09-21-08 - 5

Followups :

Lomont says you can't rely on the behavior of shifts on negative numbers. IMO if you start dealing with really weird platofrms then you need to make wrapper functions for everything and customize them for each platform.

I've been thinking about negatives and two's complement and such. It's one of those things that's common knowledge and all, but it seems new and confusing every time I think about it.

The easiest way for me to understand it is to think about ints as a ring. Unsigned ints are a ring that go from [0,FFFF] and then loop back around. Don't think about the integer line as a straight line, it's a circle like a clock. You count up to 65535 then +1 and you're back at 0. Similarly you can subtract by going backwards on the clock, so 3 - 7 = 65532. Sometimes I get confused when I'm doing subtracts with unsigned values, but of course subtracting unsigned just means walk backwards around the ring.

Now, signed ints are just exactly the same thing, but offset into the middle of the range. Note with signed ints, 0 is part of the "positive" half of the range. It helps to think of 0 as positive, and the actual middle of the range is at -0.5. So the positives are [0,32767] and negative is [-32768,-1]. This is also a ring, it loops around, when you take 32767 and add 1 you get -32768.

Conceptually you could pretend that a signed value is unsigned offset by half, like S = U - 32768. That's not how the bits are actually stored, but it gives you equivalent operations. It makes it obvious that addition and subtraction and so on produce the exact same values, because you're just offset by a constant amount around the ring.

In terms of data compression if you think about the ring, in unsigned ints, 0 and FFFF are actually neighbors, but they look very different bitwise. As an example, we can consider the JPEG-LS or PNG style bytewise delta operations :

D = A - B;
A = D + B;
These are reversible in 8 bits. That's not completely obvious. The range of D is actually [-255,255] , it would seem we've expanded our 8 bit values to 9 bits. But depending on the value of B, not all of that range is possible. A - B is the delta from B, which is the number of steps on the ring from B in the positive direction. If A is less than B, you'll loop all the way around the ring, eg. 3 - 7 = 252 steps around the ring forward. That number of steps is always in [0,255]. To undo the delta you just take the same number of steps. Obviously you don't want to transmit this because it's offset, but
D = A - B + 128;
A = D + B - 128;
is nicely centered. Again the ring wraps when A-B = 127 to 128 you get a bitwise discontinuity, but that's okay because in compression that should be very rare, and those values have equal probability.

Speaking of probabilities, I wrote a little thing before about how to shift and preserve average by making half round down and half round up. It's an exercise for the reader to make that work on negative numbers too (aka, I don't want to do it myself).

Writing with Chris it made me realize that division has a very weird property with negatives. Int division goes to zero (in C). Shifts always go "down" (towards minus infinity (on x86 anyway)). Int division basically takes the abs, does a positive divide, then puts the sign back (it's not actually done this way of course, that's just the concept). Shifts are actual bitwise register shifts, so the top sign bit gets in. What this means is that integer division actually makes you switch directions of rounding at zero. That never really seemed like a big deal to me, but man it's really weird. For example, if you have a bunch of signed ints and take the average, now you divide them all by two in ints - take the average of the new numbers and multiply that up by two. With real numbers you'd have the same value. With positive ints, what you get is everything shifted down towards zero a bit, so the result is less by 0.5. If you had a bunch of numbers whose average was zero before, the average is still zero! Because half went down and half went up!

In terms of probabilities, if you have a bunch of ints scattered evenly over the int line, and you divide them all by two - the probabilities of the result are no longer evenly distributed! The probabilty of a zero is 3/2 the probability of any other value. All values in the result come from 2 values in the source, eg. {7,6} -> 3 , except for zero which comes from 3 values! {-1,0,1} -> 0. This makes it obvious why you can never use division of negatives for wavelet transforms or any kind of reversible mapping, because you are mapping too many values to zero. BTW in a "deadzone" style quantizer we actually do this exact thing on purpose, we make the zero more likely than anything else.

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