**Description:**

Generalized
ImageJ plug-in for the calculation of Fourier Transforms. The plug-in handles
forward and inverse transformations of arbitrary-sized three-dimensional (3D)
volumes as well as single two-dimensional (2D) images. Please note that image
stacks are always considered to represent 3D volumes and NOT series of 2D
images. For a brief introduction to Fourier Transforms consult the links
provided below.

As the
Fourier Transform is separable, it is calculated in three steps, one for the
x-, y-, and z-direction, respectively. However, a true Fast Fourier Transform
(FFT) implementation is only used for those directions that are of a
power-of-two size. Otherwise, a slow Discrete Fourier Transform (DFT) is used.
For instance, if a 256x400x16 volume is to be transformed, the transformation
in x- and z-direction is done by means of a true FFT, whereas the
transformation in y-direction needs to be based on a slow DFT.

For the
output of the transformation, the user can choose from several characteristics
of the transformed data. Additionally, the user can choose whether the origin
of the Fourier domain shall be placed at the origin of or be centered in the
resultant volume/image. Placing the Fourier domain origin at the center of the
resultant volume/image is useful for visualization purposes.

**User manual:**

After
starting the plug-in, the user can specify the real and imaginary part of the
data to be transformed by choosing currently open images/stacks from the
drop-down lists. If no imaginary part is specified the imaginary part of the
data is set to zero, which is usually the case if forward transforms of spatial
images/stacks shall be computed. Two options are available:

·
*'Complex
Number Precision'*
defines whether calculations shall be based on single or double precision
complex numbers. Generally, computation based on double precision complex
numbers is more accurate. However, using single precision complex numbers
reduces memory demands and has shown to yield results almost identical to those
obtained by double precision calculations. Nevertheless, this may depend on the
actual application.

·
*'FFT
Direction'*
specifies whether a forward or inverse Fourier Transformation shall be
computed.

After
calculations have finished, a new window appears and the user can choose from
various output modalities. By clicking one of the buttons on the left, the user
can choose which characteristic of the transformed data shall be displayed.
Available characteristic are real part, imaginary part, Fourier Frequency Spectrum,
logarithmic Fourier Frequency Spectrum, Fourier Phase Spectrum, Fourier Power
Spectrum, and logarithmic Fourier Power Spectrum. The check-boxes on the right
specify whether the origin of the Fourier domain shall be placed at the origin
of or be centered in the resultant volume/image.

**Links:**

·
http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip-Fourier.html

· http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip-Properti-2.html