x_i)|)
├─ Multiply both weights and image value: weight * I(x_i)
└─ Normalize by total weight sum
🧪 6. Python Code Example (Simplified)
import cv2
import numpy as np
img = cv2.imread('image.jpg')
filtered = cv2.bilateralFilter(img, d=9, sigmaColor=75, sigmaSpace=75)
| Param | Meaning |
|---|---|
d |
Diameter of neighborhood |
sigmaColor = $\sigma_r$ |
How much color difference is tolerated |
sigmaSpace = $\sigma_s$ |
How much spatial distance is tolerated |
🧠 7. Comparison to Non-Local Means (NLM)
| Feature | Bilateral Filter | Non-Local Means (NLM) |
|---|---|---|
| Neighborhood | Local (fixed radius) | Non-local (can span image) |
| Weight based on | Spatial + Intensity difference | Patch similarity |
| Edge-preserving? | Yes | Yes |
| Texture handling | OK | Better (uses structure) |
| Computation | Fast | Slow (can be accelerated) |
| Noise removal quality | Moderate | High |
| Sensitive to parameter tuning | Moderate | High |
| Mathematical base | Point-wise similarity | Patch-wise similarity |
| Implemented in OpenCV | cv2.bilateralFilter() |
cv2.fastNlMeansDenoising() |
🧩 8. Intuition: Bilateral vs NLM
Bilateral:
“I’ll average with nearby pixels that are close in value.”
Non-Local Means:
“I’ll average with any patch in the image that looks like me, regardless of where it is.”
🧬 9. Non-Local Means (NLM) Brief Overview
Let $P(x)$ be a patch around pixel $x$. NLM is:
\[I_{\text{out}}(x) = \frac{1}{Z(x)} \sum_{x_i} I(x_i) \cdot \exp\left(-\frac{\|P(x) - P(x_i)\|^2}{h^2}\right)\]Where $h$ controls filtering strength.
🧪 NLM Python Code
dst = cv2.fastNlMeansDenoisingColored(img, None, h=10, hColor=10, templateWindowSize=7, searchWindowSize=21)
🧠 10. When to Use What?
| Scenario | Use |
|---|---|
| Fast edge-preserving smoothing | Bilateral |
| High-quality denoising of textures (but slow) | NLM |
| Real-time system | Bilateral |
| Medical, satellite, or low-light images | NLM (better structure preservation) |
✅ Summary
- Bilateral Filter: Edge-preserving blur using spatial + intensity proximity.
- NLM: Powerful denoiser using patch-based similarity, non-local in nature.
- Bilateral = simple, fast, local
- NLM = accurate, slow, global
Would you like me to walk through an end-to-end image example in NumPy, or illustrate both filters side-by-side?