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| ## AcademicFFT## A Java class to perform Fourier-related operations on discrete sequences, images, and volumes.## Philippe Thévenaz, Biomedical Imaging Group, Swiss Federal Institute of Technology LausanneFigure 1. Top: (898 x 81) grayscale image of a tree core. Middle: log-amplitude of its discrete Fourier transform. Bottom: color-coded phase of its discrete Fourier transform. ## General Description
This Java class contains methods to perform forward and backward discrete Fourier transforms and to perform Fourier-based circular convolutions. The data live in dimension one (sequences), two (images), or three (volumes). They are stored as linear arrays of
Contrarily to several other Fourier packages currently available, the data manipulated by
The algorithms deployed by this class are standard versions of the fast Fourier transform (FFT). They include mixed-radix, split-radix, coprime, and Rader decompositions, among others. They are able to handle arbitrary data lengths, memory permitting. Behind the scenes, a stage of planification is first performed to determine which combination of approaches is optimal for the current data length. This planification is performed once only; its outcome is stored for future calls, until the time comes when the class is unloaded from memory. Consequently, the very first call to an instance of an
## I. DownloadThis distribution is dated May 31, 2014. It includes the complete set of source files, along with the documentation. - compressed jar file [68,486 bytes]
- source file [622,591 bytes]
## II. User Manual
Consider a periodic space-domain sequence. It is convenient to describe it over a single period, starting at the origin, like Here are a few basic definitions. - Imaginary unit:
- Unit impulse:
- Dirac distribution:
Here are a few basic relations featuring the discrete Fourier transform. - Periodicity:
- Linearity:
- Duality (Fourier of Fourier):
- Duality (Inverse Fourier of inverse Fourier):
- Conjugates:
- Parseval:
- Hermitian symmetry (real data):
- Separate real and imaginary:
- Circular integer translation:
- Circular fractional translation:
- Harmonic modulation:
- Circular spatial convolution:
- Spatial multiplication:
- Reflection:
- Spatial up-sampling:
- Fourier up-sampling:
- Spatial padding: with
- Fourier padding (even length): with
- Fourier padding (odd length): with
- Relation to the discrete-time Fourier transform (even length): with
- Relation to the discrete-time Fourier transform (odd length): with
- Relation to the Fourier series (even length): with
- Relation to the Fourier series (odd length): with
Here are the discrete Fourier transforms of a few particular sequences. - Constant-valued spatial data:
- Constant-valued Fourier:
## III. Conditions of UseEPFL makes no warranties of any kind on this software and shall in no event be liable for damages of any kind in connection with the use and exploitation of this technology. |