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BC7 optimal solid-color blocks

November 3, 2024

That's right, it's another texture compression blog post! I'll keep
it short. By "solid-color block", I mean a 4x4 block of pixels that
all have the same color. ASTC has a dedicated encoding for these
("void-extent blocks"), BC7 does not. Therefore we have an 8-bit RGBA
input color and want to figure out how to best encode that color with
the encoding options we have. (The reason I'm writing this up now is
because it came up in a private conversation.)

In BC1/3, hitting a single color exactly is not always possible, and
we have the added complication of the decoder being under-specified.
BC7 has neither problem. If we look at our options in table 109 of
the Khronos data format spec, we see that mode 5 stores color
endpoints with 7 bits of precision per channel and alpha endpoints
with a full 8 bits, so it's a very solid candidate for getting our
8-bit colors through unharmed.

The alpha portion is trivial: we can send our alpha value as the
first endpoint and just use index 0 for the alpha portion (mode 5 is
one of the BC7 modes that code RGB and A separately, similar to BC3),
which leaves the colors. Can we use the color interpolation to turn
our 7 endpoint bits per channel into an effective 8?

We have 2 color index bits per pixel. Indices 0 and 3 are not useful
for us, they return the dequantized endpoint value, and those are 7
bits, so that only gives us 128 out of 256 possible options for each
color channel. Index 1 is interpolated between the two at a 21/64
fraction; index 2 is symmetric (exactly symmetric in BC7, unlike the
situation in BC1), i.e. it's the same as using index 1 with the two
endpoints swapped, and therefore doesn't give us any new options over
just using index 1. That means we only need to consider the case
where all index values are 1: if the value we need in a color channel
happens to be one of the 128 values we can represent directly, we
just set both endpoints to that value, otherwise we want select a
pair of endpoints so that the value we actually want is between them,
using that 21/64 interpolation factor (subject to the BC7 rounding
rules).

For BC1, at this point, we would usually build a table where we
search for ideal endpoint pairs for every possible target color. For
BC7, we can do the same thing, but it turns out we don't even need a
table. Specifically, if we build that table (breaking ties to favor
pairs that lie closer together) and look at it for a bit, it becomes
quickly clear that we can not only hit each value in [0,255] exactly,
but there's a very simple endpoint pair that works:

// Determine e0, e1 such that (BC7 interpolation formula)
//   target == (43*expand7(e0) + 21*expand7(e1) + 32) >> 6
// where expand7(x) = (x << 1) | (x >> 6)
e0 = target >> 1
e1 = ((target < 128) ? (target + 1) : (target - 1)) >> 1

And that's it. Do this for each of the R, G, B channels of the input
color, and set all the color indices to 1. As noted earlier, the A
channel is trivial since we just get to send a 8-bit value there to
begin with, so we just send it as one of the endpoints and leave all
alpha index bits at 0.

This is exact and the encoding we've used in all our shipping BC7
encoders, regular and with rate-distortion optimization both. Often
there are many other possible encodings using the other modes,
especially for particularly easy cases like all-white or all-black.
In our experiments it didn't particularly matter which encoding is
used, so we always use the above. The one thing that does matter is
that whatever choice you make should be consistent, since solid-color
blocks frequently occur in runs.

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