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Specification for WebP Lossless Bitstream
=========================================

_Jyrki Alakuijala, Ph.D., Google, Inc., 2012-06-19_

Paragraphs marked as \[AMENDED\] were amended on 2014-09-16.

Paragraphs marked as \[AMENDED2\] were amended on 2022-05-13.

Abstract
--------

WebP lossless is an image format for lossless compression of ARGB
images. The lossless format stores and restores the pixel values
exactly, including the color values for zero alpha pixels. The
format uses subresolution images, recursively embedded into the format
itself, for storing statistical data about the images, such as the used
entropy codes, spatial predictors, color space conversion, and color
table. LZ77, prefix coding, and a color cache are used for compression
of the bulk data. Decoding speeds faster than PNG have been
demonstrated, as well as 25% denser compression than can be achieved
using today's PNG format.


* TOC placeholder
{:toc}


Nomenclature
------------

ARGB
: A pixel value consisting of alpha, red, green, and blue values.

ARGB image
: A two-dimensional array containing ARGB pixels.

color cache
: A small hash-addressed array to store recently used colors, to be able
  to recall them with shorter codes.

color indexing image
: A one-dimensional image of colors that can be indexed using a small
  integer (up to 256 within WebP lossless).

color transform image
: A two-dimensional subresolution image containing data about
  correlations of color components.

distance mapping
: Changes LZ77 distances to have the smallest values for pixels in 2D
  proximity.

entropy image
: A two-dimensional subresolution image indicating which entropy coding
  should be used in a respective square in the image, i.e., each pixel
  is a meta prefix code.

prefix code
: A classic way to do entropy coding where a smaller number of bits are
  used for more frequent codes.

LZ77
: Dictionary-based sliding window compression algorithm that either
  emits symbols or describes them as sequences of past symbols.

meta prefix code
: A small integer (up to 16 bits) that indexes an element in the meta
  prefix table.

predictor image
: A two-dimensional subresolution image indicating which spatial
  predictor is used for a particular square in the image.

prefix coding
: A way to entropy code larger integers that codes a few bits of the
  integer using an entropy code and codifies the remaining bits raw.
  This allows for the descriptions of the entropy codes to remain
  relatively small even when the range of symbols is large.

scan-line order
: A processing order of pixels, left-to-right, top-to-bottom, starting
  from the left-hand-top pixel, proceeding to the right. Once a row is
  completed, continue from the left-hand column of the next row.


1 Introduction
--------------

This document describes the compressed data representation of a WebP
lossless image. It is intended as a detailed reference for WebP lossless
encoder and decoder implementation.

In this document, we extensively use C programming language syntax to
describe the bitstream, and assume the existence of a function for
reading bits, `ReadBits(n)`. The bytes are read in the natural order of
the stream containing them, and bits of each byte are read in
least-significant-bit-first order. When multiple bits are read at the
same time, the integer is constructed from the original data in the
original order. The most significant bits of the returned integer are
also the most significant bits of the original data. Thus the statement

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
b = ReadBits(2);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

is equivalent with the two statements below:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
b = ReadBits(1);
b |= ReadBits(1) << 1;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We assume that each color component (e.g. alpha, red, blue and green) is
represented using an 8-bit byte. We define the corresponding type as
uint8. A whole ARGB pixel is represented by a type called uint32, an
unsigned integer consisting of 32 bits. In the code showing the behavior
of the transformations, alpha value is codified in bits 31..24, red in
bits 23..16, green in bits 15..8 and blue in bits 7..0, but
implementations of the format are free to use another representation
internally.

Broadly, a WebP lossless image contains header data, transform
information and actual image data. Headers contain width and height of
the image. A WebP lossless image can go through four different types of
transformation before being entropy encoded. The transform information
in the bitstream contains the data required to apply the respective
inverse transforms.


2 RIFF Header
-------------

The beginning of the header has the RIFF container. This consists of the
following 21 bytes:

   1. String "RIFF"
   2. A little-endian 32 bit value of the block length, the whole size
      of the block controlled by the RIFF header. Normally this equals
      the payload size (file size minus 8 bytes: 4 bytes for the 'RIFF'
      identifier and 4 bytes for storing the value itself).
   3. String "WEBP" (RIFF container name).
   4. String "VP8L" (chunk tag for lossless encoded image data).
   5. A little-endian 32-bit value of the number of bytes in the
      lossless stream.
   6. One byte signature 0x2f.

The first 28 bits of the bitstream specify the width and height of the
image. Width and height are decoded as 14-bit integers as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int image_width = ReadBits(14) + 1;
int image_height = ReadBits(14) + 1;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The 14-bit dynamics for image size limit the maximum size of a WebP
lossless image to 16384✕16384 pixels.

The alpha_is_used bit is a hint only, and should not impact decoding.
It should be set to 0 when all alpha values are 255 in the picture, and
1 otherwise.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int alpha_is_used = ReadBits(1);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The version_number is a 3 bit code that must be set to 0. Any other value
should be treated as an error. \[AMENDED\]

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int version_number = ReadBits(3);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


3 Transformations
-----------------

Transformations are reversible manipulations of the image data that can
reduce the remaining symbolic entropy by modeling spatial and color
correlations. Transformations can make the final compression more dense.

An image can go through four types of transformation. A 1 bit indicates
the presence of a transform. Each transform is allowed to be used only
once. The transformations are used only for the main level ARGB image:
the subresolution images have no transforms, not even the 0 bit
indicating the end-of-transforms.

Typically an encoder would use these transforms to reduce the Shannon
entropy in the residual image. Also, the transform data can be decided
based on entropy minimization.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
while (ReadBits(1)) {  // Transform present.
  // Decode transform type.
  enum TransformType transform_type = ReadBits(2);
  // Decode transform data.
  ...
}

// Decode actual image data (Section 4).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If a transform is present then the next two bits specify the transform
type. There are four types of transforms.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
enum TransformType {
  PREDICTOR_TRANSFORM             = 0,
  COLOR_TRANSFORM                 = 1,
  SUBTRACT_GREEN                  = 2,
  COLOR_INDEXING_TRANSFORM        = 3,
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The transform type is followed by the transform data. Transform data
contains the information required to apply the inverse transform and
depends on the transform type. Next we describe the transform data for
different types.


### Predictor Transform

The predictor transform can be used to reduce entropy by exploiting the
fact that neighboring pixels are often correlated. In the predictor
transform, the current pixel value is predicted from the pixels already
decoded (in scan-line order) and only the residual value (actual -
predicted) is encoded. The _prediction mode_ determines the type of
prediction to use. We divide the image into squares and all the pixels
in a square use the same prediction mode.

The first 3 bits of prediction data define the block width and height in
number of bits. The number of block columns, `block_xsize`, is used in
indexing two-dimensionally.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int size_bits = ReadBits(3) + 2;
int block_width = (1 << size_bits);
int block_height = (1 << size_bits);
#define DIV_ROUND_UP(num, den) ((num) + (den) - 1) / (den))
int block_xsize = DIV_ROUND_UP(image_width, 1 << size_bits);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The transform data contains the prediction mode for each block of the
image. All the `block_width * block_height` pixels of a block use same
prediction mode. The prediction modes are treated as pixels of an image
and encoded using the same techniques described in
[Chapter 4](#image-data).

For a pixel _x, y_, one can compute the respective filter block address
by:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int block_index = (y >> size_bits) * block_xsize +
                  (x >> size_bits);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

There are 14 different prediction modes. In each prediction mode, the
current pixel value is predicted from one or more neighboring pixels
whose values are already known.

We choose the neighboring pixels (TL, T, TR, and L) of the current pixel
(P) as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
O    O    O    O    O    O    O    O    O    O    O
O    O    O    O    O    O    O    O    O    O    O
O    O    O    O    TL   T    TR   O    O    O    O
O    O    O    O    L    P    X    X    X    X    X
X    X    X    X    X    X    X    X    X    X    X
X    X    X    X    X    X    X    X    X    X    X
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

where TL means top-left, T top, TR top-right, L left pixel.
At the time of predicting a value for P, all pixels O, TL, T, TR and L
have been already processed, and pixel P and all pixels X are unknown.

Given the above neighboring pixels, the different prediction modes are
defined as follows.

| Mode   | Predicted value of each channel of the current pixel    |
| ------ | ------------------------------------------------------- |
|  0     | 0xff000000 (represents solid black color in ARGB)       |
|  1     | L                                                       |
|  2     | T                                                       |
|  3     | TR                                                      |
|  4     | TL                                                      |
|  5     | Average2(Average2(L, TR), T)                            |
|  6     | Average2(L, TL)                                         |
|  7     | Average2(L, T)                                          |
|  8     | Average2(TL, T)                                         |
|  9     | Average2(T, TR)                                         |
| 10     | Average2(Average2(L, TL), Average2(T, TR))              |
| 11     | Select(L, T, TL)                                        |
| 12     | ClampAddSubtractFull(L, T, TL)                          |
| 13     | ClampAddSubtractHalf(Average2(L, T), TL)                |


`Average2` is defined as follows for each ARGB component:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
uint8 Average2(uint8 a, uint8 b) {
  return (a + b) / 2;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The Select predictor is defined as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
uint32 Select(uint32 L, uint32 T, uint32 TL) {
  // L = left pixel, T = top pixel, TL = top left pixel.

  // ARGB component estimates for prediction.
  int pAlpha = ALPHA(L) + ALPHA(T) - ALPHA(TL);
  int pRed = RED(L) + RED(T) - RED(TL);
  int pGreen = GREEN(L) + GREEN(T) - GREEN(TL);
  int pBlue = BLUE(L) + BLUE(T) - BLUE(TL);

  // Manhattan distances to estimates for left and top pixels.
  int pL = abs(pAlpha - ALPHA(L)) + abs(pRed - RED(L)) +
           abs(pGreen - GREEN(L)) + abs(pBlue - BLUE(L));
  int pT = abs(pAlpha - ALPHA(T)) + abs(pRed - RED(T)) +
           abs(pGreen - GREEN(T)) + abs(pBlue - BLUE(T));

  // Return either left or top, the one closer to the prediction.
  if (pL < pT) {     // \[AMENDED\]
    return L;
  } else {
    return T;
  }
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The functions `ClampAddSubtractFull` and `ClampAddSubtractHalf` are
performed for each ARGB component as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// Clamp the input value between 0 and 255.
int Clamp(int a) {
  return (a < 0) ? 0 : (a > 255) ?  255 : a;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int ClampAddSubtractFull(int a, int b, int c) {
  return Clamp(a + b - c);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int ClampAddSubtractHalf(int a, int b) {
  return Clamp(a + (a - b) / 2);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

There are special handling rules for some border pixels. If there is a
prediction transform, regardless of the mode \[0..13\] for these pixels,
the predicted value for the left-topmost pixel of the image is
0xff000000, L-pixel for all pixels on the top row, and T-pixel for all
pixels on the leftmost column.

\[AMENDED2\]
Addressing the TR-pixel for pixels on the rightmost column is
exceptional. The pixels on the rightmost column are predicted by using
the modes \[0..13\] just like pixels not on the border, but the leftmost pixel
on the same row as the current pixel is instead used as the TR-pixel.


### Color Transform

\[AMENDED2\]

The goal of the color transform is to decorrelate the R, G and B values
of each pixel. Color transform keeps the green (G) value as it is,
transforms red (R) based on green and transforms blue (B) based on green
and then based on red.

As is the case for the predictor transform, first the image is divided
into blocks and the same transform mode is used for all the pixels in a
block. For each block there are three types of color transform elements.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
typedef struct {
  uint8 green_to_red;
  uint8 green_to_blue;
  uint8 red_to_blue;
} ColorTransformElement;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The actual color transformation is done by defining a color transform
delta. The color transform delta depends on the `ColorTransformElement`,
which is the same for all the pixels in a particular block. The delta is
subtracted during color transform. The inverse color transform then is just
adding those deltas.

The color transform function is defined as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
void ColorTransform(uint8 red, uint8 blue, uint8 green,
                    ColorTransformElement *trans,
                    uint8 *new_red, uint8 *new_blue) {
  // Transformed values of red and blue components
  int tmp_red = red;
  int tmp_blue = blue;

  // Applying the transform is just subtracting the transform deltas
  tmp_red  -= ColorTransformDelta(p->green_to_red_,  green);
  tmp_blue -= ColorTransformDelta(p->green_to_blue_, green);
  tmp_blue -= ColorTransformDelta(p->red_to_blue_, red);

  *new_red = tmp_red & 0xff;
  *new_blue = tmp_blue & 0xff;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`ColorTransformDelta` is computed using a signed 8-bit integer
representing a 3.5-fixed-point number, and a signed 8-bit RGB color
channel (c) \[-128..127\] and is defined as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int8 ColorTransformDelta(int8 t, int8 c) {
  return (t * c) >> 5;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A conversion from the 8-bit unsigned representation (uint8) to the 8-bit
signed one (int8) is required before calling `ColorTransformDelta()`.
It should be performed using 8-bit two's complement (that is: uint8 range
\[128-255\] is mapped to the \[-128, -1\] range of its converted int8 value).

The multiplication is to be done using more precision (with at least
16-bit dynamics). The sign extension property of the shift operation
does not matter here: only the lowest 8 bits are used from the result,
and there the sign extension shifting and unsigned shifting are
consistent with each other.

Now we describe the contents of color transform data so that decoding
can apply the inverse color transform and recover the original red and
blue values. The first 3 bits of the color transform data contain the
width and height of the image block in number of bits, just like the
predictor transform:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int size_bits = ReadBits(3) + 2;
int block_width = 1 << size_bits;
int block_height = 1 << size_bits;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The remaining part of the color transform data contains
`ColorTransformElement` instances corresponding to each block of the
image. `ColorTransformElement` instances are treated as pixels of an
image and encoded using the methods described in
[Chapter 4](#image-data).

During decoding, `ColorTransformElement` instances of the blocks are
decoded and the inverse color transform is applied on the ARGB values of
the pixels. As mentioned earlier, that inverse color transform is just
subtracting `ColorTransformElement` values from the red and blue
channels.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
void InverseTransform(uint8 red, uint8 green, uint8 blue,
                      ColorTransformElement *p,
                      uint8 *new_red, uint8 *new_blue) {
  // Transformed values of red and blue components
  int tmp_red = red;
  int tmp_blue = blue;

  // Applying inverse transform is just adding the
  // color transform deltas
  tmp_red  += ColorTransformDelta(trans->green_to_red, green);
  tmp_blue += ColorTransformDelta(trans->green_to_blue, green);
  tmp_blue += ColorTransformDelta(trans->red_to_blue, tmp_red & 0xff);

  *new_red = tmp_red & 0xff;
  *new_blue = tmp_blue & 0xff;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


### Subtract Green Transform

The subtract green transform subtracts green values from red and blue
values of each pixel. When this transform is present, the decoder needs
to add the green value to both red and blue. There is no data associated
with this transform. The decoder applies the inverse transform as
follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
void AddGreenToBlueAndRed(uint8 green, uint8 *red, uint8 *blue) {
  *red  = (*red  + green) & 0xff;
  *blue = (*blue + green) & 0xff;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This transform is redundant as it can be modeled using the color
transform, but it is still often useful. Since it can extend the
dynamics of the color transform and there is no additional data here,
the subtract green transform can be coded using fewer bits than a
full-blown color transform.


### Color Indexing Transform

If there are not many unique pixel values, it may be more efficient to
create a color index array and replace the pixel values by the array's
indices. The color indexing transform achieves this. (In the context of
WebP lossless, we specifically do not call this a palette transform
because a similar but more dynamic concept exists in WebP lossless
encoding: color cache.)

The color indexing transform checks for the number of unique ARGB values
in the image. If that number is below a threshold (256), it creates an
array of those ARGB values, which is then used to replace the pixel
values with the corresponding index: the green channel of the pixels are
replaced with the index; all alpha values are set to 255; all red and
blue values to 0.

The transform data contains color table size and the entries in the
color table. The decoder reads the color indexing transform data as
follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// 8 bit value for color table size
int color_table_size = ReadBits(8) + 1;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The color table is stored using the image storage format itself. The
color table can be obtained by reading an image, without the RIFF
header, image size, and transforms, assuming a height of one pixel and
a width of `color_table_size`. The color table is always
subtraction-coded to reduce image entropy. The deltas of palette colors
contain typically much less entropy than the colors themselves, leading
to significant savings for smaller images. In decoding, every final
color in the color table can be obtained by adding the previous color
component values by each ARGB component separately, and storing the
least significant 8 bits of the result.

The inverse transform for the image is simply replacing the pixel values
(which are indices to the color table) with the actual color table
values. The indexing is done based on the green component of the ARGB
color.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// Inverse transform
argb = color_table[GREEN(argb)];
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If the index is equal or larger than color_table_size, the argb color value
should be set to 0x00000000 (transparent black).  \[AMENDED\]

When the color table is small (equal to or less than 16 colors), several
pixels are bundled into a single pixel. The pixel bundling packs several
(2, 4, or 8) pixels into a single pixel, reducing the image width
respectively. Pixel bundling allows for a more efficient joint
distribution entropy coding of neighboring pixels, and gives some
arithmetic coding-like benefits to the entropy code, but it can only be
used when there are a small number of unique values.

`color_table_size` specifies how many pixels are combined together:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int width_bits;
if (color_table_size <= 2) {
  width_bits = 3;
} else if (color_table_size <= 4) {
  width_bits = 2;
} else if (color_table_size <= 16) {
  width_bits = 1;
} else {
  width_bits = 0;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`width_bits` has a value of 0, 1, 2 or 3. A value of 0 indicates no
pixel bundling to be done for the image. A value of 1 indicates that two
pixels are combined together, and each pixel has a range of \[0..15\]. A
value of 2 indicates that four pixels are combined together, and each
pixel has a range of \[0..3\]. A value of 3 indicates that eight pixels
are combined together and each pixel has a range of \[0..1\], i.e., a
binary value.

The values are packed into the green component as follows:

  * `width_bits` = 1: for every x value where x ≡ 0 (mod 2), a green
    value at x is positioned into the 4 least-significant bits of the
    green value at x / 2, a green value at x + 1 is positioned into the
    4 most-significant bits of the green value at x / 2.
  * `width_bits` = 2: for every x value where x ≡ 0 (mod 4), a green
    value at x is positioned into the 2 least-significant bits of the
    green value at x / 4, green values at x + 1 to x + 3 are positioned in order
    to the more significant bits of the green value at x / 4.
  * `width_bits` = 3: for every x value where x ≡ 0 (mod 8), a green
    value at x is positioned into the least-significant bit of the green
    value at x / 8, green values at x + 1 to x + 7 are positioned in order to
    the more significant bits of the green value at x / 8.


4 Image Data
------------

Image data is an array of pixel values in scan-line order.

### 4.1 Roles of Image Data

We use image data in five different roles:

  1. ARGB image: Stores the actual pixels of the image.
  1. Entropy image: Stores the
     [meta prefix codes](#decoding-of-meta-prefix-codes). The red and green
     components of a pixel define the meta prefix code used in a particular
     block of the ARGB image.
  1. Predictor image: Stores the metadata for [Predictor
     Transform](#predictor-transform). The green component of a pixel defines
     which of the 14 predictors is used within a particular block of the
     ARGB image.
  1. Color transform image. It is created by `ColorTransformElement` values
     (defined in [Color Transform](#color-transform)) for different blocks of
     the image. Each `ColorTransformElement` `'cte'` is treated as a pixel whose
     alpha component is `255`, red component is `cte.red_to_blue`, green
     component is `cte.green_to_blue` and blue component is `cte.green_to_red`.
  1. Color indexing image: An array of size `color_table_size` (up to 256
     ARGB values) storing the metadata for the
     [Color Indexing Transform](#color-indexing-transform). This is stored as an
     image of width `color_table_size` and height `1`.

### 4.2 Encoding of Image Data

The encoding of image data is independent of its role.

The image is first divided into a set of fixed-size blocks (typically 16x16
blocks). Each of these blocks are modeled using their own entropy codes. Also,
several blocks may share the same entropy codes.

**Rationale:** Storing an entropy code incurs a cost. This cost can be minimized
if statistically similar blocks share an entropy code, thereby storing that code
only once. For example, an encoder can find similar blocks by clustering them
using their statistical properties, or by repeatedly joining a pair of randomly
selected clusters when it reduces the overall amount of bits needed to encode
the image.

Each pixel is encoded using one of the three possible methods:

  1. prefix coded literal: each channel (green, red, blue and alpha) is
     entropy-coded independently;
  2. LZ77 backward reference: a sequence of pixels are copied from elsewhere
     in the image; or
  3. Color cache code: using a short multiplicative hash code (color cache
     index) of a recently seen color.

The following sub-sections describe each of these in detail.

#### 4.2.1 Prefix Coded Literals

The pixel is stored as prefix coded values of green, red, blue and alpha (in
that order). See [this section](#decoding-entropy-coded-image-data) for details.

#### 4.2.2 LZ77 Backward Reference

Backward references are tuples of _length_ and _distance code_:

  * Length indicates how many pixels in scan-line order are to be copied.
  * Distance code is a number indicating the position of a previously seen
    pixel, from which the pixels are to be copied. The exact mapping is
    described [below](#distance-mapping).

The length and distance values are stored using **LZ77 prefix coding**.

LZ77 prefix coding divides large integer values into two parts: the _prefix
code_ and the _extra bits_: the prefix code is stored using an entropy code,
while the extra bits are stored as they are (without an entropy code).

**Rationale**: This approach reduces the storage requirement for the entropy
code. Also, large values are usually rare, and so extra bits would be used for
very few values in the image. Thus, this approach results in a better
compression overall.

The following table denotes the prefix codes and extra bits used for storing
different ranges of values.

Note: The maximum backward reference length is limited to 4096. Hence, only the
first 24 prefix codes (with the respective extra bits) are meaningful for length
values. For distance values, however, all the 40 prefix codes are valid.

| Value range     | Prefix code | Extra bits |
| --------------- | ----------- | ---------- |
| 1               | 0           | 0          |
| 2               | 1           | 0          |
| 3               | 2           | 0          |
| 4               | 3           | 0          |
| 5..6            | 4           | 1          |
| 7..8            | 5           | 1          |
| 9..12           | 6           | 2          |
| 13..16          | 7           | 2          |
| ...             | ...         | ...        |
| 3072..4096      | 23          | 10         |
| ...             | ...         | ...        |
| 524289..786432  | 38          | 18         |
| 786433..1048576 | 39          | 18         |

The pseudocode to obtain a (length or distance) value from the prefix code is
as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
if (prefix_code < 4) {
  return prefix_code + 1;
}
int extra_bits = (prefix_code - 2) >> 1;
int offset = (2 + (prefix_code & 1)) << extra_bits;
return offset + ReadBits(extra_bits) + 1;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Distance Mapping:**
{:#distance-mapping}

As noted previously, distance code is a number indicating the position of a
previously seen pixel, from which the pixels are to be copied. This sub-section
defines the mapping between a distance code and the position of a previous
pixel.

The distance codes larger than 120 denote the pixel-distance in scan-line
order, offset by 120.

The smallest distance codes \[1..120\] are special, and are reserved for a close
neighborhood of the current pixel. This neighborhood consists of 120 pixels:

  * Pixels that are 1 to 7 rows above the current pixel, and are up to 8 columns
    to the left or up to 7 columns to the right of the current pixel. \[Total
    such pixels = `7 * (8 + 1 + 7) = 112`\].
  * Pixels that are in same row as the current pixel, and are up to 8 columns to
    the left of the current pixel. \[`8` such pixels\].

The mapping between distance code `i` and the neighboring pixel offset
`(xi, yi)` is as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(0, 1),  (1, 0),  (1, 1),  (-1, 1), (0, 2),  (2, 0),  (1, 2),  (-1, 2),
(2, 1),  (-2, 1), (2, 2),  (-2, 2), (0, 3),  (3, 0),  (1, 3),  (-1, 3),
(3, 1),  (-3, 1), (2, 3),  (-2, 3), (3, 2),  (-3, 2), (0, 4),  (4, 0),
(1, 4),  (-1, 4), (4, 1),  (-4, 1), (3, 3),  (-3, 3), (2, 4),  (-2, 4),
(4, 2),  (-4, 2), (0, 5),  (3, 4),  (-3, 4), (4, 3),  (-4, 3), (5, 0),
(1, 5),  (-1, 5), (5, 1),  (-5, 1), (2, 5),  (-2, 5), (5, 2),  (-5, 2),
(4, 4),  (-4, 4), (3, 5),  (-3, 5), (5, 3),  (-5, 3), (0, 6),  (6, 0),
(1, 6),  (-1, 6), (6, 1),  (-6, 1), (2, 6),  (-2, 6), (6, 2),  (-6, 2),
(4, 5),  (-4, 5), (5, 4),  (-5, 4), (3, 6),  (-3, 6), (6, 3),  (-6, 3),
(0, 7),  (7, 0),  (1, 7),  (-1, 7), (5, 5),  (-5, 5), (7, 1),  (-7, 1),
(4, 6),  (-4, 6), (6, 4),  (-6, 4), (2, 7),  (-2, 7), (7, 2),  (-7, 2),
(3, 7),  (-3, 7), (7, 3),  (-7, 3), (5, 6),  (-5, 6), (6, 5),  (-6, 5),
(8, 0),  (4, 7),  (-4, 7), (7, 4),  (-7, 4), (8, 1),  (8, 2),  (6, 6),
(-6, 6), (8, 3),  (5, 7),  (-5, 7), (7, 5),  (-7, 5), (8, 4),  (6, 7),
(-6, 7), (7, 6),  (-7, 6), (8, 5),  (7, 7),  (-7, 7), (8, 6),  (8, 7)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For example, distance code `1` indicates an offset of `(0, 1)` for the
neighboring pixel, that is, the pixel above the current pixel (0 pixel
difference in X-direction and 1 pixel difference in Y-direction). Similarly,
distance code `3` indicates left-top pixel.

The decoder can convert a distance code `i` to a scan-line order distance
`dist` as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(xi, yi) = distance_map[i]
dist = x + y * xsize
if (dist < 1) {
  dist = 1
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

where `distance_map` is the mapping noted above and `xsize` is the width of the
image in pixels.


#### 4.2.3 Color Cache Coding
{:#color-cache-code}

Color cache stores a set of colors that have been recently used in the image.

**Rationale:** This way, the recently used colors can sometimes be referred to
more efficiently than emitting them using the other two methods (described in
[4.2.1](#prefix-coded-literals) and [4.2.2](#lz77-backward-reference)).

Color cache codes are stored as follows. First, there is a 1-bit value that
indicates if the color cache is used. If this bit is 0, no color cache codes
exist, and they are not transmitted in the prefix code that decodes the green
symbols and the length prefix codes. However, if this bit is 1, the color cache
size is read next:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int color_cache_code_bits = ReadBits(4);
int color_cache_size = 1 << color_cache_code_bits;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color_cache_code_bits` defines the size of the color_cache by (1 <<
`color_cache_code_bits`). The range of allowed values for
`color_cache_code_bits` is \[1..11\]. Compliant decoders must indicate a
corrupted bitstream for other values.

A color cache is an array of size `color_cache_size`. Each entry
stores one ARGB color. Colors are looked up by indexing them by
(0x1e35a7bd * `color`) >> (32 - `color_cache_code_bits`). Only one
lookup is done in a color cache; there is no conflict resolution.

In the beginning of decoding or encoding of an image, all entries in all
color cache values are set to zero. The color cache code is converted to
this color at decoding time. The state of the color cache is maintained
by inserting every pixel, be it produced by backward referencing or as
literals, into the cache in the order they appear in the stream.


5 Entropy Code
--------------

### 5.1 Overview

Most of the data is coded using a [canonical prefix code][canonical_huff].
Hence, the codes are transmitted by sending the _prefix code lengths_, as
opposed to the actual _prefix codes_.

In particular, the format uses **spatially-variant prefix coding**. In other
words, different blocks of the image can potentially use different entropy
codes.

**Rationale**: Different areas of the image may have different characteristics.
So, allowing them to use different entropy codes provides more flexibility and
potentially better compression.

### 5.2 Details

The encoded image data consists of several parts:

  1. Decoding and building the prefix codes \[AMENDED2\]
  1. Meta prefix codes
  1. Entropy-coded image data

#### 5.2.1 Decoding and Building the Prefix Codes

There are several steps in decoding the prefix codes.

**Decoding the Code Lengths:**
{:#decoding-the-code-lengths}

This section describes how to read the prefix code lengths from the bitstream.

The prefix code lengths can be coded in two ways. The method used is specified
by a 1-bit value.

  * If this bit is 1, it is a _simple code length code_, and
  * If this bit is 0, it is a _normal code length code_.

In both cases, there can be unused code lengths that are still part of the
stream. This may be inefficient, but it is allowed by the format.

**(i) Simple Code Length Code:**

\[AMENDED2\]

This variant is used in the special case when only 1 or 2 prefix symbols are
in the range \[0..255\] with code length `1`. All other prefix code lengths
are implicitly zeros.

The first bit indicates the number of symbols:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int num_symbols = ReadBits(1) + 1;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Following are the symbol values.
This first symbol is coded using 1 or 8 bits depending on the value of
`is_first_8bits`. The range is \[0..1\] or \[0..255\], respectively.
The second symbol, if present, is always assumed to be in the range \[0..255\]
and coded using 8 bits.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int is_first_8bits = ReadBits(1);
symbol0 = ReadBits(1 + 7 * is_first_8bits);
code_lengths[symbol0] = 1;
if (num_symbols == 2) {
  symbol1 = ReadBits(8);
  code_lengths[symbol1] = 1;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Note:** Another special case is when _all_ prefix code lengths are _zeros_
(an empty prefix code). For example, a prefix code for distance can be empty
if there are no backward references. Similarly, prefix codes for alpha, red,
and blue can be empty if all pixels within the same meta prefix code are
produced using the color cache. However, this case doesn't need a special
handling, as empty prefix codes can be coded as those containing a single
symbol `0`.

**(ii) Normal Code Length Code:**

The code lengths of the prefix code fit in 8 bits and are read as follows.
First, `num_code_lengths` specifies the number of code lengths.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int num_code_lengths = 4 + ReadBits(4);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If `num_code_lengths` is > 18, the bitstream is invalid.

The code lengths are themselves encoded using prefix codes: lower level code
lengths `code_length_code_lengths` first have to be read. The rest of those
`code_length_code_lengths` (according to the order in `kCodeLengthCodeOrder`)
are zeros.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int kCodeLengthCodes = 19;
int kCodeLengthCodeOrder[kCodeLengthCodes] = {
  17, 18, 0, 1, 2, 3, 4, 5, 16, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
};
int code_length_code_lengths[kCodeLengthCodes] = { 0 };  // All zeros.
for (i = 0; i < num_code_lengths; ++i) {
  code_length_code_lengths[kCodeLengthCodeOrder[i]] = ReadBits(3);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Next, if `ReadBits(1) == 0`, the maximum number of different read symbols is
`num_code_lengths`. Otherwise, it is defined as:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int length_nbits = 2 + 2 * ReadBits(3);
int max_symbol = 2 + ReadBits(length_nbits);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A prefix table is then built from `code_length_code_lengths` and used to read
up to `max_symbol` code lengths.

  * Code \[0..15\] indicates literal code lengths.
    * Value 0 means no symbols have been coded.
    * Values \[1..15\] indicate the bit length of the respective code.
  * Code 16 repeats the previous non-zero value \[3..6\] times, i.e.,
    `3 + ReadBits(2)` times. If code 16 is used before a non-zero
    value has been emitted, a value of 8 is repeated.
  * Code 17 emits a streak of zeros \[3..10\], i.e., `3 + ReadBits(3)`
    times.
  * Code 18 emits a streak of zeros of length \[11..138\], i.e.,
    `11 + ReadBits(7)` times.

Once code lengths are read, a prefix code for each symbol type (A, R, G, B,
distance) is formed using their respective alphabet sizes:

  * G channel: 256 + 24 + `color_cache_size`
  * other literals (A,R,B): 256
  * distance code: 40

#### 5.2.2 Decoding of Meta Prefix Codes

As noted earlier, the format allows the use of different prefix codes for
different blocks of the image. _Meta prefix codes_ are indexes identifying
which prefix codes to use in different parts of the image.

Meta prefix codes may be used _only_ when the image is being used in the
[role](#roles-of-image-data) of an _ARGB image_.

There are two possibilities for the meta prefix codes, indicated by a 1-bit
value:

  * If this bit is zero, there is only one meta prefix code used everywhere in
    the image. No more data is stored.
  * If this bit is one, the image uses multiple meta prefix codes. These meta
    prefix codes are stored as an _entropy image_ (described below).

**Entropy image:**

The entropy image defines which prefix codes are used in different parts of the
image, as described below.

The first 3-bits contain the `prefix_bits` value. The dimensions of the entropy
image are derived from 'prefix_bits'.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int prefix_bits = ReadBits(3) + 2;
int prefix_xsize = DIV_ROUND_UP(xsize, 1 << prefix_bits);
int prefix_ysize = DIV_ROUND_UP(ysize, 1 << prefix_bits);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

where `DIV_ROUND_UP` is as defined [earlier](#predictor-transform).

The next bits contain an entropy image of width `prefix_xsize` and height
`prefix_ysize`.

**Interpretation of Meta Prefix Codes:**

For any given pixel (x, y), there is a set of five prefix codes associated with
it. These codes are (in bitstream order):

  * **prefix code #1**: used for green channel, backward-reference length and
    color cache
  * **prefix code #2, #3 and #4**: used for red, blue and alpha channels
    respectively.
  * **prefix code #5**: used for backward-reference distance.

From here on, we refer to this set as a **prefix code group**.

The number of prefix code groups in the ARGB image can be obtained by finding
the _largest meta prefix code_ from the entropy image:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int num_prefix_groups = max(entropy image) + 1;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
where `max(entropy image)` indicates the largest prefix code stored in the
entropy image.

As each prefix code group contains five prefix codes, the total number of
prefix codes is:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int num_prefix_codes = 5 * num_prefix_groups;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Given a pixel (x, y) in the ARGB image, we can obtain the corresponding prefix
codes to be used as follows:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
int position = (y >> prefix_bits) * prefix_xsize + (x >> prefix_bits);
int meta_prefix_code = (entropy_image[pos] >> 8) & 0xffff;
PrefixCodeGroup prefix_group = prefix_code_groups[meta_prefix_code];
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

where, we have assumed the existence of `PrefixCodeGroup` structure, which
represents a set of five prefix codes. Also, `prefix_code_groups` is an array
of `PrefixCodeGroup` (of size `num_prefix_groups`).

The decoder then uses prefix code group `prefix_group` to decode the pixel
(x, y) as explained in the [next section](#decoding-entropy-coded-image-data).

#### 5.2.3 Decoding Entropy-coded Image Data

\[AMENDED2\]

For the current position (x, y) in the image, the decoder first identifies the
corresponding prefix code group (as explained in the last section). Given the
prefix code group, the pixel is read and decoded as follows:

Read next symbol S from the bitstream using prefix code #1. Note that S is any
integer in the range `0` to
`(256 + 24 + ` [`color_cache_size`](#color-cache-code)` - 1)`.

The interpretation of S depends on its value:

  1. if S < 256
     1. Use S as the green component.
     1. Read red from the bitstream using prefix code #2.
     1. Read blue from the bitstream using prefix code #3.
     1. Read alpha from the bitstream using prefix code #4.
  1. if S >= 256 && S < 256 + 24
     1. Use S - 256 as a length prefix code.
     1. Read extra bits for length from the bitstream.
     1. Determine backward-reference length L from length prefix code and the
        extra bits read.
     1. Read distance prefix code from the bitstream using prefix code #5.
     1. Read extra bits for distance from the bitstream.
     1. Determine backward-reference distance D from distance prefix code and
        the extra bits read.
     1. Copy the L pixels (in scan-line order) from the sequence of pixels
        prior to them by D pixels.
  1. if S >= 256 + 24
     1. Use S - (256 + 24) as the index into the color cache.
     1. Get ARGB color from the color cache at that index.


6 Overall Structure of the Format
---------------------------------

Below is a view into the format in Backus-Naur form. It does not cover
all details. End-of-image (EOI) is only implicitly coded into the number
of pixels (xsize * ysize).


#### Basic Structure

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
<format> ::= <RIFF header><image size><image stream>
<image stream> ::= <optional-transform><spatially-coded image>
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


#### Structure of Transforms

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
<optional-transform> ::= (1-bit value 1; <transform> <optional-transform>) |
                         1-bit value 0
<transform> ::= <predictor-tx> | <color-tx> | <subtract-green-tx> |
                <color-indexing-tx>
<predictor-tx> ::= 2-bit value 0; <predictor image>
<predictor image> ::= 3-bit sub-pixel code ; <entropy-coded image>
<color-tx> ::= 2-bit value 1; <color image>
<color image> ::= 3-bit sub-pixel code ; <entropy-coded image>
<subtract-green-tx> ::= 2-bit value 2
<color-indexing-tx> ::= 2-bit value 3; <color-indexing image>
<color-indexing image> ::= 8-bit color count; <entropy-coded image>
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


#### Structure of the Image Data

\[AMENDED2\]

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
<spatially-coded image> ::= <color cache info><meta prefix><data>
<entropy-coded image> ::= <color cache info><data>
<color cache info> ::= 1 bit value 0 |
                       (1-bit value 1; 4-bit value for color cache size)
<meta prefix> ::= 1-bit value 0 |
                  (1-bit value 1; <entropy image>)
<data> ::= <prefix codes><lz77-coded image>
<entropy image> ::= 3-bit subsample value; <entropy-coded image>
<prefix codes> ::= <prefix code group> | <prefix code group><prefix codes>
<prefix code group> ::= <prefix code><prefix code><prefix code>
                        <prefix code><prefix code>
                        See "Interpretation of Meta Prefix Codes" to
                        understand what each of these five prefix codes are
                        for.
<prefix code> ::= <simple prefix code> | <normal prefix code>
<simple prefix code> ::= see "Simple code length code" for details
<normal prefix code> ::= <code length code>; encoded code lengths
<code length code> ::= see section "Normal code length code"
<lz77-coded image> ::= ((<argb-pixel> | <lz77-copy> | <color-cache-code>)
                       <lz77-coded image>) | ""
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A possible example sequence:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
<RIFF header><image size>1-bit value 1<subtract-green-tx>
1-bit value 1<predictor-tx>1-bit value 0<color cache info>1-bit value 0
<prefix codes><lz77-coded image>
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

[canonical_huff]: https://en.wikipedia.org/wiki/Canonical_Huffman_code