++ed by:
Dimitrios Kechagias

# NAME

Math::DCT - 1D and NxN 2D Fast Discreet Cosine Transforms (DCT-II)

# SYNOPSIS

``````  use Math::DCT qw/dct dct1d dct2d idct1d idct2d/;

# DCT of 1D array
my \$dct1d = dct([[1,2,3,4]]);
\$dct1d = dct1d([1,2,3,4]);

# DCT of 2D array
my \$dct2d = dct([[1,2],[3,4]]);
\$dct2d = dct2d([1,2,3,4]);

# iDCT of 1D and 2D array
my \$idct1d = idct1d([1,2,3,4]);
my \$idct2d = idct2d([1,2,3,4]);``````

Version 0.04

# DESCRIPTION

An unscaled DCT-II implementation for 1D and NxN 2D matrices implemented in XS. For array sizes which are a power of 2 a fast - O(n logn) for 1D, O(n² logn) for 2D - algorithm (FCT) described by Lee is used with some tweaks. In addition, an unscaled version of the specially optimized Arai, Agui, Nakajima FCT is used for 1x8, 8x8 matrices. A less optimized algorithm is used for the generic case, so any 1D or square 2D matrix can be processed (O(n²), O(n³) respectivelly).

For convenience the inverse functions are provided (inverse DCT-II, usually called iDCT, is essentially a scaled DCT-III), with an implementation equivalent to the generic DCT-II case.

The module was written for a perceptual hash project that needed 32x32 DCT-II, and on a 2.5GHz i7 2015 Macbook Pro about 18000/s per thread are processed. The common 8x8 DCT-II uses a special path, (about 380000/s on that same CPU), although for most image/video applications that require 8x8 DCT there are much faster implementations (SIMD, approximations etc) that usually produce an already scaled result for the specific application.

None of the algorithms used on this module are approximate, the test suite verifies against a naive DCT-II implementation with a tolerance of 1e-08.

# METHODS

## `dct`

``  my \$dct = dct([[1,2],[3,4]]);   # Example for 2x2 2D matrix ``

Pass an array (ref) of either a single array, or N x length-N arrays for 1D and NxN 2D DCT-II calculation respectivelly. The output will be an arrayref of array(s) with the result of the transform.

It is a convenience function with some overhead, mainly in the case of NxN arrays which have to be flattened before processing - for already flat 2D data see dct2d below.

## `dct1d`

``  my \$dct = dct1d([1,2,3]);``

Pass an array (ref) for a 1D DCT-II calculation. The output will be an arrayref with the result of the transform.

## `idct1d`

``  my \$idct = idct1d([1,2,3]);``

Pass an array (ref) for a 1D iDCT calculation. The output will be an arrayref with the result of the transform. This is essentially a DCT-III transform scaled by 2/N.

## `dct2d`

``````  my \$dct = dct2d(
[1,2,3,4],   # Arrayref containing your NxN matrix
2            # Optionally, the size N of your array (sqrt of its length)
);``````

Pass an array (ref) for a 2D DCT-II calculation. The length of the array is expected to be a square (as only NxN arrays are supported) - you can optionally pass N as the second argument to avoid a `sqrt` calculation. The output will be an arrayref with the result of the transform.

If your 2D data is available in a 1D array as is usual with most image manipulation etc cases, this function will be faster than `dct`, as the DCT calculation is anyway done on a flattened (1D) array, hence you skip the conversion.

## `idct2d`

``````  my \$idct = idct2d(
[1,2,3,4],   # Arrayref containing your NxN matrix
2            # Optionally, the size N of your array (sqrt of its length)
);``````

Pass an array (ref) for a 2D iDCT calculation. The length of the array is expected to be a square (as only NxN arrays are supported) - you can optionally pass N as the second argument to avoid a `sqrt` calculation. This is essentially a DCT-III transform scaled by 2/N. The output will be an arrayref with the result of the transform.

# USAGE NOTES

The C functions are not exported, but theoretically you could use them directly if you do your own `pack/unpack`. The fast versions for power-of-2 size arrays are `fast_dct_1d` and `fast_dct_2d`, while the generic versions are `dct_1d` and `dct_2d` (with their inverse functions being `idct_1d` and `idct_2d`). The specialized size-8 versions are `fct8_1d` and `fct8_2d`. First argument is a `char *` (use `pack "dN"`), second is the size N (except for the fct8* functions which don't need a second argument).

There is a simple benchmarking script available (`bench/benchmarking.pl`). Sample output (on an Apple M1 CPU):

``````  ** Fast 2D DCT-II (Arai et al.) **
8x8: 688910/s
** Fast 2D DCT-II (Lee) **
32x32: 34501/s
64x64: 8471/s
256x256: 438/s
** Generic 2D DCT-II **
24x24: 49898/s
48x48: 7616/s
** Generic 2D iDCT **
8x8: 516756/s
32x32: 24187/s``````

# ACKNOWLEDGEMENTS

C-code for 1D DCT was adapted from Project Nayuki and improved where possible.

# AUTHOR

Dimitrios Kechagias, `<dkechag at cpan.org>`

# BUGS

Please report any bugs or feature requests either on GitHub, or on RT (via the email `bug-math-dct at rt.cpan.org` or web interface at https://rt.cpan.org/NoAuth/ReportBug.html?Queue=Math-DCT).

I will be notified, and then you'll be notified of progress on your bug as I make changes.

# GIT

https://github.com/SpareRoom/Math-DCT