NFFT  3.4.1
Macros | Functions
NFSOFT - Nonequispaced fast SO(3) Fourier transform

This module implements nonuniform fast SO(3) Fourier transforms. More...

Macros

#define NFSOFT_NORMALIZED   (1U << 0)
 
#define NFSOFT_USE_NDFT   (1U << 1)
 
#define NFSOFT_USE_DPT   (1U << 2)
 
#define NFSOFT_MALLOC_X   (1U << 3)
 
#define NFSOFT_REPRESENT   (1U << 4)
 
#define NFSOFT_MALLOC_F_HAT   (1U << 5)
 
#define NFSOFT_MALLOC_F   (1U << 6)
 
#define NFSOFT_PRESERVE_F_HAT   (1U << 7)
 
#define NFSOFT_PRESERVE_X   (1U << 8)
 
#define NFSOFT_PRESERVE_F   (1U << 9)
 
#define NFSOFT_DESTROY_F_HAT   (1U << 10)
 
#define NFSOFT_DESTROY_X   (1U << 11)
 
#define NFSOFT_DESTROY_F   (1U << 12)
 
#define NFSOFT_NO_STABILIZATION   (1U << 13)
 
#define NFSOFT_CHOOSE_DPT   (1U << 14)
 
#define NFSOFT_SOFT   (1U << 15)
 
#define NFSOFT_ZERO_F_HAT   (1U << 16)
 
#define NFSOFT_INDEX(m, n, l, B)   (((l)+((B)+1))+(2*(B)+2)*(((n)+((B)+1))+(2*(B)+2)*((m)+((B)+1))))
 
#define NFSOFT_F_HAT_SIZE(B)   (((B)+1)*(4*((B)+1)*((B)+1)-1)/3)
 

Functions

void nfsoft_precompute (nfsoft_plan *plan)
 
fpt_set nfsoft_SO3_single_fpt_init (int l, int k, int m, unsigned int flags, int kappa)
 
void nfsoft_init (nfsoft_plan *plan, int N, int M)
 
void nfsoft_init_advanced (nfsoft_plan *plan, int N, int M, unsigned int nfsoft_flags)
 
void nfsoft_init_guru (nfsoft_plan *plan, int N, int M, unsigned int nfsoft_flags, unsigned int nfft_flags, int nfft_cutoff, int fpt_kappa)
 
void nfsoft_trafo (nfsoft_plan *plan_nfsoft)
 
void nfsoft_adjoint (nfsoft_plan *plan_nfsoft)
 
void nfsoft_finalize (nfsoft_plan *plan)
 

Detailed Description

This module implements nonuniform fast SO(3) Fourier transforms.

In the following, we abbreviate the term "nonuniform fast SO(3) Fourier transform" by NFSOFT.

Macro Definition Documentation

#define NFSOFT_NORMALIZED   (1U << 0)

By default, all computations are performed with respect to the unnormalized basis functions

\[ D_{mn}^l(\alpha,\beta,\gamma) = d^{mn}_{l}(\cos\beta) \mathrm{e}^{-\mathrm{i} m \alpha}\mathrm{e}^{-\mathrm{i} n \gamma}. \]

If this flag is set, all computations are carried out using the $L_2$- normalized basis functions

\[ \tilde D_{mn}^l(\alpha,\beta,\gamma) = \sqrt{\frac{2l+1}{8\pi^2}}d^{mn}_{l}(\cos\beta) \mathrm{e}^{-\mathrm{i} m \alpha}\mathrm{e}^{-\mathrm{i} n \gamma} \]

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 686 of file nfft3.h.

Referenced by nfsoft_adjoint(), and nfsoft_trafo().

#define NFSOFT_USE_NDFT   (1U << 1)

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

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 687 of file nfft3.h.

Referenced by nfsoft_adjoint(), and nfsoft_trafo().

#define NFSOFT_USE_DPT   (1U << 2)

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

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 688 of file nfft3.h.

#define NFSOFT_MALLOC_X   (1U << 3)

If this flag is set, the init methods (see nfsoft_init , nfsoft_init_advanced , and nfsoft_init_guru) will allocate memory and the method nfsoft_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
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 689 of file nfft3.h.

Referenced by nfsoft_finalize(), and nfsoft_init().

#define NFSOFT_REPRESENT   (1U << 4)

If this flag is set, the Wigner-D functions will be normed such that they satisfy the representation property of the spherical harmonics as defined in the NFFT software package, i.e. for every rotation matrix A with Euler angles $\alpha, \beta, \gamma$ and every unit vector x the Wigner-D functions will be normed such that

\[ \sum_{m=-l}^l D_{mn}^l(\alpha,\beta,\gamma) Y_m^l(x) = Y_n^l(A^{-1} x) \]

Author
Antje Vollrath

Definition at line 690 of file nfft3.h.

Referenced by nfsoft_adjoint(), and nfsoft_trafo().

#define NFSOFT_MALLOC_F_HAT   (1U << 5)

If this flag is set, the init methods (see nfsoft_init , nfsoft_init_advanced , and nfsoft_init_guru) will allocate memory and the method nfsoft_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
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 691 of file nfft3.h.

Referenced by nfsoft_finalize(), and nfsoft_init().

#define NFSOFT_MALLOC_F   (1U << 6)

If this flag is set, the init methods (see nfsoft_init , nfsoft_init_advanced , and nfsoft_init_guru) will allocate memory and the method nfsoft_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
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 692 of file nfft3.h.

Referenced by nfsoft_finalize(), and nfsoft_init().

#define NFSOFT_PRESERVE_F_HAT   (1U << 7)

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

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 693 of file nfft3.h.

#define NFSOFT_PRESERVE_X   (1U << 8)

If this flag is set, it is guaranteed that during an execution of nfsoft_trafo or nfsoft_adjoint the content of x remains unchanged.

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 694 of file nfft3.h.

#define NFSOFT_PRESERVE_F   (1U << 9)

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

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 695 of file nfft3.h.

#define NFSOFT_DESTROY_F_HAT   (1U << 10)

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

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 696 of file nfft3.h.

#define NFSOFT_DESTROY_X   (1U << 11)

If this flag is set, it is explicitely allowed that during an execution of nfsoft_trafo or nfsoft_adjoint the content of x may be changed.

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 697 of file nfft3.h.

#define NFSOFT_DESTROY_F   (1U << 12)

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

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 698 of file nfft3.h.

#define NFSOFT_NO_STABILIZATION   (1U << 13)

If this flag is set, the fast NFSOFT algorithms (see nfsoft_trafo, nfsoft_adjoint) will use internally the FPT algorithm without the stabilization scheme and thus making bigger errors for higher bandwidth but becoming significantly faster

Author
Antje Vollrath

Definition at line 701 of file nfft3.h.

#define NFSOFT_CHOOSE_DPT   (1U << 14)

If this flag is set, the fast NFSOFT algorithms (see nfsoft_trafo, nfsoft_adjoint) will decide whether to use the DPT or FPT algorithm depending on which is faster for the chosen orders.

not yet included in the checked-in version

Author
Antje Vollrath

Definition at line 702 of file nfft3.h.

#define NFSOFT_SOFT   (1U << 15)

If this flag is set, the fast NFSOFT algorithms (see nfsoft_trafo, nfsoft_adjoint) becomes a SOFT, i.e., we use equispaced nodes. The FFTW will be used instead of the NFFT.–>not included yet

See Also
nfsoft_init
nfsoft_init_advanced
nfsoft_init_guru
Author
Antje Vollrath

Definition at line 703 of file nfft3.h.

#define NFSOFT_ZERO_F_HAT   (1U << 16)

If this flag is set, the transform nfsoft_adjoint sets all unused entries in f_hat not corresponding to SO(3) Fourier coefficients to zero.

Author
Antje Vollrath

Definition at line 704 of file nfft3.h.

#define NFSOFT_INDEX (   m,
  n,
  l,
  B 
)    (((l)+((B)+1))+(2*(B)+2)*(((n)+((B)+1))+(2*(B)+2)*((m)+((B)+1))))

These macro expands to the index $i$ corresponding to the SO(3) Fourier coefficient $f_hat^{mn}_l$ for $l=0,...,B$, $m,n =-l,...,l$ with

Definition at line 707 of file nfft3.h.

Referenced by nfsoft_adjoint(), and nfsoft_trafo().

#define NFSOFT_F_HAT_SIZE (   B)    (((B)+1)*(4*((B)+1)*((B)+1)-1)/3)

This macro expands to the logical size of a SO(3) Fourier coefficients array for a bandwidth B.

Definition at line 709 of file nfft3.h.

Function Documentation

void nfsoft_precompute ( nfsoft_plan *  plan)

Does all node-dependent and node-independent precomputations needed for the NFSOFT.

  • plan a pointer to a nfsoft_plan structure

Definition at line 438 of file nfsoft.c.

References FG_PSI, nfft_plan::M_total, nfsoft_plan_::M_total, nfsoft_plan_::N_total, nfft_precompute_one_psi(), nfsoft_plan_::p_nfft, PRE_PSI, nfft_plan::x, and nfsoft_plan_::x.

fpt_set nfsoft_SO3_single_fpt_init ( int  l,
int  k,
int  m,
unsigned int  flags,
int  kappa 
)

Computes the FPT transform.

  • coeffs the Chebychev coefficients that should be transformed
  • set the FPT-set containing precomputed data
  • l the polynomial degree
  • k the first order
  • m the second order
  • nfsoft_flags
void nfsoft_init ( nfsoft_plan *  plan,
int  N,
int  M 
)

Creates a NFSOFT transform plan.

  • plan a pointer to a nfsoft_plan structure
  • N the bandwidth $N \in \mathbb{N}_0$
  • M the number of nodes $M \in \mathbb{N}$
Author
Antje Vollrath

Definition at line 37 of file nfsoft.c.

References nfsoft_init_advanced(), NFSOFT_MALLOC_F, NFSOFT_MALLOC_F_HAT, and NFSOFT_MALLOC_X.

void nfsoft_init_advanced ( nfsoft_plan *  plan,
int  N,
int  M,
unsigned int  nfsoft_flags 
)

Creates a NFSOFT transform plan.

  • plan a pointer to a nfsoft_plan structure
  • N the bandwidth $N \in \mathbb{N}_0$
  • M the number of nodes $M \in \mathbb{N}$
  • nfsoft_flags the NFSOFT flags
Author
Antje Vollrath

Definition at line 43 of file nfsoft.c.

References FFT_OUT_OF_PLACE, FFTW_INIT, MALLOC_F, MALLOC_F_HAT, MALLOC_X, nfsoft_init_guru(), PRE_PHI_HUT, and PRE_PSI.

Referenced by nfsoft_init().

void nfsoft_init_guru ( nfsoft_plan *  plan,
int  N,
int  M,
unsigned int  nfsoft_flags,
unsigned int  nfft_flags,
int  nfft_cutoff,
int  fpt_kappa 
)

Creates a NFSOFT transform plan.

  • plan a pointer to a nfsoft_plan structure
  • N the bandwidth $N \in \mathbb{N}_0$
  • M the number of nodes $M \in \mathbb{N}$
  • nfsoft_flags the NFSFT flags
  • nfft_flags the NFFT flags
  • fpt_kappa a parameter contolling the accuracy of the FPT
  • nfft_cutoff the NFFT cutoff parameter
Author
Antje Vollrath

Definition at line 51 of file nfsoft.c.

Referenced by nfsoft_init_advanced().

void nfsoft_trafo ( nfsoft_plan *  plan_nfsoft)

Executes a NFSOFT, i.e. computes for $m = 0,\ldots,M-1$

\[ f(g_m) = \sum_{l=0}^B \sum_{m=-l}^l \sum_{n=-l}^l \hat{f}^{mn}_l D_l^{mn}\left( \alpha_m,\beta_m,\gamma_m\right). \]

  • plan_nfsoft the plan
Author
Antje Vollrath

Definition at line 469 of file nfsoft.c.

References c2e(), nfsoft_plan_::cheby, nfft_plan::f, nfsoft_plan_::f, nfft_plan::f_hat, nfsoft_plan_::f_hat, nfsoft_plan_::flags, nfsoft_plan_::internal_fpt_set, nfsoft_plan_::M_total, nfft_plan::N_total, nfsoft_plan_::N_total, nfft_trafo(), NFSOFT_INDEX, NFSOFT_NORMALIZED, NFSOFT_REPRESENT, NFSOFT_USE_NDFT, nfsoft_plan_::p_nfft, nfsoft_plan_::wig_coeffs, and X.

void nfsoft_adjoint ( nfsoft_plan *  plan_nfsoft)

Executes an adjoint NFSOFT, i.e. computes for $l=0,\ldots,B; m,n=-l,\ldots,l$

\[ \hat{f}^{mn}_l = \sum_{m = 0}^{M-1} f(g_m) D_l^{mn}\left( \alpha_m,\beta_m,\gamma_m\right) \]

  • plan_nfsoft the plan
Author
Antje Vollrath

Definition at line 598 of file nfsoft.c.

References nfsoft_plan_::cheby, nfft_plan::f, nfsoft_plan_::f, nfft_plan::f_hat, nfsoft_plan_::f_hat, nfsoft_plan_::flags, nfsoft_plan_::internal_fpt_set, nfsoft_plan_::M_total, nfsoft_plan_::N_total, nfft_adjoint(), NFSOFT_INDEX, NFSOFT_NORMALIZED, NFSOFT_REPRESENT, NFSOFT_USE_NDFT, nfsoft_plan_::p_nfft, and nfsoft_plan_::wig_coeffs.

void nfsoft_finalize ( nfsoft_plan *  plan)

Destroys a plan.

  • plan the plan to be destroyed
Author
Antje Vollrath

Definition at line 691 of file nfsoft.c.

References nfsoft_plan_::aux, nfsoft_plan_::cheby, nfsoft_plan_::f, nfsoft_plan_::f_hat, nfsoft_plan_::flags, nfsoft_plan_::internal_fpt_set, nfft_finalize(), nfft_free(), NFSOFT_MALLOC_F, NFSOFT_MALLOC_F_HAT, NFSOFT_MALLOC_X, nfsoft_plan_::p_nfft, nfsoft_plan_::wig_coeffs, and nfsoft_plan_::x.

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