Otsu Threshold

The engine computes the threshold in four steps:

1. Histogram. Compute the intensity histogram of the input image, or of the intensity range set by the Lower and Upper limits.
2. Exhaustive search. For every possible threshold T from the minimum to the maximum intensity:
   • Split all pixels into background (intensity ≤ T) and foreground (intensity > T).
   • Compute the weight, mean and variance of each group.
   • Compute the between-class variance: σ²_B = w₁ × w₂ × (m₁ − m₂)².
3. Selection. Choose the T that maximizes σ²_B, where the two groups are most separated.
4. One value for the whole sample. The threshold draws on all fields of view, so one value applies to the entire sample or region of interest. Other engines use it: for example, Total AreaLoading... can threshold with the value this engine produces.

The mathematics. For a threshold T on an image with L gray levels, w₁(T) and w₂(T) are the probabilities of the background class (pixels ≤ T) and the foreground class (pixels > T), m₁(T) and m₂(T) are their means, and m_G is the global mean. The between-class variance is σ²_B(T) = w₁(T) × w₂(T) × [m₁(T) − m₂(T)]². Otsu proved that the threshold maximizing σ²_B minimizes the probability of classification error when the two classes have equal covariance. The search is fast: its cost grows only linearly with the number of gray levels.

Where Otsu needs help. Three conditions blur the two-peak picture the method relies on:

• Small object area. When the foreground occupies a tiny fraction of the image, such as sparse fluorescent dots on a large dark background, its peak barely influences the histogram, and Otsu may place the threshold in the middle of the background distribution.
• High noise. Noise fills the valley between the histogram peaks, so the bimodal structure is less distinct and the optimal threshold less stable.
• Non-uniform illumination. When background intensity varies across the image, no single global threshold classifies every region correctly. Adaptive (local) thresholding, which applies Otsu in small neighborhoods, addresses this case.

Otsu in the broader context. Otsu's method ignores spatial structure and classifies each pixel by intensity alone. In effect it is a Bayesian classifier with equal Gaussian variances and equal priors, working in one dimension, intensity. Approaches such as MRF-based segmentation and learned classifiers use spatial context, the tendency of neighboring pixels to belong to the same class, for better results, but Otsu's simplicity and speed keep it the standard first approach.

Creating a tissue mask from an autofluorescence channel:

1. In the autofluorescence channel, tissue is moderately bright and the glass background is dark.
2. Otsu Threshold (global) finds the tissue/glass boundary in the intensity histogram automatically.
3. Total AreaLoading... uses that threshold to create the mask that defines the analysis region. Tissue Detection does a similar job on the low-resolution preview image, with its own fragment-merging, minimum-area and contour-inflate settings.

The tissue/glass histogram is strongly bimodal with a deep valley, which is the case Otsu's method handles well.