Functions
static int	polar_grid (int T, int R, double x, double w)
	Generates the points $x_{t,j}$ with weights $w_{t,j}$ for the polar grid with angles and offsets.
static int	polar_dft (fftw_complex f_hat, int NN, fftw_complex f, int T, int R, int m)
	discrete polar FFT
static int	polar_fft (fftw_complex f_hat, int NN, fftw_complex f, int T, int R, int m)
	NFFT-based polar FFT.
static int	inverse_polar_fft (fftw_complex f, int T, int R, fftw_complex f_hat, int NN, int max_i, int m)
	inverse NFFT-based polar FFT
int	main (int argc, char **argv)
	test program for various parameters

Detailed Description

Function Documentation

static int polar_grid	(	int	T,
		int	R,
		double *	x,
		double *	w
	)

static

Generates the points $x_{t,j}$ with weights $w_{t,j}$ for the polar grid with $T$ angles and $R$ offsets.

The nodes of the polar grid lie on concentric circles around the origin. They are given for $(j,t)^{\top}\in I_R\times I_T$ by a signed radius $r_j := \frac{j}{R} \in [-\frac{1}{2},\frac{1}{2})$ and an angle $\theta_t := \frac{\pi t}{T} \in [-\frac{\pi}{2},\frac{\pi}{2})$ as

$x_{t,j} := r_j\left(\cos\theta_t, \sin\theta_t\right)^{\top}\,.$

The total number of nodes is $M=TR$ , whereas the origin is included multiple times.

Weights are introduced to compensate for local sampling density variations. For every point in the sampling set, we associate a small surrounding area. In case of the polar grid, we choose small ring segments. The area of such a ring segment around $x_{t,j}$ ( $j \ne 0$ ) is

$w_{t,j} = \frac{\pi}{2TR^2}\left(\left(|j|+\frac{1}{2}\right)^2- \left(|j|-\frac{1}{2}\right)^2\right) = \frac{\pi |j| }{TR^2}\, .$

The area of the small circle of radius $\frac{1}{2R}$ around the origin is $\frac{\pi}{4R^2}$ . Divided by the multiplicity of the origin in the sampling set, we get $w_{t,0} := \frac{\pi}{4TR^2}$ . Thus, the sum of all weights is $\frac{\pi}{4}(1+\frac{1}{R^2})$ and we divide by this value for normalization.

Definition at line 78 of file polar_fft_test.c.

References PI.

Referenced by inverse_polar_fft(), main(), polar_dft(), and polar_fft().

Functions

Detailed Description

Function Documentation