| Statistic | Formula | Primary Use |
|---|---|---|
| Min | Minimum value in window | Local background estimation, equivalent to grayscale erosion |
| Max | Maximum value in window | Local peak detection, equivalent to grayscale dilation |
| Mean | Average value in window | Adaptive thresholding, local background estimation |
| Median | Middle value in sorted window | Robust local estimate (outlier-resistant) |
| Standard Deviation | σ = √(Σ(xi−μ)²/N) | Texture detection, edge detection, feature extraction |
| Variance | σ² = Σ(xi−μ)²/N | Same applications as std dev; sometimes preferred for computation |
BOM Group
Statistical Operations
Per-pixel and regional statistical computations
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Definition
Per-pixel or regional statistical computations — min, max, mean, median, standard deviation, and variance — applied within sliding windows or labeled regions for local feature extraction, adaptive processing, and texture analysis.
Local Statistics
Sliding window computes statistics at each pixel
Adaptive Thresholding Foundation
Local mean image serves as spatially varying threshold
Texture Detection
Local variance distinguishes textured from smooth regions
Morphological Equivalence
Local min = erosion, local max = dilation
Operations Reference
Simplified
Each operation computes a statistic within a sliding window. Mean gives average brightness per neighborhood — useful as an adaptive threshold. Std dev/Variance shows how much brightness varies locally — high variance means texture or edges. Min/Max find the darkest/brightest pixel per neighborhood.
When to Use Which
Adaptive Thresholding
Local mean — Compute with a window larger than the largest object. Then compare: original > (local_mean + offset). This detects locally bright structures regardless of regional illumination variation. Solomon & Breckon describe this as using "a different threshold at each pixel location" — the local mean serves as the spatially varying background estimate.
Texture Segmentation
Local variance or std dev — Textured regions (tissue with cellular structure) have high local variance, while smooth regions (glass background, homogeneous staining) have low variance. Thresholding the variance image segments by texture.
Background Estimation
Local min — With a window larger than bright objects, local minimum captures the background underneath objects. Useful as an alternative background estimation method.
Local max — With a window larger than dark artifacts, local maximum estimates the local foreground peak intensity.
Window Size Selection
The window size is critical: too small, and the statistic reflects individual objects rather than background; too large, and genuine illumination variation within the window averages out. As a rule of thumb, the window should be 3–5× larger than the objects of interest.
Simplified
For adaptive thresholding: compute local mean with a large window, then compare each pixel to its local mean — brighter-than-local-mean = foreground. For texture detection: compute local standard deviation — high values mean textured tissue, low values mean smooth glass. Window size should be 3–5× larger than the objects of interest.