Chapter 8
Optical Correlation: Experiment
Introduction
In this chapter an in depth evaluation of the 16×16 SLM as an
optical correlator is performed
1. The relatively low
space-bandwidth product of the 16×16 device requires that
considerable attention be devoted to the choice of filter algorithm
used. Consequently, after an introduction to phase binarisation
algorithms in the first section, section two looks at
practical filter design and evaluates several novel algorithms devised
in this project. Experimental results are compared with those from
simulations in section three and the algorithms used are analysed in
detail in section four. Section five reviews the results of this chapter.
1 Phase Binarisation Algorithms
The most common approach to computer simulation of optical correlation, as
published, is to represent the input object on a 128×128 array, for
example, and Fourier
Transform this to find the spectrum which is written on an array of identical
dimensions. For the binary phase-only matched filters, the phase of
the complex number representing the spectrum is conjugated and the
binarisation algorithm under evaluation is used to binarise the phase
of each spectral data point to either zero or
p radians. The amplitude
of each point on the filter is set to unity.
This procedure is usually applied point by point to make a filter which
is used to multiply the original spectrum and obtain the auto-correlation
in the image plane.
There are very few publications dealing with the details of
filter calculation for use on actual SLMs where the information
from several spectral data pixels is required to be combined in some way
to produce a single binary phase value for each mirror.
Of note, Flannery et al have performed a most extensive study into the
application of binary phase-only correlation to machine vision [45].
Experimental work was performed with the 48×48 magneto-optic SLM,
described in chapter four, and computer simulations using a
128×128 data array. To match the size of SLM transmissive element
with the scale of the FFT, a 3×3 sub-array was averaged to
produce a single complex number, the phase of which was used as input to
a phase binarisation algorithm, actually that of Horner et al (see
following subsection).
The filter calculation process of this project combines several
published phase binarisation algorithms to form a new binarisation
procedure applicable to pixellated spatial filters. Briefly, the
algorithms used in this project shall now be described.
1.1 Commonly Used Algorithms
Binary phase-only filters are calculated from their continuous filter
counterparts by using a binarisation algorithm. If the continuous phase
filter H(
n) (where the discussion relates to a general spatial
frequency
n) is of form
where
F(
n) is the phase of the target object Fourier Transform,
then
represents the binarised phase-only version of H(
n). If the phase
of H(
n) is plotted on an Argand diagram, one must decide where to
divide the plane such that phases on either side are mapped to the poles
of either 0 or
p radians.
Figure 8.1: Threshold Decision Line.
Figure 8.1 illustrates what is referred to
by Downie et al as the `threshold decision line' [68], defined by angle
b, and table 8.1 lists the threshold angle b which is proposed by three
different `camps', those of Horner et al [33], Psaltis et al
[69] and Cottrell et al [70].
| Binarisation | b | Phase Angles ® 0 |
| HORNER et al | 0 | 0 £ q £ p |
| PSALTIS et al | [(p)/2] | -[(p)/2] £ q £ [(+p)/2] |
| COTTRELL et al | [(p)/4] | -[(3p)/4] £ q £ [(+p)/4] |
Table 1: Phase binarisation algorithms used
All three algorithms have been well documented and, as shall be
described shortly, are used as substages to a new procedure developed in
this thesis. A concise review of these algorithms may be found in an
article by Dickey, Stalker and Mason [50].
Object Symmetry
As pointed out by Dickey et al [50], the even-odd symmetry of
the target object is reflected in the REAL and IMAGINARY parts of the
Fourier Transform and thus directly affects the phase.
The degree of symmetry of the target object thus has a marked affect on the
object's phase spectrum, requiring care in the selection of filter
algorithm. This can be seen directly if the Fourier Transform is, in 1-D for
simplicity, decomposed into
Thus, for example, an entirely symmetric object (EVEN) has FIm=0. Recall
that any REAL 2-D function may be decomposed into both an entirely EVEN and
entirely ODD function by the operation
In selecting a threshold angle of b = 0, the filter of Horner
et al is effectively matched to the odd part of the object function. By this,
it is meant that a similar target object would produce the exact same binary
phase filter if the real parts of both object functions were
identical. Psaltis et al, with
b = [(p)/2], choose to match their filter with the even part
of the object function. Both these filters then may suffer from false
correlations with input objects having similar even and odd parts
respectively, as analysed by Barnes et al [71]. The binarisation algorithm of
Cottrell et al, with b = [(p)/4], attempts to combine information
from both the EVEN and ODD parts of the object function to decide the
final binarised filter phase, and indeed has been shown to reduce
false correlations in simulations.
1.2 Optimal Binary Correlation Filters
In contrast to the fixed threshold angle algorithms just described,
there exist optimisation techniques which seek to maximise certain
characteristic parameters of the correlation (peak height,
discrimination, etc.) which operate by varying the angle b. Farn
& Goodman [72] have shown that, in 1-D, the optimal correlation for an
object which is mostly EVEN corresponds to b @ [(p)/2],
as might be expected from the discussion immediately above.
Downie et al [68]
have performed an optimisation procedure on correlations which use a
2-D binary image of the NASA space shuttle as an input object, a function
which in this case was largely EVEN. Experimental
results were obtained using 128×128 magneto-optic SLMs as both
input and spatial filter. Of interest to this chapter, they found that
while in general an off-centre shift in the shuttle (making the object
function more ODD) moved the optimally calculated threshold angle away from
[(p)/2], the change in correlation peak intensity differed only
marginally from the b = [(p)/2] correlation. In fact, the
conclusion drawn from there study was that the b = [(p)/2]
threshold angle is `near optimum for BPOFs for most object functions.'
This observation
shall be returned to in the next section. However, optimisation
procedures are mentioned only for completeness and do not take any
further part in the work of this chapter, save for the observation just
made.
2 Combined Techniques
2.1 Algorithm Framework
From the initial Fourier Transform data produced by the DFT of a
specified target object, one must arrive at a single phase value of
either 0 or p radians to assign to each SLM mirror. The spectral
phase information from the DFT must somehow be combined so as to
form a single phase value which characterises the spectrum over that
mirror. This intermediate phase value is called the characteristic
phase of the mirror in this project. Following this, a single binary value
must be derived from the characteristic phase.
For the moment, let the choices open to us at each stage of the filter
calculation be stated without questioning exactly why
one might perform some actions.
- Firstly, we are free to choose whether or not to binarise the
initial phases Fij of the 9×9 DFT sub-array which
describe the spectrum over each mirror.
- Following this, one must decide upon an appropriate averaging
procedure to produce a single phase value, Fc, from this data with
which to characterise the mirror phase.
- Finally, one must select an appropriate binarisation algorithm
which operates on Fc to produce a single mirror phase Fm
with which to program the SLM.
These three stages form the basis from which all filter calculation
performed in this project stems. What effect, one might ask, do the initial
and final binarisation stages have on the performance of the
correlation filter ? It will be shown that the determination of the
characteristic phase allows an answer to this question and further
offers the widest choice of filter variability in this investigation.
In the calculation of the characteristic phase, one would like to give
more emphasis to those spectral points containing most energy. For
instance, if out of two spectral data points on a mirror one contains
106 more energy than the other, should one value the phase
contributions from each pixel equally ? In computational
determination of the characteristic phase, for instance,
the FORTRAN function `ATAN2'
determines the phase of a complex number irrespective of the magnitude
of that number. Therefore if only one `bright' data pixel truly
represented the object spectrum over that mirror the phase of this pixel
should be given higher priority in determination of the characteristic
phase. Thresholding of the initial spectrum to eliminate very low
amplitude spatial frequencies could be performed although a mirror
dependent threshold algorithm would possibly be required to counter the
fact that mirrors in the high spatial frequency region of the spectrum
generally receive less energy than those closer to the zero frequency.
Thresholding was not used in this project for this reason.
To begin with, two rather straightforward methods of determining Fc
are now described.
2.2 Basic Algorithms
A very straightforward method of forming the characteristic phase of a mirror
is the perform an intensity weighted average of Fij, followed by
binarisation using one of the three algorithms previously described.
Care must be taken in this process that the weight assigned to the
spatial offset (phase) of a component sinusoid depends only on the
intensity (amplitude squared) of that component, and is not influenced
by the representation of Fij. If a single
Fourier component of the object has wavelength l, so that
then a phase shift of ±[(p)/4] moves the component
spatially to the left or to the right by [(l)/8]. The
shift to the right (-[(p)/8]) could equally well be accomplished
if the phase were written as 2p-[(p)/8], which would result in
an intensity average heavily biased towards initially negative phases.
Therefore, in the intensity weighted average of N points,
one must ensure that -p < Fij < +p in order that two offsets of
equal magnitude but opposite direction have an equal (but opposite in
sign) effect on Fc.
Figure 8.2 plots the results of simulations using such a simple phase
averaging. Note that the Psaltis final binarisation stage does not
appear as filters using this final algorithm were found to be spatially
invariant. The ordinate plots percentage peak energy and the abscissa the
Gaussian parameter s (Chapter seven) which best befits description of
the data by a 2-D Gaussian function (smaller s means a sharper peak).
The ordinate is identical with the practical definition of optical
efficiency hP given in chapter seven.
Figure 8.2: Intensity Weighted Phase Averaging: Simulation Results
It is observed that neither the optical efficiency (percentage energy in
peak) nor peak sharpness are particularly inspiring, the reason being
the extreme simplicity of the filter algorithm. It will be
shown later in this chapter that the phase information of the
target objects used is approximated very well by a
uniform distribution on the range -p to +p. If, over each mirror,
the intensity does not vary rapidly then
a large degree of cancellation occurs in equation 8.7 so that the
distribution of Fc over the 15×15 mirror array is
confined to a rather narrow region about Fc=0.
Consequently, although the final binarisation algorithm of
Horner et al will bisect this spread (and Cottrel et al if the
distribution is wide enough), the algorithm of Psaltis et al will not.
This explains the spatial invariance found in computer
simulations of such simple filters using the Psaltis algorithm as a final
binarisation stage.
Further, consider the
case of a high intensity data point which has a phase of zero. In such a
case this point will not contribute anything to the intensity average
and thus highlights the limitations of equation 8.7 as a means of
determination of characteristic phase. As the characteristic phase was
formed by such a simple procedure these results are not of a high standard but
are included for comparison with more sophisticated
algorithms of this chapter, of which coherent summation is next to
be described.
2.3 Coherent Summation
A preferable technique for calculating the characteristic phase is known
as `coherent summation'. In this process, the complex numbers
describing the spectrum over any given mirror are, as the name suggests,
coherently added together in a process of phasor addition. The phase of the
resulting summation, Fc, may
then be binarised by one of the three algorithms of the preceding
subsection. This process has an inbuilt tendency to perform an
amplitude weighting of the phase from each data pixel.
If one considers the process of diffraction from a single
mirror only, the light field at infinity will be approximated by a phasor
addition of the complex light field over the mirror, which is where this
technique gets its name.
Figure 8.3: Determination of Characteristic Phase by Coherent Summation
This technique enables rapid, straightforward filter calculation and has
been used by Flannery et al with the 48×48 magneto-optic SLM. In
their calculations a 3×3 sub-array of the DFT was coherently
summed and the binarisation algorithm of Horner et al used to obtained
the final SLM phases Fm. In fitting this process within the
framework of general filter calculation, one would say that
- No initial data binarisation is required.
- The phase averaging procedure was one of coherent summation.
- Any final binarisation procedure may be used to obtain Fm.
Flannery et al chose to use Horner final binarisation, though it is not
stated why preference was given to this over the other two popular
algorithms. One can make accurate predictions as to the nature of the
filter produced by coherent summation by noting that the phases of each
component phasor in the coherent sum are conjugates for conjugate
mirrors. Therefore the characteristic phases of conjugate mirrors are
necessarily also conjugate. The final binarisation procedure of Horner
et al will thus map the phases of a mirror-conjugate pair to opposite
poles of zero and p resulting in a filter which is 100%
anti-symmetric. By the same reasoning, a final binarisation using the
Psaltis algorithm produces a filter of 100% symmetry about the
origin. From the threshold line of Cottrell one would expect neither a
completely symmetric nor completely anti-symmetric filter. Figure 8.4
illustrates these ideas with the filters produced for the `C' and `STAR'
objects.
Figure 8.4: Coherent Summation Filters for `C' and `STAR'
Applying a voltage to a mirror, it may be recalled, decreases the
optical path length through the liquid crystal as ne® no
and no > ne. Therefore the light from an `ON' mirror
(higher applied voltage) is spatially advanced relative to an `OFF'
mirror. However, when viewed in amplitude modulation mode a mirror with
a higher voltage appears darker (optical OFF state) and so mirrors
shown dark in the above figure have are spatially advanced by
[(l)/2] relative to mirrors shown as white.
In the next section, simulation and experimental autocorrelation results
obtained from filters produced by two different techniques are
compared. The first technique is that
of coherent summation just described. The second technique goes under
the title of the `combined' technique, whereby both initial
and final data binarisation stages are carried out. Determination of
the characteristic phase in the `combined' technique is by
intensity weighted averaging of the (binarised) initial phases.
This more general technique is under investigation in this thesis and
the results obtained with coherent summation filters are used as a benchmark
for comparison. A detailed analysis of `combined' filters is
presented in the final section.
3 Simulation and Experimental Results
Appendix nine provides a brief description of the data capture procedure
for the experimental results. Note that great care was taken in the
central placement of the objects and in noting the fluctuations of laser
intensity. All experimental results of this chapter have been corrected
for laser beam fluctuations which in any case amounted to no more than
one or two percent of the mean laser output.
There are many ways of extracting information from the experimental
results as detailed in chapter seven. Figures 8.6 to 8.8 show the
experimental auto-correlations obtained using filters produced by coherent
summation, as recorded by CCD array camera.
Each page shows the result of a single final binarisation algorithm on the
coherently summed spectra over each mirror eg. figure 8.6 shows the results
of Horner final binarisation and figure 8.7 that of using a Psaltis
final binarisation algorithm. The letters beneath each plot denote the
target object used in each case.
For comparison, figures 8.9 to
8.11 show the experimental auto-correlations obtained using combined
algorithm generated filters. In each case the spectral data over each
mirror was initially binarised by either the Horner, Psaltis or Cottrell
algorithm, the characteristic phase was then formed by an intensity
weighted average of the binarised phases2, and a final mirror phase is
chosen by means of the Psaltis algorithm operating on the characteristic
phase of each mirror.
A data thresholding operation has been performed to enhance each
picture and offset the reduction in contrast which occurs during the hardcopy
process. The thresholding operation in question is identical to that
used by Hudson et al [73] and is depicted in figure 8.5.
Figure 8.5: Image Thresholding Operation
Such operations have also been performed by Flannery et al
[44] by optimisation of their video display controls where it
is realised that basic post-processing of this kind will usually be
required in experimental correlator systems. Appendix nine provides a
comparison of the hardcopys obtained both with and without the
thresholding operation.
Figure 8.6: Coherent Summation: Horner Final Binarisation
Figure 8.7: Coherent Summation: Psaltis Final Binarisation
Figure 8.8: Coherent Summation: Cottrel Final Binarisation
Figure 8.9: Combined Algorithms: Horner Initial Binarisation, Intensity
weighted Average
Fc, Psaltis Final Binarisation
Figure 8.10: Combined Algorithms: Psaltis Initial Binarisation, Intensity
weighted Average
Fc, Psaltis Final Binarisation
Figure 8.11: Combined Algorithms: Cottrel Initial Binarisation, Intensity
weighted Average
Fc, Psaltis Final Binarisation
Observations
- Firstly, something very much like an auto-correlation peak
is found close to the centre of each image for many of the filters.
Certainly no image resembles that of the corresponding target object which
would result from a constant phase assigned to each mirror of the SLM.
In many cases the image does not contain a single peak and certainly
suffers from a considerable amount of noise even after thresholding.
- Of those filters produced via coherent summation, Psaltis final
stage binarisation produces considerably brighter, more distinct
correlation peaks. Recall the conclusions of Downie et al [68]
which also found the Psaltis final binarisation algorithm to be the most
effective.
- Filters produced with the combined technique, encompassing
intensity weighted determination of the characteristic phase,
may be characterised by noting that the auto-correlation is worst
for Horner initial binarisation, very good on the whole for Psaltis
initial binarisation, and not quite as good for Cottrell initial binarisation.
Estimation of quality is based on visual inspection of peak intensity,
number of sub-peaks and the distribution of energy between all
peak-like features in each image.
- It is observed that of the two filter computation processes used,
equivalent filters3 produced broadly similar results.
During the course of the experiment the filters were cycled through the SLM
in turn for each object so that slight object misplacement effects were
identical for each. The change in
autocorrelation observed on the video monitor in switching between a
combined filter and the equivalent coherent summation filter was
extremely slight, although a definite change was registered.
- For both combined and coherent summation filters, the
results for the `I' target object are very poor. Speculation as to the
cause of this will be returned to shortly upon a quantitative analysis of these
results and comparison with simulation. Note, however, that the experiment
was carried out on a number of different objects to reduce the danger of
carrying all one's eggs in the same basket. For example, it may be that
the high symmetry of the object reduces the algorithm efficiency as
inspection of the `H' auto-correlations might also suggest. An analysis
based on the results from two such objects would not yield results of
general validity.
Figure 8.12 plots the experimental correlation peaks arising from
Horner, Psaltis and Cottrell initial binarisation, intensity weighted
Fc and Psaltis final binarisation for the `C' target object.
This object was chosen as it is neither the best nor the worst result
and shows the clear central peak obtained from the algorithm expected to
perform best, that of Psaltis initial binarisation, where no energy
would lie were the image merely an inverted letter `C'.
Figure 8.12: 3-D Contour Graphs of Auto-correlations using the `C'
Target. (Combined Algorithm Filter Results.)
3.1 Quantitative Analysis
After initial data capture by CCD and PC based framegrabber the data was
transferred to a DEC microvax computer for analysis. A comprehensive
FORTRAN data examination program used the voltage information from a photodiode
in the object plane (illuminated alongside the object) to correct the
data for laser beam fluctuations, as detailed in appendix nine. A search
was then instigated to locate the position of the maximum pixel value.
Assuming this to be an auto-correlation peak, the best fitting Gaussian
curve is fitted to the data and the percentage peak-to-total energy of
the data is also computed. Figures 8.6 to 8.11 show that this
assumption is not valid for filters using the Horner binarisation
algorithm, whether the filter be coherent summation based or combined
algorithm based. Therefore, the scatter plots of peak sharpness s to
percentage peak energy must be viewed in relation to these visual
correlation plots.
Figure 8.13 shows scatter graphs of percentage peak energy to peak
sharpness for every filter used in this project. Both simulation
results and experimental results are displayed for convenient
cross-comparison. The array camera field of view is equivalent to
@ 1.7mm on a side so that the CCD array just picks out one
replication4
of the correlation peak in the image plane (Recall the target objects
are @ 1.9 mm on a side). Note that the areas used to collect the
energy are therefore slightly different between simulation and
experiment, but that this should cause no difficulty if it is remembered
only to compare correlation peak characterisation parameters arising from
different filters within the simulation results or the
experimental results obtained, but not between the two.
Appendix nine provides an example of the
fitted Gaussian over an auto-correlation peak so that the reader may
judge the validity of s as a peak characterisation parameter.
Figure 8.13: Scatter Graphs of % Peak Energy vs Peak Sharpness
s
Scatter Graph Observations
Coherent Summation
- In both simulation and experiment similar trends are found.
Horner final binarisation generally has the lowest peak energy but tends
also to be the sharpest. From the CCD hardcopy figures however it is clear
that filters with Horner final binarisation produce no clear correlation
peak5 and such data should be excluded from
comparison with the other final binarisation algorithms.
- Although considerable overlap exists, it is found that Cottrel
final binarisation produces a marginally less bright peak than Psaltis
final binarisation for both simulation and experimental results.
Combined Technique
- The general observations made for coherent summation also apply
for these results. However, the distinction between results from
filters using different initial data binarisation is greater than for
coherent summation.
- It is evident from the graphs that Psaltis
initial binarisation produces, on the whole, a brighter peak of
approximately equal sharpness to Cottrell initial binarisation.
A final analysis of the data is effected by performing a one-to-one
comparison of the percentage peak energy between simulation and
experiment, for each object, as shown in figure 8.14.
Figure 8.14: % Peak Energy: Direct Comparison Theory and Experiment
This method of data analysis is more direct than the scatter graph
approach and, if the Horner data is not compared with the other
binarisation algorithms, shows solid agreement for the combined technique
between simulation and
experiment. In all cases the Psaltis initial binarisation resulted in an
equal or, more usually, brighter correlation peak than that of Cottrell
initial binarisation. If one does choose to look at the results of
Horner initial binarisation one can only say that both simulation and
experiment agree the correlation resulting is extremely poor. Note
again that this is not a reflection of the actual Horner
algorithm merit, but arises from the nature of the combined
process as previously detailed.
For coherent summation, in simulations
the percentage peak energy for the object `I' is
low, but comparable with the lowest Psaltis and Cottrell percentage peak
energies of other target objects. The significantly lower peak energy
of the `I' experimental results relative to the other target objects might
suggest the actual fabricated object was somehow imperfect rather than
an algorithmic weakness. For the intensity weighted Fc
algorithm, simulations find the percentage peak energy lies
approximately midway between the highest and lowest of all simulations
(again discounting Horner initial binarisation) which again suggests the
fabricated test object suffered some imperfection during manufacture due
to the large experimental discrepancy relative to the other targets.
The results of coherent summation are again in strong agreement between
simulation and experiment. In only one case, that of the `STAR' object,
does the simulation actually predict that Cottrell final binarisation
produces a brighter peak than that of Psaltis. Experimentally the
Psaltis final binarisation is a clear winner for this object however.
If one accepts the possibility that slight object scaling errors during
manufacture, together with different random phase noise for each target
object manufactured, may cause object dependent differences between
simulation and experiment then this result should not cause too much
concern. On the contrary, it is the degree to which simulation and
experiment are in agreement which is the most significant result.
3.2 Discrimination Ability
Figure 8.15 displays the first correlation experment results ever
performed with the 16×16 SLM in FIB mode by Ranshaw.
Ranshaw's experiment used a square target object, the phase of the
Fourier Transform being already binary and avoiding the need for
extensive binarisation algorithm research.
Figure 8.15: Auto and Cross-correlation Results of Ranshaw
On the left of figure 8.15 is
the autocorrelation of the square target object and on the right the
cross-correlation with a letter `D'. The cross-correlation shows
significant reduction in the replication intensities, even though they
may only loosely be described as `peaks' in the auto-correlation. The
front face reflection is clearly visible in both photographs.
It was these photographs which prompted the Spatial Light Modulator research
part of this project, and they are included to illustrate the potential
improvement in this field by intensive algorithm research and improved
device fabrication techniques.
A limited experimental investigation into the discrimination ability of
the algorithms used on the 16×16 SLM was conducted. The best
filter for the letter `A' (Psaltis
initial and final binarisation and intensity weighted determination of
the characteristic phase) was selected whilst the target object used was
varied. Figure 8.16 depicts the
intensity distributions in the image plane for the `B', `C'
and `CND' target objects with image plane thresholding.
Figure 8.16: Discrimination for `B', `C' and `CND' Target Objects.
Combined algorithm filter for the letter `A'.
These results were typical of those generally obtained for which
simulations show contain a percentage peak energy of at most 3%.
Notice, however, that the arms of the `CND' symbol result in a strong
cross-correlation with the letter `A' in figure 8.16..
Study into filter discrimination is in itself a substantial task and was
was not pursued further in this project. The experimental results do,
however, show that the filter algorithms chosen for use with the
16×16 SLM are both capable of generating both good object
recognition characteristics and discrimination ability given the limited
space-bandwidth product of the device.
4 Analysis of Combined Technique
4.1 Initial Phase Binarisation
Initial binarisation of Fij naturally has a strong influence on the
characteristic phase produced from such data. In this subsection it will
be assertained that the characteristic phase resulting from an intensity
weighted average of the binarised initial data Fij
does, in fact, strongly represent the results of the initial
binarisation operation. Filter performance, it shall be shown, may in
part be speculated upon by consideration of the symmetry of the filter
about the origin.
It must be stressed that the analysis performed in this section relates
only to binarisation algorithms which use 0 and p radians as the
poles of the algorithm. The analysis presented is invalid should one
choose poles such as -[(p)/2] and +[(p)/2] and therefore
cannot be generalised to the case of general pole values. However, for
the specific poles chosen (which are as valid as any) the analysis
provides a detailed explanation of the filter characteristics and
provides an explanation as to the differing performance of filters
produced using the combined technique.
Over an arbitrary SLM mirror, let the subset of initial phase
information set to 0 be denoted as set `A', and those pixels set to
p radians be denoted as set `B'. The characteristic phase is most
readily formed by an intensity weighted average of the binarised initial
where Iij denotes the intensity of one pixel within the 9×9
DFT sub-array over the mirror in question. In order to proceed one
must make certain approximations about the light distribution over the
mirror, the first of which concerns the intensity. It is proposed that the
intensity be replaced by an average value over that mirror so that
Iij®[`I]. This approximation is most easily validated
in cases where the scale of Fourier Transform is such that the frequency range
covered by any mirror is relatively small, so that the intensity variations
over a mirror will also, hopefully, be small. If nA and nB denote the number of points in sets A and B
respectively the characteristic phase reduces to
Therefore, if initial data binarisation is used followed by an intensity
weighted average of the (binarised) phases, one obtains a characteristic
phase which is linearly proportional to the percentage of initial data
set to p. Whatever the criteria used by the initial binarisation
algorithm to set phase to p, it might be argued that the
characteristic phase should reflect the results of this algorithm and, with
the procedure outlined here, just such a characteristic phase occurs.
Note that the approximation of a slowly varying intensity over the
mirror is a sufficient though not a necessary condition for replacing
Iij with a mean value. If, for instance, the intensity changes rapidly
over the mirror so that
then, if dIij is a function with a mean value close to zero,
equation 8.9 remains unchanged. Sight must not be lost on the fact that
the approximations made here are but guides through the filter
calculation process.
Distribution of Fij
The second approximation of this analysis relates to determination of
nA and nB of equation 8.9, which in turn will determine the
plausibility of the explanation of the combined technique.
What would happen if nB=0, for instance ? From chapter five it
will be recalled that the importance of the phase information of a spectrum
far outweighs the amplitude information and it is also known that the average
amplitude spectrum of an input scene to an optical system is accurately
described by a Gaussian function. If the amplitude spectra of, say, 100
typical input scenes were replaced with a common suitable Gaussian function
the phase information present in the frequency plane would be enough to
allow a recognizable image of each object in the image plane.
This observation relies on the fact that the uniqueness of each object is
related to the uniqueness of its associated phase spectrum, which may
sound like stating the obvious. However, given that the phase spectra
is unique to each object, it is a logical conclusion that if one were to
pick the phase of an arbitrary spatial frequency of an arbitrary object
it could lie anywhere on the range -p to +p i.e it is a random
variable with a uniform probability distribution6.
If this is true, then
any threshold angle b chosen for an initial data binarisation
should set approximately half of the phases to 0 and half to p.
Table 8.2 lists the
percentage of Fij set to p radians for the SLM as a whole,
for just two of the seven target objects used.
| Object | Initial Binarisation | %®p |
| A | Horner at al | 49.80% |
| Psaltis et al | 49.86% |
| Cottrell et al | 49.82% |
| C | Horner et al | 49.80% |
| Psaltis et al | 49.79% |
| Cottrell at al | 49.76% |
Table 2: Percentage of Fij set to p for `A' and `C' Target Objects
In fact, every target object showed identical results to those of
this table. Percentages are always on the lower side of 50% due to a
programming decision which maps all phases which actually lie on the
threshold line to a phase of zero. Based on the results such as those of
table 8.2, the assumption of a rather uniform distribution of Fij
would seem valid for the target objects used in this project and valid
in general for almost every (non-symmetric) target object likely to be
used in a realistic correlator. The conclusion is therefore that
nB is most unlikely to be zero for all practical target objects.
In relation to the problem at hand, both sets A and B contain
approximately half of the total number of data pixels over any one mirror
which leads to the result that
Therefore, for the seven target objects used in this project, one might
expect the characteristic phases of each mirror of the SLM to be closely
distributed about the value [(p)/2].
Effects of the Characteristic Phase
Recall the three threshold angles b introduced so far. If the
characteristic phases of the SLM mirrors are closely distributed about
[(p)/2] then only the threshold line of Psaltis et al will
divide this distribution, as figure 8.17 depicts.
Figure 8.17: Characteristic Phase Distribution formed by Initial Binarisation and Intensity Averaging.
The threshold lines of Horner et al and Cottrell et al have all
characteristic phases lying in just one of their two distinguishable
regions and thus would produce filters where Fm does not vary
from mirror to mirror, as has been verified to occur in simulations
carried out for this project. To recap on the filter formation process
then,
- On any mirror the 9×9 DFT sub-array is phase binarised by
any of the three algorithms used in this chapter.
- An intensity weighted average of the binarised phases
leads to a characteristic phase Fc which is a fraction of
p.
- The characteristic phase approaches p in proportion to the
initial fraction of data which was binarised to p. Using the
assumption of an initially almost uniform distribution of Fij on
the range -p,+p one must conclude that Fc is clustered
around the value [(p)/2].
- Upon final binarisation of Fc only the algorithm of Psaltis
et al can produce a filter with mirror phases Fm which are not
all identical due to the tight clustering described above. If
Fc is such that it is approximately evenly distributed about
[(p)/2] then @ 50% of the final filter has mirror phases
Fm set to 0 and @ 50% to p.
Table 8.3 shows the extent to which this explanation is plausible by listing the
percentages of mirrors which have a final phase Fm set to p.
| Initial Binarisation | A | B | C | CND | H | I | STAR |
| Horner et al | 58.7 | 48.4 | 48.8 | 47.3 | 48.0 | 49.2 | 47.3 |
| Psaltis et al | 49.2 | 44.4 | 54.3 | 45.7 | 43.4 | 41.4 | 51.6 |
| Cottrell et al | 48.8 | 50.0 | 49.6 | 49.6 | 48.6 | 43.8 | 49.2 |
Table 3: Percentage of Mirrors set to p vs Initial Binarisation (Intensity Weighted Average then Psaltis Final Binarisation).
So far, no criterion has been established to predict how well each filter
will perform as a binary phase-only filter for optical correlation work.
It is the analysis of filter symmetry which provides clues as to filter
performance, as shall now be shown.
4.2 Filter Symmetry
Consider a single mirror M of the SLM which is represented by a
9×9 sub-array of DFT points, each with intensity Iij and
phase Fij. The symmetry of the filter as a whole may be
calculated by considering also the conjugate mirror to M which is
M*. By definition of the Fourier Transform, a point on M
with intensity Iij and phase Fij has a conjugate point on
M* with equal amplitude but conjugate phase -Fij.
Figure 8.18 depicts on an Argand diagram the phase of four data points on
both M and M*, where it is noted that phase conjugation is
equivalent to reflection about the REAL axis.
Figure 8.18: Conjugate Mirrors in the Fourier Plane.
Recall that set B denotes those points which are mapped to a phase of
p (and is used in determining Fc), and set A those points mapped
to a phase of 0. Table 8.4 defines these sets for the three
binarisation algorithms of this chapter for both M and M*,
taking the phases shown in the figure for example.
| Binarisation Algorithm | Mirror M | Mirror M* |
| Set A | Set B | Set A | Set B |
| Horner et al | 1,4 | 2,3 | 2,3 | 1,4 |
| Psaltis et al | 1,2 | 3,4 | 1,2 | 3,4 |
| Cottrell et al | 1,2,3 | 4 | 1,2,4 | 3 |
Table 4: Mapping Sets for the Binarisation Algorithms of this Chapter.
Horner Initial Binarisation
Consider the phase mapping of the Horner algorithm on the data points of
both mirrors. From table 8.4 it is apparent that set B for mirror M is
identical to set A for mirror M* and vice versa.
As only the information from set B takes part in
the formation of the characteristic phase for each mirror it is further
apparent that in its calculation Fc(M*) uses a mutually
exclusive data set than does Fc(M). Thus, from equation 8.9, the
characteristic phase of one mirror has absolutely no relation at all with the
characteristic phase of the conjugate mirror, and one might expect a
corresponding filter performance of dubious quality to result.
It is not immediately apparent that the fact that mutually exclusive data sets
are used for conjugate mirrors results in an filter which is completely
anti-symmetric. For mirror M, the characteristic phase is found from
and, if A and B again refer to the sets of mirror M, the
characteristic phase of mirror M* is
However, one may write
|
|
å
A
|
Iij = |
å
A+B
|
Iij - |
å
B
|
Iij |
| (14) |
From earlier work it is known that Fc(M) is closely distributed
about [(p)/2] so that if one writes
then
Therefore, the characteristic phases of conjugate mirrors lie on
opposite sides of the imaginary axis and will be binarised to different
poles by the Psaltis final binarisation algorithm. An entirely
anti-symmetric filter must then result no matter what target object is
used.
Psaltis Initial Binarisation
Now consider the effects of using the initial binarisation of Psaltis et
al where the threshold line coincides with the imaginary axis of figure
8.18. Set B now comprises all points to the left of the imaginary axis and
as conjugation is effected by reflection in the REAL axis, set B is
identical for both M and M*. Therefore as the characteristic
phases for a mirror and its conjugate are identical, the final
binarisation algorithm must give an identical output phase for both
mirrors and the filter will be 100% symmetrical about the origin.
Only half of the information contained in each mirror (those points with
phases to the left of the Imaginary axis in the Argand plane) is used to
form the characteristic phase but this information is the same for
conjugate mirrors.
Cottrell Initial Binarisation
Assuming a uniform distribution of phases Fij, the threshold
line of Cottrell et al results in set B of mirror M* using
50% of the information used to compute the characteristic phase
of mirror M, and 50% of previously unused data. Figure 8.19 aids this
discussion. Initial binarisation using the algorithm of Cottrell et al followed by
an intensity weighted average of the binarised phases is a procedure
which therefore lies midway between the previous two initial
binarisation algorithms in as much as analysis of the characteristic
phase is concerned.
Figure 8.19: Cottrell Initial Binarisation
twice per cycle of the lens cut-off frequency.
As such, one might expect the filter to be neither completely symmetric
nor completely anti-symmetric after the final Psaltis binarisation
algorithm is applied to the characteristic phases. Further, if the
filter resulting from Horner initial binarisation is suspected to give
poor results, and that arising from Psaltis initial binarisation rather
better results, one might expect the filter in question to lie in
between these two extremes in terms of correlation quality.
It should be
noted that expectations expounded here of filter performance are based solely
on the dependence of Fc on the choice of data sets used in its
calculation. However, the approximations and reasoning used so far
have led to an in depth analysis of filter symmetry expected to occur
from the filter creation process framework at the beginning of this
section. Figure 8.20 shows several filters calculated using this combined
algorithm technique, these being typical of the results obtained for all
test objects used. For the `star' target object the filter pattern is
also shown as it appears on the 16×16 SLM, in amplitude
mode. Remember that the top row and left column are not required for symmetric
placing of the SLM in the frequency plane, and should be excluded from
visual estimations of symmetry. The same mirror phase calculation is
performed for these mirrors (from a reduced data set) nonetheless. Note
the similarity between these filters and those as calculated by
coherent summation.
Figure 8.20: Filters for Target Objects `C' and `STAR'.
4.3 The Effect of Sample Number.
With the large variety of pixellated SLMs available, coupled with the
free choice of f-number used in the optical processor, one may arrive
at a very large number of samples being taken over each mirror if one
samples at the PSR determined interval as described in chapter 7. This has
important consequences for the combined technique which are now
spelled out in detail.
One must recall a result of the `central limit theorem', which
states that if a large population has mean mp and standard deviation
sp then the mean of a set of sample populations of size n is
follows a Gaussian distribution with mean mp and standard
deviation ss given by
One can imagine performing a very large DFT, say 1024×1024, the
statistics of which may be taken to very closely approximate those of the
population referred to above, should they be required explicitly.
It has been conjectured in this chapter
that the phases Fij follow an almost uniform distribution on the
range -p to +p, so that the population mean
of the phase information is expected to be almost zero7.
This has important
consequences which bear directly on the filter creation process as
described above.
- The mean phases of each mirror are Gaussianly
distributed about zero with a standard deviation of
[(sp)/(Ön)] where n denotes the number of pixels in
each mirror.
- As the number of DFT pixels n chosen to represent the
spectrum over each mirror increases, the mean phase of the mirror
approaches zero. This is equivalent to saying that the percentage of
data Fij lying on either side of the initial binarisation line
approaches the same
value of 50%, so that in the approximation of equation 8.9 the
resulting distribution of the characteristic phases Fc becomes
more tightly distributed about [(p)/2]. In this project, the
distribution about [(p)/2] is tight enough that nowhere does the
Cottrell binarisation line intersect it, and so only the Psaltis
threshold line, with b = [(p)/2] can bisect the distribution
to form a final filter with distinguishable mirror phases. This is
illustrated in figure 8.21.
Figure 8.21: Intersection of
Fc Distribution by Threshold Line.
- Mirrors represented by very small DFT sub-arrays, however, have a
very much wider distribution of Fc (by equation 8.10) due to the wider
difference between nA and nB. Consequently, after initial
binarisation and intensity averaging have been performed, the range of
threshold angles which will intersect the distribution of Fc is
much larger. The minimum array size nmin for which the Horner
binarisation line (b = 0) can still intersect the distribution will
depend on sp by equation 8.18. SLMs of larger array size than
the 16×16 used in this project will, for the same size of Fourier
Transform (e.g. 256×256) have a reduced density of data pixels
and filters fabricated with a wider range of final binarisation lines
than used in this project may be investigated.
5 Chapter Summary
The low space-bandwidth product of the 16×16 phase modulating SLM
demands that appropriate effort be made to devise quality auto-correlation
algorithms for its appraisal as an binary, phase-only optical
correlator. In this chapter several phase binarisation
algorithms commonly used in correlator systems have been introduced and
briefly analysed as to their dependence on target object symmetry, for
instance. A general
framework suitable for the discussion of any filter computation
procedure has been introduced, and the comparison of two such techniques
formed the basis of the experimental work of this chapter.
The `Combined Technique' devised in this project has been
shown in computer simulations to produce slightly better auto-correlations
than the technique of coherent summation, although there is little to choose
between the two experimentally. Although a complete analysis of the CA
technique has been performed, it is not the intention of this thesis to
propose that this replace the procedure of coherent summation in
auto-correlation work. Indeed, the CA technique has been the result of
allowing complete freedom of choice in the overall framework of filter
computation and it is remarkable that such an effective algorithm
should have arisen from this.
In summary, it has been found that the Psaltis final binarisation
algorithm applied to coherently summed spectral data over each mirror
produces the best auto-correlation results (as judged by peak sharpness
and peak energy) for the coherent summation filters. This is in
agreement with the results of Downie et al [68] described in
section 8.1.8. It is also found that if one initially binarises the
spectral phase data according to the Psaltis algorithm, performs an
intensity weighted average of these binarised phases to obtain a
characteristic phase Fc for each mirror, and finally binarises
Fc again according to Psaltis, one obtains a filter at least as
good, if not better, as that obtained by coherent summation.
The auto-correlations depicted in figures 8.7 and 8.10
show the results of the two best filter computation algorithms.
Footnotes:
1 In the summer of 1991 the experimental results of this
chapter were presented at the annual conference of the Society of
Photo-Instrumentation Engineers, San Diego, California. The
corresponding paper was based on an unpublished paper by Ranshaw but all
experimental procedures and results are due to the author of this
thesis. The paper, entitled `Optical correlation using a phase-only
liquid crystal over silicon spatial light modulator', may be found towards
the end of this thesis.
2A detailed analysis of
the combined algorithm technique follows the experimental results
of this chapter.
3Filters are termed
equivalent if they share the same binarisation algorithm, whether it be
used as an initial stage (intensity weighted Fc) or final stage
(coherent summation).
4The replication chosen was the first one off-axis to
avoid the glare from the front-face reflection of the SLM which produced
a constant intensity inverted image of the object.
5The `B' object is an
exception to this however.
6
A symmetric object function will, however, have an entirely REAL
Fourier Transform and as such the phases Fij over any particular
mirror must be either 0 or p (cos(Fij)=±1 when
Fij is 0 or p.) However, the Fourier synthesis
of an arbitrary input function will generally require a much wider
distribution of spatial frequency offsets (phase) than this and, with
the exception of symmetric functions and several physically unlikely
target object functions, may be approximated as a uniformly
distributed random variable on the range -p,+p.
7This
will not be true for objects exhibiting a high degree of symmetry however.
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