Chapter 4
Optical Correlation
Introduction
Chapters one, two and three of this thesis are concerned with elementary
phase filtering operations on the spatial frequency spectra of a class
of objects exhibiting spatial variations in phase only. Within each type
of phase filtering operation (Schlieren, Dark Ground, Phase Contrast etc.)
the filtering operation is independent of the particular
phase object characteristics. Commonly, the purpose of the filter is
to render the phase structure of the object visible as an intensity
variation as described in the last chapter.
The second half of this thesis concentrates on a class of phase filters
which allow much more intricate operations to be performed on the object
spectra, and are subsequently more complicated in form. Specifically,
attention is turned to the subject of spatial filters which allow certain
pattern recognition operations to be performed. The object is usually an
amplitude-only object so that whereas previously the complexity
of the problem lay in the object structure,
now the complexity lies in the structure of the filter.
Pattern Recognition
The purpose of this chapter is to introduce the general terminology
of two dimensional spatial filtering, and in particular, pattern recognition
filters. The construction and principles of operation of a two dimensional
reconfigureable Fourier plane filter (a Spatial Light Modulator or SLM) is
detailed in chapters five and six. Specifically, optical pattern recognition
experiments have been performed with this device, and form the subject
of chapters seven and eight. As such, this chapter lays an essential framework
within which the subsequent work of this project should be set against.
In the first section, the concept of two dimensional phase
filters with a frequency dependent complex transmittance is introduced.
Spatial filters which allow pattern recognition tasks to be
performed optically, such as the classical Vander Lugt matched spatial
filter, are defined. Examples of practical modern day filters (SLMs)
are given in section two together with a mathematical background to such
effects as filter pixellation. In section three a major constraint upon
many spatial light modulators, that of single parameter modulation, is
discussed with relevance to its effect on phase-only correlation filters.
Finally, section four discusses the merit of alternative correlation techniques
and their relevance to this project assessed.
1 Two Dimensional Spatial Filtering
The alteration of the spatial frequency spectrum of an object is
known as `spatial filtering'. Appendix one provides a brief review of the
standard optical processing bench on which
such operations are usually performed.
A single spatial frequency present in the object has both an associated
amplitude and
phase, which is to say a number describing the
spatial offset of that component from the origin of the object plane. In
the frequency plane, the amplitude information of a single frequency is
represented by the Point Spread Function of the transform lens, which is
usually a sharply peaked function, and the phase as a temporal
delay of the light field at that point.
1.1 Mathematics of Correlation Filters
[
Update 2006:
An excellent external link: Convolution of Two Functions (PDF)
from Will Hossack's home page, my PhD supervisor. ]
One particular optical processing operation has been studied
extensively, that of optical correlation. Consider the spatial
frequency spectrum of an object g(x,y) which is conventionally denoted
by G(nx,ny). The spectrum can be separated into a product of
two functions, one entirely REAL and describing the amplitude of
G(nx,ny) and the other complex, describing the phase
distribution so that
|
G(nx,ny) = |G(nx,ny)| ei F(nx,ny) |
| (1) |
It is well known [28] that a filter H(nx,ny) of form
|
H(nx,ny) = |G(nx,ny)| e-i F(nx,ny) |
| (2) |
will result in an image plane amplitude distribution which is the
auto-correlation of g(x,y), which will be defined
shortly (equation 4.4). Such a filter is
known as a `matched' filter, being matched to one specific object
function, in this case g(x,y). If the filter
amplitude and phase correspond to amplitude and (conjugate) phase of a
different function p(x,y) then the image plane contains the
cross-correlation of the functions g(x,y) and
p(x,y), which again will be rigorously defined shortly.
These
results are at the very heart of most pattern recognition filters, for if
it is desired that a particular object is identified in an input scene
to a processor, one way of detecting the presence of that object is to
use a spatial filter having identical amplitude and conjugate phase to
the object in the frequency plane. The function g(x,y) is called the
`target' object. Suppose an input scene to the correlator contains several
different, spatially distinct objects fi(x,y).
The image plane will contain the
cross-correlations of the target g(x,y) with the input functions
fi(x,y). Should one of the fi(x,y) prove identical to the target
function then the image plane will contain the auto-correlation of
g(x,y) instead. Mathematically, the cross-correlation of
two functions g(x,y) and f(x,y) is
|
cgf(Dx,Dy) = |
ó
õ
|
|
ó
õ
|
+¥
-¥
|
g( (x-Dx),(y-Dy) ) f*( x,y ) dx dy |
| (3) |
which is the integral of the area of overlap of the two
functions
1.
after they are spatially offset by amounts
Dx and
Dy in
the x and y directions respectively.
It is usual to normalise this function by dividing by the zero ordinate
of the autocorrelation of g(x,y), defined as
|
cgg(0,0) = |
ó
õ
|
|
ó
õ
|
+¥
-¥
|
g( x,y ) g*( x,y ) dx dy |
| (4) |
Using this convention, the cross-correlation c
gf has a maximum value
of unity occuring when f(x,y)=g(x,y).
If the power P contained in two functions g(x,y) and f(x,y) is
identical, power defined in the usual way as
|
P = |
ó
õ
|
|
ó
õ
|
+¥
-¥
|
g2(x,y) dx dy |
| (5) |
then it can be shown that the normalised auto-correlation c
gg of a
function is greatest at c
gg(0,0).
For functions of identical power, the central value of the autocorrelation can be
shown to always exceed any cross-correlation value.
As the auto-correlation function is generally peaked at the origin, the
brighter target object auto-correlation peak may be picked out from the dimmer
cross-correlation peaks as illustrated in figure 4.1.
Figure 4.1: Optical Correlation
1.2 Vander Lugt Matched Filter
The first optical implementation of matched spatial filtering was
carried out holographically by Vander Lugt [
29]. The spectrum of a
target object g(x,y) is caused to interfere with an off axis reference beam
in the Fourier Plane and the resulting interference pattern recorder on
photographic emulsion.
Once the emulsion is developed, it may be reinserted into the Fourier
plane to act as a matched spatial filter.
If the input object is
replaced by another of transmittance f(x,y), illuminated by a plane wave
and the reference beam is removed,
it can be shown [
30] that three beams emerge from
the far side of the holographic filter. One beam emerges at zero angle
to the optical system and is focussed to the origin of the image plane, but
contains no extractable information. Two further beams emerge at equal but
opposite angles from the optical axis of the system and are again focussed to
the image plane. One contains the
convolution of the target
function f(x,y) with the object function g(x,y) and the other contains the
cross-correlation of the two functions. Until the advent of fast,
reconfigurable spatial filters (to be described in section 4.2) much
work on optical correlation utilised this experimental arrangement. See,
for instance, the work of Casasent & Furman [
31].
Although the matched spatial filter produces the highest
signal-to-noise ratio (SNR, to be discussed shortly) in the image
plane [
32], there exist several disadvantages to its use in practice.
Section 4.2 lists some of the drawbacks of this technique and describes
alternative methods of implementing correlation optically. All
of these, however, are modifications of the basic Vander Lugt matched
spatial filter described here.
Figure 4.2: Vander Lugt Holographic Filter
Improved Discrimination
For many objects however the difference in brightness between the
autocorrelation and cross-correlation arising from use of a matched
spatial filter is quite small. It is thus
difficult to discriminate between a
recognised target by the
brightness of the correlation peak compared with the surrounding
cross-correlation peaks. Also, the peak need not necessarily be very
sharp, so that even the auto-correlation may be a broad, dim function.
It has been shown however that discrimination is greatly improved if the
input scene has undergone an edge enhancing operation before being used
as input to the correlator (as commonly performed by the subtraction of
the mean value of the object signal throughout the function, for example).
The low spatial frequencies present in the object spectrum, which `fill
in' large, fairly uniform areas of the object, are suppressed in
such an operation in deference to the high spatial frequencies which
define the edges of the object. Research performed by Horner and
Bartelt [33] also indicates that binarisation of the input scene
combined with a filter matched to the binary object, rather than a
continuous tone object, results in a brighter correlation peak value and
an improved signal to noise ratio (SNR). The precise specification of
what the SNR measures, together with a parameter known as the optical
efficiency hH, are defined as follows:
- The SNR used was defined as the ratio of the peak correlation value
to the rms noise outside of a 50% of peak threshold level.
- HORNER efficiency hH is defined as the ratio of the energy
within the 50% of peak threshold level to the total energy falling on the
output plane of the correlator.
Table 4.1 compares the correlation peak intensity (as a fraction of that
obtained by a classical matched filter), SNR and HORNER efficiency for
several combinations of input and filter type. Subscripts CTS and
BIN refer to the object type from which the filter was made. These
results are from Horner and Bartelt, where the input scene was a doll's
face, all results being computer simulations.
| Input Object | | FilterCTS | FilterBIN |
|
|
| Continuous | SNR | 4.1 | 4.1 |
| R02 | 1.0 | 1.3 |
| hH | 6.3% | 3.1% |
| Binary | SNR | 4.1 | 7.1 |
| R02 | 1.1 | 3.4 |
| hH | 1.5% | 0.7% |
Table 1: Matched Filter: Three Parameters Affected by Input / Filter Type
Thus the highest SNR ratio and intensity of autocorrelation peak occur
when a filter is made from a binarised input scene and the same binary
object is presented at the input. In this case however optical efficiency
decreases because although the peak is made sharper and higher, less energy
goes into the peak region as a whole than it does with a classical
matched filter using a continuous object.
In section 4.3.3, the same three parameters used here are again
compared but for a filter which is matched only to the phase
of the target object. It will be argued that a phase-only filter should
give much improved results and a similar table to that shown above is
presented, again quoted from Horner and Bartelt, to quantify this
improvement.
2 Practical Filters
In practice, the matched spatial filters of Vander Lugt suffer from the
severe drawback of lengthy preparation time and the need for a
physically new filter for each target object. A significant increase in the
speed of the whole process may be achieved by encoding several matched
filters on a single piece of film. Such a filter is known as a
`frequency-multiplexed' or `composite' filter [34], [35].
To avoid overlapping of
the autocorrelations in the image plane, the
holographically stored matched spectra of a number of different input
objects are recorded using a plane
wave reference beam with an object dependent angle to the optical axis.
An important strength of this technique is the ability to
store, on a single filter, a number of filters matched to both scaled
and rotated versions of a single target object. Using computer generated
holograms, this field has been investigated by Leger and Lee [36].
A practical use of an optical correlator has been suggested by Johnson
[37]
which serves to highlight this problem. If a small video camera views
the entrance to a doorway it can send signals which, if suitably
displayed, can be used as the input scene to a compact optical
processor (The means whereby this might be effected are discussed shortly).
The frequency spectrum of the input scene can then be filtered so as to
produce a correlation peak whenever a particular face appears at the
input to the system. This has obvious security implications. Due to the
large number of variations in subject distance from camera, aspect
ratios of the face and facial expressions, it would be required to scan
through a number of filters in rapid succession for each time selected
input scene, each filter pre-stored and calculated to recognise the
face at various distances, etc. Holographic filters could be used
if mechanically replaced rapidly enough but owing to the large number
envisaged this scheme would be severely limited in the number of
different people it could `recognise', even if frequency-multiplexing
were used.
Spatial Light Modulators
It would clearly be advantageous to update the filter pattern without
physical removal of the filter. To date, a large number of devices
exist which can act as reprogrammable spatial filters, and are known as
`Spatial Light Modulators' or `SLM's. In general, a spatial light modulator
is a device which may be used to impress information onto a wavefront,
so that the information may represent an input image scene as an
amplitude variation or a spatial filter as phase variations, for
example.
Actual spatial light modulators are constrained by
many factors to perform modulation of either amplitude or phase,
but not both together. However, it is a frequent occurence for a phase
modulator to also, sometimes unavoidably, introduce a small degree of
amplitude modulation as well and vice versa for an amplitude modulator.
These are aberrations of the filter and can usually be tolerated if
small enough, but have led to some authors referring to specific SLMs as
either `phase-mostly' or `amplitude-mostly' modulators [38],
[39].
A further consideration in physical devices is that the modulation parameter
frequently takes one or other of only two values, and such
devices are said to have a binary mode of operation. For amplitude-only
modulators this is usually adequate if it is desired either to pass or
block a group of spatial frequencies, for example. Of far more concern
is the effect this has on phase modulating filters. This forms the
subject of section 4.3.3.
SLMs are subclassified as either pixellated, a necessity arising from
the need to assign a stored memory location value to a particular
position on the surface of the modulator, memory locations being
discrete, or non-pixellated. The process of loading the information
onto the SLM is known as `addressing'. Pixellated devices are usually
electrically addressed whereas non-pixellated devices are
commonly addressed optically, though this is a generalisation.
4.1 Optical Addressing
As an example of what is meant by optical addressing, it will prove
instructive to examine the operation of an SLM known as the `Hughes
Liquid Crystal Light Valve' [40],
[41], the operation of
which is depicted in figure 4.3.
Figure 4.3: Operation of the Hughes LC Light Valve SLM
In this device, an image is focussed onto the front face of the SLM
using an incoherent beam of light. A photoconductive layer of cadmium
sulphide behind the glass substrate experiences a decrease in resistance
as light falls on the surface, causing the voltage dropped across a
layer of nematic liquid crystal to decrease. The light modulating effect
is actually quite involved and uses a phenomena known as the hybrid
field effect, but the specific effect is not of concern here. A coherent beam
of light is reflected off the back face of the device and as it travels
through the liquid crystal layer experiences a spatially varying optical
effect which is primarily one of polarisation guidance in this case. By
this means, the information of an incoherent signal is imprinted onto the
wavefront of a coherent signal which can be used, for example, as the
input stage to an optical processor.
4.2 Pixellated Devices
Whilst optically addressed devices are suitable for incoherent to
coherent conversion in the input plane to an optical processor, they are
of limited practical use as frequency filters with one notable exception
- the Joint Transform correlator (See section 4.4).
Most filters are calculated computationally and pixellation of the devices
allows interfacing of the SLM to a computer for filter update.
Consequently, most SLMs are pixellated devices requiring a discrete number of
values to describe the modulation characteristics of each pixel. Figure 4.4
illustrates the general characteristics of a pixellated spatial light
modulator in the frequency plane.
Figure 4.4: Characteristics of a Pixellated Spatial Light Modulator
The pixels may either be transparent (a transmissive SLM) or reflecting
(reflection mode SLM), in which case the reflected beam is isolated by
using a beamsplitter placed in front of the device.
The finite extent of the filter results in a maximum frequency passed by
the device, each pixel covering a range of spatial frequencies which it
ideally modulates in an identical fashion. If `FL' denotes the
focal length of the lenses used in the processor, `xT' the physical
distance along the x-axis of the frequency plane then the
frequency-distance relationship has been given as
The highest spatial frequency passed by the SLM with a width `W' is
then
If the pixels are separated by a physical distance `dp', the
corresponding separation in frequency can be found. According to the
`sampling theorem' [] a sampling in frequency space at interval
dn results in a replicated image of the object, each
replication separated by a distance [ 1/(2dn)]. To avoid
overlapping images the object width OW must not exceed this value,
so that
and
which in a pixellated device is merely the number of pixels along any
one axis of the SLM, generally assumed here to be square. This result
is known as the `space-bandwidth product' (or SBP), and states that a gain in
image resolution (by passing higher frequencies) can be obtained but
only at the expense of the maximum object size usable as input, if
aliasing in the image plane is to be avoided.
This result shall be of central importance in chapter 7 and shall be
discussed with reference to a specific optical processing system, and is
noted here as a figure of merit of an SLM. Notably, the greater the
number of pixels the better becomes the SBP becomes allowing either
improved image resolution or larger object dimensions for the same
degree of resolution.
4.3 Electrical Addressing
One commercially available electrically addressed SLM
is known as the Litton magneto-optic SLM or
LIGHT-MOD [42], [43] as it is more commonly known, utilises the effects of Faraday
rotation as a light modulation mechanism. The LIGHT-MOD is a 48×48
pixellated transmissive array2, the pixels defined by the areas of intersect of
a network of `drive lines' in which currents are caused to flow. By
suitably addressing the device, the resulting currents in the drive line
network cause the magnetisation vector in the region of a pixel to lie
in one of two possible directions. Consequently, a linearly polarised
light field normally incident on the array has its polarisation vector
rotated in either a positive or negative sense according to the magnetic
field on each pixel by the Faraday effect. By a suitable arrangement of
polarisers before and after the SLM either binary amplitude or binary
phase modulation may be achieved.
This device has been used extensively in experimental studies, most
notably by Flannery et al [44] in 1986 where the results of an initial
investigation into optical binary phase-only filtering were published.
`Excellent agreement' between computer simulation and the experimental
correlations was found and a photograph was presented showing two bright,
distinct spots of light in the output plane of the optical correlation
bench. Further experimental correlation results from a very thorough
investigation were published in 1988 where an impressive agreement with
predictions from simulations was obtained [45].
The space-bandwidth product (as determined by the number of pixels
along one axis of the filter) of this device is a factor of three
times higher than that of the 16×16 array used in this project.
Thus the results of Flannery
et al, particularly those of reference [44], may be used as a
benchmark for comparison with those of this project.
More shall be said of this in chapter eight.
4.4 Information Function
In the mathematical description of a pixellated filter it is often
useful to form an expression describing the modulation parameter of each
pixel of the SLM. Individual pixel modulation parameter settings
are described by an information function v(x,y), the local value
of which is identical to the modulation parameter at any given pixel
location. Consider the description of an SLM used in the object plane:
the array of pixels is commonly described by a Dirac comb
function [], convolved with a pixel function describing the shape of each
pixel (assumed not to vary over the SLM). Reducing to one dimension for
simplicity, and assuming the filter is so large that the summation may
be extended to ±¥, the light modulation performed by the SLM can
be written as
|
t(x) = [ v(x) |
+¥
å
n=-¥
|
d(x - nD) ] * P(x) |
| (10) |
which has Fourier Transform [17]
This particular function3
gets broader as the pixel width a becomes
narrower, and in general the envelope function attenuates the higher order
replications caused by pixellation of the filter.
The Fourier Transform of equation 4.10 is, ignoring the envelope function,
given by
|
T(n) = V(n) * |
+¥
å
n=-¥
|
d(n - |
n
D
|
) |
| (13) |
Optimum Sampling
Note that if the bandlimit of the information function is too wide
( > [ 1/(D)]) then the
spectral replicas overlap and aliasing occurs, so that high frequencies from
the outer replications wrongly appear in lower frequency locations of the
zero order replication. (Figure 4.5).
Figure 4.5: Aliasing arising from too low a sampling interval
This occurs if the SLM sampling
interval D is too large as may be seen from equation 4.13. `Optimum
sampling' refers to the situation where the spectral replications just
touch at their peripheries, so that if nL is the highest
frequency required in the information function then
3 Single Parameter Correlation Filters
The general characteristics of practical filters (pixellation,
bandwidth, etc.) have been introduced. The primary limitation of many
pixellated SLMs is the binary mode of operation.
As stated in chapter one, it is a primary objective of this project to
demonstrate the capabilities of a low space-bandwidth-product,
pixellated, binary-mode SLM as an optical correlator. Therefore it is
required now to ascertain the likely performance of correlation filters
which have only two phase values allowed - binary
phase-only filters or BPOFs, as they are commonly known.
Specifically there are three points to be addressed.
- Given that the vast majority of SLMs available can function either
as binary amplitude-only or binary phase-only filters,
which modulation parameter should be chosen to represent a pixellated
correlation filter ?
- The classical `matched' correlation filter requires both amplitude and
phase filtering operations to be performed. Whichever modulation parameter is
chosen, how will the filter perform with only one modulation parameter ?
- What effect on filter performance does the quantisation of the
modulation parameter to only two values have ?
3.1 Choice of Modulation Parameter
Consider an object centered in the object plane of an optical processor.
At the image plane it is desired to obtain the cross-correlation of
this object with some target object, which is a sharply peaked function
centered about the origin of the image plane. The phase of the filter
serves to cancel out the phase of the object spectrum so that no phase
variations exist over the frequency plane. As such, the complex light
field immediately behind the filter behaves like a plane wave but with a
spatially varying amplitude over the wavefront. This wave is focussed
down to the center of the image plane and the resulting sharply peaked
image plane light distribution is the cross-correlation of the object
with the target object from which the filter was calculated.
It is likely that sharper still focussing would occur if the amplitude
over the surface of the plane-type wave was uniform, so that the light
field immediately after the filter perfectly resembles a plane wave in
both amplitude and phase.
Looking at the process in another way, setting the phase of the
spectrum to be zero for each and every Fourier component means that the
spatial offset of each spatial frequency in the image is zero. Thus if
the image amplitude g(xi,yi) is described as a Fourier integral
|
g(xi,yi) = |
ó
õ
|
|
ó
õ
|
+¥
-¥
|
G(nx,ny) eiF(nx,ny) dnx dny |
| (15) |
As such, it
would seem that the phase information of the filter is primarily
responsible for the correlation process. Indeed, the phase spectrum of
an object is unique, whereas the amplitude spectrum need not be [24].
Correlation performed with an amplitude-only filter is compared to that
from phase-only filters in reference [46] where it is also concluded that
an amplitude-only filter is virtually useless.
Having established which parameter is the more important, the effect of
dropping the other modulation parameter - amplitude - is now
considered. This will be aided by studying some particular types of
phase filter.
3.2 Phase Only Filters
Actual correlation filters commonly differ from the matched filter
thus far introduced. The matched filter, as described by equation 4.2, has an
amplitude equal to that of the Fourier Transform of the target object.
The phase-only filter however (POF) has an amplitude everywhere equal to unity,
and is of form
This filter may be considered to be a product of two filters, one the
matched spatial filter and the other having an amplitude transmittance
of [ 1/(|G(nx,ny)|)]. As such, the second filter in the
product emphasises the lower amplitude (and commonly high spatial
frequency) components of the spectrum and acts in a similar manner to a
high pass filter. The sharpness of the correlation peak derives mainly
from high spatial frequencies in the image, and consequently the
correlation sharpness is greatly improved. Some quantitative results
will be quoted later in this section. Notice how the elimination of the
amplitude modulation is expected to improve the filter performance
rather than detract from it4.
As mentioned earlier, practical filters
(SLMs) most commonly modulate only a single parameter. Further, this
parameter frequently is allowed to take on just one of two possible
values. From the discussion above the choice of modulation parameter in
such a device should be phase rather than amplitude in a practical
optical correlation system. Knowing the relative importance of the phase
information of a spectrum, it is important that a theoretical basis be
laid which shows that binarisation of phase is an allowable procedure in
the first place. Such an analysis has already been published
and is summarised here.
Phase Quantisation
As mentioned earlier, the vast majority of phase modulating SLMs cannot
perform continuous phase modulation. Their modulation capabilities are
quantised, frequently to only two levels, and a proper
understanding of the effects of phase quantisation is thus required.
The effects of phase quantisation of the Fourier Transform on the image
has been analysed in detail by Goodman and Silvestri [].
They found that the
image was described by a series of terms relating to the number of phase
quantisation levels. Specifically, if N denotes the number of
quantisation levels and g¢(x) the actual image amplitude obtained then
|
g¢(x) = |
+¥
å
m=-¥
|
sinc( m + |
1
N
|
) gm(x) |
| (17) |
|
|
|
|
|
ó
õ
|
+¥
-¥
|
|G(n)| ei (Nm+1) F(n) |
| |
|
| (18) |
|
This work forms the basis of a theoretical explanation
as to why binary phase-only filters still work. The ideal Fourier
Transform of a continuous phase-only filter should resemble the object
function from which the filter was calculated. In reducing the allowable
phase values to only two the above analysis reveals that the primary
image still exists, though attenuated, together with a number of
unwanted (modified) images superimposed.
In the next subsection, simulations performed by Horner et al
indirectly provide a quantitative comparison of the effects of these
`false' images on the correlation arising from a binarised phase filter
with the correlation arising from a continuous phase-only filter.
3.3 Binary Phase Only filters (BPOFs)
Binarisation of the filter phase must be done in many spatial light
modulators due to their binary mode of operation, more of which shall be
discussed later. A very large number of binarisation algorithms have
been proposed, the aim of which is to produce a single number with
which to characterise the phase of a single SLM pixel. This number most
commonly takes the value 0 or p radians, arising partly because
some SLMs can only retard one pixel by p radians relative to another.
In order that a binary phase value be calculated, the
target object is modelled on a computer and the Fourier Transform taken.
The phases F(nx,ny) of all points lying in the physical
extent of any pixel are binarised according to a relation such as
The performance of matched filters, phase-only filters and BPOFs has
been studied by Horner et al [] to whom the above
binarisation criteria is attributed. Choosing a 2-D object, the face of a
doll, Horner has investigated the performance of each filter and characterised
this by peak intensity, optical efficiency and signal to noise ratio in the
same manner as for the classical matched filter of section 4.1.2. The peak
intensity was measured in units whereby the matched spatial filter gives
a peak intensity of unity. For both a continuous tone and binary input object,
table 4.2 quotes Horner's results for the matched (MF), phase-only
(POF) and binary phase-only (BPOF) filters.
| MF | POF | BPOF |
| FilterCTS, Object Continuous | 1 | 125 | 36 |
| FilterBIN, Object Binary | 3.4 | 1191 | 471 |
Table 2: Correlation Peak Intensities relative to the Matched filter
These results suggest that phase only filters far outperform, with
respect to peak intensity, the classical matched spatial filter. Also
the BPOF, though not as good as the POF, produces correlation peaks
several hundred times brighter than the classical matched filter if the
target object is binarised and the filter made from the binary object
also. Binary phase-only filters have recently been given serious consideration
[48] as a means of guiding remote spacecraft to their landing
site, specifically the Mars Rover Sample Return mission of NASA
[49]. Although
the correlation is proposed to be performed electronically rather than
optically, the BPOF guidance technique has been demonstrated to match
the best tracking algorithms to date and thus illustrates the power of
this class of filter.
3.4 Binary Phase Filter Spectra
The complete analysis of Goodman and Silvestri may be complemented in
this chapter by a related analysis on the spectra of binary phase
objects and filters. In this case the information function v(x,y) of
section 4.2.4 is phase only so that a binary phase filter would be
represented by
where f(x,y) is a two dimensional, REAL binary function.
It will now be shown that the spectrum
of v(x,y) does not change form between describing a binary amplitude
object or a binary phase object.
The analysis is performed in 1-D for ease of discussion though the
generalisation to 2-D is immediately apparent. Let f(x) be a
REAL, binary function with minimum zero and maximum unity as shown in
figure 4.6. This function might be the 1-D representation of a binary
amplitude pattern displayed on an SLM, for instance.
Figure 4.6: A 1-D REAL, Binary Function f(x)
The function comprises a set of rectangle functions placed along the
x-axis, though these need not be placed at regular intervals nor do
they need to have identical widths for this analysis. One may
now describe a binary phase object g(x) with phase retardance
a by
Using a Taylor expansion of the exponential and using the fact that
f(x)k=f(x), f(x)=1 or 0 only, it is straightforward to show that
|
|
|
|
cos(af(x) ) + i sin( af(x) ) |
| |
|
| 1 + ( |
a2
2!
|
- |
a4
4!
|
+¼ ) + i sin(a) f(x) |
| (21) |
|
|
|
|
|
1 + [ cos(a)-1 ] f(x) + i sin(a) f(x) |
| |
|
| (22) |
|
|
G(n) = d(n) + [ eia-1 ]F(n) |
| (23) |
Therefore, the intensity of the phase filter spectrum is given by
|
|
|
|
|
1
2
|
( 1+cos(a) ) F(n = 0) |
| (24) | |
|
| |
|
| (25) |
|
- The spectrum of an arbitrarily shaped binary phase object is
identical in form to that of a binary amplitude object, save for a
complex multipicative constant.
- Variations of the phase retardance a serve only to vary the
intensity of the zero frequency relative to the rest of the spectrum as
a whole.
Binary phase objects thus do not suffer from the effects of ghost
spectral orders as defined in chapter two. This effectively defines the
useful region of the Bessel function analysis to non-binary phase
objects. However, for the sake of completeness, the area of overlap
between the two regimes has been examined and is briefly presented here.
Using the Bessel function convolution program, the properties of the spectrum
resulting when the initial Fourier series coefficients are those of a
square wave phase object are compared with the predictions of equation 4.44.
Each Fourier coefficient will give rise to a Bessel comb, but from the
work above it is known that no light must be found outwith
the confines of the equivalent amplitude object spectrum. Therefore,
the Fourier coefficients must be such that cancellation of the ghost
orders occurs.
Twenty-one coefficients of the square wave series were used in the
program and the amplitude of the spectrum at the frequency origin
recorded for several phase modulation depths of the square wave which the
coefficients describe. Figure 4.7 plots the theoretical spectral amplitude
at n = 0, from equation 4.41, together with the measured amplitude from the
Bessel function program.
Figure 4.7: Amplitude at
n = 0, Theory and Bessel Function Program Results.
As can be seen, excellent overall agreement between the Bessel function
program and the theoretical expression is found. Further, the percentage
of light energy outwith the region of the 21 Fourier coefficients was
never observed to exceed 2% of the total energy of the spectrum
indicating that cancellation of the ghost orders did indeed occur.
For a phase retardance of a = p, a 1-D binary phase object with
equal areas of 0 and p retardance should cause complete
cancellation of the light field at the frequency plane origin.
The evolution of the convolution process which leads to this
cancellation in the Bessel function program is shown in figure 4.8, where
the ordinate records the spectral amplitude after the N'th convolution
stage.
Figure 4.8: Evolution of Zero Frequency Cancellation
It is observed that cancellation occurs rapidly after just a few
convolution stages have taken place, so that although the Bessel
function analysis can be used in the analysis of binary phase filters,
more convenient analysis techniques exist and the Bessel function
analysis ends its usefulness for this project here.
Rigorous Imaging Equation
As shown in chapters two and three, a fully analytical solution for the
spectrum of a general phase object is difficult to obtain, yet alone an
expression for the image intensity. For a binary phase object, however,
a fully analytical expression for the image intensity after a phase
contrast operation on the spectrum may be deduced and is included here
for completeness.
Let F(n) be split into a two spatially exclusive components
where Fs(n) is the whole spectrum minus the zero frequency
component. Then, if E(a)=ei a-1, equation 4.40 may be written
|
G(n) = [ 1 + E(a) F(0) ] d(n) + E(a) Fs(n) |
| (27) |
|
Gp(n) = i [ 1 + E(a) F(0) ] d(n) + E(a) Fs(n) |
| (28) |
The image field is given by the Fourier Transform of equation 4.45. If we
set
then the image field gp(x) which results is of form
|
|
|
|
[ fs(x) (cos(a)-1) - F(0) sin(a) ] |
| |
|
| i [1 + F(0) cos(a) - F(0) + fs(x) sin(a) ] |
| (30) |
|
|
|
|
|
2f(-x) [ sin(a) + 2 A F(0) - A ] - 4 A F2(0) |
| |
|
| (31) |
|
It should be noted that
- The analysis is applicable to 2-D phase objects although it is
conducted in 1-D.
- Binary phase objects need not be periodic for this analysis to apply.
- For a 1-D square wave phase grating of total width D
having a total of n rectangle functions of phase retardance
a and width W, the variable F(0) reduces to [ nW/D].
In the limit of small phase, equation 4.48 gives the intensity over a phase
pixel as
|
gp2(x) @ 1 + 2a [1-F(0)] + ¼ |
| (32) |
The full analytical analysis presented here shows that not only does the
intensity of a phase pixel depend on the phase retardance of that pixel
(as the Taylor expansion shows) but also on F(0) which is proportional
to the mean of f(x). This prediction has been verified by comparing the
analytical prediction of pixel intensity of equation 4.48 with that
obtained by computer simulation where the spectrum was computed using a
Fast Fourier Transform algorithm. The results are shown in figure 4.9.
Figure 4.9: Pixel Intensity of a Binary Phase Object after Positive
Phase Contrast Filtering: Theory and Simulation
This short subsection has shown binary phase objects to be tractable to
analytical solution in as far as certain phase visualisation operations
are concerned. Further, the results presented here may be used as a basis
of phase calibration for certain 2-D transmissive binary phase only filters
if the variable F(0) is replaced by the fill factor of the
device5.
3.5 Noise Effects
Input scenes to a correlator usually suffer from a degree of noise, defined
generally as small, random
variations in amplitude transmission. Such noise will have an associated power
spectrum. `White' noise, for example, is defined as having an equally
strong power spectrum at all spatial frequencies. The effect of a
phase-only filter is to pass all frequency components without any
attenuation of amplitude. Clearly, if the bandwidth of the input signal
is much less than the pass-band of the spatial filter then if white
noise is present a large amount of noise is passed to the image
plane with no additional increase in signal (the object spectrum) as
illustrated in figure 4.10.
Figure 4.10: Choice of Passband for Noise Reduction
A trade off between the high spatial frequency components and the amount
of noise let through in passing those frequencies should be made in
calculation of an optimum passband for the phase-only filter. Several
papers have been written on this subject [50], and it is
mentioned here only in passing.
In computer simulations of correlations without input noise, the `signal
to noise ratio' (SNR )
defines noise as anything which is not part of the correlation peak but
is present in the image. Horner, for example, uses a 50% of peak
value threshold [33] to define what is noise in the image. Examination of
filter performance without noise is useful in providing a clear, basic
figure of comparison between filters. Table 4.3 again quotes from Horner
the SNR and
optical efficiency (as defined in section 4.1.2) for the classical matched
filter, phase-only filter and the binary-phase-only filter. This
time, only those results obtained from using both a binary object and
the filters made from this object are tabled as this was the
experimental situation realised in chapter seven of this thesis.
| MF | POF | BPOF |
| SNR | 7:1 | 62:1 | 36:1 |
| hH | 0.7% | 19% | 7.5% |
Table 3: SNR and HORNER Efficiency, Three Types of Filter.
The signal to noise ratio, as might be expected, is best for the POF and
not quite so good for the BPOF. Figures for both these filters however
are very much better than for the classical matched filter even though
it is operating as best it ever will, on a binary object.
The conclusions of this section are
- Phase-only filters result in an enormous increase in peak
intensity (and, though results are not shown here, peak sharpness also),
and signal to noise ratio of the resulting correlation.
- Binary phase-only filters have a reduced performance over their
continuous counterparts but nonetheless are much superior to matched
spatial filters with regards to the same parameters.
- Optical efficiency - the fractional percentage of energy in the
region of the peak to the total image plane energy - is much improved
in both POFs and BPOFs over that obtained by matched spatial filtering.
These conclusions have been verified in numerous publications. It is not
the intention of this section to review the very large field of optical
phase-only correlation, rather to provide an introduction to the
subject and quote the major results which allow the use of current
binary spatial light modulators as phase filters.
4 Alternative Correlation Techniques
It should be said that there are other techniques available for
performing optical correlation than the one described here. Most notable
is a technique known as `Joint Transform Correlation', where both the
target object and and input scene are displayed side by side in the
object plane. The resulting intensity distribution in the
frequency plane is an interference pattern between the individual
Fourier Transforms of the input functions, and is used as input to a
spatial light modulator. The input transmittance can be written as
|
t(x,y) = p(x,y+b) + h(x,y-b) |
| (34) |
|
[ g(x,y)*h(x,y) ] ** d(x, y-2b) + [ h(x,y)*g(x,y) ] ** d(x,y+2b) |
| (35) |
Figure 4.11: Joint Transform Correlator Configuration
A most concise article on this subject is provided by Gregory and
Loudin [51], who have successfully demonstrated this technique by using a
modified liquid crystal television (LCTV) as the spatial light modulator. The
LCTV provides the necessary continuous representation of the filter
interference pattern, which binary mode devices clearly cannot. Several
advantages of this technique over the `classical' one described in this
thesis are presented, but perhaps most notable is the absence of
computation required to find the filter pattern as it automatically
appears as an interference pattern in the frequency plane. If one SLM is
used to input the object scene and target, and another to display the
intensity distribution of the resulting Fourier Transform patterns then
correlation may be performed at video frame rates.
Although it is also possible to binarise the filter pattern (and thus
make use of the binary mode devices available within the Applied Optics
Group at Edinburgh) it was decided not to pursue this technique as a
possible filtering algorithm for use in this device.
As discussed in section 4.4.2 (space-bandwidth product), there exists a
maximum object field size for most real spatial filters in current use. For
the particular device used in this thesis (the subject of chapters 5 & 6)
this size is very small indeed, so that the whole object space can
readily accommodate only one (small) scene of limited resolution. This
unfortunately means that the recent field of Joint Transform Correlation
cannot be investigated experimentally. It is suggested that this form
of correlator be investigated within this Group with SLMs having a much
larger number of pixels, this directly increasing the space-bandwidth product.
4.1 Amplitude-Modulated Phase-Only Filter
Even more recently than the subject of Joint Transform Correlation, a
new correlation filter has been proposed which combines both amplitude
and phase modulation to give a significant improvement over even the
phase-only filter described in section 4.3.1.
Awwal et al [] propose an amplitude-modulated phase-only filter
(AMPOF). If the target object spectrum is described by
G(nx,ny) which has a phase distribution characterised by
F(nx,ny), the AMPOF is given by
|
H(nx,ny) = D |
e-i F(nx,ny)
|G(nx,ny)| + a
|
|
| (36) |
Table 4.4, taken from [52], compares the percentage energy in the
correlation peak relative to the energy in the image plane as a whole,
the normalised peak power (NPP), for the AMPOF and the POF, results
quoted from AWWAL et al. The input and
filter combinations are varied so as to
compare both auto-correlation performance and filter discrimination for
both filter types. The results were obtained from computer simulation
using letters defined on a 64×64 object data grid.
It is observed that the AMPOF results in much more energy being
channeled into the correlation peak, and the percentage change in
discrimination is much greater than for the POF.
| Input | Filter | POF | AMPOF |
| E | E | 57 | 134 |
| F | F | 62 | 131 |
| G | G | 70 | 163 |
| 0 | 0 | 55 | 144 |
| E | F | 41 | 81 |
| F | E | 44 | 83 |
| G | O | 52 | 105 |
| O | G | 44 | 99 |
Table 4: Normalised Peak Power for AMPOF and POF filters
The high performance obtained by using the AMPOF, as explained in their
original paper, is roughly as follows. It can be shown [x] that the
auto-correlation Cgg(D) of a function g(x) is given by
|
Cgg(D) = |
ó
õ
|
+¥
-¥
|
|G(n)|2 einD dn |
| (37) |
At the time of publication, the authors intended to study the effects of
binarisation on the filter performance.
Although this type of filter is obviously very powerful, it relies on a
filter capable of modulating both amplitude and phase.
Cascading of two devices each operating in different modes (amplitude
and phase) is possible though the alignment is liable to be an area of
considerable experimental difficulty. This type of filter was not
chosen for use with the SLMs available primarily because the project was
already well underway before the publication of the original paper.
4.2 Review
This chapter has introduced the subject of optical correlation and the
associated mathematics involved. The realisation of practical spatial
filters is frequently constrained by the fact that only one parameter
describing the complex light field (amplitude or phase) can be modulated
by any one device. In summary,
- The general characteristics of practical `pixellated' filters have been
introduced, the principle limitation of which is that the product of
input object size with the highest spatial frequency passed by the
device is a constant.
- Where a single parameter only can be modulated in correlation
filters, that parameter should be phase and in fact correlation filters
employing phase-only modulation actually
operate much better than the classical matched spatial filter.
- Binarisation of the phase value is normally required by most phase
modulating devices (SLMs), which is a special type of phase
quantisation. However, quantisation of the filter phase to two states
does reduce the correlation peak intensity among other things, but still
provides a much improved correlation over the classical matched filter.
- Alternative correlation techniques combining both amplitude
and phase modulation offer much improvement over phase-only filtering
but are unsuitable for this project.
The subject of this chapter will form a background for the experimental
work on optical correlation performed with a spatial light modulator
designed within the Applied Optics Group at Edinburgh University. The
design, operation and assembly of this device shall form the bulk of the
next two chapters.
Footnotes:
1The conjugate of the second
function may be dropped if both functions are entirely REAL.
2Larger array sizes are also
available to date though the 48×48 device was the first to become
commercially available.
3The `sinc' function is defined here,
according to Gaskill [17], as sinc(x)=[(sin(p x))/(p x)].
4The effects of noise on phase-only
filters is considered in section 4.3.3
5A square pixel of side a has, if pixels are spaced d
apart, a fill factor of order ([ a/d])2 for instance.
6Indeed, this fact is fundamental to the
operation of all matched correlator systems.
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