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+% -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
+%!TEX root = Vorbis_I_spec.tex
+\section{Floor type 1 setup and decode} \label{vorbis:spec:floor1}
+
+\subsection{Overview}
+
+Vorbis floor type one uses a piecewise straight-line representation to
+encode a spectral envelope curve. The representation plots this curve
+mechanically on a linear frequency axis and a logarithmic (dB)
+amplitude axis. The integer plotting algorithm used is similar to
+Bresenham's algorithm.
+
+
+
+\subsection{Floor 1 format}
+
+\subsubsection{model}
+
+Floor type one represents a spectral curve as a series of
+line segments.  Synthesis constructs a floor curve using iterative
+prediction in a process roughly equivalent to the following simplified
+description:
+
+\begin{itemize}
+ \item  the first line segment (base case) is a logical line spanning
+from x_0,y_0 to x_1,y_1 where in the base case x_0=0 and x_1=[n], the
+full range of the spectral floor to be computed.
+
+\item the induction step chooses a point x_new within an existing
+logical line segment and produces a y_new value at that point computed
+from the existing line's y value at x_new (as plotted by the line) and
+a difference value decoded from the bitstream packet.
+
+\item floor computation produces two new line segments, one running from
+x_0,y_0 to x_new,y_new and from x_new,y_new to x_1,y_1. This step is
+performed logically even if y_new represents no change to the
+amplitude value at x_new so that later refinement is additionally
+bounded at x_new.
+
+\item the induction step repeats, using a list of x values specified in
+the codec setup header at floor 1 initialization time.  Computation
+is completed at the end of the x value list.
+
+\end{itemize}
+
+
+Consider the following example, with values chosen for ease of
+understanding rather than representing typical configuration:
+
+For the below example, we assume a floor setup with an [n] of 128.
+The list of selected X values in increasing order is
+0,16,32,48,64,80,96,112 and 128.  In list order, the values interleave
+as 0, 128, 64, 32, 96, 16, 48, 80 and 112.  The corresponding
+list-order Y values as decoded from an example packet are 110, 20, -5,
+-45, 0, -25, -10, 30 and -10.  We compute the floor in the following
+way, beginning with the first line:
+
+\begin{center}
+\includegraphics[width=8cm]{floor1-1}
+\captionof{figure}{graph of example floor}
+\end{center}
+
+We now draw new logical lines to reflect the correction to new_Y, and
+iterate for X positions 32 and 96:
+
+\begin{center}
+\includegraphics[width=8cm]{floor1-2}
+\captionof{figure}{graph of example floor}
+\end{center}
+
+Although the new Y value at X position 96 is unchanged, it is still
+used later as an endpoint for further refinement.  From here on, the
+pattern should be clear; we complete the floor computation as follows:
+
+\begin{center}
+\includegraphics[width=8cm]{floor1-3}
+\captionof{figure}{graph of example floor}
+\end{center}
+
+\begin{center}
+\includegraphics[width=8cm]{floor1-4}
+\captionof{figure}{graph of example floor}
+\end{center}
+
+A more efficient algorithm with carefully defined integer rounding
+behavior is used for actual decode, as described later.  The actual
+algorithm splits Y value computation and line plotting into two steps
+with modifications to the above algorithm to eliminate noise
+accumulation through integer roundoff/truncation.
+
+
+
+\subsubsection{header decode}
+
+A list of floor X values is stored in the packet header in interleaved
+format (used in list order during packet decode and synthesis).  This
+list is split into partitions, and each partition is assigned to a
+partition class.  X positions 0 and [n] are implicit and do not belong
+to an explicit partition or partition class.
+
+A partition class consists of a representation vector width (the
+number of Y values which the partition class encodes at once), a
+'subclass' value representing the number of alternate entropy books
+the partition class may use in representing Y values, the list of
+[subclass] books and a master book used to encode which alternate
+books were chosen for representation in a given packet.  The
+master/subclass mechanism is meant to be used as a flexible
+representation cascade while still using codebooks only in a scalar
+context.
+
+\begin{Verbatim}[commandchars=\\\{\}]
+
+  1) [floor1\_partitions] = read 5 bits as unsigned integer
+  2) [maximum\_class] = -1
+  3) iterate [i] over the range 0 ... [floor1\_partitions]-1 \{
+
+        4) vector [floor1\_partition\_class\_list] element [i] = read 4 bits as unsigned integer
+
+     \}
+
+  5) [maximum\_class] = largest integer scalar value in vector [floor1\_partition\_class\_list]
+  6) iterate [i] over the range 0 ... [maximum\_class] \{
+
+        7) vector [floor1\_class\_dimensions] element [i] = read 3 bits as unsigned integer and add 1
+	8) vector [floor1\_class\_subclasses] element [i] = read 2 bits as unsigned integer
+        9) if ( vector [floor1\_class\_subclasses] element [i] is nonzero ) \{
+
+             10) vector [floor1\_class\_masterbooks] element [i] = read 8 bits as unsigned integer
+
+           \}
+
+       11) iterate [j] over the range 0 ... (2 exponent [floor1\_class\_subclasses] element [i]) - 1 \{
+
+             12) array [floor1\_subclass\_books] element [i],[j] =
+                 read 8 bits as unsigned integer and subtract one
+           \}
+      \}
+
+ 13) [floor1\_multiplier] = read 2 bits as unsigned integer and add one
+ 14) [rangebits] = read 4 bits as unsigned integer
+ 15) vector [floor1\_X\_list] element [0] = 0
+ 16) vector [floor1\_X\_list] element [1] = 2 exponent [rangebits];
+ 17) [floor1\_values] = 2
+ 18) iterate [i] over the range 0 ... [floor1\_partitions]-1 \{
+
+       19) [current\_class\_number] = vector [floor1\_partition\_class\_list] element [i]
+       20) iterate [j] over the range 0 ... ([floor1\_class\_dimensions] element [current\_class\_number])-1 \{
+             21) vector [floor1\_X\_list] element ([floor1\_values]) =
+                 read [rangebits] bits as unsigned integer
+             22) increment [floor1\_values] by one
+           \}
+     \}
+
+ 23) done
+\end{Verbatim}
+
+An end-of-packet condition while reading any aspect of a floor 1
+configuration during setup renders a stream undecodable.  In addition,
+a \varname{[floor1\_class\_masterbooks]} or
+\varname{[floor1\_subclass\_books]} scalar element greater than the
+highest numbered codebook configured in this stream is an error
+condition that renders the stream undecodable.  Vector
+[floor1\_x\_list] is limited to a maximum length of 65 elements; a
+setup indicating more than 65 total elements (including elements 0 and
+1 set prior to the read loop) renders the stream undecodable.  All
+vector [floor1\_x\_list] element values must be unique within the
+vector; a non-unique value renders the stream undecodable.
+
+\subsubsection{packet decode} \label{vorbis:spec:floor1-decode}
+
+Packet decode begins by checking the \varname{[nonzero]} flag:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [nonzero] = read 1 bit as boolean
+\end{Verbatim}
+
+If \varname{[nonzero]} is unset, that indicates this channel contained
+no audio energy in this frame.  Decode immediately returns a status
+indicating this floor curve (and thus this channel) is unused this
+frame.  (A return status of 'unused' is different from decoding a
+floor that has all points set to minimum representation amplitude,
+which happens to be approximately -140dB).
+
+
+Assuming \varname{[nonzero]} is set, decode proceeds as follows:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [range] = vector \{ 256, 128, 86, 64 \} element ([floor1\_multiplier]-1)
+  2) vector [floor1\_Y] element [0] = read \link{vorbis:spec:ilog}{ilog}([range]-1) bits as unsigned integer
+  3) vector [floor1\_Y] element [1] = read \link{vorbis:spec:ilog}{ilog}([range]-1) bits as unsigned integer
+  4) [offset] = 2;
+  5) iterate [i] over the range 0 ... [floor1\_partitions]-1 \{
+
+       6) [class] = vector [floor1\_partition\_class]  element [i]
+       7) [cdim]  = vector [floor1\_class\_dimensions] element [class]
+       8) [cbits] = vector [floor1\_class\_subclasses] element [class]
+       9) [csub]  = (2 exponent [cbits])-1
+      10) [cval]  = 0
+      11) if ( [cbits] is greater than zero ) \{
+
+             12) [cval] = read from packet using codebook number
+                 (vector [floor1\_class\_masterbooks] element [class]) in scalar context
+          \}
+
+      13) iterate [j] over the range 0 ... [cdim]-1 \{
+
+             14) [book] = array [floor1\_subclass\_books] element [class],([cval] bitwise AND [csub])
+             15) [cval] = [cval] right shifted [cbits] bits
+	     16) if ( [book] is not less than zero ) \{
+
+	           17) vector [floor1\_Y] element ([j]+[offset]) = read from packet using codebook
+                       [book] in scalar context
+
+                 \} else [book] is less than zero \{
+
+	           18) vector [floor1\_Y] element ([j]+[offset]) = 0
+
+                 \}
+          \}
+
+      19) [offset] = [offset] + [cdim]
+
+     \}
+
+ 20) done
+\end{Verbatim}
+
+An end-of-packet condition during curve decode should be considered a
+nominal occurrence; if end-of-packet is reached during any read
+operation above, floor decode is to return 'unused' status as if the
+\varname{[nonzero]} flag had been unset at the beginning of decode.
+
+
+Vector \varname{[floor1\_Y]} contains the values from packet decode
+needed for floor 1 synthesis.
+
+
+
+\subsubsection{curve computation} \label{vorbis:spec:floor1-synth}
+
+Curve computation is split into two logical steps; the first step
+derives final Y amplitude values from the encoded, wrapped difference
+values taken from the bitstream.  The second step plots the curve
+lines.  Also, although zero-difference values are used in the
+iterative prediction to find final Y values, these points are
+conditionally skipped during final line computation in step two.
+Skipping zero-difference values allows a smoother line fit.
+
+Although some aspects of the below algorithm look like inconsequential
+optimizations, implementors are warned to follow the details closely.
+Deviation from implementing a strictly equivalent algorithm can result
+in serious decoding errors.
+
+{\em Additional note:} Although \varname{[floor1\_final\_Y]} values in
+the prediction loop and at the end of step 1 are inherently limited by
+the prediction algorithm to [0, \varname{[range]}), it is possible to
+  abuse the setup and codebook machinery to produce negative or
+  over-range results.  We suggest that decoder implementations guard
+  the values in vector \varname{[floor1\_final\_Y]} by clamping each
+  element to [0, \varname{[range]}) after step 1.  Variants of this
+    suggestion are acceptable as valid floor1 setups cannot produce
+    out of range values.
+
+\begin{description}
+\item[step 1: amplitude value synthesis]
+
+Unwrap the always-positive-or-zero values read from the packet into
++/- difference values, then apply to line prediction.
+
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [range] = vector \{ 256, 128, 86, 64 \} element ([floor1\_multiplier]-1)
+  2) vector [floor1\_step2\_flag] element [0] = set
+  3) vector [floor1\_step2\_flag] element [1] = set
+  4) vector [floor1\_final\_Y] element [0] = vector [floor1\_Y] element [0]
+  5) vector [floor1\_final\_Y] element [1] = vector [floor1\_Y] element [1]
+  6) iterate [i] over the range 2 ... [floor1\_values]-1 \{
+
+       7) [low\_neighbor\_offset] = \link{vorbis:spec:low:neighbor}{low\_neighbor}([floor1\_X\_list],[i])
+       8) [high\_neighbor\_offset] = \link{vorbis:spec:high:neighbor}{high\_neighbor}([floor1\_X\_list],[i])
+
+       9) [predicted] = \link{vorbis:spec:render:point}{render\_point}( vector [floor1\_X\_list] element [low\_neighbor\_offset],
+				      vector [floor1\_final\_Y] element [low\_neighbor\_offset],
+                                      vector [floor1\_X\_list] element [high\_neighbor\_offset],
+				      vector [floor1\_final\_Y] element [high\_neighbor\_offset],
+                                      vector [floor1\_X\_list] element [i] )
+
+      10) [val] = vector [floor1\_Y] element [i]
+      11) [highroom] = [range] - [predicted]
+      12) [lowroom]  = [predicted]
+      13) if ( [highroom] is less than [lowroom] ) \{
+
+            14) [room] = [highroom] * 2
+
+          \} else [highroom] is not less than [lowroom] \{
+
+            15) [room] = [lowroom] * 2
+
+          \}
+
+      16) if ( [val] is nonzero ) \{
+
+            17) vector [floor1\_step2\_flag] element [low\_neighbor\_offset] = set
+            18) vector [floor1\_step2\_flag] element [high\_neighbor\_offset] = set
+            19) vector [floor1\_step2\_flag] element [i] = set
+            20) if ( [val] is greater than or equal to [room] ) \{
+
+                  21) if ( [highroom] is greater than [lowroom] ) \{
+
+                        22) vector [floor1\_final\_Y] element [i] = [val] - [lowroom] + [predicted]
+
+		      \} else [highroom] is not greater than [lowroom] \{
+
+                        23) vector [floor1\_final\_Y] element [i] = [predicted] - [val] + [highroom] - 1
+
+                      \}
+
+                \} else [val] is less than [room] \{
+
+                    24) if ([val] is odd) \{
+
+                        25) vector [floor1\_final\_Y] element [i] =
+                            [predicted] - (([val] + 1) divided by  2 using integer division)
+
+                      \} else [val] is even \{
+
+                        26) vector [floor1\_final\_Y] element [i] =
+                            [predicted] + ([val] / 2 using integer division)
+
+                      \}
+
+                \}
+
+          \} else [val] is zero \{
+
+            27) vector [floor1\_step2\_flag] element [i] = unset
+            28) vector [floor1\_final\_Y] element [i] = [predicted]
+
+          \}
+
+     \}
+
+ 29) done
+
+\end{Verbatim}
+
+
+
+\item[step 2: curve synthesis]
+
+Curve synthesis generates a return vector \varname{[floor]} of length
+\varname{[n]} (where \varname{[n]} is provided by the decode process
+calling to floor decode).  Floor 1 curve synthesis makes use of the
+\varname{[floor1\_X\_list]}, \varname{[floor1\_final\_Y]} and
+\varname{[floor1\_step2\_flag]} vectors, as well as [floor1\_multiplier]
+and [floor1\_values] values.
+
+Decode begins by sorting the scalars from vectors
+\varname{[floor1\_X\_list]}, \varname{[floor1\_final\_Y]} and
+\varname{[floor1\_step2\_flag]} together into new vectors
+\varname{[floor1\_X\_list]'}, \varname{[floor1\_final\_Y]'} and
+\varname{[floor1\_step2\_flag]'} according to ascending sort order of the
+values in \varname{[floor1\_X\_list]}.  That is, sort the values of
+\varname{[floor1\_X\_list]} and then apply the same permutation to
+elements of the other two vectors so that the X, Y and step2\_flag
+values still match.
+
+Then compute the final curve in one pass:
+
+\begin{Verbatim}[commandchars=\\\{\}]
+  1) [hx] = 0
+  2) [lx] = 0
+  3) [ly] = vector [floor1\_final\_Y]' element [0] * [floor1\_multiplier]
+  4) iterate [i] over the range 1 ... [floor1\_values]-1 \{
+
+       5) if ( [floor1\_step2\_flag]' element [i] is set ) \{
+
+             6) [hy] = [floor1\_final\_Y]' element [i] * [floor1\_multiplier]
+ 	     7) [hx] = [floor1\_X\_list]' element [i]
+             8) \link{vorbis:spec:render:line}{render\_line}( [lx], [ly], [hx], [hy], [floor] )
+             9) [lx] = [hx]
+	    10) [ly] = [hy]
+          \}
+     \}
+
+ 11) if ( [hx] is less than [n] ) \{
+
+        12) \link{vorbis:spec:render:line}{render\_line}( [hx], [hy], [n], [hy], [floor] )
+
+     \}
+
+ 13) if ( [hx] is greater than [n] ) \{
+
+            14) truncate vector [floor] to [n] elements
+
+     \}
+
+ 15) for each scalar in vector [floor], perform a lookup substitution using
+     the scalar value from [floor] as an offset into the vector \link{vorbis:spec:floor1:inverse:dB:table}{[floor1\_inverse\_dB\_static\_table]}
+
+ 16) done
+
+\end{Verbatim}
+
+\end{description}