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Versions: 00 01 02 draft-terriberry-netvc-codingtools

Network Working Group                                      T. Terriberry
Internet-Draft                                       Mozilla Corporation
Intended status: Informational                             March 9, 2015
Expires: September 10, 2015


             Coding Tools for a Next Generation Video Codec
                    draft-terriberry-codingtools-02

Abstract

   This document proposes a number of coding tools that could be
   incorporated into a next-generation video codec.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

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   This Internet-Draft will expire on September 10, 2015.

Copyright Notice

   Copyright (c) 2015 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
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   described in the Simplified BSD License.






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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 . . . . . . . . . . . . . . .   4
     2.3.  Simple Experiment . . . . . . . . . . . . . . . . . . . .   8
   3.  Reversible Integer Transforms . . . . . . . . . . . . . . . .   8
     3.1.  Lifting Steps . . . . . . . . . . . . . . . . . . . . . .   9
     3.2.  4-Point Transform . . . . . . . . . . . . . . . . . . . .  11
     3.3.  Larger Transforms . . . . . . . . . . . . . . . . . . . .  14
     3.4.  Walsh-Hadamard Transforms . . . . . . . . . . . . . . . .  15
   4.  Development Repository  . . . . . . . . . . . . . . . . . . .  17
   5.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  17
   6.  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . .  17
   7.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  17
     7.1.  Informative References  . . . . . . . . . . . . . . . . .  17
     7.2.  URIs  . . . . . . . . . . . . . . . . . . . . . . . . . .  18
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  18

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
   an eventual working group, if formed, 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.
   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



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   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
   same set of drawbacks apply to the DCT token partitions in
   VP8 [RFC6386].








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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.

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



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   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
   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:



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   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.  For example, let

                                      k-1
                                      __
                              fl[k] = \  f[i]
                                      /_
                                      i=0

       be the cumulative frequency of all symbol values less than k and

                                     ( 0, k <= i
                           e[i][k] = <
                                     ( 1, k > i

       be the elementary change in the cumulative frequency count fl[k]
       caused by adding 1 to f[i].  Then one possible update formula
       after decoding the value i is

              fl[k]' = fl[k] - floor(D*fl[k]) + k + F*e[i][k]

       where D is a negative power of two chosen such that
       floor(D*ft) == (N + F) .  This ensures that ft == fl[N] == fl[N]'



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       is a constant.  This requires O(N) operations, but the arithmetic
       is very simple (given the freedom to choose D and F, and to some
       extent N), and trivially parallelizable, in SIMD or otherwise.
       The downside is the addition of the value k at each step.  This
       is necessary to ensure that the probability of an individual
       symbol (fl[k+1] - fl[k]) is never reduced to zero.  However it is
       equivalent to mixing in a uniform distribution with counts that
       are otherwise an exponential moving average.  That means that ft
       and F must be sufficiently large, or there will be an adverse
       impact on coding efficiency.  The upside is that F*e[i] may be
       replaced by any monotonically non-decreasing vector whose Nth
       element is F.  That is, instead of just incrementing the
       probability of symbol i, it can increase the probability of
       values that are highly correlated with i.  E.g., this allows
       decoding value i to apply a small probability increase to the
       neighboring values (i - 1) and (i + 1), in addition to a large
       probability increase to the value i.  This may help, for example,
       in motion vector coding, and is much more sensible than the
       approach taken with binary context modeling, which often does
       things like "increase the probability of all even values when
       decoding a 6" because the same context is always used to code the
       least significant bit.

   The other approach is to change the function used to partition the
   set of valid code points so that it does not need a division, even
   when ft is not a power of two.  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.
   Assume that ft <= R < 2*ft (this is easy to enforce with the normal
   rescaling operations used with frequency counts).  Then one possible
   partition function is

                     r[k] = fl[k] + min(fl[k], R - ft)

   so that the new range after coding symbol k is
   [L + r[k], L + r[k+1]).

   This is a variation of the partition function proposed by [SM98].
   The size of the new partition (r[k+1] - r[k]) is no longer truly
   proportional to R*p[k].  It can be off by up to a factor of 2,
   implying a peak error as large as one bit per symbol.  However, if
   the probabilities are accurate and the symbols being coded are
   independent, the average inefficiency introduced will be as low as
   log2(log2(e)*2/e) ~= 0.0861 bits per symbol.  This error can, of
   course, be reduced by coding fewer symbols with larger alphabets.  In
   practice the overhead is roughly equal to the overhead introduced by
   other approximate arithmetic coders like H.264's CABAC.




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2.3.  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.

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




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   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

                              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.



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   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
   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



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   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:

                                             _
                                 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



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   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:

   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:





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   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

   [ 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]






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   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
   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



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   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.

   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 */





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   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 */

   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



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   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.

   [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.





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   [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.

   [SM98]     Stuiver, L. and A. Moffat, "Piecewise Integer Mapping for
              Arithmetic Coding", Proc. of the 17th IEEE Data
              Compression Conference (DCC'98) pp. 1--10, March/April
              1998.

   [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

   [4] https://xiph.org/daala/

Author's Address

   Timothy B. Terriberry
   Mozilla Corporation
   331 E. Evelyn Avenue
   Mountain View, CA  94041
   USA

   Phone: +1 650 903-0800
   Email: tterribe@xiph.org





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