Convolution via multiplication in the DCT domain. More...

Detailed Description

Convolution via multiplication in the DCT domain.

Via the convolution-multiplication property, discrete cosine transforms (DCTs) are an effective way to implement Gaussian convolution. We follow Martucci's use of DCTs to perform convolution with half-sample symmetric boundary handling,

$G_\sigma\!\stackrel{\text{sym}}{*}\!f=\mathcal{C}_\mathrm{2e}^{-1}\bigl (\mathcal{C}_\mathrm{1e}(G_\sigma)\cdot\mathcal{C}_\mathrm{2e}(f)\bigr),$

where $\mathcal{C}_\mathrm{1e}$ and $\mathcal{C}_\mathrm{2e}$ denote respectively the DCT-I and DCT-II transforms of the same period length. This DCT-based convolution is equivalent to (but is faster and uses less memory) than FFT-based convolution with the symmetrized signal $(f_0,\ldots,f_{N-1},f_{N-1},\ldots,f_0)$ .

The FFTW library is used to compute the DCT transforms.

The process to use these functions is the following:

dct_precomp() or dct_precomp_image() to set up the convolution
dct_gaussian_conv() to perform the convolution itself (it may be called multiple times if desired)
dct_free() to clean up

Example: dct_coeffs c;

dct_precomp(&c, dest, src, N, stride, sigma);

dct_gaussian_conv(c);

dct_free(&c);

Reference

S. Martucci, "Symmetric convolution and the discrete sine and cosine transforms," IEEE Transactions on Signal Processing SP-42, pp. 1038-1051, 1994. http://dx.doi.org/10.1109/78.295213

Data Structures
struct	dct_coeffs_
	FFTW plans and coefficients for DCT-based Gaussian convolution. More...

Typedefs
typedef struct dct_coeffs_	dct_coeffs
	FFTW plans and coefficients for DCT-based Gaussian convolution.

Functions
int	dct_precomp (dct_coeffs c, num dest, const num *src, long N, long stride, double sigma)
	DCT precomputations for Gaussian convolution. More...

int	dct_precomp_image (dct_coeffs c, num dest, const num *src, int width, int height, int num_channels, double sigma)
	DCT precomputations for 2D Gaussian convolution. More...

void	dct_free (dct_coeffs *c)
	Release FFTW plans associated with a dct_coeffs struct. More...

void	dct_gaussian_conv (dct_coeffs c)
	Perform DCT-based Gaussian convolution. More...

Function Documentation

void dct_free ( dct_coeffs * c )

Release FFTW plans associated with a dct_coeffs struct.

Parameters

c	dct_coeffs created by dct_precomp() or dct_precomp_image()

Definition at line 203 of file gaussian_conv_dct.c.

void dct_gaussian_conv ( dct_coeffs c )

Perform DCT-based Gaussian convolution.

Parameters

c	dct_coeffs created by dct_precomp() or dct_precomp_image()

This routine performs 1D and 2D Gaussian convolution with symmetric boundary handling using DCT transforms,

$G_\sigma\!\stackrel{\text{sym}}{*}\!f=\mathcal{C}_\mathrm{2e}^{-1}\bigl (\mathcal{C}_\mathrm{1e}(G_\sigma)\cdot\mathcal{C}_\mathrm{2e}(f)\bigr),$

where $\mathcal{C}_\mathrm{1e}$ and $\mathcal{C}_\mathrm{2e}$ denote respectively the DCT-I and DCT-II transforms of the same period length.

In one dimension, the DCT-I transform of the Gaussian $\mathcal{C}_\mathrm{1e}(G_\sigma)$ is

$\mathcal{C}_\mathrm{1e}(G_\sigma)_k = \exp\bigl(-2\pi^2\sigma^2(\tfrac{k}{2N})^2\bigr).$

The DCT-II transforms of the signal are performed using the FFTW library, and the plans are prepared by dct_precomp() or dct_precomp_image().

Definition at line 156 of file gaussian_conv_dct.c.

int dct_precomp	(	dct_coeffs *	c,
		num *	dest,
		const num *	src,
		long	N,
		long	stride,
		double	sigma
	)

DCT precomputations for Gaussian convolution.

Parameters

c	dct_coeffs pointer to hold precomputations
dest	output convolved data
src	data to be convolved, modified in-place if src = dest
N	number of samples
stride	stride between successive samples
sigma	standard deviation of the Gaussian in pixels

Returns: 1 on success, 0 on failure

This routine performs precomputations for 1D DCT-based Gaussian convolution, to be used in dct_gaussian_conv(). FFTW transform plans are prepared for the forward and inverse DCT-II transform and the value $\alpha = (\sigma\pi/N)^2 / 2$ is precomputed.

The convolution can be performed in-place by setting src = dest (the source array is overwritten with the result).

Note: For use of the num typedef with FFTW's APIs, the macro FFT(functionname) expands to fftw_functionname if num is double or fftwf_functionname if num is float.

Definition at line 53 of file gaussian_conv_dct.c.

int dct_precomp_image	(	dct_coeffs *	c,
		num *	dest,
		const num *	src,
		int	width,
		int	height,
		int	num_channels,
		double	sigma
	)

DCT precomputations for 2D Gaussian convolution.

Parameters

c	dct_coeffs pointer to hold precomputations
dest	output convolved image
src	input image, modified in-place if src = dest
width	image width
height	image height
num_channels	number of image channels
sigma	standard deviation of the Gaussian in pixels

Returns: 1 on success, 0 on failure

Similar to dct_precomp(), this routine performs precomputations for 2D DCT-based Gaussian convolution, to be used in dct_gaussian_conv().

The convolution can be performed in-place by setting src = dest (the source array is overwritten with the result).

Definition at line 100 of file gaussian_conv_dct.c.

Detailed Description

Data Structures

Typedefs

Functions

Function Documentation