Histogram as probability density (Gonzalez & Woods): Divide each bin count by the total, p(r_k) = n_k/N, and the histogram approximates the probability density of pixel (or cell) intensities. Every threshold method rests on that reading. Otsu's method, for one, treats the histogram as a mixture of two distributions and finds where they separate best. The histogram is the empirical estimate of the measurement's distribution.
Histogram equalization (Gonzalez & Woods): Equalization spreads the histogram across all available gray levels, using the cumulative distribution function as the transform: s_k = CDF(r_k). It helps when you compare samples with different staining intensity ranges, because it normalizes the dynamic range. The price is that it distorts the proportional relationship between intensity and expression.
Gray levels and noise (Pawley): The number of meaningful gray levels depends on the signal-to-noise ratio: g = 1 + SNR. With 100 photons per pixel, SNR = 10, which gives about 11 distinguishable levels. Below 25 photons per pixel, the histogram has only about 5 meaningful bins. If a measurement is too imprecise to fill more than a few bins, no histogram of it can show fine population structure.
Bimodality as a test: Two genuine peaks mean two distinct populations, which is the basis for reliable binary gating. Hartigan's dip test checks this formally: if the histogram is significantly bimodal (p below 0.05), the two populations are statistically distinguishable and a threshold between them is meaningful. If it is not, a binary gate imposes an artificial division on a continuous distribution.
The histogram is the measured distribution of your values, and Otsu's method, like every threshold method, finds the natural boundaries in it. How many meaningful bins it has depends on the measurement's precision, which depends on how many photons were collected. Two well-separated peaks are the best case for gating; one featureless peak means the marker may not divide the cells into useful categories.
