Gwyddion – Free SPM (AFM, SNOM/NSOM, STM, MFM, …) data analysis software

inttrans

inttrans — FFT and other integral transforms

Functions

gint	gwy_fft_find_nice_size ()
void	gwy_data_line_fft ()
void	gwy_data_line_part_fft ()
void	gwy_data_line_fft_raw ()
void	gwy_data_line_zoom_fft ()
void	gwy_data_field_zoom_fft ()
void	gwy_data_field_1dfft ()
void	gwy_data_field_area_1dfft ()
void	gwy_data_field_1dfft_raw ()
void	gwy_data_field_2dfft ()
void	gwy_data_field_area_2dfft ()
void	gwy_data_field_2dfft_raw ()
void	gwy_data_field_2dfft_humanize ()
void	gwy_data_field_2dfft_dehumanize ()
void	gwy_data_field_fft_postprocess ()
void	gwy_data_field_fft_filter_1d ()
void	gwy_data_line_fft_window ()
void	gwy_data_field_fft_window ()
void	gwy_data_field_fft_window_1d ()
void	gwy_data_field_cwt ()

Includes

#include <libprocess/gwyprocess.h>

Description

There are two main groups of FFT functions.

High-level functions such as gwy_data_field_2dfft(), gwy_data_line_fft() can perform windowing, leveling and other pre- and postprocessing. This makes them suitable for calculation of spectral densities and other statistical characteristics.

Low-level functions have raw appended to their name: gwy_data_field_2dfft_raw(), gwy_data_line_fft_raw(). They perform no other operations on the data beside the transform itself. This makes them suitable for applications where both forward and inverse transform is performed.

Both types of functions wrap FFTW3 routines.

Functions

gwy_fft_find_nice_size ()

gint
gwy_fft_find_nice_size (gint size);

Finds a nice-for-FFT array size.

Here ‘nice’ means three properties are guaranteed: it is greater than or equal to size ; it can be directly used with current FFT backend without scaling (since 2.8 this is true for any size); and the transform is fast, i.e. the number is highly factorable.

To be compatible with Gwyddion <= 2.7 one has to pass only data fields and lines with sizes returned by this function to raw integral transforms. Otherwise this function is mainly useful if you extend and pad the input data for other reasons and thus have the freedom to choose a convenient transform size.

Parameters

size

Transform size.

Returns

A nice FFT array size.

gwy_data_line_fft ()

void
gwy_data_line_fft (GwyDataLine *rsrc,
                   GwyDataLine *isrc,
                   GwyDataLine *rdest,
                   GwyDataLine *idest,
                   GwyWindowingType windowing,
                   GwyTransformDirection direction,
                   GwyInterpolationType interpolation,
                   gboolean preserverms,
                   gint level);

Calculates Fast Fourier Transform of a data line.

A windowing or data leveling can be applied if requested.

Parameters

rsrc	Real input data line.
isrc	Imaginary input data line.
rdest	Real output data line. It will be resized to the size of the input data line.
idest	Imaginary output data line. It will be resized to the size of the input data line.
windowing	Windowing type to use.
direction	FFT direction.
interpolation	Interpolation type. Ignored since 2.8 as no resampling is performed.
preserverms	`TRUE` to preserve RMS value while windowing.
level	0 to perform no leveling, 1 to subtract mean value, 2 to subtract line (the number can be interpreted as the first polynomial degree to keep, but only the enumerated three values are available).

gwy_data_line_part_fft ()

void
gwy_data_line_part_fft (GwyDataLine *rsrc,
                        GwyDataLine *isrc,
                        GwyDataLine *rdest,
                        GwyDataLine *idest,
                        gint from,
                        gint len,
                        GwyWindowingType windowing,
                        GwyTransformDirection direction,
                        GwyInterpolationType interpolation,
                        gboolean preserverms,
                        gint level);

Calculates Fast Fourier Transform of a part of a data line.

A windowing or data leveling can be applied if requested.

Parameters

rsrc	Real input data line.
isrc	Imaginary input data line. Since 2.7 it can be `NULL` for real-to-complex transforms.
rdest	Real output data line, it will be resized to `len` .
idest	Imaginary output data line, it will be resized to `len` .
from	The index in input lines to start from (inclusive).
len	Lenght of data line part, it must be at least 2.
windowing	Windowing type to use.
direction	FFT direction.
interpolation	Interpolation type. Ignored since 2.8 as no resampling is performed.
preserverms	`TRUE` to preserve RMS value while windowing.
level	0 to perform no leveling, 1 to subtract mean value, 2 to subtract line (the number can be interpreted as the first polynomial degree to keep, but only the enumerated three values are available).

gwy_data_line_fft_raw ()

void
gwy_data_line_fft_raw (GwyDataLine *rsrc,
                       GwyDataLine *isrc,
                       GwyDataLine *rdest,
                       GwyDataLine *idest,
                       GwyTransformDirection direction);

Calculates Fast Fourier Transform of a data line.

No leveling, windowing nor scaling is performed.

The normalisation of FFT is symmetrical, so transformations in both directions are unitary.

Since 2.8 the dimensions need not to be from the set of sizes returned by gwy_fft_find_nice_size().

Parameters

rsrc	Real input data line.
isrc	Imaginary input data line. Since 2.7 it can be `NULL` for real-to-complex transform.
rdest	Real output data line. It will be resized to the size of the input data line.
idest	Imaginary output data line. It will be resized to the size of the input data line.
direction	FFT direction.

Since: 2.1

gwy_data_line_zoom_fft ()

void
gwy_data_line_zoom_fft (GwyDataLine *rsrc,
                        GwyDataLine *isrc,
                        GwyDataLine *rdest,
                        GwyDataLine *idest,
                        gint m,
                        gdouble f0,
                        gdouble f1);

Computes Zoom FFT of a data line.

The output is DFTs, but computed for an arbitrary linear sequence of frequencies. The frequencies do not have to be in any relation to the data sampling step.

The first item of output corresponds exactly to f0 and the last exactly to f1 . So the frequency sampling step will be (f1 − f0 )/(m − 1), instead of the more usual division by m . To follow the usual Gwyddion conventions, the output data line real size will be (f1 − f0 )/(m − 1)*m . If it seems confusing, just take the output as indexed by integers and work with that.

Frequency step of one corresponds to the normal DFT frequency step. Therefore, passing f0 =0, f1 =n -1 (where rsrc has n points) and m =n reproduces the usual DFT, except more slowly. The result is normalised as raw FFT and the units of the output data lines are unchanged.

The transform direction is always forward. Windowing or other preprocessing need to be done separately beforehand. They would be usually once, but followed by any number of (Zoom) FFTs.

Parameters

rsrc	Real input data line.
isrc	Imaginary input data line. It can be `NULL` for real-to-complex transform.
rdest	Real output data line. It will be resized to `m` samples.
idest	Imaginary output data line. It will be resized to `m` samples.
f0	The first spatial frequency, measured in DFT frequency steps.
f1	The last spatial frequency, measured in DFT frequency steps.
m	The number of frequencies to compute. It must be at least 2.

Since: 2.61

gwy_data_field_zoom_fft ()

void
gwy_data_field_zoom_fft (GwyDataField *rsrc,
                         GwyDataField *isrc,
                         GwyDataField *rdest,
                         GwyDataField *idest,
                         gint mx,
                         gint my,
                         gdouble fx0,
                         gdouble fy0,
                         gdouble fx1,
                         gdouble fy1);

Computes Zoom FFT of a data field.

The output is DFTs, but computed for an arbitrary 2D Cartesian grid of frequencies along x and y . The frequencies do not have to be in any relation to the data sampling step.

The top-left pixel of output corresponds exactly to (fx0 ,fy0 ) and the bottom right exactly to (fx1 ,fy1 ). So the frequency sampling steps will be (fx1 − fx0 )/(mx − 1) and (fy1 − fy0 )/(my − 1), instead of the more usual division by mx and my . To follow the usual Gwyddion conventions, the output data field real size will be (fx1 − fx0 )/(mx − 1)*mx along x , and similarly along y . If it seems confusing, just take the output as indexed by integers and work with that.

Frequency step of one corresponds to the normal DFT frequency step. Therefore, passing fx0 =0, fx1 =xres –1, fy0 =0, fy1 =yres –1 (where rsrc has xres × yres points), mx =xres and my =yres reproduces the usual DFT, except more slowly. The result is normalised as raw FFT and the units of the output data fields are unchanged.

The transform direction is always forward. Windowing or other preprocessing need to be done separately beforehand. They would be usually once, but followed by any number of (Zoom) FFTs.

Parameters

rsrc	Real input data field.
isrc	Imaginary input data field. It can be `NULL` for real-to-complex transform.
rdest	Real output data field. It will be resized to `mx` × `my` samples.
idest	Imaginary output data field. It will be resized to `mx` × `my` samples.
fx0	The first horizontal spatial frequency, measured in DFT frequency steps.
fy0	The first vertical spatial frequency, measured in DFT frequency steps.
fx1	The last horizontal spatial frequency, measured in DFT frequency steps.
fy1	The last vetical spatial frequency, measured in DFT frequency steps.
mx	The number of horizontal frequencies to compute. It must be at least 2.
my	The number of vertical frequencies to compute. It must be at least 2.

Since: 2.62

gwy_data_field_1dfft ()

void
gwy_data_field_1dfft (GwyDataField *rin,
                      GwyDataField *iin,
                      GwyDataField *rout,
                      GwyDataField *iout,
                      GwyOrientation orientation,
                      GwyWindowingType windowing,
                      GwyTransformDirection direction,
                      GwyInterpolationType interpolation,
                      gboolean preserverms,
                      gint level);

Transforms all rows or columns in a data field with Fast Fourier Transform.

If requested a windowing and/or leveling is applied to preprocess data to obtain reasonable results.

Parameters

rin	Real input data field.
iin	Imaginary input data field. It can be `NULL` for real-to-complex transform which can be somewhat faster than complex-to-complex transform.
rout	Real output data field, it will be resized to area size.
iout	Imaginary output data field, it will be resized to area size.
orientation	Orientation: pass `GWY_ORIENTATION_HORIZONTAL` to transform rows, `GWY_ORIENTATION_VERTICAL` to transform columns.
windowing	Windowing type.
direction	FFT direction.
interpolation	Interpolation type. Ignored since 2.8 as no resampling is performed.
preserverms	`TRUE` to preserve RMS while windowing.
level	0 to perform no leveling, 1 to subtract mean value, 2 to subtract line (the number can be interpreted as the first polynomial degree to keep, but only the enumerated three values are available).

gwy_data_field_area_1dfft ()

void
gwy_data_field_area_1dfft (GwyDataField *rin,
                           GwyDataField *iin,
                           GwyDataField *rout,
                           GwyDataField *iout,
                           gint col,
                           gint row,
                           gint width,
                           gint height,
                           GwyOrientation orientation,
                           GwyWindowingType windowing,
                           GwyTransformDirection direction,
                           GwyInterpolationType interpolation,
                           gboolean preserverms,
                           gint level);

Transforms all rows or columns in a rectangular part of a data field with Fast Fourier Transform.

If requested a windowing and/or leveling is applied to preprocess data to obtain reasonable results.

Parameters

rin	Real input data field.
iin	Imaginary input data field. It can be `NULL` for real-to-complex transform which can be somewhat faster than complex-to-complex transform.
rout	Real output data field, it will be resized to area size.
iout	Imaginary output data field, it will be resized to area size.
col	Upper-left column coordinate.
row	Upper-left row coordinate.
width	Area width (number of columns), must be at least 2 for horizontal transforms.
height	Area height (number of rows), must be at least 2 for vertical transforms.
orientation	Orientation: pass `GWY_ORIENTATION_HORIZONTAL` to transform rows, `GWY_ORIENTATION_VERTICAL` to transform columns.
windowing	Windowing type.
direction	FFT direction.
interpolation	Interpolation type. Ignored since 2.8 as no resampling is performed.
preserverms	`TRUE` to preserve RMS while windowing.
level	0 to perform no leveling, 1 to subtract mean value, 2 to subtract lines (the number can be interpreted as the first polynomial degree to keep, but only the enumerated three values are available).

gwy_data_field_1dfft_raw ()

void
gwy_data_field_1dfft_raw (GwyDataField *rin,
                          GwyDataField *iin,
                          GwyDataField *rout,
                          GwyDataField *iout,
                          GwyOrientation orientation,
                          GwyTransformDirection direction);

Transforms all rows or columns in a data field with Fast Fourier Transform.

No leveling, windowing nor scaling is performed.

The normalisation of FFT is symmetrical, so transformations in both directions are unitary.

Since 2.8 the dimensions need not to be from the set of sizes returned by gwy_fft_find_nice_size().

Parameters

rin	Real input data field.
iin	Imaginary input data field. It can be `NULL` for real-to-complex transform.
rout	Real output data field, it will be resized to `rin` size.
iout	Imaginary output data field, it will be resized to `rin` size.
orientation	Orientation: pass `GWY_ORIENTATION_HORIZONTAL` to transform rows, `GWY_ORIENTATION_VERTICAL` to transform columns.
direction	FFT direction.

Since: 2.1

gwy_data_field_2dfft ()

void
gwy_data_field_2dfft (GwyDataField *rin,
                      GwyDataField *iin,
                      GwyDataField *rout,
                      GwyDataField *iout,
                      GwyWindowingType windowing,
                      GwyTransformDirection direction,
                      GwyInterpolationType interpolation,
                      gboolean preserverms,
                      gint level);

Calculates 2D Fast Fourier Transform of a rectangular a data field.

If requested a windowing and/or leveling is applied to preprocess data to obtain reasonable results.

Lateral dimensions, offsets and units are unchanged. See gwy_data_field_fft_postprocess() for that.

Parameters

rin	Real input data field.
iin	Imaginary input data field. It can be `NULL` for real-to-complex transform which can be somewhat faster than complex-to-complex transform.
rout	Real output data field, it will be resized to area size.
iout	Imaginary output data field, it will be resized to area size.
windowing	Windowing type.
direction	FFT direction.
interpolation	Interpolation type.
preserverms	`TRUE` to preserve RMS while windowing.
level	0 to perform no leveling, 1 to subtract mean value, 2 to subtract plane (the number can be interpreted as the first polynomial degree to keep, but only the enumerated three values are available).

gwy_data_field_area_2dfft ()

void
gwy_data_field_area_2dfft (GwyDataField *rin,
                           GwyDataField *iin,
                           GwyDataField *rout,
                           GwyDataField *iout,
                           gint col,
                           gint row,
                           gint width,
                           gint height,
                           GwyWindowingType windowing,
                           GwyTransformDirection direction,
                           GwyInterpolationType interpolation,
                           gboolean preserverms,
                           gint level);

Calculates 2D Fast Fourier Transform of a rectangular area of a data field.

If requested a windowing and/or leveling is applied to preprocess data to obtain reasonable results.

Parameters

rin	Real input data field.
iin	Imaginary input data field. It can be `NULL` for real-to-complex transform which can be somewhat faster than complex-to-complex transform.
rout	Real output data field, it will be resized to area size.
iout	Imaginary output data field, it will be resized to area size.
col	Upper-left column coordinate.
row	Upper-left row coordinate.
width	Area width (number of columns), must be at least 2.
height	Area height (number of rows), must be at least 2.
windowing	Windowing type.
direction	FFT direction.
interpolation	Interpolation type. Ignored since 2.8 as no resampling is performed.
preserverms	`TRUE` to preserve RMS while windowing.
level	0 to perform no leveling, 1 to subtract mean value, 2 to subtract plane (the number can be interpreted as the first polynomial degree to keep, but only the enumerated three values are available).

gwy_data_field_2dfft_raw ()

void
gwy_data_field_2dfft_raw (GwyDataField *rin,
                          GwyDataField *iin,
                          GwyDataField *rout,
                          GwyDataField *iout,
                          GwyTransformDirection direction);

Calculates 2D Fast Fourier Transform of a data field.

No leveling, windowing nor scaling is performed.

The normalisation of FFT is symmetrical, so transformations in both directions are unitary.

Since 2.8 the dimensions need not to be from the set of sizes returned by gwy_fft_find_nice_size().

Lateral dimensions, offsets and units are unchanged. See gwy_data_field_fft_postprocess() for that.

Since 2.53 iout can be NULL for complex-to-real transforms. Note that this means Hermitean symmetry of the input data is assumed, i.e. about half of the input is ignored. If you want to extract the real part of a complex transform, you must pass a non-NULL iout .

Parameters

rin	Real input data field.
iin	Imaginary input data field. It can be `NULL` for real-to-complex transform.
rout	Real output data field, it will be resized to `rin` size.
iout	Imaginary output data field, it will be resized to `rin` size.
direction	FFT direction. It should be `GWY_TRANSFORM_DIRECTION_FORWARD` for real-to-complex transforms and `GWY_TRANSFORM_DIRECTION_BACKWARD` for complex-to-real transforms.

Since: 2.1

gwy_data_field_2dfft_humanize ()

void
gwy_data_field_2dfft_humanize (GwyDataField *data_field);

Rearranges 2D FFT output to a human-friendly form.

Top-left, top-right, bottom-left and bottom-right sub-rectangles are swapped to obtain a humanized 2D FFT output with (0,0) in the centre.

More precisely, for even field dimensions the equally-sized blocks starting with the Nyquist frequency and with the zero frequency (constant component) will exchange places. For odd field dimensions, the block containing the zero frequency is one item larger and the constant component will actually end up in the exact centre.

Also note if both dimensions are even, this function is involutory and identical to gwy_data_field_2dfft_dehumanize(). However, if any dimension is odd, gwy_data_field_2dfft_humanize() and gwy_data_field_2dfft_dehumanize() are different, therefore they must be paired properly.

Parameters

data_field

A data field.

gwy_data_field_2dfft_dehumanize ()

void
gwy_data_field_2dfft_dehumanize (GwyDataField *data_field);

Rearranges 2D FFT output back from the human-friendly form.

Top-left, top-right, bottom-left and bottom-right sub-rectangles are swapped to reshuffle a humanized 2D FFT output back into the natural positions.

See gwy_data_field_2dfft_humanize() for discussion.

Parameters

data_field

A data field.

Since: 2.8

gwy_data_field_fft_postprocess ()

void
gwy_data_field_fft_postprocess (GwyDataField *data_field,
                                gboolean humanize);

Updates units, dimensions and offsets for a 2D FFT-processed field.

The field is expected to have dimensions and units of the original direct-space data. The lateral units and resolutions are updated to correspond to its Fourier transform.

The real dimensions are set for spatial frequencies, not wavevectors. For wavevector lateral coordinates, mutiply all real dimensions and offsets by 2*G_PI.

If humanize is TRUE gwy_data_field_2dfft_humanize() is applied to the field data and the lateral offsets are set accordingly. Otherwise the offsets are cleared.

Value units are kept intact.

Parameters

data_field	A data field.
humanize	`TRUE` to rearrange data to have the frequency origin in the centre.

Since: 2.38

gwy_data_field_fft_filter_1d ()

void
gwy_data_field_fft_filter_1d (GwyDataField *data_field,
                              GwyDataField *result_field,
                              GwyDataLine *weights,
                              GwyOrientation orientation,
                              GwyInterpolationType interpolation);

Performs 1D FFT filtering of a data field.

Parameters

data_field	A data field to filter.
result_field	A data field to store the result to. It will be resampled to `data_field` 's size.
weights	Filter weights for the lower half of the spectrum (the other half is symmetric). Its size can be arbitrary, it will be interpolated.
orientation	Filter direction.
interpolation	The interpolation to use for resampling.

gwy_data_line_fft_window ()

void
gwy_data_line_fft_window (GwyDataLine *line,
                          GwyWindowingType windowing);

Performs windowing of a data line in preparation for FFT.

Parameters

line	A one-dimensional data line.
windowing	Windowing type to use.

Since: 2.62

gwy_data_field_fft_window ()

void
gwy_data_field_fft_window (GwyDataField *field,
                           GwyWindowingType windowing);

Performs two-dimensional windowing of a data field in preparation for 2D FFT.

The same windowing function is used row-wise and column-wise.

Parameters

field	A two-dimensional data field.
windowing	Windowing type to use.

Since: 2.62

gwy_data_field_fft_window_1d ()

void
gwy_data_field_fft_window_1d (GwyDataField *field,
                              GwyOrientation orientation,
                              GwyWindowingType windowing);

Performs row-wise or column-wise windowing of a data field in preparation for 1D FFT.

Parameters

field	A two-dimensional data field.
orientation	Windowing orientation (the same as corresponding FFT orientation).
windowing	Windowing type to use.

Since: 2.62

gwy_data_field_cwt ()

void
gwy_data_field_cwt (GwyDataField *data_field,
                    GwyInterpolationType interpolation,
                    gdouble scale,
                    Gwy2DCWTWaveletType wtype);

Computes a continuous wavelet transform (CWT) at given scale and using given wavelet.

Parameters

data_field	A data field.
interpolation	Interpolation type. Ignored since 2.8 as no resampling is performed.
scale	Wavelet scale.
wtype	Wavelet type.