--- /dev/null
+% -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
+%!TEX root = Vorbis_I_spec.tex
+% $Id$
+\section{Residue setup and decode} \label{vorbis:spec:residue}
+
+\subsection{Overview}
+
+A residue vector represents the fine detail of the audio spectrum of
+one channel in an audio frame after the encoder subtracts the floor
+curve and performs any channel coupling. A residue vector may
+represent spectral lines, spectral magnitude, spectral phase or
+hybrids as mixed by channel coupling. The exact semantic content of
+the vector does not matter to the residue abstraction.
+
+Whatever the exact qualities, the Vorbis residue abstraction codes the
+residue vectors into the bitstream packet, and then reconstructs the
+vectors during decode. Vorbis makes use of three different encoding
+variants (numbered 0, 1 and 2) of the same basic vector encoding
+abstraction.
+
+
+
+\subsection{Residue format}
+
+Residue format partitions each vector in the vector bundle into chunks,
+classifies each chunk, encodes the chunk classifications and finally
+encodes the chunks themselves using the the specific VQ arrangement
+defined for each selected classification.
+The exact interleaving and partitioning vary by residue encoding number,
+however the high-level process used to classify and encode the residue
+vector is the same in all three variants.
+
+A set of coded residue vectors are all of the same length. High level
+coding structure, ignoring for the moment exactly how a partition is
+encoded and simply trusting that it is, is as follows:
+
+\begin{itemize}
+\item Each vector is partitioned into multiple equal sized chunks
+according to configuration specified. If we have a vector size of
+\emph{n}, a partition size \emph{residue\_partition\_size}, and a total
+of \emph{ch} residue vectors, the total number of partitioned chunks
+coded is \emph{n}/\emph{residue\_partition\_size}*\emph{ch}. It is
+important to note that the integer division truncates. In the below
+example, we assume an example \emph{residue\_partition\_size} of 8.
+
+\item Each partition in each vector has a classification number that
+specifies which of multiple configured VQ codebook setups are used to
+decode that partition. The classification numbers of each partition
+can be thought of as forming a vector in their own right, as in the
+illustration below. Just as the residue vectors are coded in grouped
+partitions to increase encoding efficiency, the classification vector
+is also partitioned into chunks. The integer elements of each scalar
+in a classification chunk are built into a single scalar that
+represents the classification numbers in that chunk. In the below
+example, the classification codeword encodes two classification
+numbers.
+
+\item The values in a residue vector may be encoded monolithically in a
+single pass through the residue vector, but more often efficient
+codebook design dictates that each vector is encoded as the additive
+sum of several passes through the residue vector using more than one
+VQ codebook. Thus, each residue value potentially accumulates values
+from multiple decode passes. The classification value associated with
+a partition is the same in each pass, thus the classification codeword
+is coded only in the first pass.
+
+\end{itemize}
+
+
+\begin{center}
+\includegraphics[width=\textwidth]{residue-pack}
+\captionof{figure}{illustration of residue vector format}
+\end{center}
+
+
+
+\subsection{residue 0}
+
+Residue 0 and 1 differ only in the way the values within a residue
+partition are interleaved during partition encoding (visually treated
+as a black box--or cyan box or brown box--in the above figure).
+
+Residue encoding 0 interleaves VQ encoding according to the
+dimension of the codebook used to encode a partition in a specific
+pass. The dimension of the codebook need not be the same in multiple
+passes, however the partition size must be an even multiple of the
+codebook dimension.
+
+As an example, assume a partition vector of size eight, to be encoded
+by residue 0 using codebook sizes of 8, 4, 2 and 1:
+
+\begin{programlisting}
+
+ original residue vector: [ 0 1 2 3 4 5 6 7 ]
+
+codebook dimensions = 8 encoded as: [ 0 1 2 3 4 5 6 7 ]
+
+codebook dimensions = 4 encoded as: [ 0 2 4 6 ], [ 1 3 5 7 ]
+
+codebook dimensions = 2 encoded as: [ 0 4 ], [ 1 5 ], [ 2 6 ], [ 3 7 ]
+
+codebook dimensions = 1 encoded as: [ 0 ], [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ]
+
+\end{programlisting}
+
+It is worth mentioning at this point that no configurable value in the
+residue coding setup is restricted to a power of two.
+
+
+
+\subsection{residue 1}
+
+Residue 1 does not interleave VQ encoding. It represents partition
+vector scalars in order. As with residue 0, however, partition length
+must be an integer multiple of the codebook dimension, although
+dimension may vary from pass to pass.
+
+As an example, assume a partition vector of size eight, to be encoded
+by residue 0 using codebook sizes of 8, 4, 2 and 1:
+
+\begin{programlisting}
+
+ original residue vector: [ 0 1 2 3 4 5 6 7 ]
+
+codebook dimensions = 8 encoded as: [ 0 1 2 3 4 5 6 7 ]
+
+codebook dimensions = 4 encoded as: [ 0 1 2 3 ], [ 4 5 6 7 ]
+
+codebook dimensions = 2 encoded as: [ 0 1 ], [ 2 3 ], [ 4 5 ], [ 6 7 ]
+
+codebook dimensions = 1 encoded as: [ 0 ], [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ]
+
+\end{programlisting}
+
+
+
+\subsection{residue 2}
+
+Residue type two can be thought of as a variant of residue type 1.
+Rather than encoding multiple passed-in vectors as in residue type 1,
+the \emph{ch} passed in vectors of length \emph{n} are first
+interleaved and flattened into a single vector of length
+\emph{ch}*\emph{n}. Encoding then proceeds as in type 1. Decoding is
+as in type 1 with decode interleave reversed. If operating on a single
+vector to begin with, residue type 1 and type 2 are equivalent.
+
+\begin{center}
+\includegraphics[width=\textwidth]{residue2}
+\captionof{figure}{illustration of residue type 2}
+\end{center}
+
+
+\subsection{Residue decode}
+
+\subsubsection{header decode}
+
+Header decode for all three residue types is identical.
+\begin{programlisting}
+ 1) [residue\_begin] = read 24 bits as unsigned integer
+ 2) [residue\_end] = read 24 bits as unsigned integer
+ 3) [residue\_partition\_size] = read 24 bits as unsigned integer and add one
+ 4) [residue\_classifications] = read 6 bits as unsigned integer and add one
+ 5) [residue\_classbook] = read 8 bits as unsigned integer
+\end{programlisting}
+
+\varname{[residue\_begin]} and
+\varname{[residue\_end]} select the specific sub-portion of
+each vector that is actually coded; it implements akin to a bandpass
+where, for coding purposes, the vector effectively begins at element
+\varname{[residue\_begin]} and ends at
+\varname{[residue\_end]}. Preceding and following values in
+the unpacked vectors are zeroed. Note that for residue type 2, these
+values as well as \varname{[residue\_partition\_size]}apply to
+the interleaved vector, not the individual vectors before interleave.
+\varname{[residue\_partition\_size]} is as explained above,
+\varname{[residue\_classifications]} is the number of possible
+classification to which a partition can belong and
+\varname{[residue\_classbook]} is the codebook number used to
+code classification codewords. The number of dimensions in book
+\varname{[residue\_classbook]} determines how many
+classification values are grouped into a single classification
+codeword. Note that the number of entries and dimensions in book
+\varname{[residue\_classbook]}, along with
+\varname{[residue\_classifications]}, overdetermines to
+possible number of classification codewords.
+If \varname{[residue\_classifications]}\^{}\varname{[residue\_classbook]}.dimensions
+exceeds \varname{[residue\_classbook]}.entries, the
+bitstream should be regarded to be undecodable.
+
+Next we read a bitmap pattern that specifies which partition classes
+code values in which passes.
+
+\begin{programlisting}
+ 1) iterate [i] over the range 0 ... [residue\_classifications]-1 {
+
+ 2) [high\_bits] = 0
+ 3) [low\_bits] = read 3 bits as unsigned integer
+ 4) [bitflag] = read one bit as boolean
+ 5) if ( [bitflag] is set ) then [high\_bits] = read five bits as unsigned integer
+ 6) vector [residue\_cascade] element [i] = [high\_bits] * 8 + [low\_bits]
+ }
+ 7) done
+\end{programlisting}
+
+Finally, we read in a list of book numbers, each corresponding to
+specific bit set in the cascade bitmap. We loop over the possible
+codebook classifications and the maximum possible number of encoding
+stages (8 in Vorbis I, as constrained by the elements of the cascade
+bitmap being eight bits):
+
+\begin{programlisting}
+ 1) iterate [i] over the range 0 ... [residue\_classifications]-1 {
+
+ 2) iterate [j] over the range 0 ... 7 {
+
+ 3) if ( vector [residue\_cascade] element [i] bit [j] is set ) {
+
+ 4) array [residue\_books] element [i][j] = read 8 bits as unsigned integer
+
+ } else {
+
+ 5) array [residue\_books] element [i][j] = unused
+
+ }
+ }
+ }
+
+ 6) done
+\end{programlisting}
+
+An end-of-packet condition at any point in header decode renders the
+stream undecodable. In addition, any codebook number greater than the
+maximum numbered codebook set up in this stream also renders the
+stream undecodable. All codebooks in array [residue\_books] are
+required to have a value mapping. The presence of codebook in array
+[residue\_books] without a value mapping (maptype equals zero) renders
+the stream undecodable.
+
+
+
+\subsubsection{packet decode}
+
+Format 0 and 1 packet decode is identical except for specific
+partition interleave. Format 2 packet decode can be built out of the
+format 1 decode process. Thus we describe first the decode
+infrastructure identical to all three formats.
+
+In addition to configuration information, the residue decode process
+is passed the number of vectors in the submap bundle and a vector of
+flags indicating if any of the vectors are not to be decoded. If the
+passed in number of vectors is 3 and vector number 1 is marked 'do not
+decode', decode skips vector 1 during the decode loop. However, even
+'do not decode' vectors are allocated and zeroed.
+
+Depending on the values of \varname{[residue\_begin]} and
+\varname{[residue\_end]}, it is obvious that the encoded
+portion of a residue vector may be the entire possible residue vector
+or some other strict subset of the actual residue vector size with
+zero padding at either uncoded end. However, it is also possible to
+set \varname{[residue\_begin]} and
+\varname{[residue\_end]} to specify a range partially or
+wholly beyond the maximum vector size. Before beginning residue
+decode, limit \varname{[residue\_begin]} and
+\varname{[residue\_end]} to the maximum possible vector size
+as follows. We assume that the number of vectors being encoded,
+\varname{[ch]} is provided by the higher level decoding
+process.
+
+\begin{programlisting}
+ 1) [actual\_size] = current blocksize/2;
+ 2) if residue encoding is format 2
+ 3) [actual\_size] = [actual\_size] * [ch];
+ 4) [limit\_residue\_begin] = maximum of ([residue\_begin],[actual\_size]);
+ 5) [limit\_residue\_end] = maximum of ([residue\_end],[actual\_size]);
+\end{programlisting}
+
+The following convenience values are conceptually useful to clarifying
+the decode process:
+
+\begin{programlisting}
+ 1) [classwords\_per\_codeword] = [codebook\_dimensions] value of codebook [residue\_classbook]
+ 2) [n\_to\_read] = [limit\_residue\_end] - [limit\_residue\_begin]
+ 3) [partitions\_to\_read] = [n\_to\_read] / [residue\_partition\_size]
+\end{programlisting}
+
+Packet decode proceeds as follows, matching the description offered earlier in the document.
+\begin{programlisting}
+ 1) allocate and zero all vectors that will be returned.
+ 2) if ([n\_to\_read] is zero), stop; there is no residue to decode.
+ 3) iterate [pass] over the range 0 ... 7 {
+
+ 4) [partition\_count] = 0
+
+ 5) while [partition\_count] is less than [partitions\_to\_read]
+
+ 6) if ([pass] is zero) {
+
+ 7) iterate [j] over the range 0 .. [ch]-1 {
+
+ 8) if vector [j] is not marked 'do not decode' {
+
+ 9) [temp] = read from packet using codebook [residue\_classbook] in scalar context
+ 10) iterate [i] descending over the range [classwords\_per\_codeword]-1 ... 0 {
+
+ 11) array [classifications] element [j],([i]+[partition\_count]) =
+ [temp] integer modulo [residue\_classifications]
+ 12) [temp] = [temp] / [residue\_classifications] using integer division
+
+ }
+
+ }
+
+ }
+
+ }
+
+ 13) iterate [i] over the range 0 .. ([classwords\_per\_codeword] - 1) while [partition\_count]
+ is also less than [partitions\_to\_read] {
+
+ 14) iterate [j] over the range 0 .. [ch]-1 {
+
+ 15) if vector [j] is not marked 'do not decode' {
+
+ 16) [vqclass] = array [classifications] element [j],[partition\_count]
+ 17) [vqbook] = array [residue\_books] element [vqclass],[pass]
+ 18) if ([vqbook] is not 'unused') {
+
+ 19) decode partition into output vector number [j], starting at scalar
+ offset [limit\_residue\_begin]+[partition\_count]*[residue\_partition\_size] using
+ codebook number [vqbook] in VQ context
+ }
+ }
+
+ 20) increment [partition\_count] by one
+
+ }
+ }
+ }
+
+ 21) done
+
+\end{programlisting}
+
+An end-of-packet condition during packet decode is to be considered a
+nominal occurrence. Decode returns the result of vector decode up to
+that point.
+
+
+
+\subsubsection{format 0 specifics}
+
+Format zero decodes partitions exactly as described earlier in the
+'Residue Format: residue 0' section. The following pseudocode
+presents the same algorithm. Assume:
+
+\begin{itemize}
+\item \varname{[n]} is the value in \varname{[residue\_partition\_size]}
+\item \varname{[v]} is the residue vector
+\item \varname{[offset]} is the beginning read offset in [v]
+\end{itemize}
+
+
+\begin{programlisting}
+ 1) [step] = [n] / [codebook\_dimensions]
+ 2) iterate [i] over the range 0 ... [step]-1 {
+
+ 3) vector [entry\_temp] = read vector from packet using current codebook in VQ context
+ 4) iterate [j] over the range 0 ... [codebook\_dimensions]-1 {
+
+ 5) vector [v] element ([offset]+[i]+[j]*[step]) =
+ vector [v] element ([offset]+[i]+[j]*[step]) +
+ vector [entry\_temp] element [j]
+
+ }
+
+ }
+
+ 6) done
+
+\end{programlisting}
+
+
+
+\subsubsection{format 1 specifics}
+
+Format 1 decodes partitions exactly as described earlier in the
+'Residue Format: residue 1' section. The following pseudocode
+presents the same algorithm. Assume:
+
+\begin{itemize}
+\item \varname{[n]} is the value in
+\varname{[residue\_partition\_size]}
+\item \varname{[v]} is the residue vector
+\item \varname{[offset]} is the beginning read offset in [v]
+\end{itemize}
+
+
+\begin{programlisting}
+ 1) [i] = 0
+ 2) vector [entry\_temp] = read vector from packet using current codebook in VQ context
+ 3) iterate [j] over the range 0 ... [codebook\_dimensions]-1 {
+
+ 4) vector [v] element ([offset]+[i]) =
+ vector [v] element ([offset]+[i]) +
+ vector [entry\_temp] element [j]
+ 5) increment [i]
+
+ }
+
+ 6) if ( [i] is less than [n] ) continue at step 2
+ 7) done
+\end{programlisting}
+
+
+
+\subsubsection{format 2 specifics}
+
+Format 2 is reducible to format 1. It may be implemented as an additional step prior to and an additional post-decode step after a normal format 1 decode.
+
+
+Format 2 handles 'do not decode' vectors differently than residue 0 or
+1; if all vectors are marked 'do not decode', no decode occurrs.
+However, if at least one vector is to be decoded, all the vectors are
+decoded. We then request normal format 1 to decode a single vector
+representing all output channels, rather than a vector for each
+channel. After decode, deinterleave the vector into independent vectors, one for each output channel. That is:
+
+\begin{enumerate}
+ \item If all vectors 0 through \emph{ch}-1 are marked 'do not decode', allocate and clear a single vector \varname{[v]}of length \emph{ch*n} and skip step 2 below; proceed directly to the post-decode step.
+ \item Rather than performing format 1 decode to produce \emph{ch} vectors of length \emph{n} each, call format 1 decode to produce a single vector \varname{[v]} of length \emph{ch*n}.
+ \item Post decode: Deinterleave the single vector \varname{[v]} returned by format 1 decode as described above into \emph{ch} independent vectors, one for each outputchannel, according to:
+ \begin{programlisting}
+ 1) iterate [i] over the range 0 ... [n]-1 {
+
+ 2) iterate [j] over the range 0 ... [ch]-1 {
+
+ 3) output vector number [j] element [i] = vector [v] element ([i] * [ch] + [j])
+
+ }
+ }
+
+ 4) done
+ \end{programlisting}
+
+\end{enumerate}
+
+
+
+
+
+
+
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