Image arithmetic (Gonzalez & Woods): Arithmetic operations on images are foundational tools. Addition averages multiple acquisitions to improve SNR (noise is uncorrelated; signal adds constructively). Subtraction reveals changes (difference images) or removes unwanted components. Division produces ratio images that normalize for variation in one component. These operations are pixel-wise — they treat each pixel independently, which is appropriate when the operation being modeled (signal combination, normalization) is also pixel-wise.
Signal combination theory (Vetterli): In signal processing, combining signals (addition, subtraction) is well-understood. Addition in the spatial domain corresponds to addition in the frequency domain — the spectrum of the sum is the sum of the spectra. This means adding two channels preserves the frequency content of both. Subtraction of a low-frequency component (background) from the original preserves only the high-frequency content (biological features) — the same principle as background removal but applied channel-to-channel rather than within a single channel.
Hanrahan's complementary filters: Hanrahan notes that low-pass(f) + high-pass(f) = f — complementary filters always sum to the original. In virtual channel operations, if you split a signal into components (by subtraction), the components always sum back to the original. This conservation property ensures that virtual channel arithmetic doesn't create or destroy total signal — it redistributes it between channels.
When division is dangerous: Dividing by a channel with values near zero produces extremely large values — mathematically correct but biologically meaningless. Adding a small constant to the denominator (e.g., Ch1 / (Ch2 + 10)) prevents division by zero while preserving the ratio for well-expressed markers. This is standard practice in ratiometric imaging.
Virtual channels use the same image arithmetic principles as background removal and spectral unmixing — adding signals combines them, subtracting removes unwanted components, and dividing normalizes for variation. The key caution is division by near-zero values, which produces extreme outliers. Adding a small constant to the denominator prevents this artifact.
