10/23/2018

Oodle 2.7.3 on the Nintendo Switch

It's been a while since I reported Switch performance, and we've made a lot of improvements to ARM performance since then (both for ARM32 and 64, for Switch, iOS, and Mobile), and of course added Leviathan. Time for an update!

I compare Oodle against the zlib implementation provided in the Nintendo SDK (nn deflate). Deflate is run at level 9, Oodle at level 8. I'm timing decode speed only.

On mixed content test file lzt99 :

Plot of log compresion ratio vs. log decode speed :

Raw numbers :

lzt99      : nn_deflate-l9 : 1.883 to 1 :   71.70 MB/s
lzt99      : Leviathan-z8  : 2.773 to 1 :  217.26 MB/s
lzt99      : Kraken-z8     : 2.655 to 1 :  282.96 MB/s
lzt99      : Mermaid-z8    : 2.437 to 1 :  526.64 MB/s
lzt99      : Selkie-z8     : 1.943 to 1 :  971.69 MB/s

Leviathan is about 3X faster to decode than zlib/deflate, with way more compression. The rest of the Oodle compressor family provides even faster decode speeds with lower compression ratios. The fastest, Oodle Selkie, is similar compression ratio to zlib/deflate but more than 10X faster to decode.

details :

This is with Switch SDK 5.5, clang-5.0.1 ; the nn:deflate speed has gotten a little worse since the last time I measured it (was 74.750 MB/s). The compression ratio of nn:deflate is the same. For reference, old numbers are here , for Oodle 2.6.0 and 2.4.2. For example you can see that LZNA was at 24 MB/s with compression just slightly below Leviathan. Leviathan is truly a remarkable beast.

10/22/2018

recip_arith without unused range

Finishing up, today we'll go through how to make recip_arith take advantage of the un-mapped portion of the coding range, and see some more connections to past work.

recip_arith uses only the top bits of "range" in the map, rounding down the cdf to range scaling factor when it converts to fixed point. Thus it always maps less than range.

The classic "range coder" does something similar. With the map :

uint32_t r_norm = ac->range >> cdf_bits;

forward(cdf) = cdf * r_norm;
the bottom "cdf_bits" of range are not used in this map, so a small portion of range is unused.

The standard way to make use of this extra range is to assign to the last symbol. Most of the older papers show this as the default. They'll do something like :

(range coder "map end")

uint32_t r_norm = ac->range >> cdf_bits;

forward(cdf) :
if ( cdf == cdf_tot ) return range;
else return cdf * r_norm;

that is, explicitly taking the last interval and making it go all the way to range. You may see this in other "range coder" implementations.

I do not do this in my range coder. The benefit is around 0.001 bpb (or less), and it does cost some speed. The benefit depends on whether you can give that extra range to a useful symbol. Ideally you would give it to the most probable symbol. What you are doing is reducing the code length of one symbol, you get the most benefit by reducing the code length of the symbol that occurs most often. The higher the probability of the MPS, the more benefit you will get, assuming you put the MPS at the end of the alphabet where it gets this range. (conversely if the end of the alphabet is a symbol that never/rarely occurs, you get no benefit).

I call this variant "map end". In recip_arith the tradeoff is a little more biased towards doing "map end", because the unmapped portion of range is larger.

(note that the "map end" excess range is less than the amount of range assigned per cdf; that is, it's equivalent to increasing that integer symbol frequency by some fractional amount, so it's not a huge distortion of the symbol probabilities)

But in recip_arith, just mapping that extra range to one symbol is more of a probability distortion than it is in the range coder, because the extra space is larger.

We can do something better. The scaling factor from cdf to range in recip_arith is the *floor* to fixed point. When range is near the top of that fractional truncation, you would much rather round up. But you can't round up because that would lead to using more than all of range, which is uncodeable :

int range_clz = clz32(range);
uint32_t r_top_down = range >> (32 - range_clz - RECIP_ARITH_TABLE_BITS);
uint32_t r_norm_down = r_top_down << (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits);

uint32_t r_top_up = r_top_down+1;
uint32_t r_norm_up = r_top_up << (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits);
r_norm_up == r_norm_down + (1 << (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits));

r_norm_down * cdf_tot <= range
r_norm_up * cdf_tot > range

if you just use r_norm_down , you use too little of range

unmapped_range = range - r_norm_down * cdf_tot = range & ( (1 << (32 - range_clz - RECIP_ARITH_TABLE_BITS)) - 1 )

conversely if you used r_norm_up for scaling, you would use too much range
the amount is the ~ bit reverse of those bottom bits
So inspired by SM98 we can use a mix of round-down & round-up of the scaling factor, such that all of range is covered.

What we want is to use round_up as much as possible, because it gives lower codelens, and then for enough symbols to avoid over-using range, switch to round_down. Just like SM98 we can find a threshold such that we place the transmission so that all of range is used.

The nice way to think about it conceptually is that we construct two maps : we use r_norm_down as the scaling factor from the left side of range (starting at 0), and we use r_norm_up as the scaling factor from the right side, and we switch over whether they cross.

The implementation is simple :

r_norm_down & r_norm_up as above

uint32_t map_up_excess = (r_norm_up << cdf_bits) - range;

forward(cdf) :

uint32_t cdf_down = cdf * r_norm_down;
int64_t cdf_up = cdf * r_norm_up - (int64_t) map_up_excess; // signed

ret = MAX( cdf_down, cdf_up );
and this can be simplified further with some algebra which I won't show.

Perhaps a picture is clearer :

I call this map "recip_arith down/up". The decoder is straightforward except for one point :

The reciprocal for both r_top and (r_top+1) have to be looked up. range is conceptually in [1,2) in fixed point, r_top can be exactly 1 but never 2. (r_top+1) can be exactly 2. The reciprocal table needs one more entry than (1< The coding loss of "recip_arith down/up" (at RECIP_ARITH_TABLE_BITS = 8) is extremely low. It's typically better than the "range coder" map, and comparable to the CACM87 map.

"recip_arith down/up" is even extremely good at RECIP_ARITH_TABLE_BITS = 4. That might be interesting for hardware implementations, because it means very tiny tables can be used, and the multipliers needed (for encoding) are also very low bit count.

Note that if you reduce RECIP_ARITH_TABLE_BITS all the way down to 1, then r_top_down is always just 1, and r_top_up is just 2. Then "recip_arith down/up" is choosing between a 1X and 2X mapping. This is exactly SM98 !

At RECIP_ARITH_TABLE_BITS = 2, then "recip_arith down/up" is choosing 1X or 1.5X or 2X. (the top bits are either 10 or 11 , we're getting only 1 fractional bit of range, the top bit is always on). This is exactly the same as the "reduced overhead" coder of Daala. (see OD_EC_REDUCED_OVERHEAD, in entcode.h)

OBJ2
range coder:                    246,814 ->   193,172 =  6.261 bpb =  1.278 to 1 
recip_arith (8 bits):           246,814 ->   193,282 =  6.265 bpb =  1.277 to 1 
recip_arith down/up (8 bits):   246,814 ->   193,171 =  6.261 bpb =  1.278 to 1 
recip_arith down/up (4 bits):   246,814 ->   193,240 =  6.264 bpb =  1.277 to 1 
recip_arith down/up (3 bits):   246,814 ->   193,436 =  6.270 bpb =  1.276 to 1 

PAPER3
range coder:                     46,526 ->    27,133 =  4.665 bpb =  1.715 to 1 
recip_arith (8 bits):            46,526 ->    27,156 =  4.669 bpb =  1.713 to 1 
recip_arith down/up (8 bits):    46,526 ->    27,133 =  4.665 bpb =  1.715 to 1 
recip_arith down/up (4 bits):    46,526 ->    27,127 =  4.664 bpb =  1.715 to 1 
recip_arith down/up (3 bits):    46,526 ->    27,155 =  4.669 bpb =  1.713 to 1 

PIC
H : 1.21464
range coder:                    513,216 ->    78,408 =  1.222 bpb =  6.545 to 1 
recip_arith (8 bits):           513,216 ->    78,651 =  1.226 bpb =  6.525 to 1 
recip_arith down/up (8 bits):   513,216 ->    78,408 =  1.222 bpb =  6.545 to 1 
recip_arith down/up (4 bits):   513,216 ->    78,395 =  1.222 bpb =  6.547 to 1 
recip_arith down/up (3 bits):   513,216 ->    78,806 =  1.228 bpb =  6.512 to 1 

PROGL
range coder:                     71,646 ->    42,723 =  4.770 bpb =  1.677 to 1 
recip_arith (8 bits):            71,646 ->    42,757 =  4.774 bpb =  1.676 to 1 
recip_arith down/up (8 bits):    71,646 ->    42,721 =  4.770 bpb =  1.677 to 1 
recip_arith down/up (4 bits):    71,646 ->    42,724 =  4.771 bpb =  1.677 to 1 
recip_arith down/up (3 bits):    71,646 ->    42,731 =  4.771 bpb =  1.677 to 1 

TRANS
range coder:                     93,695 ->    64,806 =  5.533 bpb =  1.446 to 1 
recip_arith (8 bits):            93,695 ->    64,851 =  5.537 bpb =  1.445 to 1 
recip_arith down/up (8 bits):    93,695 ->    64,806 =  5.533 bpb =  1.446 to 1 
recip_arith down/up (4 bits):    93,695 ->    64,820 =  5.535 bpb =  1.445 to 1 
recip_arith down/up (3 bits):    93,695 ->    64,884 =  5.540 bpb =  1.444 to 1 

recip_arith down/up (4 bits) has very low coding loss, it's very close to the standard range coder. (note that "range coder" here is not doing "map end" which would improve it slightly, particularly if end was assigned to the MPS, particularly so on "PIC" which is the only file where range coder noticeably misses the entropy)

Base recip_arith loss vs range coder is exactly 0.004 bpb in all cases here. This is with cdf_bits = 13 and a crappy frequency normalization heuristic, so there are some anomalies where eg. recip_arith down/up 4 bits gets more compression than 8 bits. recip_arith vanilla version at 4 bits gets very poor; coding loss is comparable to SM98, around 1.0%

recip_arith down/up is probably not practical for software implementation, but might be interesting for hardware. It's also interesting to me as a theoretical superset that connects recip_arith to SM98.

Working through recip_arith

Time to look at the actual recip_arith map and how it works.

recip_arith takes the top N bits of range (always a high 1 bit, and then N-1 fractional bits in the fixed point view) and uses them to scale the (power of 2 total) CDF up to the arithmetic domain.

The recip_arith forward map is :

int range_clz = clz32(range);
uint32_t r_top = range >> (32 - range_clz - RECIP_ARITH_TABLE_BITS);
uint32_t r_norm = r_top << (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits);

forward(cdf) = cdf * r_norm;

or

forward(cdf) = (cdf * r_top) << (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits);
r_norm is the scaling factor, close to (range/cdf_tot), differing in that all bits below RECIP_ARITH_TABLE_BITS are set to zero. Recall the standard "range coder" map is :
uint32_t r_norm = ac->range >> cdf_bits;

forward(cdf) = cdf * r_norm;
the only difference in recip_arith is that we are only using the top RECIP_ARITH_TABLE_BITS of "range" in the scaling. This is a floor of the ideal scaling factor, and means we use less of range than we would like.

Note that the "range coder" itself only uses (r_bits - cdf_bits) in its scaling factor, so it relies on r_bits reasonably larger than cdf_bits. This is in contract to CACM87 or full precision scaling.

The recip_arith map can be thought of as starting with the "identity map" from the last post, and then adding fractional bits to the fixed point in [1,2) , possibly 1.5X, 1.25X, etc. refining the scaling factor.

The point of course is to able to invert it. The naive inverse would be :

inverse(code) = code / r_norm;
which works, but has the divide we are trying to avoid. Our idea is that because r_top is small, we can use it to look up a reciprocal multiply. How should we do that? There are two choices, where we invert the steps of the forward map one by one, in reverse order :
forward(cdf) = (cdf * r_top) << (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits);

inverse(code) = (code >> (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits)) / r_top;

or

forward(cdf) = (cdf << (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits)) * r_top;

inverse(code) = (code / r_top) >> (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits);
if you actually do the divide, these are the same, but with the reciprocal multiply they are not. The issue is our reciprocal will only be exactly equal to the divide for some number of numerator bits. If we used the second form, we would need (code / r_top) , and "code" can be very large (24-31 bits or 56-63 bits). To do that reciprocal exactly would require an intermediate multiply larger than 64 bits.

Therefore we need the first form : first shift down code to only the necessary bits to resolve cdf boundaries, then do the divide :

inverse(code) :

uint32_t code_necessary_bits = code >> (32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits);

ret = code_necessary_bits / r_top;

ret = ( code_necessary_bits * recip_table[r_top] ) >> RECIP_ARITH_NUMERATOR_BITS;
(for 32-bit systems you might want to choose RECIP_ARITH_NUMERATOR_BITS = 32 so that this is a 32x32 -> hi word multiply; on 64-bit that's moot)

"code_necessary_bits" has its top bit at cdf_bits + RECIP_ARITH_TABLE_BITS ; in practice this is something like 14 + 8 = 22 . This is just enough bits that after dividing by r_top , you still resolve every cdf value exactly. In order for recip_table[] to have an exact reciprocal, it needs to be 1 more bit than the numerator, eg. cdf_bits + RECIP_ARITH_TABLE_BITS + 1. This means the intermediate that we need to multiply easily fits in 64 bits.

This acts as the limit on RECIP_ARITH_TABLE_BITS, aside from the consideration of how much space in L1 you can afford to give it.

With RECIP_ARITH_TABLE_BITS = 1 , recip_arith is the "identity map". With RECIP_ARITH_TABLE_BITS = 8, coding loss is very low (0.1%).

The fraction of "range" that can be wasted by the recip_arith map is 1/2^RECIP_ARITH_TABLE_BITS , which means the coding loss is :

!
coding loss @ maximum ~= -log2( 1 - 1/2^RECIP_ARITH_TABLE_BITS )

(eg. at RECIP_ARITH_TABLE_BITS = 8 this is 0.00565 bits)
The maximum occurs when all bits uner the top RECIP_ARITH_TABLE_BITS are 1; when they're all 0 this map is exact and there's no coding loss. The real world coding loss seems to be about 75% of the above formula.

Note that in a standard "range coder" you are already approximating to only use the bits of range between r_bits and cdf_bits; so eg. r_bits in 24-31 , cdf_bits of 14, means you are using 10-17 top bits of range in r_norm (the scaling factor). recip_arith just always uses 8 (for example) rather than the variable 10-17.

Note that recip_arith does not map all of range. We'll talk about variants that do use the whole range in the next post. (The standard range coder also doesn't map all of range). This means there are "code" values which the decoder cannot handle through its normal paths, where "code" is out of the mapped range. A valid stream produced by the encoder will never lead to those values being seen in the decoder, but a corrupt/attack/fuzzy stream could have them. Therefore a safe decoder must detect when code is out of the mapped range and at least handle it well enough not to crash. Testing and returning failure is an efficient way to handle it since it's just a branch that's never taken.

In review : the fully general CACM87 map uses 2 divides to encode, and 3 to decode. (encode is 2 forward maps, decode is 1 inverse map + 2 forward maps). By choosing cdf_tot to be a power of 2 we can easily eliminate the divide in the forward map. The divide in the inverse map is harder to remove, but we do it by reducing the number of bits used so that we can use a table of reciprocal multiplies.

The decoder critical path of instructions in recip_arith is : clz -> shift -> table lookup -> multiply . On modern desktops this is about the same speed or just a little faster than a divide.

10/21/2018

Arithmetic coder division-free maps

We have now narrowed in on the key issue that forces us to use division in arithmetic coders : the inverse map from arithmetic domain to CDF domain. Let's now start to look at division free options for that map.

Often henceforth I will talk about the map after normalizing the ratio to [1,2) by shifting CDFs. That is :


while cdf_tot*2 < range
    cdf's *= 2

same as :

left shift cdf's to put the top bit of CDF aligned with the top bit of range

Recall that in a typical implementation "range" has its top bit go from position 24-31. cdf_tot = (1< We now think in terms of a fixed point fraction. cdf_tot after normalizing is "one" , and "range" is in [1,2) , that is :

one   = 1.00000000000000000

range = 1.01001010101100000

essentially all we're getting at here is that in the scaling from CDF domain to "arithmetic domain", the binary powers of 2 to get us as close to range as possible are the easy part. We just shift the CDF's up. The fractional part of range in [1,2) is the hard part that makes us need a multiply/divide.

And that leads us directly to our first division free map :


identity map : aka "just shift" map :

r_bits = bsr(range) = position of top bit of range
shift = r_bits - cdf_shift

forward(cdf) = cdf << shift

inverse(code) = code >> shift

If we look at how this map performs for range in [1,2) , when range is near 1 ("one" in fixed point, that is, range nearly 1< (aside : in the ancient days, Rissanen actually used the "identity map")

The problem with this map is that it is failing to scale up cdf to use all of range when range is over 1. Obviously we can just check for that :


"one or 1.5 map" :

If range is in [1,1.5) , scale by 1X
if range is in [1.5,2) , scale by 1.5X 

forward(cdf) :

cdf_norm = cdf << shift;
ret = cdf_norm;

if ( range (fixed point) >= 1.5 ) (test 0.1 bit)
  ret += cdf_norm >> 1;

inverse(code) :

if ( range (fixed point) >= 1.5 )
  ret = (code * 2)/3;
else
  ret = code;

so we are now testing 1 fractional bit of range below the leading bit. This divides the "range" interval into two scaling zones. We could obviously test further fractional bits of range to divide the range into scaling zones like :

test 2 fractional bits of range (below top 1) :  [ 1X  ,  1.25X , 1.5X , 1.75X ]

but that's just the same as doing the normal range coder scaling, but only using the top 3 bits of range (top one + 2 fractional bits). And that is the same as the recip_arith forward map.

Note that in the "one or 1.5 map" it may look like we are still doing a divide to get from arithmetic domain back to cdf domain. But it's divide by the constant 3, which is implemented by the compiler (or us) as a reciprocal multiply. As we use more (fractional) bits of range, we need various other divides by constants (1/5,1/7,etc.) which are just reciprocal multiplies. Rather than branching, we can just use a table and take the top bits of range to look them up. This leads us directly to recip_arith, which we will flush out in the next post.

In these maps we are always using less than the full range. eg. say we do the "2 fractional bits of range" , if range is in the [1.25,1.5) fixed point interval, we will use 1.25X to scale from CDF to arithmetic domain. That is the correct scaling when range is the bottom of [1.25,1.5) but does not use all of range when range is larger. The reason is we can't do something like use a scaling factor that's the middle of the bucket (1.375), since when range is low that would give us forward(cdf) > range , which is uncodeable. Therefore we must also use the round-down or truncation of range to fewer top bits. This approach of using some number of fractional bits of range means that the map never makes use of all of range; as you add more bits, you can use more and more of range, but you are slowly approaching the limit from below.

There's an alternative approach due to Stuiver & Moffat ("Piecewise Integer Mapping for Arithmetic Coding"), commonly called SM98.

The SM98 map says : consider range and CDF normalized, so range is in [1,2) fixed point. If we just scale CDF by 1X everywhere ("identity map" above) we are not using all of range. We can't *ever* scale CDF by 2X uniformly, because range is strictly < 2 , that would make forward(cdf_tot) exceed range. What we can do is scale CDF by 2X for a portion of the interval, and 1X for the remainder, so that we use all of range :


Choose some CDF threshold t such that when we make the map :

cdf < t -> scale by 1X
cdf >= t -> scale by 2X

then we use the whole range in our map, eg. forward(cdf_tot) = range

The map is :

forward(cdf) :
if ( cdf < t ) ret = cdf
else ret = t + (cdf - t)*2 = cdf*2 - t

t = cdf_tot*2 - range

note that cdf_tot is normalized to be equal to range's top bit here,
so 't' is the same as "2 - range" in fixed point
that's the same as the ~ bit inverse of range's fractional bits

forward(cdf) :
if ( cdf < t ) ret = cdf;
else ret = range - (cdf_tot - cdf)*2;

forward(cdf) :
ret = MAX( cdf , range - (cdf_tot - cdf)*2 )

This final form with the branchless MAX is nicer for implementation, but it's also an alternate way to see the map. What we're doing is a 1X map for the early cdf's, starting at the left side of the arithmetic range. If we stuck with that map the whole way, it would not reach the end of range (and thus waste coding quality). We're simultaneously doing a 2X mapping of the late CDF's, starting at the *right* side of the arithmetic range. If we stuck with that map the whole way, it would overshoot zero and thus not be codeable. Where those two maps cross, we switch between them, thus using the more generous 2X mapping as much as possible.

(the inverse map is done similarly, just with >>1 instead of *2)

So, the SM98 mapping uses all of range, thus does not introduce coding loss due to failure to use all of range. It does, however, not assign intervals propertional to the probability.

When range is near 1, SM98 does the 1X map nearly everywhere, so its scaling is correct. When range is near 2, SM98 does the 2x map nearly everwhere, so again there is little coding loss. Intervals are proportional to the probability. The problem is when range is in the middle. Then some symbols will get a 1X scaling, and others will get a 2X scaling, distorting the probabilities, causing coding loss.

(aside: I did an experiment and confirmed that you can compensate for this somewhat with a skewed probability estimate. SM98 in this form gives too much code space to later symbols (you can of course choose to reverse that). To compensate you should increase the frequency more when you see early symbols than when you see later ones, so that the probability estimate is skewed in the opposite way of the coder. This does in fact cut the coding loss of SM98, roughly in half, from about 1.0% to 0.5%. Note this is totally evil and I'm not actually recommending this, just writing it down for the record.)

And I'll finish with a drawing :

Teaser : you can of course combine the ideas of "fractional bits of range" map and the SM98 map. When range is in the interval [1,1.5) you could use a 1X scaling for low CDF and 1.5X scaling for high CDF; in the [1.5,2) interval use an SM98 threshold to split the CDF's into intervals with 1.5X and 2X scaling. This was tried by Daala as the "reduced overhead" coder. We will come back to this later.

Oh, and another connection :

If you did the "fractional bits of range" scaling thing; first 1 bit giving you a 1.5X zone, then two bits adding 1.25X and 1.75X zones, etc. If you keep doing that all the way down, in a range coder framework you are trying to compute (range / cdf_tot). That means you need to look at (r_bits - cdf_bits). If you simply keep testing that number of bits and adding in the forward() map - you will wind up with the full range coder map.

That process is the Moffat-Neal-Witten DCC95 multiplication free coder. In that context you might want to choose r_bits = 14 or 15, bit renormalization. cdf_bits = 11 or so. The difference (r_bits - cdf_bits) is your coding precision (larger = less coding loss), and it's also the number of times you have to test bits of r and possibly do (ret += cdf>>n) in the forward map.

ADD :

I brought up thinking of range normalized to [1,2) as a conceptual aid, but that can also be a good implementation choice, particularly for coders like SM98 where you spend most of your time doing clz to find the top bit position of range. Instead of letting range float, like in Michael Schindler range coder 24-31 bits, you instead keep range pegged at 31 bits. That lets you avoid the clz to find the top bit, at the cost of doing a clz after encoding to find out how much range has shrunk to normalize it back up.

Now you might think this requires bit renorm, but it does not. Instead you can still do byte renorm, and keep a count of the number of spare bits at the *bottom*. So you are still doing 24-31 bit renorm, but the space is at the bottom instead of the top.

This implementation style is not shown in recip_arith for clarity, but I figure I better mention everything so Google can't patent it.

10/18/2018

About arithmetic coders and recip_arith in particular

An arithmetic coder encodes a sequence of symbols as an infinite precision number. You divide the range of that number (I will call this the "arithmetic domain") into successively smaller intervals; each interval corresponds to a symbol. After encoding a symbol, your current coding interval shrinks to the low/high range of that symbol, and you encode further symbols within that range. The minimum number of bits required to distinguish the desired sequence from another is the code length of the sequence.

If you make each symbol's interval proportional to the probability of that symbol, the arithmetic coder can produce a code length equal to the entropy. (we're still assuming infinite precision encoding). If your estimated probabilities do not match the true symbol probabilities (they never do) you have some loss due to modeling. The difference between the arithmetic coder's output length and the sum of -log2(P) for all model probabilities is the coding loss.

In order to do arithmetic coding in finite precision, we track the low/high interval of the arithmetic coder. As the top bits (or bytes) of low & high become equal, we stream them out. This is like "zooming in" on a portion of the infinite number line where the top & bottom of the interval are in the same binary power of two buckets.

We must then confront the "underflow problem". That is, sometimes (high - low) ("range") can get very small, but the top bits of high and low never match, eg. if they happen to straddle a binary power of 2. They can be something like


high = 1000000000aaaaa..
low  = 0111111111bbbbb..

There are several solutions to the underflow problem. For example CACM87 "bit plus follow" or the MACM / VirtQ approach which you can read about elsewhere, also the "just force a shrink" method (see links at end).

The method I use in "recip_arith" rather hides the underflow problem. Rather than checking for the top bits (bytes) of low & high being equal, we simply zoom in regardless. The renormalization step is :

    while ( ac->range < (1<<24) )
    {
        *(ac->ptr)++ = (uint8_t)(ac->low>>24);
        ac->low <<= 8;
        ac->range <<= 8;
    }
low and range are 32 bit here, when range is less than 2^24 the top byte of low & high is the same and can be shifted out, *except* in the underflow case. In the underflow case, we could have the top byte of low is = FF , and high is actually 100 with an implicit bit above the 32nd bit (eg. low + range exceeds 2^32). What we do is go ahead and output the FF, then if we later find that we made a mistake we correct it by propagating a carry into the already sent bits.

(note that you could do range += FF here for slightly less coding loss, but the difference is small; the actual "high" of our arithmetic interval is 1 above range, range can approach that infinitely closely from below but never touch it; the coding interval is [low,high) inclusive on the bottom & exclusive on the top. Coders that don't quite get this right have a lot of +1/-1 adjustments around low & high)

Renormalization means we can send an infinite length number while only working on a finite precision portion of that number down in the active range of bits at the bottom. Renormalization also means that "range" is kept large enough to be able to distinguish symbols with only integer subdivision of the range, which we shall now address. Renormalization in and of itself does not introduce any coding loss; it is perfectly accurate (though failing to add FF is coding loss, and schemes like the "force shrink" method or "just dont renormalize" method of fpaq0p do contribute to coding loss).

The other way we must adapt to finite precision is the division of the interval into ranges proportional to symbol probabilities. The infinite precision refinement would be (real numbers!) :

arithmetic_low += CDF_low * arithmetic_range / CDF_sum;

arithmetic_range *= CDF_freq / CDF_sum;

(real numbers, no floor divide)

(CDF = cumulative distribution function, aka cumulative probability, sum of previous symbol frequencies)
CDF_freq = CDF_high - CDF_low for the current symbol ; CDFs in [0,CDF_sum]
We don't want to do real numbers, so we just approximate them with integer math. But how exactly?

The crucial distinguishing aspect of an arithmetic coder is how you map the CDF domain to the arithmetic domain

The CDF domain is controlled by you; you have modeled probabilities somehow. The CDF domain always starts at 0 and goes to CDF_sum, which is under your control. In the decoder, you must search in the CDF domain to find what symbol is specified. Working in the CDF domain is easy. In contrast, the arithmetic interval is always changing; "low/range" is being shrunk by coding, and then zoomed in again by renormalization.

The forward map takes you from CDF domain to arithmetic domain. Adding on the arithmetic "low" is trivial and we will not include it in the map. The crucial thing is just scaling by (arithmetic_range / CDF_sum).

We can now write a very general arithmetic encoder :

arithmetic_low += forward(CDF_low,arithmetic_range,CDF_sum);

arithmetic_range = forward(CDF_high,arithmetic_range,CDF_sum) - forward(CDF_low,arithmetic_range,CDF_sum);
our "forward" map will be working on integers. Some properties forward must have :
forward(x) should be monotonic

forward(x+1) > forward(x) strictly  (so that range can never shrink to zero)

this may only be true for arithmetic_range > CDF_sum or some similar constraint

forward(0) >= 0
forward(CDF_sum) <= arithmetic_range

forward map of the CDF end points does not need to hit the end points of range, but it must be within them
(failure to use all of range does contribute to coding loss)
The coding loss of our approximation is caused by the difference in forward(high) - forward(low) and the ideal scaling (which should be proportional to range & symbol probability).

The integer forward map with lowest coding loss is the "CACM87 map" :


forward(cdf,range,cdf_sum) = ( cdf * range ) / cdf_sum;

this is now integers (eg. floor division)

CACM87 has
forward(cdf_sum) = range ; eg. it uses the full range

coding loss is just due to the floor division not exactly matching the real number divide. (note that you might be tempted to say, hey add (cdf_sum/2) to get a rounded integer division instead of floor; the exact form here is needed to be able to construct an inverse map with the right properties, which we will get to later).

Sketch of full arithmetic coding process

A quick overview of what the decoder has to do. Most of the decoder just replicates the same work as the encoder; it narrows the arithmetic interval in exactly the same way. Rather than streaming out bytes in renormalization, the decoder streams them in. The decoder sees the arithmetic code value that the encoder sent, to some precision ahead. It needs enough bits fetched to be able to resolve the correct symbol (to tell which CDF bin is selected).

In implementation, rather than track the low/high arithmetic interval and the arithmetic number within that interval, we instead just track (arithmetic - low), the offset inside the interval. I call this the "code" in the decoder.

The decoder needs an extra step that the encoder doesn't do : given the current "code" , figure out what symbol that specifies. To do that, we have to take the "code" (in the arithmetic interval domain), map it back to CDF domain, then scan the CDF intervals to find which symbol's bin it falls in.

To do so requires an "inverse" map (arithmetic domain -> CDF domain), the opposite of the "forward" map (CDF -> arithmetic) we just introduced.

A full general purpose (multi-symbol) arithmetic coder is :


(in integers now)

Encode :

look up CDF of the symbol you want to encode
map CDF interval to range inverval :

lo = forward(cdf_low,range,cdf_sum);
hi = forward(cdf_high,range,cdf_sum);

arithmetic_low += lo;
arithmetic_range = hi - lo;

propagate carry in "arithmetic_low" if necessary
renormalize if necessary

Decode :

take current arithmetic "code"
map it back to CDF domain :

target = inverse(arithmetic_code,range,cdf_sum);

find symbol from CDF target such that :
CDF_low <= target < CDF_high

rest proceeds like encoder:

lo = forward(cdf_low,range,cdf_sum);
hi = forward(cdf_high,range,cdf_sum);

arithmetic_code -= lo;
arithmetic_range = hi - lo;

renormalize if necessary

The encoder needs "forward" twice, the decoder needs "forward" twice plus "inverse" once.

Naive implementation of forward & inverse both need division, which would mean 2 and 3 divides for encode & decode, respectively.

The inverse map and when you don't need it

First of all, why do you need the inverse map, and when do you not need it?

One common case where you don't need the inverse map at all is binary arithmetic coding. In that case it is common to just do the forward map and resolve the symbol in arithmetic domain, rather than CDF domain.

That is :


binary decoder :

arithetmetic_code is known

map the threshold between bits 0 & 1 to arithmetic domain :

arihmetic_mid = forward(cdf_min,range,cdf_sum);

find bin in arithmetic domain :

symbol = arithetmetic_code >= arithmetic_mid;

lo/hi = { 0 , arithmetic_mid , range }

(in the binary case we also only need one forward map, not two, since one of the end points is always 0 or range).

Now, you can do the same technique for small alphabet multi-symbol, for 4 or 8 or 16 symbols (in SIMD vectors); rather than make a CDF target to look up the symbol, instead take all the symbol CDF's and scale them into arithmetic domain. In practice this means a bunch of calls to "forward" (ie. a bunch of multiplies) rather than one call to "inverse" (a divide).

But for full alphabet (ie 8 bit, 256 symbol), you don't want to scale all the CDF's. (exception: Fenwick Tree style descent). Typically you want a static (or semi-static, defsum) probability model, then you can do the CDF -> symbol lookup using just a table. In that case we can construct the symbol lookup in CDF domain, we need the map from arithmetic domain back to CDF domain.

The inverse map must have properties :


assume range >= cdf_sum
so the forward map is a stretch , inverse map is a contraction

should invert exactly at the CDF end points :

y = forward(x);
inverse(y) == x

the CDF buckets should map to the lower CDF :

lo = forward(x)
hi = forward(x+1)

(hi > lo , it can be much greater than 1 apart)

inverse( anything in [lo,hi) ) = x

in hand wavey terms, you need inverse to act like floor division. eg :

CDF domain [012]  -> arithmetic domain [00011122]

For example, for the CACM87 forward map we used above, the inverse is :
CACM87

forward(cdf,range,cdf_sum) = ( cdf * range ) / cdf_sum;

inverse(code,range,cdf_sum) = ( code * cdf_sum + cdf_sum-1 ) / range;

(integers, floor division)
In most general form, both forward & inverse map require division. The forward map is easy to make divide-free, but the inverse map not so.

Getting rid of division and cdf_sum power of 2

We'll now start getting rid of pesky division.

The first thing we can do, which we will adopt henceforth, is to choose cdf_sum to be a power of 2. We can choose our static model to normalize the cdf sum to a power of 2, or with adaptive modeling use a scheme that maintains power of 2 sums. (defsum, constant sum shift, sliding window, etc.)


cdf_sum = 1<<cdf_shift;

CACM87 forward(cdf,range,cdf_sum) = ( cdf * range ) >> cdf_shift;

So we have eliminated division from the forward map, but it remains in the inverse map. (the problem is that the CDF domain is our "home" which is under our control, while the arithmetic domain is constantly moving around, stretching and shrinking, as the "range" interval is modified).

We're fundamentally stuck needing something like "1 / range" , which is the whole crux of the problem that recip_arith is trying to attack.

I think we'll come back to that next time, as we've come far enough.

While I'm going into these forward/inverse maps lets go ahead and mention the standard "range coder" :


range coder :

forward(cdf,range,cdf_sum) = cdf * ( range / cdf_sum );

inverse(code,range,cdf_sum) = code / ( range / cdf_sum );

(integer, floor division)

or with power of 2 cdf_sum and in a more C way :

r_norm = range >> cdf_shift;

forward(cdf,range,cdf_sum) = cdf * r_norm;

inverse(code,range,cdf_sum) = code / r_norm;

where I'm introducing "r_norm" , which is just the ratio between "range" and "cdf_sum" , or the scaling from CDF to arithmetic domain.

Historically, the "range coder" map was attractive (vs CACM87) because it allowed 32 bit arithmetic state. In the olden days we needed to do the multiply and stay in 32 bits. In CACM87 you have to do (cdf * range) in the numerator, so each of those was limited to 16 bits. Because the arithmetic state (code/range) was only 16 bits, you had to do bit renormalization (in order to keep range large enough to do large CDF sums (actually byte renormalization was done as far back as 1984, in which case CDF sum was capped at 8 bits and only binary arithmetic coding could be done)). By adopting the "range coder" map, you could put the arithmetic state in 32 bits and still use just 32 bit registers. That meant byte renormalization was possible.

So, with modern processors with 64-bit registers there's actually very little advantage to the "range coder" map over the CACM87 map.

The range coder map has some coding loss. The fundamental reason is that the forward() map scaling is not exactly (Probability * range). Another way to think of that is that not all of range is used. In the "range coder" :


forward(cdf_sum) = cdf_sum * r_norm = r_norm << cdf_shift = (range >> cdf_shift) << cdf_shift

forward(cdf_sum) = range with bottom cdf_shift bits turned off

unused region is (range % cdf_sum)

The coding loss is small in practice (because we ensure that range is much larger than cdf_sum). In typical use, range is 24-31 bits and cdf_shift is in 11-14 , then the coding loss is on the order of 0.001 bpb. You can make the coding loss of the range coder arbitrarily small by using larger range (eg. 56-63 bits) with small cdf_sum.

The "range coder" map is actually much simpler than the CACM87 map. It simply takes each integer step in the CDF domain, and turns that into a step of "r_norm" in the arithmetic domain. The inverse map then just does the opposite, each run of "r_norm" steps in the arithmetic domain maps to a single integer step in the CDF domain.

.. and that's enough background for now.

The entry page for recip_arith : cbloom rants A Multi-Symbol Division Free Arithmetic Coder with Low Coding Loss using Reciprocal Multiplication

Links :

cbloom rants 10-05-08 Rant on New Arithmetic Coders
cbloom rants 10-06-08 Followup on the Russian Range Coder
cbloom rants 10-07-08 A little more on arithmetic coding ...
cbloom rants 10-10-08 On the Art of Good Arithmetic Coder Use
cbloom rants 08-16-10 - Range Coder Revisited .. oh wait, nevermind
cbloom rants 09-16-10 - Modern Arithmetic Coding from 1987
cbloom rants 10-08-08 Arithmetic coders throw away accuracy in lots of little places.
cbloom rants 08-11-10 - Huffman - Arithmetic Equivalence
On the Overhead of Range Coders

A Multi-Symbol Division Free Arithmetic Coder with Low Coding Loss using Reciprocal Multiplication

A standard multi-symbol arithmetic coder necessarily requires a costly divide to map the arithmetic code target back to a cdf (even when you have chosen the cdf sum to be a power of 2).

Doing binary arithmetic coding without a divide is trivial. Previous methods for division free multi-symbol arithmetic coding have been at the cost of low coding precision, that is coding in more bits than -log2 of the symbol probabilities (for example see DCC95 and SM98).

code is here :

recip_arith on github

recip_arith is public domain.

Our method uses an approximate map from the cdf interval to the range interval, scaling by only a few top bits of range. This approximate map introduces coding loss, but that can be made quite small with just 8 bits from range.

The advantage of this approximate map is that it can be exactly inverted using a table of reciprocal multiplies :

x / y == (x * recip_table[y]) >> 32

recip_table[y] = ((1<<32) + y-1) / y
x/y in integers is the floor divide, and recip_table is the ceil reciprocal. (see Alverson, "Integer Division using reciprocals"). The choice of a 32 bit numerator is somewhat arbitrary, but we do want the reciprocals to fit in 32 bit words to minimize the size of recip_table in L1 cache.

Crucially because only the top 8 bits of "range" are used in the forward map, then "y" needed in the divide is only 8 bits, so the recip_table can be quite small. Furthermore, 8 bits + 24 bits of cdf in the numerator "x" can be inverted exactly.

Coding Efficiency

The coding loss of "recip_arith" with 8 bit tables is reliably under 0.005 bpb (around 0.1%) , in constrast to SM98 where the coding loss can be 10X higher (0.05 bpb or 1.0%)

Calgary Corpus file "news" , len=377,109

cdf_bits = 13  (symbol frequencies sum to 1<<13)

from smallest to largest output :

recip_arith coder (down/up 8-bit) :                                244,641 = 5.190 bpb
cacm87:                                                            244,642 = 5.190 bpb
range coder:                                                       244,645 = 5.190 bpb
recip_arith coder (with end of range map) :                        244,736 = 5.192 bpb
recip_arith coder:                                                 244,825 = 5.194 bpb
recip_arith coder (down/up 2-bit) (aka Daala "reduced overhead") : 245,488 = 5.208 bpb
SM98 :                                                             245,690 = 5.212 bpb
(the down/up and "end of range map" variants will be discussed later)

The crucial step of recip_arith is the way the arithmetic coding interval is shrunk to encode a symbol.

The standard range coder does :

    uint32_t r_norm = ac->range >> cdf_bits;

    ac->low += cdf_low * r_norm;
    ac->range = cdf_freq * r_norm;
while recip_arith does :
    uint32_t range = ac->range;

    int range_clz = clz32(range);
    uint32_t r_top = range >> (32 - range_clz - RECIP_ARITH_TABLE_BITS);
    uint32_t r_norm = r_top << ( 32 - range_clz - RECIP_ARITH_TABLE_BITS - cdf_bits);
            
    ac->low += cdf_low * r_norm;
    ac->range = cdf_freq * r_norm;
where "r_top" is just the highest RECIP_ARITH_TABLE_BITS bits of range, and r_norm is those bits back in their original position, then shifted down by cdf_bits. In the end the way the interval is shrunk is exactly the same as the range coder, except that all bits below the top RECIP_ARITH_TABLE_BITS of "r_norm" are turned off.

Is it useful ?

It's unclear if there are really killer practical advantages to recip_arith at the moment. On many modern high end CPUs, 32 bit divides are pretty fast, so if you just take recip_arith and use it to replace a standard range coder you might not see much difference. On CPUs with slow divides there is a more significant advantage.

recip_arith provides an interesting continuum that connects very low precision coders like SM98 to the full precision range coder. The "up/down" variant with a very small reciprocal table (4 bits?) might be interesting for hardware implementations, where division is not desirable.

recip_arith works with 64-bit coding state (the standard range coder would require a 64-bit divide, which is often much slower than a 32-bit divide), which can provide advantages. recip_arith also works with the encoder & decoder not in lock step; eg. you could encode with 32-bit state and decode with 64-bit state, allowing you independence of the bitstream & implementations, which is very desirable; not all arithmetic coders have this property.

I think primarily it is theoretically interesting at the moment, and it remains to be seen if that turns into a compelling practical application.

What's Next

I'll be doing a couple of followup posts going into the details of how/why this works. We'll talk about the theory of arithmetic coders and how to think about them. Then we'll look at the up/down and "map end" variants.

Index of posts

recip_arith on github
1. cbloom rants About arithmetic coders and recip_arith in particular
2. cbloom rants Arithmetic coder division-free maps
3. cbloom rants Working through recip_arith
4. cbloom rants recip_arith without unused range

10/13/2018

If you have had difficulty sending me email

My email has been extremely flaky over the last few months due to problems at dreamhost (who host cbloom.com).

If you sent me an email and I did not reply, it's possible I didn't get it. Check your spam for bounced emails, since in my experience many undelivered mail responses incorrectly wind up in spam folders these days.

I will be changing to a new host (because dreamhost sucks and has consistently failed to fix their email issues). There may be more missed emails in the transition.

If you want to send me something reliably, please try carrier pigeon or can on string, since technology has only gotten worse since then. (grumble grumble, we all as an industry fucking epically suck and computers are monumentally awful and depressing, grumble grumble)


Transition in progress... looks like I'll be moving cbloom.com web hosting off dreamhost as well. Let me know if you see any problems.


... done. Now on fastmail.

10/01/2018

Oodle Lossless Image

We have released Oodle Lossless Image Compression (aka OLI) v1.0

OLI is built on the blazing fast Oodle Data Compression engine, with a new very efficient image specific front end. OLI has a very simple native API, and it also has drop in look-alike APIs to replace stb_image or libpng, so it's very easy to integrate.

OLI gets much more compression than PNG. Its compression ratio is similar to higher compression codecs like webp-ll or FLIF. The big advantage of OLI is its blazing fast decode speed. OLI decodes way faster than any of the competition (3-10X faster), so you can compress big images smaller and load them faster.

(OLI is mostly by Jon Olick who did an awesome job on it)

OLI is currently for full color lossless images only. It's not for 3d game textures, it doesn't do BCn GPU compressed textures, it doesn't do things like half-float normal maps. It's for 24 bit RGB and 32 bit RGBA images, the kind of things you would have used PNG for before. OLI currently has only limited support for low color images (eg. palettized, 1-bit, and gray scale images). It's early days for OLI still and that support will get better, particularly if we hear from customers who need it.

For more about OLI visit the RAD Game Tools web site.

Image sizes on my "good" test set, smallest size for each image in bold :

oli webpll pngcrush png
4069138464_193981e0e2_o 3140944 3319222 4019282 4025911
9xkFD 2524927 2772626 3850985 3856541
A-Girl-With-Kind-Eyes 1212598 1307518 1568931 1620105
Beach_Hurricane_WaLp_TW2011 1970283 1997532 2543245 2606013
Beach_Starfish_and_Conch_Shell_Wp_TW 2266789 2299242 2591019 2701580
CliffSideA01_cm0001 3475407 3340540 4302971 4309175
CliffSideA03_cm0001 3626390 3560460 4697715 4704495
Girl_in_white 1253577 1313662 1726569 1729053
IMG_0060 2324810 2601634 3767787 3773211
IMG_0152 2166388 2389052 3668094 3673386
IMG_0155 1526228 1613594 2369898 2373318
IMG_0195 1998851 2115548 2910248 2914448
IMG_0339 1815852 2021904 3153675 3158211
IMG_0341 1196634 1267144 1865148 1867836
LOVE1 1170354 1217124 1598967 1601271
Lily+Allen+smile 1156702 1291656 1850890 1853554
PDI_1200 921293 959710 1260621 1262433
POMSep2007 1061705 1066320 1520546 1522742
Portal2Win 2992069 2904416 3695920 3701248
Spring_Beauty 1749534 1788812 2041309 2118280
Sunset_Rocky_Beach_Walp_TW 2700270 2665954 3137206 3192653
ainokura_bibble 817631 817516 1085462 1087022
artificial 721090 578668 845572 846796
big_building 3772059 3795954 4569456 4579407
big_tree 3476811 3466330 3747685 3753085
bmw 1408411 1429148 1790859 1793439
bridge 1401722 1400592 1535838 1538094
c77b75083b34 2516018 2657536 3818229 3823737
cathedral 1914743 1926612 2212063 2215255
church-in-the-prairie 1955457 1984848 2973585 2977869
cool-orb 379704 326258 492951 493671
deer 4753833 4454234 4757131 4763983
flower_foveon 1701007 1672546 2194479 2213494
hdr 1588182 1610458 1942456 2011759
highclass_by_DivineError_lossless 591236 588602 848270 849494
leaves_iso_200 2537504 2613286 3120814 3125314
moses 2137994 2183616 3220571 3225215
mysoup 2473925 2717346 3631056 3636300
nails2 2181876 2300032 3228688 3233344
nightshot_iso_100 2023458 1948254 2293315 2397815
phoenix 110389 59800 83692 131474
pulchra2.1 2421925 2536570 3482309 3487337
record_only 789644 790842 1211780 1282677
school-shoot-moodboard-number-1 1587878 1638888 2199389 2202557
space-desktop 561563 545412 724571 725615
spider_web 2119921 2203508 2603659 2715256
windmill_evsm 1470104 1487660 2090287 2093311
yahoo 1999722 1700342 2710221 2849650
total 91665412 93248528 121555414 122618434

oli is oli_enc --super
webpll is cwebp -lossless -z 9
pngcrush is default options
png is made by bmp2png


Oodle is an SDK for high performance lossless data compression. For more about Oodle, or licensing inquiries, visit the RAD Game Tools web site. This is my personal blog where I post supplemental material about Oodle.

old rants