Statistical Measures Commonly Used in Computer Vision
📊 Why Statistics Matter in Computer Vision
Computer vision systems rely heavily on statistics to:
- Handle noise and uncertainty
- Make decisions from pixel distributions
- Design robust algorithms
Most classical CV algorithms are statistical at their core.
📐 Mean (Average)
Definition
\(\mu = \frac{1}{N} \sum_{i=1}^{N} x_i\)
Usage
- Global / local brightness
- Adaptive thresholding
- Integral image based methods
Characteristics
- Simple and fast
- Sensitive to outliers
📉 Variance
Definition
\(\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2\)
Usage
- Texture strength
- Focus measure
- Otsu thresholding
Characteristics
- Measures spread
- Noise sensitive
📏 Standard Deviation
Definition
\(\sigma = \sqrt{\sigma^2}\)
Usage
- Adaptive thresholding (Niblack, Sauvola)
- Noise estimation
Characteristics
- Same unit as data
- Intuitive interpretation
📊 Histogram
Definition
Frequency distribution of pixel intensities.
Usage
- Global thresholding
- Histogram equalization
- Otsu / Entropy methods
Characteristics
- Captures global distribution
- Loses spatial information
🧠 Entropy
Definition
\(H = - \sum_i p(i) \log p(i)\)
Usage
- Maximum entropy thresholding
- Texture complexity
- Information-based segmentation
Characteristics
- Measures uncertainty
- Robust to illumination changes
🔗 Covariance
Definition
\(\text{Cov}(X,Y) = E[(X-\mu_X)(Y-\mu_Y)]\)
Usage
- PCA
- Feature correlation
- Motion analysis
Characteristics
- Directional relationship
- Scale dependent
🔄 Correlation Coefficient
Definition
\(r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}\)
Usage
- Template matching
- Stereo correspondence
Characteristics
- Normalized
- Range [-1, 1]
📌 Median
Definition
Middle value of sorted data.
Usage
- Median filtering
- Background modeling
Characteristics
- Robust to outliers
- Slower than mean
🧮 Percentile / Quantile
Definition
Value below which a percentage of data falls.
Usage
- Contrast stretching
- Robust thresholding
Characteristics
- Resistant to noise
- Distribution-aware
⚖️ Skewness
Definition
\(\text{Skewness} = E\left[ \left( \frac{x-\mu}{\sigma} \right)^3 \right]\)
Usage
- Histogram shape analysis
- Illumination bias detection
Characteristics
- Detects asymmetry
- Sensitive to noise
📈 Kurtosis
Definition
\(\text{Kurtosis} = E\left[ \left( \frac{x-\mu}{\sigma} \right)^4 \right]\)
Usage
- Texture classification
- Outlier detection
Characteristics
- Measures tail heaviness
- Sensitive to extreme values
🛡️ Robust Statistics (CV Practice)
| Statistic | Robust to Outliers | Typical Use |
|---|---|---|
| Mean | ❌ | Fast estimation |
| Median | ✅ | Noise removal |
| MAD | ✅ | Robust spread |
| Percentile | ✅ | Contrast control |
🧠 Practical Insight
Typical CV pipeline:
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Image
↓
Local / Global Statistics
↓
Decision (Threshold / Model)
↓
Geometry / Logic
Understanding which statistic to use is often more important than the algorithm itself.
🎯 Takeaway
Statistics are the decision engine of computer vision.
Choosing the right measure:
- Improves robustness
- Reduces parameter tuning
- Explains algorithm behavior
Classical CV = Statistics + Geometry + Logic 🚀