$Id: 08-residue.xml 13159 2007-06-21 05:22:35Z xiphmont $

# Residue setup and decode

## 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.

## 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:

- Each vector is partitioned into multiple equal sized chunks according to configuration specified. If we have a vector size of n, a partition size residue_partition_size, and a total of ch residue vectors, the total number of partitioned chunks coded is n/residue_partition_size*ch. It is important to note that the integer division truncates. In the below example, we assume an example residue_partition_size of 8.

- 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.

- 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.

[illustration of residue vector format]

## 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:

                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 ]

It is worth mentioning at this point that no configurable value in the residue coding setup is restricted to a power of two.

## 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:

                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 ]

## 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 ch passed in vectors of length n are first interleaved and flattened into a single vector of length ch*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.

[illustration of residue type 2]

## Residue decode

### header decode

Header decode for all three residue types is identical.

      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

`[residue_begin]` and `[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 `[residue_begin]` and ends at `[residue_end]`. Preceding and following values in the unpacked vectors are zeroed. Note that for residue type 2, these values as well as `[residue_partition_size]`apply to the interleaved vector, not the individual vectors before interleave. `[residue_partition_size]` is as explained above, `[residue_classifications]` is the number of possible classification to which a partition can belong and `[residue_classbook]` is the codebook number used to code classification codewords. The number of dimensions in book `[residue_classbook]` determines how many classification values are grouped into a single classification codeword.

Next we read a bitmap pattern that specifies which partition classes code values in which passes.

      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

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):

      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

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.

### 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 `[residue_begin]` and `[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 `[residue_begin]` and `[residue_end]` to specify a range partially or wholly beyond the maximum vector size. Before beginning residue decode, limit `[residue_begin]` and `[residue_end]` to the maximum possible vector size as follows. We assume that the number of vectors being encoded, `[ch]` is provided by the higher level decoding process.

      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]);

The following convenience values are conceptually useful to clarifying the decode process:

      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]

Packet decode proceeds as follows, matching the description offered earlier in the document.

      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

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.

### 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:

- `[n]` is the value in `[residue_partition_size]`

- `[v]` is the residue vector

- `[offset]` is the beginning read offset in [v]

     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

### 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:

- `[n]` is the value in `[residue_partition_size]`

- `[v]` is the residue vector

- `[offset]` is the beginning read offset in [v]

     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

### 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:

1. If all vectors 0 through ch-1 are marked 'do not decode', allocate and clear a single vector `[v]`of length ch*n and skip step 2 below; proceed directly to the post-decode step.

2. Rather than performing format 1 decode to produce ch vectors of length n each, call format 1 decode to produce a single vector `[v]` of length ch*n.

3. Post decode: Deinterleave the single vector `[v]` returned by format 1 decode as described above into ch independent vectors, one for each outputchannel, according to:

         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