Histogram

The Histogram engine visualizes the distribution of any per-cell measurement:

1. Measurement selection — Choose any column from the measurement table: marker intensity, area, compactness, derived value.
2. Binning — The measurement range is divided into bins of equal width. Each bin counts the number of cells with measurement values falling in that range.
3. Display — Bar height represents the count (or frequency) of cells in each bin. Optional overlays: fitted distributions, gate positions, population coloring by phenotype.
4. Statistics — Compute and display mean, median, standard deviation, coefficient of variation, percentiles, and modality (number of peaks).

Histogram as probability density (Gonzalez & Woods): The normalized histogram p(r_k) = n_k/N approximates the probability density function of pixel (or cell) intensities. This statistical interpretation underpins all threshold-based analysis: Otsu's method treats the histogram as a mixture of two probability distributions and finds their optimal separation. The histogram is therefore not just a visualization tool — it is the empirical estimate of the measurement's statistical distribution.

Histogram equalization: Gonzalez & Woods describe histogram equalization as a transform that spreads the histogram to use all available gray levels. The transform is the cumulative distribution function (CDF): s_k = CDF(r_k). This is relevant to tissue analysis when comparing samples with different staining intensity ranges — equalization normalizes the dynamic range, though it also distorts the proportional relationship between intensity and expression.

Gray levels and noise (Pawley): The number of meaningful gray levels in a histogram depends on SNR: g = 1 + SNR. With 100 photons per pixel, SNR = 10, yielding ~11 distinguishable levels. Below 25 photons/pixel, the histogram has only ~5 meaningful bins. This sets a fundamental floor on the resolution of any histogram-based analysis — if the measurement doesn't have enough precision to fill more than a few bins, the histogram cannot reveal fine population structure.

Bimodality as diagnostic: A truly bimodal histogram indicates two distinct populations — the basis for reliable binary gating. The "dip test" (Hartigan's) provides a formal statistical test for bimodality: if the histogram is significantly bimodal (p < 0.05), the two populations are statistically distinguishable, and a threshold between them is meaningful. If the histogram is not significantly bimodal, forcing a binary gate creates an artificial division of a continuous distribution.

Evaluating Ki-67 staining quality before analysis:

• Good staining: Bimodal histogram with clear valley at intensity 80 — negative population peaks at 30, positive population peaks at 150. Gate at 80 cleanly separates the two.
• Poor staining: Unimodal histogram peaking at 50 with a long right tail — no clear positive population. Forcing a gate at any point divides a continuous distribution arbitrarily.
• Too much background: Bimodal but with the negative peak at 120 (high background) and positive peak at 180 — populations overlap extensively. Background Removal needed before meaningful gating.

The histogram reveals the problem before you invest time in analysis. A quick histogram check is the single best quality control step in any tissue analysis workflow.