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What are the differences between the compositor's Shadow Pass and Diffuse Direct Passes in Cycles? I know one is that the Shadow Pass only responds to lamps and not mesh emitters. But even on renders that only use point lamps, it still seems to be different.

What other differences are there?

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The Shadow Pass is not as interested in the gradient intensity of the reflected light (like DiffDir) as it is in where the emitted light is not directly reflected back to the camera. This probably involves a threshold, even though I haven't looked at the code. From my tests, it seems that once the reflected light falls below a certain point, only then does it become non-white on the shadow pass. Whereas the purpose of the diffuse pass is to show the gradient direct light intensity:

enter image description here

Also, when you add another type of shader into the scene, the similarities between DiffDir and Shadow continue to diverge since the shadow pass is affected by them as well. An interesting phenomenon occurs when the Glossy Shader is at 0.0 roughness... it becomes totally black. This is because of what I said earlier about direct light reflection. As you probably know, when glossy is perfectly reflective, no off-axis light will be reflected back to the camera. Therefore, you get nothing on the shadow pass. But if you give it some roughness, it shows up. That's because it's reflecting enough directly emitted light to breach the threshold. The center cube below has a glossy shader applied:

enter image description here

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Diffuse Pass Vs Shadow Pass.blend

Here is a simple setup up for exploring the differences. There are two scenes in the blend file. One uses a white small point light and the other, a white world light. There are 8 basic materials, Diffuse, 50% Transparent, 100% Transparent (as based on the Value parameter in the HSV of the color of the shader being set to 0.5 and 1), Glossy, Emission, "Low Density" Volumetric, High Density Volumetric, and Translucent. I would try different things while constantly observing/toggling between the two passes.

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