FQ - Filter - Quick Fourier, High, low pass, etc. ||

PURPOSE

Applies Fourier filters to 2-D or 3-D images. Images need not have power-of-two dimensions (see notes). Padding with the average is applied during filtration.

SEE ALSO

FQ NP	[Filter - Quick Fourier, High, low pass, etc, No Padding ||]
FF	[Fourier Filter]
FP	[Fourier interpolation ||]
FT	[Fourier Transform ||]
FF PLOT	[Fourier Filter - design filter]

USAGE

.OPERATION: FQ

.INPUT FILE: PIC001
[Enter name of input file containing real image.]

.OUTPUT FILE: FIC001
[Enter name of output file containing filtered real image.]

1: LOW-PASS, 2: HIGH-PASS,
3: GAUSS LOW-PASS, 4: GAUSS HIGH-PASS,
5: FERMI LOW-PASS, 6: FERMI HIGH-PASS,
7: BUTER. LOW-PASS, 8: BUTER. HIGH-PASS

.Filter type (1-9): 7
[Enter filter option code.]

Option "1" - Low-pass truncation. Filter is "top-hat" function that truncates Fourier transform at spatial frequency RAD.

Option "2" - High-pass truncation. Filter is inverse "top-hat" function that passes Fourier transform beyond spatial frequency radius RAD.

Option "3" - Gaussian low-pass. Filter is Gaussian function EXP(-f**2/(2.*RAD**2)), where F is the frequency.

Option "4" - Gaussian high-pass. Filter is complement of Gaussian function: 1.0-EXP(-F**2/(2.*RAD**2)).

Option "5" - Fermi low-pass. Filter is 1/(1+EXP[(F-RAD)/T]) which negotiates between "top-hat" and Gaussian characteristics, depending on the value of T (see below).

Option "6" - Fermi high-pass. As low-pass, but T is replaced by -T.

Option "7" - Butterworth low-pass. Filter is: 1/(SQRT(1+F/RAD)**(2*ORDER))
In the actual implementation of the filter the parameters, RAD and ORDER are calculated from the pass-band frequency and cut-off frequency specified by the user. Frequencies below pass band are preserved, frequencies above stop band are removed, with a smooth transition in between (pass band > stop band). See NOTE 3 below.

Option "8" - Butterworth high-pass. Filter is: 1-(1/(SQRT(1+F/RAD)**(2*ORDER)))

For options 1-6:

.FILTER RADIUS: 0.12
[The FILTER RADIUS can be given either in absolute units or pixel units. If answer is > 1.0 it is treated as given in pixel units. If filter function radius is given in frequency units, they should be in the range 0.0<=f<=0.5. For an explanation of frequence units see FF PLOT.]

For options 7-8:

.PASS-BAND AND STOP-BAND FREQUENCY: 0.1,0.2
[Enter filter function radii in frequency units.]

For the Fermi filters [(5) and (6)], the temperature parameter 'T 'will be solicited:

.TEMPERATURE (0=CUTOFF): .3
[Roughly within this reciprocal distance (in terms of frequency units), the filter falls off.]

NOTES

1. 'FQ' has similar functions to 'FF' except that:
   a) It accepts only real space images (2- & 3-D) and leaves the filtered image real,
   b). it runs only in core, so it is much faster than the 'FF' operation.
   

2. The FILTER RADIUS can be given either in absolute units or pixel units. If answer is > 1.0 it is treated as given in pixel units.

3. In the Butterworth filter the ORDER determines the filter fall off and RAD corresponds to the cut-off radius. In the program RAD and ORDER are calculated from the parameters specified by the user using following equations:
   RAD = fp/((eps)**(2/ORDER))
   ORDER = [2*log(eps/sqrt(a**2-1))]/[log(fp/fs)]
   where fp and fs are the pass-band and stop-band frequencies and parameters "eps" and "a" are given by 0.882 and 10.624 resp. Note that fp and fs deviate from 1.0 and 0.0 by about 0.2 and 0.09 respectively (for a low-pass filter.) For reference see FF PLOT.

4. Implemented by: Paul Penczek.

SUBROUTINES

FOUR1A, FQ_Q, FQ3_P

CALLER

FOUR1