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+% -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
+%!TEX root = Vorbis_I_spec.tex
+\section{Probability Model and Codebooks} \label{vorbis:spec:codebook}
+
+\subsection{Overview}
+
+Unlike practically every other mainstream audio codec, Vorbis has no
+statically configured probability model, instead packing all entropy
+decoding configuration, VQ and Huffman, into the bitstream itself in
+the third header, the codec setup header.  This packed configuration
+consists of multiple 'codebooks', each containing a specific
+Huffman-equivalent representation for decoding compressed codewords as
+well as an optional lookup table of output vector values to which a
+decoded Huffman value is applied as an offset, generating the final
+decoded output corresponding to a given compressed codeword.
+
+\subsubsection{Bitwise operation}
+The codebook mechanism is built on top of the vorbis bitpacker. Both
+the codebooks themselves and the codewords they decode are unrolled
+from a packet as a series of arbitrary-width values read from the
+stream according to \xref{vorbis:spec:bitpacking}.
+
+
+
+
+\subsection{Packed codebook format}
+
+For purposes of the examples below, we assume that the storage
+system's native byte width is eight bits.  This is not universally
+true; see \xref{vorbis:spec:bitpacking} for discussion
+relating to non-eight-bit bytes.
+
+\subsubsection{codebook decode}
+
+A codebook begins with a 24 bit sync pattern, 0x564342:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+byte 0: [ 0 1 0 0 0 0 1 0 ] (0x42)
+byte 1: [ 0 1 0 0 0 0 1 1 ] (0x43)
+byte 2: [ 0 1 0 1 0 1 1 0 ] (0x56)
+\end{Verbatim}
+
+16 bit \varname{[codebook\_dimensions]} and 24 bit \varname{[codebook\_entries]} fields:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+
+byte 3: [ X X X X X X X X ]
+byte 4: [ X X X X X X X X ] [codebook\_dimensions] (16 bit unsigned)
+
+byte 5: [ X X X X X X X X ]
+byte 6: [ X X X X X X X X ]
+byte 7: [ X X X X X X X X ] [codebook\_entries] (24 bit unsigned)
+
+\end{Verbatim}
+
+Next is the \varname{[ordered]} bit flag:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+
+byte 8: [               X ] [ordered] (1 bit)
+
+\end{Verbatim}
+
+Each entry, numbering a
+total of \varname{[codebook\_entries]}, is assigned a codeword length.
+We now read the list of codeword lengths and store these lengths in
+the array \varname{[codebook\_codeword\_lengths]}. Decode of lengths is
+according to whether the \varname{[ordered]} flag is set or unset.
+
+\begin{itemize}
+\item
+  If the \varname{[ordered]} flag is unset, the codeword list is not
+  length ordered and the decoder needs to read each codeword length
+  one-by-one.
+
+  The decoder first reads one additional bit flag, the
+  \varname{[sparse]} flag.  This flag determines whether or not the
+  codebook contains unused entries that are not to be included in the
+  codeword decode tree:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+byte 8: [             X 1 ] [sparse] flag (1 bit)
+\end{Verbatim}
+
+  The decoder now performs for each of the \varname{[codebook\_entries]}
+  codebook entries:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+
+  1) if([sparse] is set) \{
+
+         2) [flag] = read one bit;
+         3) if([flag] is set) \{
+
+              4) [length] = read a five bit unsigned integer;
+              5) codeword length for this entry is [length]+1;
+
+            \} else \{
+
+              6) this entry is unused.  mark it as such.
+
+            \}
+
+     \} else the sparse flag is not set \{
+
+        7) [length] = read a five bit unsigned integer;
+        8) the codeword length for this entry is [length]+1;
+
+     \}
+
+\end{Verbatim}
+
+\item
+  If the \varname{[ordered]} flag is set, the codeword list for this
+  codebook is encoded in ascending length order.  Rather than reading
+  a length for every codeword, the encoder reads the number of
+  codewords per length.  That is, beginning at entry zero:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [current\_entry] = 0;
+  2) [current\_length] = read a five bit unsigned integer and add 1;
+  3) [number] = read \link{vorbis:spec:ilog}{ilog}([codebook\_entries] - [current\_entry]) bits as an unsigned integer
+  4) set the entries [current\_entry] through [current\_entry]+[number]-1, inclusive,
+    of the [codebook\_codeword\_lengths] array to [current\_length]
+  5) set [current\_entry] to [number] + [current\_entry]
+  6) increment [current\_length] by 1
+  7) if [current\_entry] is greater than [codebook\_entries] ERROR CONDITION;
+    the decoder will not be able to read this stream.
+  8) if [current\_entry] is less than [codebook\_entries], repeat process starting at 3)
+  9) done.
+\end{Verbatim}
+
+\end{itemize}
+
+After all codeword lengths have been decoded, the decoder reads the
+vector lookup table.  Vorbis I supports three lookup types:
+\begin{enumerate}
+\item
+No lookup
+\item
+Implicitly populated value mapping (lattice VQ)
+\item
+Explicitly populated value mapping (tessellated or 'foam'
+VQ)
+\end{enumerate}
+
+
+The lookup table type is read as a four bit unsigned integer:
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [codebook\_lookup\_type] = read four bits as an unsigned integer
+\end{Verbatim}
+
+Codebook decode precedes according to \varname{[codebook\_lookup\_type]}:
+\begin{itemize}
+\item
+Lookup type zero indicates no lookup to be read.  Proceed past
+lookup decode.
+\item
+Lookup types one and two are similar, differing only in the
+number of lookup values to be read.  Lookup type one reads a list of
+values that are permuted in a set pattern to build a list of vectors,
+each vector of order \varname{[codebook\_dimensions]} scalars.  Lookup
+type two builds the same vector list, but reads each scalar for each
+vector explicitly, rather than building vectors from a smaller list of
+possible scalar values.  Lookup decode proceeds as follows:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [codebook\_minimum\_value] = \link{vorbis:spec:float32:unpack}{float32\_unpack}( read 32 bits as an unsigned integer)
+  2) [codebook\_delta\_value] = \link{vorbis:spec:float32:unpack}{float32\_unpack}( read 32 bits as an unsigned integer)
+  3) [codebook\_value\_bits] = read 4 bits as an unsigned integer and add 1
+  4) [codebook\_sequence\_p] = read 1 bit as a boolean flag
+
+  if ( [codebook\_lookup\_type] is 1 ) \{
+
+     5) [codebook\_lookup\_values] = \link{vorbis:spec:lookup1:values}{lookup1\_values}(\varname{[codebook\_entries]}, \varname{[codebook\_dimensions]} )
+
+  \} else \{
+
+     6) [codebook\_lookup\_values] = \varname{[codebook\_entries]} * \varname{[codebook\_dimensions]}
+
+  \}
+
+  7) read a total of [codebook\_lookup\_values] unsigned integers of [codebook\_value\_bits] each;
+     store these in order in the array [codebook\_multiplicands]
+\end{Verbatim}
+\item
+A \varname{[codebook\_lookup\_type]} of greater than two is reserved
+and indicates a stream that is not decodable by the specification in this
+document.
+
+\end{itemize}
+
+
+An 'end of packet' during any read operation in the above steps is
+considered an error condition rendering the stream undecodable.
+
+\paragraph{Huffman decision tree representation}
+
+The \varname{[codebook\_codeword\_lengths]} array and
+\varname{[codebook\_entries]} value uniquely define the Huffman decision
+tree used for entropy decoding.
+
+Briefly, each used codebook entry (recall that length-unordered
+codebooks support unused codeword entries) is assigned, in order, the
+lowest valued unused binary Huffman codeword possible.  Assume the
+following codeword length list:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+entry 0: length 2
+entry 1: length 4
+entry 2: length 4
+entry 3: length 4
+entry 4: length 4
+entry 5: length 2
+entry 6: length 3
+entry 7: length 3
+\end{Verbatim}
+
+Assigning codewords in order (lowest possible value of the appropriate
+length to highest) results in the following codeword list:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+entry 0: length 2 codeword 00
+entry 1: length 4 codeword 0100
+entry 2: length 4 codeword 0101
+entry 3: length 4 codeword 0110
+entry 4: length 4 codeword 0111
+entry 5: length 2 codeword 10
+entry 6: length 3 codeword 110
+entry 7: length 3 codeword 111
+\end{Verbatim}
+
+
+\begin{note}
+Unlike most binary numerical values in this document, we
+intend the above codewords to be read and used bit by bit from left to
+right, thus the codeword '001' is the bit string 'zero, zero, one'.
+When determining 'lowest possible value' in the assignment definition
+above, the leftmost bit is the MSb.
+\end{note}
+
+It is clear that the codeword length list represents a Huffman
+decision tree with the entry numbers equivalent to the leaves numbered
+left-to-right:
+
+\begin{center}
+\includegraphics[width=10cm]{hufftree}
+\captionof{figure}{huffman tree illustration}
+\end{center}
+
+
+As we assign codewords in order, we see that each choice constructs a
+new leaf in the leftmost possible position.
+
+Note that it's possible to underspecify or overspecify a Huffman tree
+via the length list.  In the above example, if codeword seven were
+eliminated, it's clear that the tree is unfinished:
+
+\begin{center}
+\includegraphics[width=10cm]{hufftree-under}
+\captionof{figure}{underspecified huffman tree illustration}
+\end{center}
+
+
+Similarly, in the original codebook, it's clear that the tree is fully
+populated and a ninth codeword is impossible.  Both underspecified and
+overspecified trees are an error condition rendering the stream
+undecodable.
+
+Codebook entries marked 'unused' are simply skipped in the assigning
+process.  They have no codeword and do not appear in the decision
+tree, thus it's impossible for any bit pattern read from the stream to
+decode to that entry number.
+
+\paragraph{Errata 20150226: Single entry codebooks}
+
+A 'single-entry codebook' is a codebook with one active codeword
+entry. A single-entry codebook may be either a fully populated
+codebook with only one declared entry, or a sparse codebook with only
+one entry marked used. The Vorbis I spec provides no means to specify
+a codeword length of zero, and as a result, a single-entry codebook is
+inherently malformed because it is underpopulated.  The original
+specification did not address directly the matter of single-entry
+codebooks; they were implicitly illegal as it was not possible to
+write such a codebook with a valid tree structure.
+
+In r14811 of the libvorbis reference implementation, Xiph added an
+additional check to the codebook implementation to reject
+underpopulated Huffman trees. This change led to the discovery of
+single-entry books used 'in the wild' when the new, stricter checks
+rejected a number of apparently working streams.
+
+In order to minimize breakage of deployed (if technically erroneous)
+streams, r16073 of the reference implementation explicitly
+special-cased single-entry codebooks to tolerate the single-entry
+case.  Commit r16073 also added the following to the specification:
+
+\blockquote{\sout{Take special care that a codebook with a single used
+    entry is handled properly; it consists of a single codework of
+    zero bits and ’reading’ a value out of such a codebook always
+    returns the single used value and sinks zero bits.
+}} 
+
+The intent was to clarify the spec and codify current practice.
+However, this addition is erroneously at odds with the intent of preserving
+usability of existing streams using single-entry codebooks, disagrees
+with the code changes that reinstated decoding, and does not address how
+single-entry codebooks should be encoded.
+
+As such, the above addition made in r16037 is struck from the
+specification and replaced by the following:
+
+\blockquote{It is possible to declare a Vorbis codebook containing a
+  single codework entry.  A single-entry codebook may be either a
+  fully populated codebook with \varname{[codebook\_entries]} set to
+  1, or a sparse codebook marking only one entry used.  Note that it
+  is not possible to also encode a \varname{[codeword\_length]} of
+  zero for the single used codeword, as the unsigned value written to
+  the stream is \varname{[codeword\_length]-1}.  Instead, encoder
+  implementations should indicate a \varname{[codeword\_length]} of 1
+  and 'write' the codeword to a stream during audio encoding by
+  writing a single zero bit.
+
+  Decoder implementations shall reject a codebook if it contains only
+  one used entry and the encoded \varname{[codeword\_length]} of that
+  entry is not 1.  'Reading' a value from single-entry codebook always
+  returns the single used codeword value and sinks one bit.  Decoders
+  should tolerate that the bit read from the stream be '1' instead of
+  '0'; both values shall return the single used codeword.}
+
+\paragraph{VQ lookup table vector representation}
+
+Unpacking the VQ lookup table vectors relies on the following values:
+\begin{programlisting}
+the [codebook\_multiplicands] array
+[codebook\_minimum\_value]
+[codebook\_delta\_value]
+[codebook\_sequence\_p]
+[codebook\_lookup\_type]
+[codebook\_entries]
+[codebook\_dimensions]
+[codebook\_lookup\_values]
+\end{programlisting}
+
+\bigskip
+
+Decoding (unpacking) a specific vector in the vector lookup table
+proceeds according to \varname{[codebook\_lookup\_type]}.  The unpacked
+vector values are what a codebook would return during audio packet
+decode in a VQ context.
+
+\paragraph{Vector value decode: Lookup type 1}
+
+Lookup type one specifies a lattice VQ lookup table built
+algorithmically from a list of scalar values.  Calculate (unpack) the
+final values of a codebook entry vector from the entries in
+\varname{[codebook\_multiplicands]} as follows (\varname{[value\_vector]}
+is the output vector representing the vector of values for entry number
+\varname{[lookup\_offset]} in this codebook):
+
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [last] = 0;
+  2) [index\_divisor] = 1;
+  3) iterate [i] over the range 0 ... [codebook\_dimensions]-1 (once for each scalar value in the value vector) \{
+
+       4) [multiplicand\_offset] = ( [lookup\_offset] divided by [index\_divisor] using integer
+          division ) integer modulo [codebook\_lookup\_values]
+
+       5) vector [value\_vector] element [i] =
+            ( [codebook\_multiplicands] array element number [multiplicand\_offset] ) *
+            [codebook\_delta\_value] + [codebook\_minimum\_value] + [last];
+
+       6) if ( [codebook\_sequence\_p] is set ) then set [last] = vector [value\_vector] element [i]
+
+       7) [index\_divisor] = [index\_divisor] * [codebook\_lookup\_values]
+
+     \}
+
+  8) vector calculation completed.
+\end{Verbatim}
+
+
+
+\paragraph{Vector value decode: Lookup type 2}
+
+Lookup type two specifies a VQ lookup table in which each scalar in
+each vector is explicitly set by the \varname{[codebook\_multiplicands]}
+array in a one-to-one mapping.  Calculate [unpack] the
+final values of a codebook entry vector from the entries in
+\varname{[codebook\_multiplicands]} as follows (\varname{[value\_vector]}
+is the output vector representing the vector of values for entry number
+\varname{[lookup\_offset]} in this codebook):
+
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [last] = 0;
+  2) [multiplicand\_offset] = [lookup\_offset] * [codebook\_dimensions]
+  3) iterate [i] over the range 0 ... [codebook\_dimensions]-1 (once for each scalar value in the value vector) \{
+
+       4) vector [value\_vector] element [i] =
+            ( [codebook\_multiplicands] array element number [multiplicand\_offset] ) *
+            [codebook\_delta\_value] + [codebook\_minimum\_value] + [last];
+
+       5) if ( [codebook\_sequence\_p] is set ) then set [last] = vector [value\_vector] element [i]
+
+       6) increment [multiplicand\_offset]
+
+     \}
+
+  7) vector calculation completed.
+\end{Verbatim}
+
+
+
+
+
+
+
+
+
+\subsection{Use of the codebook abstraction}
+
+The decoder uses the codebook abstraction much as it does the
+bit-unpacking convention; a specific codebook reads a
+codeword from the bitstream, decoding it into an entry number, and then
+returns that entry number to the decoder (when used in a scalar
+entropy coding context), or uses that entry number as an offset into
+the VQ lookup table, returning a vector of values (when used in a context
+desiring a VQ value). Scalar or VQ context is always explicit; any call
+to the codebook mechanism requests either a scalar entry number or a
+lookup vector.
+
+Note that VQ lookup type zero indicates that there is no lookup table;
+requesting decode using a codebook of lookup type 0 in any context
+expecting a vector return value (even in a case where a vector of
+dimension one) is forbidden.  If decoder setup or decode requests such
+an action, that is an error condition rendering the packet
+undecodable.
+
+Using a codebook to read from the packet bitstream consists first of
+reading and decoding the next codeword in the bitstream. The decoder
+reads bits until the accumulated bits match a codeword in the
+codebook.  This process can be though of as logically walking the
+Huffman decode tree by reading one bit at a time from the bitstream,
+and using the bit as a decision boolean to take the 0 branch (left in
+the above examples) or the 1 branch (right in the above examples).
+Walking the tree finishes when the decode process hits a leaf in the
+decision tree; the result is the entry number corresponding to that
+leaf.  Reading past the end of a packet propagates the 'end-of-stream'
+condition to the decoder.
+
+When used in a scalar context, the resulting codeword entry is the
+desired return value.
+
+When used in a VQ context, the codeword entry number is used as an
+offset into the VQ lookup table.  The value returned to the decoder is
+the vector of scalars corresponding to this offset.