Help for OTF1

 
PURPOSE:
 
   OTF1 is a VICAR applications program which performs one-dimensional
Fast Fourier Transformations in order to compute Optical Transfer
Functions (OTF) from degraded edges in images or from either a tabulated
real function, or a line spread function using parameter inputs. OTF1 is
able to compute the entire optical transfer function (not just the MTF)
from digital picture data with greater accuracy and ease than other methods
available. This technique for computing imaging system MTF's is preferable
to previously used techniques which invloved the imaging of sine wave targets.
 
   The input to OTF1 may be either image data or data specified by parameters.
OTF1 allows OTF's to be computed from such operational image data as planet
limbs or other step functions. OTF1 will accept a real string of values (DATA)
for which the Fourier transform is desired, or a line spread function (LSF)
for which the OTF is desired. A self test routine is invoked by SINCTST which
creates a SINC function and branches to the LSF processor.
 
   To aid in the preparation of filter weights for MTF enhancement, a provision
has been added to output the Real and Imaginary Fourier Transform components.
The output is 124 triplets of Real*4 numbers organized as [Frequency,Real,
Imaginary], [Frequency,Real,Imaginary], from 1.0 cycles/sample to 0.484
cycles/sample. The output does not extend beyond 0.484 cycles/sample due to
limitations of the Sampling Theorem used for LSF resampling (see last para-
graph of Method below).
 EXECUTION:
 
   OTF1 INP  OUT  PLOT  PARAMS
 
 where: INP is an optional input image.
 
        OUT is an optional output file with fourier transform data.
 
        PLOT is an optional plot of the output.
 
             under Tutor.
METHOD:
 
   The data are normally in an input image and are processed line by line.
Each line is left adjusted into an array of dimension equal to the smallest
power of two which contain an entire input line. The data is then different-
iated to produce a Line Spread Function (LSF). For direct input of a Line
Spread Function, the parameters LSF is followed by the LSF values (Real*4).
Processing will commence with the LSF ready for fillin to the nearest power
of two using the MEAN, REFLECT, or FILLIN parameters. For direct input of
a theortical "real" edge or mathematical function, the parameter DATA is
followed by a string of values (Real*4) and will be processed as thought it
was a real one-line image input. SINCTST will cause a SINC function to be
formed and processed as though it were the LSF of an edge.
 
  The Maximum point in the LSF is located and the LSF truncated to 30 points
on either side of the LSF Maximum. If there are less than 30 points to either
side the option exists to fill in extra points by replecating the LSF mean,
by reflection, or with zeros. This LSF is then left justified and the Sampling
Theorem applied to produce a 256 point power spectrum. In addition the
resampled LSF is folded about the maximum, with points from the maximum to the
upper end placed from i=1 to the midpoint, and points from the maximum to the
lower end placed from near the midpoint to 1=256. This array is then in the
proper form upon which the subroutine FFTT can be used to perform an inverse
transform. The real and imaginary components of the transform for each line
are accumulated element by element, and these operations continued until all
the image lines have been processed. From the accumulated real and imaginary
transform components the following are computed and printed.
   Given a complex FT element "a+ib" the output will consist of:
 
(1) The spatial frequency in cycles/sample.
(2) The actual frequency in cycles/unit of time (INTERVAL).
(3) a=Real part of FT.
(4) b=Imaginary part of FT.
(5) SQRT(a**2+b**2)=The amplitude of the FT. This is also the MTF if the
    input data is an LSF.
(6) (a**2+b**2)=The power spectrum or intensity of the FT.
(7) The Normalized amplitude of the MTF which will be plotted.
(8) ACRTAN(b/a)=The phase angle Omega of the FT measured in radians
    which will also be plotted
(9) W=Omega/2*pi*F=The displacement in the same units as INTERVAL
    (the default of which is samples) of each sine wave component in
    the image from its true position in the object being imaged. F is
    in units of 1/INTERVAL. This quantity is meaningful if the input
    data is an LSF and if OTF has been specified.
If PLOT was specified the Normalized MTF Amplitude and Phase are processed
into an output data set with Fortran I/O which can be printed to produce a
Printronix like plot. In this case the line printer plot is disabled.
 
The default is to generate on the line printer a display of the pertinent
components of the OTF in a crude but rapidly analyzable format. Each
curve's data points are represented by a letter with the following code:
 
   A   Amplitude of the FT
   R   Real portion of the FT
   I   Imaginary portion of the FT
   W   Wavelength shift of the FT.
The FT is computed from
 
           N-1
    FT =SUM    [DATA *Exp(-2*PI*inm/N)]                   (1)
           n=0      n
 
where N is the data string length and  0<m<(N-1).
 
   The OTF section of the program is designed to generate an LSF positioned
in such a way as to eliminate both the effects of sampling and the bias
introduced into the phase OMEGA due to the arbitrary positioning of the LSF
data in the input array. Users of FFTT have been forced to obtain clean
OTF's by using symmetrical LSF's positoned half at the beginning of the
data string and half at the end. This was inconvenient and impossible if
the LSF was asymmetrical. In the latter event all they could do was settle
for the MTF, never knowing what the OTF was like. In a situation where the
OTF has gone negative, the MTF is dangerous to use as a restoration function
because it will boost the negative amplitudes the wrong way. The OTF allows
the user to visualize the scrambling of spatial frequencies which occur when
the LSF is asymmetrical, and also to see the sign of the OTF (given by the
real part of the FT).
 
   There are two problems associated with generating unbiased OTF's. They
can be solved simultaneously by use of the Sampling Theorem. If the LSF is
computed from a degraded edge composed of digital data, it is clear that
(particularly for narrow LSF's) the position of the sampled points is going
to influence the symmetry of the LSF. Clearly, trying to dissect several
LSF's into two halves (each different) and store them at opposite ends of
the data string before the FT operation, is going to produce different OTF's
each time. Because the data is band limited, at least at the Nyquist Frequency,
the MTF's generated in the above case will all be about the same. The problem
is to eliminate the effects of sampling. The Sampling Theorem is used by OTF1
to resample the acquired LSF about its center of weight. At this point, if the
LSF is asymmetrical, the effect is real.
   In computing the weight center of the LSF a first moments algorithm is
used:
 
              n=N                  n=N
    CENTER=SUM   [(n-1)*DATA ]/ SUM   [DATA ]               (2)
              n=1           n      n=1     n
 
The Data are then resampled about the CENTER position using
 
            n=+INF
    INT =SUM      [DATA *((SIN(x-n)*PI)/((x-n)*PI))]          (3)
       x    n=-INF     n
 
where x is some integer distance away from CENTER. The data can then be
split about the located center and positioned properly in the array to be
transformed.
 
   The use of many LSF's to produce one OTF is done by averaging the
transform components. Averaging MTF's (if that's what you want) can be
erroneous because even OTF's, which are in reality positive, can be
caused to go negative if one of the LSF's is afflicted with noise. Since
the MTF is the modulus of the OTF, the MTF is always positive. Great care
should be taken in inspecting the LSF's computed from pictures before the
resampling step to be sure they look decent. Averaged runs of 50 or more
LSF's should yield OTF's which are accurate to within a few percent. This
is an order of magnitude better than the sine wave methods used in the past.
 
   Because the Sampling Theorem summation limits must be finite (in this
case +-128) the theorem is blind to the spatial frequencies above 0.484
cycles per sample. The OTF values above this frequency are best obtained
by extrapolating from values below 0.484 to the Nyquest limit.
 RESTRICTIONS:
 
  1. Input picture record size <= 2048 samples.
 
  2. LSF or DATA input elements <= 256.
 PROGRAM HISTORY:

 05  AUG   1997...T.Huang.......Ported from VAX/VMS to UNIX and ALPHA/VMS.
                                Combined the changes made by C. Avis with
                                the existing MIPS version of OTF1. 
                                Fixed problems caused by subroutine FFTT, by
                                reducing the precision in calculation with 
                                imaginary part of a complex number for PHASE.
                                (REF: fr90520) 
 28  SEP   1995...C.C.AVIS......Added option for no Phase to table
 28  JUN   1995...C.C.AVIS......Added header to ASCII table output
 27  JUN   1995...C.C.AVIS......Modified test file to use perfect ramp case
 23  FEB   1994...C.C.Avis......1.Add tabular output, property label support,
                                2.Improved test and help files.
              ?...J.J.Lorre.....1.Scaling of phase plots.
 11  AUG   1988...F.M.MOSS......1.Add warning message for the line without edge.
                                2.Fourier Transform will not apply to the line
                                  without edge.
                                3.Print out the total number of lines processed.
                                4.Fix the bug to avoid calling XVWRIT if there
                                  is no output required. (FR 35605)
  7  MAY   1987...F.M.MOSS......1.modify the code to handle size field correctly
                                2.modify output's XVOPEN and XVWRIT calling
                                  parameters
                                3.put "NONORMAL" keyword in the source if
                                  the input is a line spread function
  15 NOV   1985...F.M.MOSS......1.modify the plot for phase angle...
                                (the axis range from -1 to 1
                                 instead of from -3 to 3)
                                2.add a new param "PZOOM" so the
                                  user can either zoom up (PZOOM >1.)
                                  or zoom down (PZOOM <1.) the phase
                                  angle
  06 MAY   1985...M.E.MORRILL...ADD OUTPUT FOR F.T. COMP.
  02 MAY   1985...M.E.MORRILL...REMOVE VAX/FFT FACTOR OF 2
                                & CLEAN UP NORMALIZATIONS
  30 APR   1985...M.E.MORRILL...SAMPLING THEOREM "BUGS"
  17 APR   1985...M.E.MORRILL...MINOR FIXES TO ADD QUANTITAVE
                                ESTIMATE OF OTF CURVE.
  06 DEC   1984...M.E.MORRILL...CODE MODIFIED TO CONFORM TO
                                THEORY FROM R.NATHAN AND TO
                                REMOVE FILTER OPERATIONS.
  06 NOV   1984...M.E.MORRILL...CONVERTED TO VAX-VICAR*2.
  02 FEB   1983...E.P.KORMSO....REVISION TO PRINTRONIX PLOTING.
  10 DEC   1982...E.P.KORSMO....PRINTRONIX PLOT PACKAGE ADDED.
  27 JUN   1975...D.A.HASS......CONVERSION TO 360/OS.
  10 MAR   1975...J.J.LORRE.....RE-RELEASE.
  16 MAY   1974...J.J.LORRE.....ORIGINAL RELEASE.
 
CURRENT COGNIZANT PROGRAMMER:  J J Lorre