inverse: Compute inverse Fourier transform of an image.

Package: fourier

Usage

inverse input output

Description

This task computes the inverse Fourier transform of a 1- or 2-dimensional image. The input may consist of both the real and imaginary parts of an image, or either part independently.

The 'forward' task does not normalize its output, but the 'inverse' task normalizes by dividing by the number of pixels. Applying 'forward' and then 'inverse' therefore returns an image which is the same as the original, except for roundoff errors.

The output coordinate parameters will be determined from the input parameters and from the header keywords 'OCRPIX1' and 'OCRVAL1' in the input image, if present; these keywords were written by the 'forward' task, if it was used.

For 2-D transforms, this task has the option of using scratch images for intermediate results. Using scratch images may take longer, but it allows the task to function even with limited memory.

Parameters

input [file name]: Name of the input data file. See also 'inreal' and 'inimag', which specify whether the real and imaginary parts are to be read. If both real and imaginary parts are to be read, the letters "r" and "i" will be appended to 'input' to form the names of the images for real and imaginary parts respectively.

output [file name]: Name of the output data file. Both a real and an imaginary part will be created, but only the parts specified by the 'outreal' and 'outimag' will actually be saved. If both real and imaginary parts are to be saved, the letters "r" and "i" will be appended to 'output' to form the names of the images for real and imaginary parts respectively.

(inreal = yes) [boolean]: Use the real part of the input data file? If this is set to yes, the real part must exist.

(inimag = yes) [boolean]: Use the imaginary part of the input data file? If this is set to yes, the imaginary part must exist.

(outreal = yes) [boolean]: Save the real part of the output data file?

(outimag = yes) [boolean]: Save the imaginary part of the output data file?

(coord_shift = no) [boolean]: If 'coord_shift=yes', then the phase of the input image will be modified so as to shift the coordinate origin of the output image back to its original position, as given by the value of 'OCRPIX1' (and 'OCRPIX2' for a 2-D image). This is appropriate if the input image was derived from the output from 'forward', and 'forward.coord_shift=yes' was used. See the help for 'coord_shift' for the 'forward' task for more information. If 'coord_shift=no', then the coordinates will not affect the value of the inverse transform. The coordinate parameters are, however, still updated. For 2-D images, 'coord_shift' and 'inmemory' must not both be set to yes.

(decenter = yes) [boolean]: The 'forward' task has the option of shifting the coordinate origin of the output image to the center, rather than leaving it at the first pixel. If this was done, then it is important to correct for that shift before doing an inverse transform. Setting 'decenter=yes' allows this correction. If 'forward.center=no' was used instead, then leaving 'inverse.decenter=yes' is harmless; therefore, the 'decenter' parameter should almost always be set to yes. The decentering is done using the value of the 'CRPIX1' keyword, which specifies the location of the coordinate origin in the Fourier domain. There is no option to center the output of the 'inverse' task, since that would not normally make sense.

(inmemory = yes) [boolean]: For a two-dimensional input image, if 'inmemory = yes' the image will be read into a complex array, the inverse Fourier transform will be performed on that array in-memory, and the array will be written to the output image. This requires one complex word for each pixel. The complex array must fit entirely in memory (i.e. no paging) because when performing the inverse Fourier transform the array is accessed both by rows and by columns. If 'inmemory = no', see the description of 'len_blk'. The parameters 'inmemory' and 'coord_shift' may not both be set to yes for 2-D images. For 1-D images, 'inmemory' is ignored.

(len_blk = 256) [integer]: Length of block for transposing images. For 2-dimensional input images, if 'inmemory = no' this task transposes each image into scratch images before computing the inverse Fourier transform of the second axis. This parameter is the length of the side of a square region that is transposed in one step. The I/O buffers for scratch images can take a lot of memory if 'len_blk' is large, e.g., about 8 megabytes for 'len_blk = 512'. If you get out-of-memory errors, you should flush the process cache (flprcache), reduce the size of 'len_blk' and try again. This parameter is ignored for 1-D images or if 'inmemory = yes'.

(verbose = yes) [boolean]: Print input and output image names? Setting 'verbose=yes' shows you the actual names of the image headers, including the "r" & "i" suffixes for real & imaginary parts.

(ftpairs = fourier$ftpairs.dat) [file name]: Name of the file that defines the type of coordinate in a transform pair. For example, "LAMBDA", "WAVENUMB".

Examples

1. Take the inverse Fourier transform of the images "tr" and "ti" (i.e., the real and imaginary parts) and put the output real part in an image called "civ"--the imaginary part is discarded.

fo> inverse t civ outimag=no

Bugs

If the task fails due to lack of memory or disk space, for example, the output image and temporary images that were created are not deleted.

References

Bracewell, R.N.: "The Fourier Transform and Its Applications," McGraw-Hill Publishing Co., New York, 1986.

The implementation of the inverse Fourier transform in the 'inverse' task differs from the definition given in Bracewell in that the output from this task is normalized by dividing by the total number of pixels. Bracewell includes the normalization in the forward transform instead.

For a 1-D array G[f], the inverse Fourier transform g[t] is

g[t] = (1/N) * sum of G[f] * exp (2*pi*i * t * f / N)
from f=0 to f=N-1,

where the indexes f and t run from 0 to N-1. For a 2-D array, a 1-D transform is done for each row, and then the 1-D transform is done for each column.