Color Separation

Color Separation applies color deconvolution in optical density (OD) space:

1. OD conversion — The RGB image is converted to optical density: OD = −log(I/I₀), where I is the pixel intensity and I₀ is the illumination intensity (blank slide). This linearizes the relationship between stain concentration and measured signal.
2. Stain vector definition — Each stain has a characteristic color vector in RGB-OD space. Hematoxylin absorbs primarily in the red and green channels; DAB absorbs primarily in the blue channel. These vectors are measured from pure-stain regions or use preset values.
3. Matrix decomposition — At each pixel, the OD vector is decomposed as a linear combination of stain vectors: OD_pixel = c₁×v₁ + c₂×v₂ + c₃×v₃. The coefficients c₁, c₂, c₃ represent the concentration of each stain at that pixel.
4. Output — Separate grayscale images for each stain component, where pixel intensity reflects the local concentration of that specific stain.

Color spaces for histology: Gonzalez & Woods describe RGB as a perceptually nonlinear space that is "inherently difficult for humans to reason about because not related to natural colour perception." Raw RGB values don't directly correspond to stain concentrations because the relationship between absorption and detected intensity is logarithmic (Beer-Lambert law). Converting to optical density (OD = −log(I/I₀)) linearizes this relationship, making stain separation a simple linear algebra problem.

Beer-Lambert law: Pawley explains contrast formation in transmitted light microscopy through absorption: I_transmitted = I₀ × e^(−εcl), where ε is the extinction coefficient, c is the concentration, and l is the path length (section thickness). In OD space: OD = εcl. This linearity means that doubling the stain concentration doubles the OD — a property that is essential for quantitative analysis. In the RGB domain, the same doubling has a nonlinear effect.

Why three stains maximum: Standard RGB imaging captures three channels (red, green, blue). Color deconvolution with three stain vectors requires inverting a 3×3 matrix — exactly solvable. With more than three stains, the system becomes underdetermined in standard RGB. This is why H&E (2 stains + residual), H-DAB (2 stains + residual), and Masson's trichrome (3 stains) are naturally suited to RGB color deconvolution. Multiplex IHC with more than 3 chromogens requires multispectral imaging (more channels) rather than standard RGB.

HSV alternative: Solomon & Breckon note that HSV color space "allows separation of colour from lighting." For some applications, converting to HSV and analyzing the hue channel provides a simpler (if less rigorous) stain separation — hue distinguishes blue (hematoxylin) from brown (DAB) regardless of staining intensity. However, this approach doesn't provide quantitative stain concentrations and breaks down when stains have similar hues.

Quantifying DAB-stained PD-L1 in an IHC section:

1. Color Separation with H-DAB preset vectors decomposes the image into hematoxylin and DAB channels
2. The hematoxylin channel serves as input for nuclear detection (nuclei appear bright)
3. The DAB channel is thresholded to identify PD-L1-positive regions
4. Per-cell DAB intensity is measured using Standard Measurements on the nuclear/cytoplasmic compartments

Without color separation, brown DAB and blue hematoxylin would be mixed in every channel, making it impossible to accurately measure either stain independently.