NFSFT

This module implements nonuniform fast spherical Fourier transforms. More...


Modules
	NFSFT: Functions related to associated Legendre
	functions/polynomials
	NFSFT: Direct algorithms
	Public API
Data Structures
struct	nfsft_plan_
	Structure for a NFSFT transform plan. More...
Defines
#define	NFSFT_NORMALIZED (1U << 0)
	By default, all computations are performed with respect to the unnormalized basis functions $\tilde{Y}_k^n(\vartheta,\varphi) = P_k^{\|n\|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}.$ If this flag is set, all computations are carried out using the - normalized basis functions $Y_k^n(\vartheta,\varphi) = \sqrt{\frac{2k+1}{4\pi}} P_k^{\|n\|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}.$ .
#define	NFSFT_USE_NDFT (1U << 1)
	If this flag is set, the fast NFSFT algorithms (see nfsft_trafo, nfsft_adjoint) will use internally the exact but usually slower direct NDFT algorithm in favor of fast but approximative NFFT algorithm.
#define	NFSFT_USE_DPT (1U << 2)
	If this flag is set, the fast NFSFT algorithms (see nfsft_trafo, nfsft_adjoint) will use internally the usually slower direct DPT algorithm in favor of the fast FPT algorithm.
#define	NFSFT_MALLOC_X (1U << 3)
	If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array `x` for you.
#define	NFSFT_MALLOC_F_HAT (1U << 5)
	If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array `f_hat` for you.
#define	NFSFT_MALLOC_F (1U << 6)
	If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array `f` for you.
#define	NFSFT_PRESERVE_F_HAT (1U << 7)
	If this flag is set, it is guaranteed that during an execution of ndsft_trafo or nfsft_trafo the content of `f_hat` remains unchanged.
#define	NFSFT_PRESERVE_X (1U << 8)
	If this flag is set, it is guaranteed that during an execution of ndsft_trafo, nfsft_trafo or ndsft_adjoint, nfsft_adjoint the content of `x` remains unchanged.
#define	NFSFT_PRESERVE_F (1U << 9)
	If this flag is set, it is guaranteed that during an execution of ndsft_adjoint or nfsft_adjoint the content of `f` remains unchanged.
#define	NFSFT_DESTROY_F_HAT (1U << 10)
	If this flag is set, it is explicitely allowed that during an execution of ndsft_trafo or nfsft_trafo the content of `f_hat` may be changed.
#define	NFSFT_DESTROY_X (1U << 11)
	If this flag is set, it is explicitely allowed that during an execution of ndsft_trafo, nfsft_trafo or ndsft_adjoint, nfsft_adjoint the content of `x` may be changed.
#define	NFSFT_DESTROY_F (1U << 12)
	If this flag is set, it is explicitely allowed that during an execution of ndsft_adjoint or nfsft_adjoint the content of `f` may be changed.
#define	NFSFT_NO_DIRECT_ALGORITHM (1U << 13)
	If this flag is set, the transforms ndsft_trafo and ndsft_adjoint do not work.
#define	NFSFT_NO_FAST_ALGORITHM (1U << 14)
	If this flag is set, the transforms nfsft_trafo and nfsft_adjoint do not work.
#define	NFSFT_ZERO_F_HAT (1U << 16)
	If this flag is set, the transforms nfsft_adjoint and ndsft_adjoint set all unused entries in `f_hat` not corresponding to spherical Fourier coefficients to zero.
#define	NFSFT_INDEX(k, n, plan) ((2(plan)->N+2)((plan)->N-n+1)+(plan)->N+k+1)
	This helper macro expands to the index corresponding to the spherical Fourier coefficient for $0 \le k \le N$ , $-k \le n \le k$ with $(N+2)(N-n+1)+N+k+1$ .
#define	NFSFT_F_HAT_SIZE(N) ((2N+2)(2*N+2))
	This helper macro expands to the logical size of a spherical Fourier coefficients array for a bandwidth N.
Typedefs
typedef nfsft_plan_	nfsft_plan
	Structure for a NFSFT transform plan.
Functions
void	nfsft_init (nfsft_plan *plan, int N, int M)
	Creates a transform plan.
void	nfsft_init_advanced (nfsft_plan *plan, int N, int M, unsigned int nfsft_flags)
	Creates a transform plan.
void	nfsft_init_guru (nfsft_plan *plan, int N, int M, unsigned int nfsft_flags, int nfft_flags, int nfft_cutoff)
	Creates a transform plan.
void	nfsft_precompute (int N, double kappa, unsigned int nfsft_flags, unsigned int fpt_flags)
	Performes precomputation up to the next power of two with respect to a given bandwidth $N \in \mathbb{N}_2$ .
void	nfsft_forget ()
	Forgets all precomputed data.
void	ndsft_trafo (nfsft_plan *plan)
	Executes a direct NDSFT, i.e.
void	ndsft_adjoint (nfsft_plan *plan)
	Executes a direct adjoint NDSFT, i.e.
void	nfsft_trafo (nfsft_plan *plan)
	Executes a NFSFT, i.e.
void	nfsft_adjoint (nfsft_plan *plan)
	Executes an adjoint NFSFT, i.e.
void	nfsft_finalize (nfsft_plan *plan)
	Destroys a plan.
void	nfsft_precompute_x (nfsft_plan *plan)

Detailed Description

This module implements nonuniform fast spherical Fourier transforms.

In the following, we abbreviate the term "nonuniform fast spherical Fourier transform" by NFSFT.

Preliminaries

This section summarises basic definitions and properties related to spherical Fourier transforms.

Every point in $\mathbb{R}^3$ can be described in spherical coordinates by a vector $(r,\vartheta,\varphi)^{\mathrm{T}}$ with the radius $r \in \mathbb{R}^{+}$ and two angles $\vartheta \in [0,\pi]$ , $\varphi \in [-\pi,\pi)$ . We denote by $\mathbb{S}^2$ the two-dimensional unit sphere embedded into $\mathbb{R}^3$ , i.e.

$\mathbb{S}^2 := \left\{\mathbf{x} \in \mathbb{R}^{3}:\; \|\mathbf{x}\|_2=1\right\}$

and identify a point from $\mathbb{S}^2$ with the corresponding vector $(\vartheta,\varphi)^{\mathrm{T}}$ . The spherical coordinate system is illustrated in the following figure:

For consistency with the other modules and the conventions used there, we also use swapped scaled spherical coordinates $x_1 := \frac{\varphi}{2\pi}$ , $x_2 := \frac{\vartheta}{2\pi}$ and identify a point from $\mathbb{S}^2$ with the vector $\mathbf{x} := \left(x_1,x_2\right) \in [-\frac{1}{2}, \frac{1}{2}) \times [0,\frac{1}{2}]$ .

Legendre Polynomials

The Legendre polynomials $P_k : [-1,1] \rightarrow \mathbb{R}$, $k \in \mathbb{N}_{0}$ as classical orthogonal polynomials are given by their corresponding Rodrigues formula

$P_k(t) := \frac{1}{2^k k!} \frac{\text{d}^k}{\text{d} t^k} \left(t^2-1\right)^k.$

The corresponding three-term recurrence relation is

$(k+1)P_{k+1}(t) = (2k+1) x P_{k}(t) - k P_{k-1}(t) \quad (k \in \mathbb{N}_0).$

With

$\left< f,g \right>_{\text{L}^2\left([-1,1]\right)} := \int_{-1}^{1} f(t) g(t) \text{d} t$

being the usual $\text{L}^2\left([-1,1]\right)$ inner product, the Legendre polynomials obey the orthogonality condition

$\left< P_k,P_l \right>_{\text{L}^2\left([-1,1]\right)} = \frac{2}{2k+1} \delta_{k,l}.$

Remarks:

The normalisation constant $c_k := \sqrt{\frac{2k+1}{2}}$ renders the scaled Legendre polynomials $c_k P_k$

orthonormal with respect to the induced $\text{L}^2\left([-1,1]\right)$ norm

$\|f\|_{\text{L}^2\left([-1,1]\right)} := \left(<f,f>_{\text{L}^2\left([-1,1]\right)}\right)^{1/2} = \left(\int_{-1}^{1} |f(t)|^2 \; \text{d} t\right)^{1/2}.$

Associated Legendre Functions

The associated Legendre functions $P_k^n : [-1,1] \rightarrow \mathbb{R}$ , $n \in \mathbb{N}_0$ , $k \ge n$ are defined by

$P_k^n(t) := \left(\frac{(k-n)!}{(k+n)!}\right)^{1/2} \left(1-t^2\right)^{n/2} \frac{\text{d}^n}{\text{d} t^n} P_k(t).$

For $n = 0$ , they coincide with the Legendre polynomials, i.e. $P_k^0 = P_k$ . The associated Legendre functions obey the three-term recurrence relation

$P_{k+1}^n(t) = v_{k}^n t P_k^n(t) + w_{k}^n P_{k-1}^n(t) \quad (k \ge n),$

with $P_{n-1}^n(t) := 0$ , $P_{n}^n(t) := \frac{\sqrt{(2n)!}}{2^n n!} \left(1-t^2\right)^{n/2}$ , and

$v_{k}^n := \frac{2k+1}{((k-n+1)(k+n+1))^{1/2}}\; ,\qquad w_{k}^n := - \frac{((k-n)(k+n))^{1/2}}{((k-n+1)(k+n+1))^{1/2}}.$

For fixed $n$ , the set $\left\{P_k^n:\: k \ge n\right\}$ forms a complete set of orthogonal functions in $\text{L}^2\left([-1,1]\right)$ with

$\left< P_k^n,P_l^n \right>_{\text{L}^2\left([-1,1]\right)} = \frac{2}{2k+1} \delta_{k,l} \quad (0 \le n \le k,l).$

Remarks:

The normalisation constant $c_k = \sqrt{\frac{2k+1}{2}}$ renders the scaled associated Legendre functions $c_k P_k^n$

orthonormal with respect to the induced $\text{L}^2\left([-1,1]\right)$ norm

$\|f\|_{\text{L}^2\left([-1,1]\right)} := \left(<f,f>_{\text{L}^2\left([-1,1]\right)}\right)^{1/2} = \left(\int_{-1}^{1} |f(t)|^2 \; \text{d} t\right)^{1/2}.$

Spherical Harmonics

The standard orthogonal basis of spherical harmonics for $\text{L}^2 \left(\mathbb{S}^2\right)$ with yet unnormalised basis functions $\tilde{Y}_k^n : \mathbb{S}^2 \rightarrow \mathbb{C}$ is given by

$\tilde{Y}_k^n(\vartheta,\varphi) := P_k^{|n|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}$

with the usual $\text{L}^2\left(\mathbb{S}^2\right)$ inner product

$\left< f,g \right>_{\mathrm{L}^2\left(\mathbb{S}^2\right)} := \int_{\mathbb{S}^2} f(\vartheta,\varphi) \overline{g(\vartheta,\varphi)} \: \mathrm{d} \mathbf{\xi} := \int_{-\pi}^{\pi} \int_{0}^{\pi} f(\vartheta,\varphi) \overline{g(\vartheta,\varphi)} \sin \vartheta \; \mathrm{d} \vartheta \; \mathrm{d} \varphi.$

The normalisation constant $c_k^n := \sqrt{\frac{2k+1}{4\pi}}$ renders the scaled basis functions

$Y_k^n(\vartheta,\varphi) := c_k^n P_k^{|n|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}$

orthonormal with respect to the induced $\text{L}^2\left(\mathbb{S}^2 \right)$ norm

$\|f\|_{\text{L}^2\left(\mathbb{S}^2\right)} = \left(<f,f>_{\text{L}^2\left(\mathbb{S}^2\right)}\right)^{1/2} = \left(\int_{-\pi}^{\pi} \int_{0}^{\pi} |f(\vartheta,\varphi)|^2 \sin \vartheta \; \mathrm{d} \vartheta \; \mathrm{d} \varphi\right)^{1/2}.$

A function $f \in \mathrm{L}^2\left(\mathbb{S}^2\right)$ has the orthogonal expansion

$f = \sum_{k=0}^{\infty} \sum_{n=-k}^{k} \hat{f}(k,n) Y_k^n,$

where the coefficients $\hat{f}(k,n) := \left< f, Y_k^{n} \right>_{\mathrm{L}^2\left(\mathbb{S}^2\right)}$ are the spherical Fourier coefficients and the equivalence is understood in the $\mathrm{L}^2$ -sense.

Nonuniform Fast Spherical Fourier Transforms

This section describes the input and output relation of the spherical Fourier transform algorithms and the layout of the corresponding plan structure.

Nonuniform Discrete Spherical Fourier Transform

The nonuniform discrete spherical Fourier transform (NDSFT) is defined as follows:

$\begin{array}{rcl} \text{\textbf{Input}} & : & \text{coefficients } \hat{f}(k,n) \in \mathbb{C} \text{ for } k=0,\ldots,N,\;n=-k, \ldots,k,\; N \in \mathbb{N}_0,\\[1ex] & & \text{arbitrary nodes } \mathbf{x}(m) \in [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}] \text{ for } m=0,\ldots,M-1, M \in \mathbb{N}. \\[1ex] \text{\textbf{Task}} & : & \text{evaluate } f(m) := f\left( \mathbf{x}(m)\right) = \sum_{k=0}^N \sum_{n=-k}^k \hat{f}_k^n Y_k^n\left(\mathbf{x}(m)\right) \text{ for } m=0,\ldots,M-1. \\[1ex] \text{\textbf{Output}} & : & \text{coefficients } f(m) \in \mathbb{C} \text{ for } m=0,\ldots,M-1.\\ \end{array}$

Adjoint Nonuniform Discrete Spherical Fourier Transform

The adjoint nonuniform discrete spherical Fourier transform (adjoint NDSFT) is defined as follows:

$\begin{array}{rcl} \text{\textbf{Input}} & : & \text{coefficients } f(m) \in \mathbb{C} \text{ for } m=0,\ldots,M-1, M \in \mathbb{N},\\ & & \text{arbitrary nodes } \mathbf{x}(m) \in [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}] \text{ for } m=0,\ldots,M-1, N \in \mathbb{N}_0.\\[1ex] \text{\textbf{Task}} & : & \text{evaluate } \hat{f}(k,n) := \sum_{m=0}^{M-1} f(m) \overline{Y_k^n\left(\mathbf{x}(m)\right)}cd Do \text{ for } k=0,\ldots,N,\;n=-k,\ldots,k.\\[1ex] \text{\textbf{Output}} & : & \text{coefficients } \hat{f}(k,n) \in \mathbb{C} \text{ for } k=0,\ldots,N,\;n=-k,\ldots,k.\\[1ex] \end{array}$

Data Layout

This section describes the public layout of the nfsft_plan structure which contains all data for the computation of the aforementioned spherical Fourier transforms. The structure contains private (no read or write allowed), public read-only (only read access permitted), and public read-write (read and write access allowed) members. In the following, we indicate read and write access by read and write. The public members are structured as follows:

N_total (read) The total number of components in f_hat. If the bandwidth is $N \in \mathbb{N}_0$ , the total number of components in f_hat is N_total .
M_total (read) the total number of samples
f_hat (read-write) The flattened array of spherical Fourier coefficents. The array has length such that valid indices $i \in \mathbb{N}_0$ for array access f_hat [ ] are $i=0,1,\ldots,(2N+2)^2-1$ . However, only read and write access to indices corresponding to spherical Fourier coefficients $\hat{f}(k,n)$ is defined. The index corresponding to the spherical Fourier coefficient $\hat{f}(k,n)$ with $0 \le k \le M$ , $-k \le n \le k$ is . For convenience, the helper macro NFSFT_INDEX(k,n) provides the necessary index calculations such that one can write f_hat[ NFSFT_INDEX( )] = ... to access the component corresponding to $\hat{f}(k,n)$ . The data layout is due to implementation details.
f (read-write) the array of coefficients for $m=0,\ldots,M-1$ such that f[ ] =
N (read) the bandwidth $N \in \mathbb{N}_0$
x the array of nodes $\mathbf{x}(m) \in [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}]$ for $m = 0, \ldots,M-1$ such that f[ ] = and f[ ] =

Good to know...

When using the routines of this module you should bear in mind the following:

The bandwidth $N_{\text{max}}$ up to which precomputation is performed is always chosen as the next power of two with respect to the specified maximum bandwidth.
By default, the NDSFT transforms (see ndsft_trafo, nfsft_trafo) are allowed to destroy the input f_hat while the input x is preserved. On the contrary, the adjoint NDSFT transforms (see ndsft_adjoint, nfsft_adjoint) do not destroy the input f and x by default. The desired behaviour can be assured by using the NFSFT_PRESERVE_F_HAT, NFSFT_PRESERVE_X, NFSFT_PRESERVE_F and NFSFT_DESTROY_F_HAT, NFSFT_DESTROY_X, NFSFT_DESTROY_F flags.

Define Documentation

#define NFSFT_DESTROY_F (1U << 12)

If this flag is set, it is explicitely allowed that during an execution of ndsft_adjoint or nfsft_adjoint the content of f may be changed.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_DESTROY_F_HAT (1U << 10)

If this flag is set, it is explicitely allowed that during an execution of ndsft_trafo or nfsft_trafo the content of f_hat may be changed.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_DESTROY_X (1U << 11)

If this flag is set, it is explicitely allowed that during an execution of ndsft_trafo, nfsft_trafo or ndsft_adjoint, nfsft_adjoint the content of x may be changed.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_MALLOC_F (1U << 6)

If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array f for you.
Otherwise, you have to assure by yourself that f points to an array of proper size before excuting a transform and you are responsible for freeing the corresponding memory before program termination.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_MALLOC_F_HAT (1U << 5)

If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array f_hat for you.
Otherwise, you have to assure by yourself that f_hat points to an array of proper size before excuting a transform and you are responsible for freeing the corresponding memory before program termination.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_MALLOC_X (1U << 3)

If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array x for you.
Otherwise, you have to assure by yourself that x points to an array of proper size before excuting a transform and you are responsible for freeing the corresponding memory before program termination.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_NO_DIRECT_ALGORITHM (1U << 13)

If this flag is set, the transforms ndsft_trafo and ndsft_adjoint do not work.
Setting this flag saves some memory for precomputed data.

See also:
nfsft_precompute
ndsft_trafo
ndsft_adjoint

Author:
Jens Keiner

#define NFSFT_NO_FAST_ALGORITHM (1U << 14)

If this flag is set, the transforms nfsft_trafo and nfsft_adjoint do not work.
Setting this flag saves memory for precomputed data.

See also:
nfsft_precompute
nfsft_trafo
nfsft_adjoint

Author:
Jens Keiner

#define NFSFT_NORMALIZED (1U << 0)

By default, all computations are performed with respect to the unnormalized basis functions
$\tilde{Y}_k^n(\vartheta,\varphi) = P_k^{|n|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}.$
If this flag is set, all computations are carried out using the $L_2$ - normalized basis functions
$Y_k^n(\vartheta,\varphi) = \sqrt{\frac{2k+1}{4\pi}} P_k^{|n|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}.$
.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_PRESERVE_F (1U << 9)

If this flag is set, it is guaranteed that during an execution of ndsft_adjoint or nfsft_adjoint the content of f remains unchanged.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_PRESERVE_F_HAT (1U << 7)

If this flag is set, it is guaranteed that during an execution of ndsft_trafo or nfsft_trafo the content of f_hat remains unchanged.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_PRESERVE_X (1U << 8)

If this flag is set, it is guaranteed that during an execution of ndsft_trafo, nfsft_trafo or ndsft_adjoint, nfsft_adjoint the content of x remains unchanged.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_USE_DPT (1U << 2)

If this flag is set, the fast NFSFT algorithms (see nfsft_trafo, nfsft_adjoint) will use internally the usually slower direct DPT algorithm in favor of the fast FPT algorithm.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

Warning:
This feature is not implemented yet!

#define NFSFT_USE_NDFT (1U << 1)

If this flag is set, the fast NFSFT algorithms (see nfsft_trafo, nfsft_adjoint) will use internally the exact but usually slower direct NDFT algorithm in favor of fast but approximative NFFT algorithm.

See also:
nfsft_init
nfsft_init_advanced
nfsft_init_guru

Author:
Jens Keiner

#define NFSFT_ZERO_F_HAT (1U << 16)

If this flag is set, the transforms nfsft_adjoint and ndsft_adjoint set all unused entries in f_hat not corresponding to spherical Fourier coefficients to zero.

Author:
Jens Keiner

Function Documentation

void ndsft_adjoint ( nfsft_plan * plan )

Executes a direct adjoint NDSFT, i.e.
computes for $k=0,\ldots,N; n=-k,\ldots,k$
$\hat{f}(k,n) = \sum_{m = 0}^{M-1} f(m) Y_k^n\left(2\pi x_1(m), 2\pi x_2(m)\right).$

plan the plan

Author:
Jens Keiner

void ndsft_trafo ( nfsft_plan * plan )

Executes a direct NDSFT, i.e.
computes for $m = 0,\ldots,M-1$
$f(m) = \sum_{k=0}^N \sum_{n=-k}^k \hat{f}(k,n) Y_k^n\left(2\pi x_1(m), 2\pi x_2(m)\right).$

plan the plan

Author:
Jens Keiner

void nfsft_adjoint ( nfsft_plan * plan )

Executes an adjoint NFSFT, i.e.
computes for $k=0,\ldots,N; n=-k,\ldots,k$
$\hat{f}(k,n) = \sum_{m = 0}^{M-1} f(m) Y_k^n\left(2\pi x_1(m), 2\pi x_2(m)\right).$

plan the plan

Author:
Jens Keiner

void nfsft_finalize ( nfsft_plan * plan )

Destroys a plan.

plan the plan to be destroyed

Author:
Jens Keiner

void nfsft_forget ( )

Forgets all precomputed data.

Author:
Jens Keiner

void nfsft_init ( nfsft_plan * plan,

int N,

int M

)

Creates a transform plan.

plan a pointer to a nfsft_plan structure

N the bandwidth $N \in \mathbb{N}_0$

M the number of nodes $M \in \mathbb{N}$

Author:
Jens Keiner

void nfsft_init_advanced ( nfsft_plan * plan,

int N,

int M,

unsigned int nfsft_flags

)

Creates a transform plan.

plan a pointer to a
nfsft_plan
structure

N the bandwidth $N \in \mathbb{N}_0$

M the number of nodes $M \in \mathbb{N}$

nfsft_flags the NFSFT flags

Author:
Jens Keiner

void nfsft_init_guru ( nfsft_plan * plan,

int N,

int M,

unsigned int flags,

int nfft_flags,

int nfft_cutoff

)

Creates a transform plan.

Todo:
Replace 4*plan->N by next_power_of_2(2*this->n).

Todo:
NFSFT: Check NFFT flags.

Todo:
Add precomputation if neccessary and possible.

void nfsft_precompute ( int N,

double kappa,

unsigned int nfsft_flags,

unsigned int fpt_flags

)

Performes precomputation up to the next power of two with respect to a given bandwidth $N \in \mathbb{N}_2$ .

Todo:
Change to functions which compute only for fixed order n.

void nfsft_trafo ( nfsft_plan * plan )

Executes a NFSFT, i.e.
computes for $m = 0,\ldots,M-1$
$f(m) = \sum_{k=0}^N \sum_{n=-k}^k \hat{f}(k,n) Y_k^n\left(2\pi x_1(m), 2\pi x_2(m)\right).$

plan the plan

Author:
Jens Keiner

Generated on Wed May 10 20:39:40 2006 for NFFT by

1.4.4

NFSFT

Modules

Data Structures

Defines

Typedefs

Functions

Detailed Description

Define Documentation

Function Documentation