$Id: 07-floor1.xml 7186 2004-07-20 07:19:25Z xiphmont $

# Floor type 1 setup and decode

## 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.

## Floor 1 format

### 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:

- 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.

- 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.

- 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.

- 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.

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:

[graph of example floor]

We now draw new logical lines to reflect the correction to new_Y, and iterate for X positions 32 and 96:

[graph of example floor]

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:

[graph of example floor]
[graph of example floor]

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.

### 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.

      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 ([j] + [floor1_values]) =
                     read [rangebits] bits as unsigned integer
                 22) increment [floor1_values] by one
               }
         }

     23) done

An end-of-packet condition while reading any aspect of a floor 1 configuration during setup renders a stream undecodable. In addition, a `[floor1_class_masterbooks]` or `[floor1_subclass_books]` scalar element greater than the highest numbered codebook configured in this stream is an error condition that renders the stream undecodable.

#### packet decode

Packet decode begins by checking the `[nonzero]` flag:

      1) [nonzero] = read 1 bit as boolean

If `[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 `[nonzero]` is set, decode proceeds as follows:

      1) [range] = vector { 256, 128, 86, 64 } element ([floor1_multiplier]-1)
      2) vector [floor1_Y] element [0] = read ilog([range]-1) bits as unsigned integer
      3) vector [floor1_Y] element [1] = read 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

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 `[nonzero]` flag had been unset at the beginning of decode.

Vector `[floor1_Y]` contains the values from packet decode needed for floor 1 synthesis.

#### curve computation

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.

##### step 1: amplitude value synthesis

Unwrap the always-positive-or-zero values read from the packet into +/- difference values, then apply to line prediction.

      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] = low_neighbor([floor1_X_list],[i])
           8) [high_neighbor_offset] = high_neighbor([floor1_X_list],[i])

           9) [predicted] = 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) [root] = [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

##### step 2: curve synthesis

Curve synthesis generates a return vector `[floor]` of length `[n]` (where `[n]` is provided by the decode process calling to floor decode). Floor 1 curve synthesis makes use of the `[floor1_X_list]`, `[floor1_final_Y]` and `[floor1_step2_flag]` vectors, as well as [floor1_multiplier] and [floor1_values] values.

Decode begins by sorting the scalars from vectors `[floor1_X_list]`, `[floor1_final_Y]` and `[floor1_step2_flag]` together into new vectors `[floor1_X_list]'`, `[floor1_final_Y]'` and `[floor1_step2_flag]'` according to ascending sort order of the values in `[floor1_X_list]`. That is, sort the values of `[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:

      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]' is set ) {

                 6) [hy] = [floor1_final_Y]' element [i] * [floor1_multiplier]
     	     7) [hx] = [floor1_X_list]' element [i]
                 8) render_line( [lx], [ly], [hx], [hy], [floor] )
                 9) [lx] = [hx]
    	    10) [ly] = [hy]
              }
         }

     11) if ( [hx] is less than [n] ) {

            12) 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 [floor1_inverse_dB_static_table]

     16) done