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Special Imaging Techniques
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Spatial Resolution
Suppose we want to compare two imaging systems, with the goal of
determining which has the best spatial resolution. In other words, we want to
know which system can detect the smallest object. To simplify things, we
would like the answer to be a single number for each system. This allows a
direct comparison upon which to base design decisions. Unfortunately, a single
parameter is not always sufficient to characterize all the subtle aspects of
imaging. This is complicated by the fact that spatial resolution is limited by
two distinct but interrelated effects: sample spacing and sampling aperture
size. This section contains two main topics: (1) how a single parameter can
best be used to characterize spatial resolution, and (2) the relationship between
sample spacing and sampling aperture size.
Figure 25-1a shows profiles from three circularly symmetric PSFs: the
pillbox, the Gaussian, and the exponential. These are representative of the
PSFs commonly found in imaging systems. As described in the last chapter,
the pillbox can result from an improperly focused lens system. Likewise,
the Gaussian is formed when random errors are combined, such as viewing
stars through a turbulent atmosphere. An exponential PSF is generated
when electrons or x-rays strike a phosphor layer and are converted into
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The Scientist and Engineer's Guide to Digital Signal Processing
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a. PSF
Spatial frequency (lp per unit distance)
0 0.5 1 1.5 2
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b. MTF
FIGURE 25-1
FWHM versus MTF. Figure (a) shows profiles of three PSFs commonly found in imaging systems: (P) pillbox,
(G) Gaussian, and (E) exponential. Each of these has a FWHM of one unit. The corresponding MTFs are
shown in (b). Unfortunately, similar values of FWHM do not correspond to similar MTF curves.
Amplitude
Amplitude
light. This is used in radiation detectors, night vision light amplifiers, and CRT
displays. The exact shape of these three PSFs is not important for this
discussion, only that they broadly represent the PSFs seen in real world
applications.
The PSF contains complete information about the spatial resolution. To express
the spatial resolution by a single number, we can ignore the shape of the PSF
and simply measure its width. The most common way to specify this is by the
Full-Width-at-Half-Maximum (FWHM) value. For example, all the PSFs in
(a) have an FWHM of 1 unit.
Unfortunately, this method has two significant drawbacks. First, it does not
match other measures of spatial resolution, including the subjective judgement
of observers viewing the images. Second, it is usually very difficult to directly
measure the PSF. Imagine feeding an impulse into an imaging system; that is,
taking an image of a very small white dot on a black background. By
definition, the acquired image will be the PSF of the system. The problem is,
the measured PSF will only contain a few pixels, and its contrast will be low.
Unless you are very careful, random noise will swamp the measurement. For
instance, imagine that the impulse image is a 512×512 array of all zeros except
for a single pixel having a value of 255. Now compare this to a normal image
where all of the 512×512 pixels have an average value of about 128. In loose
terms, the signal in the impulse image is about 100,000 times weaker than a
normal image. No wonder the signal-to-noise ratio will be bad; there's hardly
any signal!
A basic theme throughout this book is that signals should be understood in the
domain where the information is encoded. For instance, audio signals should
be dealt with in the frequency domain, while image signals should be handled
in the spatial domain. In spite of this, one way to measure image resolution is
by looking at the frequency response. This goes against the fundamental
Chapter 25- Special Imaging Techniques 425
Pixel number
0 60 120 180 240
0
50
100
150
200
250
Pixel number
0 60 120 180 240
0
50
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150
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250
a. Example profile at 12 lp/mm
b. Example profile at 3 lp/mm
40
20
10
5
2
15
30
7
4
3
FIGURE 25-2
Line pair gauge. The line pair gauge is
a tool used to measure the resolution of
imaging systems. A series of black and
white ribs move together, creating a
continuum of spatial frequencies. The
resolution of a system is taken as the
frequency where the eye can no longer
distinguish the individual ribs. This
example line pair gauge is shown
several times larger than the calibrated
scale indicates.
line pairs / mm
Pixel value Pixel value
philosophy of this book; however, it is a common method and you need to
become familiar with it.
Taking the two-dimensional Fourier transform of the PSF provides the twodimensional
frequency response. If the PSF is circularly symmetric, its
frequency response will also be circularly symmetric. In this case, complete
information about the frequency response is contained in its profile. That is,
after calculating the frequency domain via the FFT method, columns 0 to N/2
in row 0 are all that is needed. In imaging jargon, this display of the frequency
response is called the Modulation Transfer Function (MTF). Figure 25-1b
shows the MTFs for the three PSFs in (a). In cases where the PSF is not
circularly symmetric, the entire two-dimensional frequency response contains
information. However, it is usually sufficient to know the MTF curves in the
vertical and horizontal directions (i.e., columns 0 to N/2 in row 0, and rows 0
to N/2 in column 0). Take note: this procedure of extracting a row or column
from the two-dimensional frequency spectrum is not equivalent to taking the
one-dimensional FFT of the profiles shown in (a). We will come back to this
issue shortly. As shown in Fig. 25-1, similar values of FWHM do not
correspond to similar MTF curves.
Figure 25-2 shows a line pair gauge, a device used to measure image
resolution via the MTF. Line pair gauges come in different forms depending
on the particular application. For example, the black and white pattern shown
in this figure could be directly used to test video cameras. For an x-ray
imaging system, the ribs might be made from lead, with an x-ray transparent
material between. The key feature is that the black and white lines have a
closer spacing toward one end. When an image is taken of a line pair gauge,
the lines at the closely spaced end will be blurred together, while at the other
end they will be distinct. Somewhere in the middle the lines will be just barely
separable. An observer looks at the image, identifies this location, and reads
the corresponding resolution on the calibrated scale.
426 The Scientist and Engineer's Guide to Digital Signal Processing
The way that the ribs blur together is important in understanding the
limitations of this measurement. Imagine acquiring an image of the line
pair gauge in Fig. 25-2. Figures (a) and (b) show examples of the profiles
at low and high spatial frequencies. At the low frequency, shown in (b),
the curve is flat on the top and bottom, but the edges are blurred, At the
higher spatial frequency, (a), the amplitude of the modulation has been
reduced. This is exactly what the MTF curve in Fig. 25-1b describes:
higher spatial frequencies are reduced in amplitude. The individual ribs
will be distinguishable in the image as long as the amplitude is greater than
about 3% to 10% of the original height. This is related to the eye's ability
to distinguish the low contrast difference between the peaks and valleys in
the presence of image noise.
A strong advantage of the line pair gauge measurement is that it is simple and
fast. The strongest disadvantage is that it relies on the human eye, and
therefore has a certain subjective component. Even if the entire MTF curve is
measured, the most common way to express the system resolution is to quote
the frequency where the MTF is reduced to either 3%, 5% or 10%.
Unfortunately, you will not always be told which of these values is being used;
product data sheets frequently use vague terms such as "limiting resolution."
Since manufacturers like their specifications to be as good as possible
(regardless of what the device actually does), be safe and interpret these
ambiguous terms to mean 3% on the MTF curve.
A subtle point to notice is that the MTF is defined in terms of sine waves,
while the line pair gauge uses square waves. That is, the ribs are uniformly
dark regions separated by uniformly light regions. This is done for
manufacturing convenience; it is very difficult to make lines that have a
sinusoidally varying darkness. What are the consequences of using a square
wave to measure the MTF? At high spatial frequencies, all frequency
components but the fundamental of the square wave have been removed. This
causes the modulation to appear sinusoidal, such as is shown in Fig. 25-2a. At
low frequencies, such as shown in Fig. 25-2b, the wave appears square. The
fundamental sine wave contained in a square wave has an amplitude of 4/B ’
1.27 times the amplitude of the square wave (see Table 13-10). The result: the
line pair gauge provides a slight overestimate of the true resolution of the
system, by starting with an effective amplitude of more than pure black to pure
white. Interesting, but almost always ignored.
Since square waves and sine waves are used interchangeably to measure the
MTF, a special terminology has arisen. Instead of the word "cycle," those in
imaging use the term line pair (a dark line next to a light line). For example,
a spatial frequency would be referred to as 25 line pairs per millimeter,
instead of 25 cycles per millimeter.
The width of the PSF doesn't track well with human perception and is
difficult to measure. The MTF methods are in the wrong domain for
understanding how resolution affects the encoded information. Is there a
more favorable alternative? The answer is yes, the line spread function
(LSF) and the edge response. As shown in Fig. 25-3, the line spread
Chapter 25- Special Imaging Techniques 427
a. Line Spread Function (LSF) b. Edge Response
90%
50%
Full Width at
Half Maximum
(FWHM)
10% to 90%
Edge response
10%
FIGURE 25-3
Line spread function and edge response. The line spread function (LSF) is the derivative of the edge response.
The width of the LSF is usually expressed as the Full-Width-at-Half-Maximum (FWHM). The width of the
edge response is usually quoted by the 10% to 90% distance.
function is the response of the system to a thin line across the image.
Similarly, the edge response is how the system responds to a sharp straight
discontinuity (an edge). Since a line is the derivative (or first difference) of an
edge, the LSF is the derivative (or first difference) of the edge response. The
single parameter measurement used here is the distance required for the edge
response to rise from 10% to 90%.
There are many advantages to using the edge response for measuring resolution.
First, the measurement is in the same form as the image information is encoded.
In fact, the main reason for wanting to know the resolution of a system is to
understand how the edges in an image are blurred. The second advantage is
that the edge response is simple to measure because edges are easy to generate
in images. If needed, the LSF can easily be found by taking the first difference
of the edge response.
The third advantage is that all common edges responses have a similar shape,
even though they may originate from drastically different PSFs. This is shown
in Fig. 25-4a, where the edge responses of the pillbox, Gaussian, and
exponential PSFs are displayed. Since the shapes are similar, the 10%-90%
distance is an excellent single parameter measure of resolution. The fourth
advantage is that the MTF can be directly found by taking the one-dimensional
FFT of the LSF (unlike the PSF to MTF calculation that must use a twodimensional
Fourier transform). Figure 25-4b shows the MTFs corresponding
to the edge responses of (a). In other words, the curves in (a) are converted
into the curves in (b) by taking the first difference (to find the LSF), and then
taking the FFT.
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Limiting resolution
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FIGURE 25-4
Edge response and MTF. Figure (a) shows the edge responses of three PSFs: (P) pillbox, (G) Gaussian, and
(E) exponential. Each edge response has a 10% to 90% rise distance of 1 unit. Figure (b) shows the
corresponding MTF curves, which are similar above the 10% level. Limiting resolution is a vague term
indicating the frequency where the MTF has an amplitude of 3% to 10%.
Amplitude
Amplitude
The fifth advantage is that similar edge responses have similar MTF curves, as
shown in Figs. 25-4 (a) and (b). This allows us to easily convert between the
two measurements. In particular, a system that has a 10%-90% edge response
of x distance, has a limiting resolution (10% contrast) of about 1 line pair per
x distance. The units of the "distance" will depend on the type of system being
dealt with. For example, consider three different imaging systems that have
10%-90% edge responses of 0.05 mm, 0.2 milliradian and 3.3 pixels. The
10% contrast level on the corresponding MTF curves will occur at about: 20
lp/mm, 5 lp/milliradian and 0.33 lp/pixel, respectively.
Figure 25-5 illustrates the mathematical relationship between the PSF and the
LSF. Figure (a) shows a pillbox PSF, a circular area of value 1, displayed as
white, surrounded by a region of all zeros, displayed as gray. A profile of the
PSF (i.e., the pixel values along a line drawn across the center of the image)
will be a rectangular pulse. Figure (b) shows the corresponding LSF. As
shown, the LSF is mathematically equal to the integrated profile of the PSF.
This is found by sweeping across the image in some direction, as illustrated by
the rays (arrows). Each value in the integrated profile is the sum of the pixel
values along the corresponding ray.
In this example where the rays are vertical, each point in the integrated profile
is found by adding all the pixel values in each column. This corresponds to the
LSF of a line that is vertical in the image. The LSF of a line that is horizontal
in the image is found by summing all of the pixel values in each row. For
continuous images these concepts are the same, but the summations are
replaced by integrals.
As shown in this example, the LSF can be directly calculated from the PSF.
However, the PSF cannot always be calculated from the LSF. This is because
the PSF contains information about the spatial resolution in all directions,
while the LSF is limited to only one specific direction. A system
Chapter 25- Special Imaging Techniques 429
a. Point Spread Function
b. "Integrated" profile of
the PSF (the LSF)
FIGURE 25-5
Relationship between the PSF and LSF. A
pillbox PSF is shown in (a). Any row or
column through the white center will be a
rectangular pulse. Figure (b) shows the
corresponding LSF, equivalent to an
integrated profile of the PSF. That is, the
LSF is found by sweeping across the
image in some direction and adding
(integrating) the pixel values along each
ray. In the direction shown, this is done
by adding all the pixels in each column.
has only one PSF, but an infinite number of LSFs, one for each angle. For
example, imagine a system that has an oblong PSF. This makes the spatial
resolution different in the vertical and horizontal directions, resulting in the
LSF being different in these directions. Measuring the LSF at a single
angle does not provide enough information to calculate the complete PSF
except in the special instance where the PSF is circularly symmetric.
Multiple LSF measurements at various angles make it possible to calculate
a non-circular PSF; however, the mathematics is quite involved and usually
not worth the effort. In fact, the problem of calculating the PSF from a
number of LSF measurements is exactly the same problem faced in
computed tomography, discussed later in this chapter.
As a practical matter, the LSF and the PSF are not dramatically different for
most imaging systems, and it is very common to see one used as an
approximation for the other. This is even more justifiable considering that
there are two common cases where they are identical: the rectangular PSF has
a rectangular LSF (with the same widths), and the Gaussian PSF has a
Gaussian LSF (with the same standard deviations).
These concepts can be summarized into two skills: how to evaluate a
resolution specification presented to you, and how to measure a resolution
specification of your own. Suppose you come across an advertisement
stating: "This system will resolve 40 line pairs per millimeter." You
should interpret this to mean: "A sinusoid of 40 lp/mm will have its
amplitude reduced to 3%-10% of its true value, and will be just barely
visible in the image." You should also do the mental calculation that 40
lp/mm @ 10% contrast is equal to a 10%-90% edge response of 1/(40
lp/mm) = 0.025 mm. If the MTF specification is for a 3% contrast level,
the edge response will be about 1.5 to 2 times wider.
When you measure the spatial resolution of an imaging system, the steps are
carried out in reverse. Place a sharp edge in the image, and measure the
430 The Scientist and Engineer's Guide to Digital Signal Processing
resulting edge response. The 10%-90% distance of this curve is the best single
parameter measurement of the system's resolution. To keep your boss and the
marketing people happy, take the first difference of the edge response to find
the LSF, and then use the FFT to find the MTF.