$Id: 03-codebook.xml 7186 2004-07-20 07:19:25Z xiphmont $

# Probability Model and Codebooks

## 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.

### 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 .

## 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 for discussion relating to non-eight-bit bytes.

### codebook decode

A codebook begins with a 24 bit sync pattern, 0x564342:

    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)

16 bit `[codebook_dimensions]` and 24 bit `[codebook_entries]` fields:

    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)

Next is the `[ordered]` bit flag:

    byte 8: [               X ] [ordered] (1 bit)

Each entry, numbering a total of `[codebook_entries]`, is assigned a codeword length. We now read the list of codeword lengths and store these lengths in the array `[codebook_codeword_lengths]`. Decode of lengths is according to whether the `[ordered]` flag is set or unset.

- If the `[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 `[sparse]` flag. This flag determines whether or not the codebook contains unused entries that are not to be included in the codeword decode tree:

      byte 8: [             X 1 ] [sparse] flag (1 bit)

  The decoder now performs for each of the `[codebook_entries]` codebook entries:

        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;

           }

- If the `[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:

        1) [current_entry] = 0;
        2) [current_length] = read a five bit unsigned integer and add 1;
        3) [number] = read 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.

After all codeword lengths have been decoded, the decoder reads the vector lookup table. Vorbis I supports three lookup types:

1. No lookup

2. Implicitly populated value mapping (lattice VQ)

3. Explicitly populated value mapping (tessellated or 'foam' VQ)

The lookup table type is read as a four bit unsigned integer:

      1) [codebook_lookup_type] = read four bits as an unsigned integer

Codebook decode precedes according to `[codebook_lookup_type]`:

- Lookup type zero indicates no lookup to be read. Proceed past lookup decode.

- 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 `[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:

        1) [codebook_minimum_value] = float32_unpack( read 32 bits as an unsigned integer)
        2) [codebook_delta_value] = 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] = lookup1_values([codebook_entries], [codebook_dimensions] )

        } else {

           6) [codebook_lookup_values] = [codebook_entries] * [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]

- A `[codebook_lookup_type]` of greater than two is reserved and indicates a stream that is not decodable by the specification in this document.

An 'end of packet' during any read operation in the above steps is considered an error condition rendering the stream undecodable.

#### Huffman decision tree representation

The `[codebook_codeword_lengths]` array and `[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:

    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

Assigning codewords in order (lowest possible value of the appropriate length to highest) results in the following codeword list:

    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

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.

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:

[huffman tree illustration]

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:

[underspecified huffman tree illustration]

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.

#### VQ lookup table vector representation

Unpacking the VQ lookup table vectors relies on the following values:

    the [codebook_multiplicands] array
    [codebook_minimum_value]
    [codebook_delta_value]
    [codebook_sequence_p]
    [codebook_lookup_type]
    [codebook_entries]
    [codebook_dimensions]
    [codebook_lookup_values]

Decoding (unpacking) a specific vector in the vector lookup table proceeds according to `[codebook_lookup_type]`. The unpacked vector values are what a codebook would return during audio packet decode in a VQ context.

##### 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 `[codebook_multiplicands]` as follows (`[value_vector]` is the output vector representing the vector of values for entry number `[lookup_offset]` in this codebook):

      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.

##### 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 `[codebook_multiplicands]` array in a one-to-one mapping. Calculate [unpack] the final values of a codebook entry vector from the entries in `[codebook_multiplicands]` as follows (`[value_vector]` is the output vector representing the vector of values for entry number `[lookup_offset]` in this codebook):

      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.

## 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.