I'm curious to know why you want to use EXRs in your work if you don't even know what they are? What work are you doing?
EXR files are were developed for VFX pipelines. They have the ability to store arbitrary shader channels in a single file - so all map channels of a shader can be read from that single file. It is especially powerful in compositing.
Scanline files are stored line by line and tiled EXRs are stored in buckets. So a tiled EXR is an atlas that can load/unload mipped buckets of the file to save texture memory. Vray, for example can convert all the texture maps in your shaders to these tiled atlases(large VFX projects could have 1000s of 4/8K texture files)
16/32bit EXRs are files with half and full floating point data respectively. Floating point images can store exponentially more value data than LDR/regular texture files. For example: 32bit displacement maps or high dynamic range images(EXRs are commonly used in the same way as .HDR files for environment lighting)
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