Internet DRAFT - draft-terriberry-netvc-codingtools
draft-terriberry-netvc-codingtools
netvc T. Terriberry
Internet-Draft N. Egge
Intended status: Informational Mozilla Corporation
Expires: October 26, 2017 April 24, 2017
Coding Tools for a Next Generation Video Codec
draft-terriberry-netvc-codingtools-02
Abstract
This document proposes a number of coding tools that could be
incorporated into a next-generation video codec.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Entropy Coding . . . . . . . . . . . . . . . . . . . . . . . 2
2.1. Non-binary Arithmetic Coding . . . . . . . . . . . . . . 4
2.2. Non-binary Context Modeling . . . . . . . . . . . . . . . 5
2.3. Dyadic Adaptation . . . . . . . . . . . . . . . . . . . . 6
2.4. Simplified Partition Function . . . . . . . . . . . . . . 9
2.5. Context Adaptation . . . . . . . . . . . . . . . . . . . 11
2.5.1. Implicit Adaptation . . . . . . . . . . . . . . . . . 11
2.5.2. Explicit Adaptation . . . . . . . . . . . . . . . . . 12
2.5.3. Early Adaptation . . . . . . . . . . . . . . . . . . 12
2.6. Simple Experiment . . . . . . . . . . . . . . . . . . . . 13
3. Reversible Integer Transforms . . . . . . . . . . . . . . . . 14
3.1. Lifting Steps . . . . . . . . . . . . . . . . . . . . . . 14
3.2. 4-Point Transform . . . . . . . . . . . . . . . . . . . . 17
3.3. Larger Transforms . . . . . . . . . . . . . . . . . . . . 20
3.4. Walsh-Hadamard Transforms . . . . . . . . . . . . . . . . 20
4. Development Repository . . . . . . . . . . . . . . . . . . . 22
5. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 22
6. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 22
7. References . . . . . . . . . . . . . . . . . . . . . . . . . 22
7.1. Informative References . . . . . . . . . . . . . . . . . 22
7.2. URIs . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 24
1. Introduction
One of the biggest contributing factors to the success of the
Internet is that the underlying protocols are implementable on a
royalty-free basis. This allows them to be implemented widely and
easily distributed by application developers, service operators, and
end users, without asking for permission. In order to produce a
next-generation video codec that is competitive with the best patent-
encumbered standards, yet avoids patents which are not available on
an open-source compatible, royalty-free basis, we must use old coding
tools in new ways and develop new coding tools. This draft documents
some of the tools we have been working on for inclusion in such a
codec. This is early work, and the performance of some of these
tools (especially in relation to other approaches) is not yet fully
known. Nevertheless, it still serves to outline some possibilities
that NETVC could consider.
2. Entropy Coding
The basic theory of entropy coding was well-established by the late
1970's [Pas76]. Modern video codecs have focused on Huffman codes
(or "Variable-Length Codes"/VLCs) and binary arithmetic coding.
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Huffman codes are limited in the amount of compression they can
provide and the design flexibility they allow, but as each code word
consists of an integer number of bits, their implementation
complexity is very low, so they were provided at least as an option
in every video codec up through H.264. Arithmetic coding, on the
other hand, uses code words that can take up fractional parts of a
bit, and are more complex to implement. However, the prevalence of
cheap, H.264 High Profile hardware, which requires support for
arithmetic coding, shows that it is no longer so expensive that a
fallback VLC-based approach is required. Having a single entropy-
coding method simplifies both up-front design costs and
interoperability.
However, the primary limitation of arithmetic coding is that it is an
inherently serial operation. A given symbol cannot be decoded until
the previous symbol is decoded, because the bits (if any) that are
output depend on the exact state of the decoder at the time it is
decoded. This means that a hardware implementation must run at a
sufficiently high clock rate to be able to decode all of the symbols
in a frame. Higher clock rates lead to increased power consumption,
and in some cases the entropy coding is actually becoming the
limiting factor in these designs.
As fabrication processes improve, implementers are very willing to
trade increased gate count for lower clock speeds. So far, most
approaches to allowing parallel entropy coding have focused on
splitting the encoded symbols into multiple streams that can be
decoded independently. This "independence" requirement has a non-
negligible impact on compression, parallelizability, or both. For
example, H.264 can split frames into "slices" which might cover only
a small subset of the blocks in the frame. In order to allow
decoding these slices independently, they cannot use context
information from blocks in other slices (harming compression). Those
contexts must adapt rapidly to account for the generally small number
of symbols available for learning probabilities (also harming
compression). In some cases the number of contexts must be reduced
to ensure enough symbols are coded in each context to usefully learn
probabilities at all (once more, harming compression). Furthermore,
an encoder must specially format the stream to use multiple slices
per frame to allow any parallel entropy decoding at all. Encoders
rarely have enough information to evaluate this "compression
efficiency" vs. "parallelizability" trade-off, since they don't
generally know the limitations of the decoders for which they are
encoding. That means there will be many files or streams which could
have been decoded if they were encoded with different options, but
which a given decoder cannot decode because of bad choices made by
the encoder (at least from the perspective of that decoder). The
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same set of drawbacks apply to the DCT token partitions in
VP8 [RFC6386].
2.1. Non-binary Arithmetic Coding
Instead, we propose a very different approach: use non-binary
arithmetic coding. In binary arithmetic coding, each decoded symbol
has one of two possible values: 0 or 1. The original arithmetic
coding algorithms allow a symbol to take on any number of possible
values, and allow the size of that alphabet to change with each
symbol coded. Reasonable values of N (for example, N <= 16) offer
the potential for a decent throughput increase for a reasonable
increase in gate count for hardware implementations.
Binary coding allows a number of computational simplifications. For
example, for each coded symbol, the set of valid code points is
partitioned in two, and the decoded value is determined by finding
the partition in which the actual code point that was received lies.
This can be determined by computing a single partition value (in both
the encoder and decoder) and (in the decoder) doing a single
comparison. A non-binary arithmetic coder partitions the set of
valid code points into multiple pieces (one for each possible value
of the coded symbol). This requires the encoder to compute two
partition values, in general (for both the upper and lower bound of
the symbol to encode). The decoder, on the other hand, must search
the partitions for the one that contains the received code point.
This requires computing at least O(log N) partition values.
However, coding a parameter with N possible values with a binary
arithmetic coder requires O(log N) symbols in the worst case (the
only case that matters for hardware design). Hence, this does not
represent any actual savings (indeed, it represents an increase in
the number of partition values computed by the encoder). In
addition, there are a number of overheads that are per-symbol, rather
than per-value. For example, renormalization (which enlarges the set
of valid code points after partitioning has reduced it too much),
carry propagation (to deal with the case where the high and low ends
of a partition straddle a bit boundary), etc., are all performed on a
symbol-by-symbol basis. Since a non-binary arithmetic coder codes a
given set of values with fewer symbols than a binary one, it incurs
these per-symbol overheads less often. This suggests that a non-
binary arithmetic coder can actually be more efficient than a binary
one.
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2.2. Non-binary Context Modeling
The other aspect that binary coding simplifies is probability
modeling. In arithmetic coding, the size of the sets the code points
are partitioned into are (roughly) proportional to the probability of
each possible symbol value. Estimating these probabilities is part
of the coding process, though it can be cleanly separated from the
task of actually producing the coded bits. In a binary arithmetic
coder, this requires estimating the probability of only one of the
two possible values (since the total probability is 1.0). This is
often done with a simple table lookup that maps the old probability
and the most recently decoded symbol to a new probability to use for
the next symbol in the current context. The trade-off, of course, is
that non-binary symbols must be "binarized" into a series of bits,
and a context (with an associated probability) chosen for each one.
In a non-binary arithmetic coder, the decoder must compute at least
O(log N) cumulative probabilities (one for each partition value it
needs). Because these probabilities are usually not estimated
directly in "cumulative" form, this can require computing (N - 1)
non-cumulative probability values. Unless N is very small, these
cannot be updated with a single table lookup. The normal approach is
to use "frequency counts". Define the frequency of value k to be
f[k] = A*<the number of times k has been observed> + B
where A and B are parameters (usually A=2 and B=1 for a traditional
Krichevsky-Trofimov estimator). The resulting probability, p[k], is
given by
N-1
__
ft = \ f[k]
/_
k=0
f[k]
p[k] = ----
ft
When ft grows too large, the frequencies are rescaled (e.g., halved,
rounding up to prevent reduction of a probability to 0).
When ft is not a power of two, partitioning the code points requires
actual divisions (see [RFC6716] Section 4.1 for one detailed example
of exactly how this is done). These divisions are acceptable in an
audio codec like Opus [RFC6716], which only has to code a few
hundreds of these symbols per second. But video requires hundreds of
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thousands of symbols per second, at a minimum, and divisions are
still very expensive to implement in hardware.
There are two possible approaches to this. One is to come up with a
replacement for frequency counts that produces probabilities that sum
to a power of two. Some possibilities, which can be applied
individually or in combination:
1. Use probabilities that are fixed for the duration of a frame.
This is the approach taken by VP8, for example, even though it
uses a binary arithmetic coder. In fact, it is possible to
convert many of VP8's existing binary-alphabet probabilities into
probabilities for non-binary alphabets, an approach that is used
in the experiment presented at the end of this section.
2. Use parametric distributions. For example, DCT coefficient
magnitudes usually have an approximately exponential
distribution. This distribution can be characterized by a single
parameter, e.g., the expected value. The expected value is
trivial to update after decoding a coefficient. For example
E[x[n+1]] = E[x[n]] + floor(C*(x[n] - E[x[n]]))
produces an exponential moving average with a decay factor of
(1 - C). For a choice of C that is a negative power of two
(e.g., 1/16 or 1/32 or similar), this can be implemented with two
adds and a shift. Given this expected value, the actual
distribution to use can be obtained from a small set of pre-
computed distributions via a lookup table. Linear interpolation
between these pre-computed values can improve accuracy, at the
cost of O(N) computations, but if N is kept small this is
trivially parallelizable, in SIMD or otherwise.
3. Change the frequency count update mechanism so that ft is
constant. This approach is described in the next section.
2.3. Dyadic Adaptation
The goal with context adaptation using dyadic probabilities is to
maintain the invariant that the probabilities all sum to a power of
two before and after adaptation. This can be achieved with a special
update function that blends the cumulative probabilities of the
current context with a cumulative distribution function where the
coded symbol has probability 1.
Suppose we have model for a given context that codes 8 symbols with
the following probabilities:
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+------+------+------+------+------+------+------+------+
| p[0] | p[1] | p[2] | p[3] | p[4] | p[5] | p[6] | p[7] |
+------+------+------+------+------+------+------+------+
| 1/8 | 1/8 | 3/16 | 1/16 | 1/16 | 3/16 | 1/8 | 1/8 |
+------+------+------+------+------+------+------+------+
Then the cumulative distribution function is:
CDF
1 + +------+
| |
| +------+
| |
3/4 + +------+
| |
| |
| +------+
1/2 + +------+
| +------+
| |
| |
1/4 + +------+
| |
+------+
|
0 +------+------+------+------+------+------+------+------+ Bin
fl[1] fl[2] fl[3] fl[4] fl[5] fl[6] fl[7] fl[8]
Suppose we code symbol 3 and wish to update the context model so that
this symbol is now more likely. This can be done by blending the CDF
for the current context with a CDF that has symbol 3 with likelihood
1.
CDF
1 + +----------------------------------+
| |
| |
| |
0 +------+------+------+------+------+------+------+------+ Bin
fl[1] fl[2] fl[3] fl[4] fl[5] fl[6] fl[7] fl[8]
Given an adaptation rate g between 0 and 1, and assuming ft = 2^4 =
16, what we are computing is:
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+------+------+------+------+------+------+------+------+
| 2 | 4 | 7 | 8 | 9 | 12 | 14 | 16 | * (1 - g)
+------+------+------+------+------+------+------+------+
+
+------+------+------+------+------+------+------+------+
| 0 | 0 | 0 | 16 | 16 | 16 | 16 | 16 | * g
+------+------+------+------+------+------+------+------+
In order to prevent the probability of any one symbol from going to
zero, the blending functions above and below the coded symbol are
adjusted so that no adjacent cumulative probabilities are the same.
Let M be the alphabet size and 1/2^r be the adaptation rate:
( fl[i] - floor((fl[i] + 2^r - i - 1)/2^r), i <= coded symbol
fl[i] = <
( fl[i] - floor((fl[i] + M - i - ft)/2^r), i > coded symbol
Applying these formulas to the example CDF where M = 8 with
adaptation rate 1/2^16 gives the updated CDF:
+------+------+------+------+------+------+------+------+
| 1 | 3 | 6 | 9 | 10 | 13 | 15 | 16 |
+------+------+------+------+------+------+------+------+
Looking at the graph of the CDF we see that the likelihood for symbol
3 has gone up from 1/16 to 3/16, dropping the likelihood of all other
symbols to make room.
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CDF
1 + +------+
| +------+
| |
| +------+
3/4 + |
| |
| +------+
| +------+
1/2 + |
| |
| +------+
| |
1/4 + |
| +------+
| |
+------+
0 +------+------+------+------+------+------+------+------+ Bin
fl[1] fl[2] fl[3] fl[4] fl[5] fl[6] fl[7] fl[8]
2.4. Simplified Partition Function
Let the range of valid code points in the current arithmetic coder
state be [L, L + R), where L is the lower bound of the range and R is
the number of valid code points. The goal of the arithmetic coder is
to partition this interval proportional to the probability of each
symbol. When using dyadic probabilities, the partition point in the
range corresponding to a given CDF value can be determined via
fl[k]*R
u[k] = floor ( ------- )
ft
Since ft is a power of two, this may be implemented using a right
shift by T bits in place of the division:
u[k] = (fl[k]*R) >> T
The latency of the multiply still dominates the hardware timing.
However, we can reduce this latency by using a smaller multiply, at
the cost of some accuracy in the partition. We cannot, in general,
reduce the size of fl[k], since this might send a probability to zero
(i.e., cause u[k] to have the same value as u[k+1]). On the other
hand, we know that the top bit of R is always 1, since it gets
renormalized with every symbol that is encoded. Suppose R contains
16 bits and that T is at least 8. Then we can greatly reduce the
size of the multiply by using the formula
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( (fl[k]*(R >> 8)) >> (T - 8), 0 <= k < M
u[k] = <
( R, k == M
The special case for k == M is required because, with the general
formula, u[M] no longer exactly equals R. Without the special case
we would waste some amount of code space and require the decoder to
check for invalid streams. This special case slightly inflates the
probability of the last symbol. Unfortunately, in codecs the usual
convention is that the last symbol is the least probable, while the
first symbol (e.g., 0) is the most probable. That maximizes the
coding overhead introduced by this approximation error. To minimize
it, we instead add all of the accumulated error to the first symbol
by using a variation of the above update formula:
( 0, k == 0
u[k] = <
( R - (((ft - fl[k])*(R >> 8)) >> (T - 8)), 0 < k <= M
This also aids the software decoder search, since it can prime the
search loop with the special case, instead of needing to check for it
on every iteration of the loop. It is easier to incorporate into a
SIMD search as well. It does, however, add two subtractions. Since
the encoder always operates on the difference between two partition
points, the first subtraction (involving R) can be eliminated.
Similar optimizations can eliminate this subtraction in the decoder
by flipping its internal state (measuring the distance of the encoder
output from the top of the range instead of the bottom). To avoid
the other subtraction, we can simply use "inverse CDFs" that natively
store ifl[k] = (ft - fl[k]) instead of fl[k]. This produces the
following partition function:
( R, k == 0
R - u[k] = <
( (ifl[k]*(R >> 8)) >> (T - 8), 0 < k <= M
The reduction in hardware latency can be as much as 20%, and the
impact on area is even larger. The overall software complexity
overhead is minimal, and the coding efficiency overhead due to the
approximation is about 0.02%. We could have achieved the same
efficiency by leaving the special case on the last symbol and
reversing the alphabet instead of inverting the probabilities.
However, reversing the alphabet at runtime would have required an
extra subtraction (or more general re-ordering requires a table
lookup). That may be avoidable in some cases, but only by
propagating the reordering alphabet outside of the entropy coding
machinery, requiring changes to every coding tool and potentially
leading to confusion. CDFs, on the other hand, are already a
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somewhat abstract representation of the underlying probabilities used
for computational efficiency reasons. Generalizing these to "inverse
CDFs" is a straightforward change that only affects probability
initialization and adaptation, without impacting the design of other
coding tools.
2.5. Context Adaptation
The dyadic adaptation scheme described in Section 2.3 implements a
low-complexity IIR filter for the steady-state case where we only
want to adapt the context CDF as fast as the 1/2^r adaptation rate.
In many cases, for example when coding symbols at the start of a
video frame, only a limited number of symbols have been seen per
context. Using this steady-state adaptation scheme risks adapting
too slowly and spending too many bits to code symbols with incorrect
probability estimates. In other video codecs, this problem is
reduced by either implicitly or explicitly allowing for mechanisms to
set the initial probability models for a given context.
2.5.1. Implicit Adaptation
One implicit way to use default probabilities is to simply require as
a normative part of the decoder that some specific CDFs are used to
initialize each context. A representative set of inputs is run
through the encoder and a frequency based probability model is
computed and reloaded at the start of every frame. This has the
advantage of having zero bitstream overhead and is optimal for
certain stationary symbols. However for other non-stationary
symbols, or highly content dependent contexts where the sample input
is not representative, this can be worse than starting with a flat
distribution as it now takes even longer to adapt to the steady-
state. Moreover the amount of hardware area required to store
initial probability tables for each context goes up with the number
of contexts in the codec.
Another implicit way to deal with poor initial probabilities is
through backward adaptation based on the probability estimates from
the previous frame. After decoding a frame, the adapted CDFs for
each context are simply kept as-is and not reset to their defaults.
This has the advantage of having no bitstream overhead, and tracking
to certain content types closely as we expect frames with similar
content at similar rates, to have well correlated CDFs. However,
this only works when we know there will be no bitstream errors due to
the transport layer, e.g., TCP or HTTP. In low delay use cases
(video on demand, live streaming, video conferencing), implicit
backwards adaptation is avoided as it risks desynchronizing the
entropy decoder state and permanently losing the video stream.
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2.5.2. Explicit Adaptation
For codecs that include the ability to update the probability models
in the bitstream, it is possible to explicitly signal a starting CDF.
The previously described implicit backwards adaptation is now
possible by simply explicitly coding a probability update for each
frame. However, the cost of signaling the updated CDF must be
overcome by the savings from coding with the updated CDF. Blindly
updating all contexts per frame may work at high rates where the size
of the CDFs is small relative to the coded symbol data. However at
low rates, the benefit of using more accurate CDFs is quickly
overcome by the cost of coding them, which increases with the number
of contexts.
More sophisticated encoders can compute the cost of coding a
probability update for a given context, and compare it to the size
reduction achieved by coding symbols with this context. Here all
symbols for a given frame (or tile) are buffered and not serialized
by the entropy coder until the end of the frame (or tile) is reached.
Once the end of the entropy segment has been reached, the cost in
bits for coding symbols with both the default probabilities and the
proposed updated probabilities can be measured and compared.
However, note that with the symbols already buffered, rather than
consider the context probabilities from the previous frame, a simple
frequency based probability model can be computed and measured.
Because this probability model is computed based on the symbols we
are about to code this technique is called forward adaptation. If
the cost in bits to signal and code with this new probability model
is less than that of using the default then it is used. This has the
advantage of only ever coding a probability update if it is an
improvement and producing a bitstream that is robust to errors, but
requires an entire entropy segments worth of symbols be cached.
2.5.3. Early Adaptation
We would like to take advantage of the low-cost multi-symbol CDF
adaptation described in Section 2.3 without in the broadest set of
use cases. This means the initial probability adaptation scheme
should support low-delay, error-resilient streams that efficiently
implemented in both hardware and software. We propose an early
adaptation scheme that supports this goal.
At the beginning of a frame (or tile), all CDFs are initialized to a
flat distribution. For a given multi-symbol context with M potential
symbols, assume that the initial dyadic CDF is initialized so that
each symbol has probability 1/M. For the first M coded symbols, the
CDF is updated as follows:
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a[c,M] = ft/(M + c)
( fl[i] - floor((fl[i] - i)*a/ft), i <= coded symbol
fl[i] = <
( fl[i] - floor((fl[i] + M - i - ft)*a/ft), i > coded symbol
where c goes from 0 to M-1 and is the running count of the number of
symbols coded with this CDF. Note that for a fixed CDF precision (ft
is always a power of two) and a maximum number of possible symbols M,
the values of a[c,M] can be stored in a M*(M+1)/2 element table,
which is 136 entries when M = 16.
2.6. Simple Experiment
As a simple experiment to validate the non-binary approach, we
compared a non-binary arithmetic coder to the VP8 (binary) entropy
coder. This was done by instrumenting vp8_treed_read() in libvpx to
dump out the symbol decoded and the associated probabilities used to
decode it. This data only includes macroblock mode and motion vector
information, as the DCT token data is decoded with custom inline
functions, and not vp8_treed_read(). This data is available at [1].
It includes 1,019,670 values encode using 2,125,995 binary symbols
(or 2.08 symbols per value). We expect that with a conscious effort
to group symbols during the codec design, this average could easily
be increased.
We then implemented both the regular VP8 entropy decoder (in plain C,
using all of the optimizations available in libvpx at the time) and a
multisymbol entropy decoder (also in plain C, using similar
optimizations), which encodes each value with a single symbol. For
the decoder partition search in the non-binary decoder, we used a
simple for loop (O(N) worst-case), even though this could be made
constant-time and branchless with a few SIMD instructions such as (on
x86) PCMPGTW, PACKUSWB, and PMOVMASKB followed by BSR. The source
code for both implementations is available at [2] (compile with
-DEC_BINARY for the binary version and -DEC_MULTISYM for the non-
binary version).
The test simply loads the tokens, and then loops 1024 times encoding
them using the probabilities provided, and then decoding them. The
loop was added to reduce the impact of the overhead of loading the
data, which is implemented very inefficiently. The total runtime on
a Core i7 from 2010 is 53.735 seconds for the binary version, and
27.937 seconds for the non-binary version, or a 1.92x improvement.
This is very nearly equal to the number of symbols per value in the
binary coder, suggesting that the per-symbol overheads account for
the vast majority of the computation time in this implementation.
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3. Reversible Integer Transforms
Integer transforms in image and video coding date back to at least
1969 [PKA69]. Although standards such as MPEG2 and MPEG4 Part 2
allow some flexibility in the transform implementation,
implementations were subject to drift and error accumulation, and
encoders had to impose special macroblock refresh requirements to
avoid these problems, not always successfully. As transforms in
modern codecs only account for on the order of 10% of the total
decoder complexity, and, with the use of weighted prediction with
gains greater than unity and intra prediction, are far more
susceptible to drift and error accumulation, it no longer makes sense
to allow a non-exact transform specification.
However, it is also possible to make such transforms "reversible", in
the sense that applying the inverse transform to the result of the
forward transform gives back the original input values, exactly.
This gives a lossy codec, which normally quantizes the coefficients
before feeding them into the inverse transform, the ability to scale
all the way to lossless compression without requiring any new coding
tools. This approach has been used successfully by JPEG XR, for
example [TSSRM08].
Such reversible transforms can be constructed using "lifting steps",
a series of shear operations that can represent any set of plane
rotations, and thus any orthogonal transform. This approach dates
back to at least 1992 [BE92], which used it to implement a four-point
1-D Discrete Cosine Transform (DCT). Their implementation requires
6 multiplications, 10 additions, 2 shifts, and 2 negations, and
produces output that is a factor of sqrt(2) larger than the
orthonormal version of the transform. The expansion of the dynamic
range directly translates into more bits to code for lossless
compression. Because the least significant bits are usually very
nearly random noise, this scaling increases the coding cost by
approximately half a bit per sample.
3.1. Lifting Steps
To demonstrate the idea of lifting steps, consider the two-point
transform
___
[ y0 ] / 1 [ 1 1 ] [ x0 ]
[ ] = / --- [ ] [ ]
[ y1 ] v 2 [ -1 1 ] [ x1 ]
This can be implemented up to scale via
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y0 = x0 + x1
y1 = 2*x1 - y0
and reversed via
x1 = (y0 + y1) >> 1
x0 = y0 - x1
Both y0 and y1 are too large by a factor of sqrt(2), however.
It is also possible to implement any rotation by an angle t,
including the orthonormal scale factor, by decomposing it into three
steps:
cos(t) - 1
u0 = x0 + ---------- * x1
sin(t)
y1 = x1 + sin(t)*u0
cos(t) - 1
y0 = u0 + ---------- * y1
sin(t)
By letting t=-pi/4, we get an implementation of the first transform
that includes the scaling factor. To get an integer approximation of
this transform, we need only replace the transcendental constants by
fixed-point approximations:
u0 = x0 + ((27*x1 + 32) >> 6)
y1 = x1 - ((45*u0 + 32) >> 6)
y0 = u0 + ((27*y1 + 32) >> 6)
This approximation is still perfectly reversible:
u0 = y0 - ((27*y1 + 32) >> 6)
x1 = y1 + ((45*u0 + 32) >> 6)
x0 = u0 - ((27*x1 + 32) >> 6)
Each of the three steps can be implemented using just two ARM
instructions, with constants that have up to 14 bits of precision
(though using fewer bits allows more efficient hardware
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implementations, at a small cost in coding gain). However, it is
still much more complex than the first approach.
We can get a compromise with a slight modification:
y0 = x0 + x1
y1 = x1 - (y0 >> 1)
This still only implements the original orthonormal transform up to
scale. The y0 coefficient is too large by a factor of sqrt(2) as
before, but y1 is now too small by a factor of sqrt(2). If our goal
is simply to (optionally quantize) and code the result, this is good
enough. The different scale factors can be incorporated into the
quantization matrix in the lossy case, and the total expansion is
roughly equivalent to that of the orthonormal transform in the
lossless case. Plus, we can perform each step with just one ARM
instruction.
However, if instead we want to apply additional transformations to
the data, or use the result to predict other data, it becomes much
more convenient to have uniformly scaled outputs. For a two-point
transform, there is little we can do to improve on the three-
multiplications approach above. However, for a four-point transform,
we can use the last approach and arrange multiple transform stages
such that the "too large" and "too small" scaling factors cancel out,
producing a result that has the true, uniform, orthonormal scaling.
To do this, we need one more tool, which implements the following
transform:
___
[ y0 ] / 1 [ cos(t) -sin(t) ] [ 1 0 ] [ x0 ]
[ ] = / --- [ ] [ ] [ ]
[ y1 ] v 2 [ sin(t) cos(t) ] [ 0 2 ] [ x1 ]
This takes unevenly scaled inputs, rescales them, and then rotates
them. Like an ordinary rotation, it can be reduced to three lifting
steps:
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_
2*cos(t) - v2
u0 = x0 + ------------- * x1
sin(t)
___
/ 1
y1 = x1 + / --- * sin(t)*u0
v 2
_
cos(t) - v2
y0 = u0 + ----------- * y1
sin(t)
As before, the transcendental constants may be replaced by fixed-
point approximations without harming the reversibility property.
3.2. 4-Point Transform
Using the tools from the previous section, we can design a reversible
integer four-point DCT approximation with uniform, orthonormal
scaling. This requires 3 multiplies, 9 additions, and 2 shifts (not
counting the shift and rounding offset used in the fixed-point
multiplies, as these are built into the multiplier). This is
significantly cheaper than the [BE92] approach, and the output
scaling is smaller by a factor of sqrt(2), saving half a bit per
sample in the lossless case. By comparison, the four-point forward
DCT approximation used in VP9, which is not reversible, uses
6 multiplies, 6 additions, and 2 shifts (counting shifts and rounding
offsets which cannot be merged into a single multiply instruction on
ARM). Four of its multipliers also require 28-bit accumulators,
whereas this proposal can use much smaller multipliers without giving
up the reversibility property. The total dynamic range expansion is
1 bit: inputs in the range [-256,255) produce transformed values in
the range [-512,510). This is the smallest dynamic range expansion
possible for any reversible transform constructed from mostly-linear
operations. It is possible to make reversible orthogonal transforms
with no dynamic range expansion by using "piecewise-linear"
rotations [SLD04], but each step requires a large number of
operations in a software implementation.
Pseudo-code for the forward transform follows:
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Input: x0, x1, x2, x3
Output: y0, y1, y2, y3
/* Rotate (x3, x0) by -pi/4, asymmetrically scaled output. */
t3 = x0 - x3
t0 = x0 - (t3 >> 1)
/* Rotate (x1, x2) by pi/4, asymmetrically scaled output. */
t2 = x1 + x2
t2h = t2 >> 1
t1 = t2h - x2
/* Rotate (t2, t0) by -pi/4, asymmetrically scaled input. */
y0 = t0 + t2h
y2 = y0 - t2
/* Rotate (t3, t1) by 3*pi/8, asymmetrically scaled input. */
t3 = t3 - (45*t1 + 32 >> 6)
y1 = t1 + (21*t3 + 16 >> 5)
y3 = t3 - (71*y1 + 32 >> 6)
Even though there are three asymmetrically scaled rotations by pi/4,
by careful arrangement we can share one of the shift operations (to
help software implementations: shifts by a constant are basically
free in hardware). This technique can be used to even greater effect
in larger transforms.
The inverse transform is constructed by simply undoing each step in
turn:
Input: y0, y1, y2, y3
Output: x0, x1, x2, x3
/* Rotate (y3, y1) by -3*pi/8, asymmetrically scaled output. */
t3 = y3 + (71*y1 + 32 >> 6)
t1 = y1 - (21*t3 + 16 >> 5)
t3 = t3 + (45*t1 + 32 >> 6)
/* Rotate (y2, y0) by pi/4, asymmetrically scaled output. */
t2 = y0 - y2
t2h = t2 >> 1
t0 = y0 - t2h
/* Rotate (t1, t2) by -pi/4, asymmetrically scaled input. */
x2 = t2h - t1
x1 = t2 - x2
/* Rotate (x3, x0) by pi/4, asymmetrically scaled input. */
x0 = t0 - (t3 >> 1)
x3 = x0 - t3
Although the right shifts make this transform non-linear, we can
compute "basis functions" for it by sending a vector through it with
a single value set to a large constant (256 was used here), and the
rest of the values set to zero. The true basis functions for a four-
point DCT (up to five digits) are
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[ y0 ] [ 0.50000 0.50000 0.50000 0.50000 ] [ x0 ]
[ y1 ] = [ 0.65625 0.26953 -0.26953 -0.65625 ] [ x1 ]
[ y2 ] [ 0.50000 -0.50000 -0.50000 0.50000 ] [ x2 ]
[ y3 ] [ 0.27344 -0.65234 0.65234 -0.27344 ] [ x3 ]
The corresponding basis functions for our reversible, integer DCT,
computed using the approximation described above, are
[ y0 ] [ 0.50000 0.50000 0.50000 0.50000 ] [ x0 ]
[ y1 ] = [ 0.65328 0.27060 -0.27060 -0.65328 ] [ x1 ]
[ y2 ] [ 0.50000 -0.50000 -0.50000 0.50000 ] [ x2 ]
[ y3 ] [ 0.27060 -0.65328 0.65328 -0.27060 ] [ x3 ]
The mean squared error (MSE) of the output, compared to a true DCT,
can be computed with some assumptions about the input signal. Let G
be the true DCT basis and G' be the basis for our integer
approximation (computed as described above). Then the error in the
transformed results is
e = G.x - G'.x = (G - G').x = D.x
where D = (G - G') . The MSE is then [Que98]
1 1
- * E[e^T.e] = - * E[x^T.D^T.D.x]
N N
1
= - * E[tr(D.x.x^T.D^T)]
N
1
= - * E[tr(D.Rxx.D^T)]
N
where Rxx is the autocorrelation matrix of the input signal.
Assuming the input is a zero-mean, first-order autoregressive (AR(1))
process gives an autocorrelation matrix of
|i - j|
Rxx[i,j] = rho
for some correlation coefficient rho. A value of rho = 0.95 is
typical for image compression applications. Smaller values are more
normal for motion-compensated frame differences, but this makes
surprisingly little difference in transform design. Using the above
procedure, the theoretical MSE of this approximation is 1.230E-6,
which is below the level of the truncation error introduced by the
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right shift operations. This suggests the dynamic range of the input
would have to be more than 20 bits before it became worthwhile to
increase the precision of the constants used in the multiplications
to improve accuracy, though it may be worth using more precision to
reduce bias.
3.3. Larger Transforms
The same techniques can be applied to construct a reversible eight-
point DCT approximation with uniform, orthonormal scaling using
15 multiplies, 31 additions, and 5 shifts. It is possible to reduce
this to 11 multiplies and 29 additions, which is the minimum number
of multiplies possible for an eight-point DCT with uniform
scaling [LLM89], by introducing a scaling factor of sqrt(2), but this
harms lossless performance. The dynamic range expansion is 1.5 bits
(again the smallest possible), and the MSE is 1.592E-06. By
comparison, the eight-point transform in VP9 uses 12 multiplications,
32 additions, and 6 shifts.
Similarly, we have constructed a reversible sixteen-point DCT
approximation with uniform, orthonormal scaling using 33 multiplies,
83 additions, and 16 shifts. This is just 2 multiplies and
2 additions more than the (non-reversible, non-integer, but uniformly
scaled) factorization in [LLM89]. By comparison, the sixteen-point
transform in VP9 uses 44 multiplies, 88 additions, and 18 shifts.
The dynamic range expansion is only 2 bits (again the smallest
possible), and the MSE is 1.495E-5.
We also have a reversible 32-point DCT approximation with uniform,
orthonormal scaling using 87 multiplies, 215 additions, and
38 shifts. By comparison, the 32-point transform in VP9 uses
116 multiplies, 194 additions, and 66 shifts. Our dynamic range
expansion is still the minimal 2.5 bits, and the MSE is 8.006E-05
Code for all of these transforms is available in the development
repository listed in Section 4.
3.4. Walsh-Hadamard Transforms
These techniques can also be applied to constructing Walsh-Hadamard
Transforms, another useful transform family that is cheaper to
implement than the DCT (since it requires no multiplications at all).
The WHT has many applications as a cheap way to approximately change
the time and frequency resolution of a set of data (either individual
bands, as in the Opus audio codec, or whole blocks). VP9 uses it as
a reversible transform with uniform, orthonormal scaling for lossless
coding in place of its DCT, which does not have these properties.
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Applying a 2x2 WHT to a block of 2x2 inputs involves running a
2-point WHT on the rows, and then another 2-point WHT on the columns.
The basis functions for the 2-point WHT are, up to scaling, [1, 1]
and [1, -1]. The four variations of a two-step lifer given in
Section 3.1 are exactly the lifting steps needed to implement a 2x2
WHT: two stages that produce asymmetrically scaled outputs followed
by two stages that consume asymmetrically scaled inputs.
Input: x00, x01, x10, x11
Output: y00, y01, y10, y11
/* Transform rows */
t1 = x00 - x01
t0 = x00 - (t1 >> 1) /* == (x00 + x01)/2 */
t2 = x10 + x11
t3 = (t2 >> 1) - x11 /* == (x10 - x11)/2 */
/* Transform columns */
y00 = t0 + (t2 >> 1) /* == (x00 + x01 + x10 + x11)/2 */
y10 = y00 - t2 /* == (x00 + x01 - x10 - x11)/2 */
y11 = (t1 >> 1) - t3 /* == (x00 - x01 - x10 + x11)/2 */
y01 = t1 - y11 /* == (x00 - x01 + x10 - x11)/2 */
By simply re-ordering the operations, we can see that there are two
shifts that may be shared between the two stages:
Input: x00, x01, x10, x11
Output: y00, y01, y10, y11
t1 = x00 - x01
t2 = x10 + x11
t0 = x00 - (t1 >> 1) /* == (x00 + x01)/2 */
y00 = t0 + (t2 >> 1) /* == (x00 + x01 + x10 + x11)/2 */
t3 = (t2 >> 1) - x11 /* == (x10 - x11)/2 */
y11 = (t1 >> 1) - t3 /* == (x00 - x01 - x10 + x11)/2 */
y10 = y00 - t2 /* == (x00 + x01 - x10 - x11)/2 */
y01 = t1 - y11 /* == (x00 - x01 + x10 - x11)/2 */
By eliminating the double-negation of x11 and re-ordering the
additions to it, we can see even more operations in common:
Input: x00, x01, x10, x11
Output: y00, y01, y10, y11
t1 = x00 - x01
t2 = x10 + x11
t0 = x00 - (t1 >> 1) /* == (x00 + x01)/2 */
y00 = t0 + (t2 >> 1) /* == (x00 + x01 + x10 + x11)/2 */
t3 = x11 + (t1 >> 1) /* == x11 + (x00 - x01)/2 */
y11 = t3 - (t2 >> 1) /* == (x00 - x01 - x10 + x11)/2 */
y10 = y00 - t2 /* == (x00 + x01 - x10 - x11)/2 */
y01 = t1 - y11 /* == (x00 - x01 + x10 - x11)/2 */
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Simplifying further, the whole transform may be computed with just
7 additions and 1 shift:
Input: x00, x01, x10, x11
Output: y00, y01, y10, y11
t1 = x00 - x01
t2 = x10 + x11
t4 = (t2 - t1) >> 1 /* == (-x00 + x01 + x10 + x11)/2 */
y00 = x00 + t4 /* == (x00 + x01 + x10 + x11)/2 */
y11 = x11 - t4 /* == (x00 - x01 - x10 + x11)/2 */
y10 = y00 - t2 /* == (x00 + x01 - x10 - x11)/2 */
y01 = t1 - y11 /* == (x00 - x01 + x10 - x11)/2 */
This is a significant savings over other approaches described in the
literature, which require 8 additions, 2 shifts, and
1 negation [FOIK99] (37.5% more operations), or 10 additions,
1 shift, and 2 negations [TSSRM08] (62.5% more operations). The same
operations can be applied to compute a 4-point WHT in one dimension.
This implementation is used in this way in VP9's lossless mode.
Since larger WHTs may be trivially factored into multiple smaller
WHTs, the same approach can implement a reversible, orthonormally
scaled WHT of any size (2**N)x(2**M), so long as (N + M) is even.
4. Development Repository
The tools presented here were developed as part of Xiph.Org's Daala
project. They are available, along with many others in greater and
lesser states of maturity, in the Daala git repository at [3]. See
[4] for more information.
5. IANA Considerations
This document has no actions for IANA.
6. Acknowledgments
Thanks to Nathan Egge, Gregory Maxwell, and Jean-Marc Valin for their
assistance in the implementation and experimentation, and in
preparing this draft.
7. References
7.1. Informative References
[RFC6386] Bankoski, J., Koleszar, J., Quillio, L., Salonen, J.,
Wilkins, P., and Y. Xu, "VP8 Data Format and Decoding
Guide", RFC 6386, November 2011.
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[RFC6716] Valin, JM., Vos, K., and T. Terriberry, "Definition of the
Opus Audio Codec", RFC 6716, September 2012.
[BE92] Bruekers, F. and A. van den Enden, "New Networks for
Perfect Inversion and Perfect Reconstruction", IEEE
Journal on Selected Areas in Communication 10(1):129--137,
January 1992.
[FOIK99] Fukuma, S., Oyama, K., Iwahashi, M., and N. Kambayashi,
"Lossless 8-point Fast Discrete Cosine Transform Using
Lossless Hadamard Transform", Technical Report The
Institute of Electronics, Information, and Communication
Engineers of Japan, October 1999.
[LLM89] Loeffler, C., Ligtenberg, A., and G. Moschytz, "Practical
Fast 1-D DCT Algorithms with 11 Multiplications", Proc.
Acoustics, Speech, and Signal Processing (ICASSP'89) vol.
2, pp. 988--991, May 1989.
[Pas76] Pasco, R., "Source Coding Algorithms for Fast Data
Compression", Ph.D. Thesis Dept. of Electrical
Engineering, Stanford University, May 1976.
[PKA69] Pratt, W., Kane, J., and H. Andrews, "Hadamard Transform
Image Coding", Proc. IEEE 57(1):58--68, Jan 1969.
[Que98] de Queiroz, R., "On Unitary Transform Approximations",
IEEE Signal Processing Letters 5(2):46--47, Feb 1998.
[SLD04] Senecal, J., Lindstrom, P., and M. Duchaineau, "An
Improved N-Bit to N-Bit Reversible Haar-Like Transform",
Proc. of the 12th Pacific Conference on Computer Graphics
and Applications (PG'04) pp. 371--380, October 2004.
[TSSRM08] Tu, C., Srinivasan, S., Sullivan, G., Regunathan, S., and
H. Malvar, "Low-complexity Hierarchical Lapped Transform
for Lossy-to-Lossless Image Coding in JPEG XR/HD Photo",
Applications of Digital Image Processing XXXI vol 7073,
August 2008.
7.2. URIs
[1] https://people.xiph.org/~tterribe/daala/ec_test0/ec_tokens.txt
[2] https://people.xiph.org/~tterribe/daala/ec_test0/ec_test.c
[3] https://git.xiph.org/daala.git
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[4] https://xiph.org/daala/
Authors' Addresses
Timothy B. Terriberry
Mozilla Corporation
331 E. Evelyn Avenue
Mountain View, CA 94041
USA
Phone: +1 650 903-0800
Email: tterribe@xiph.org
Nathan E. Egge
Mozilla Corporation
331 E. Evelyn Avenue
Mountain View, CA 94041
USA
Phone: +1 650 903-0800
Email: negge@xiph.org
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