VII. FOURIER TRANSFORM APPLICATIONS p. 1 of 27 04/11/16 7.2 TWO DIMENSIONAL FOURIER TRANSFORM APPLICATIONS IN IMAGING The first aspect of application of Fourier transformation to imaging is that an image has two dimensions, unlike the one dimension of a single recording of neural activity. Therefore we need to extend the 1D Fourier transform in Equation (6.4) to a 2D version. This can be done as follows. Let’s start with an image ) , ( y x I and create the Fourier transform with respect to x and associated spatial frequency u. Then we take that result and perform a Fourier transform with respect to y and its associated spatial frequency v. These two steps give: By combining the two integrals we get ) , ( y x I F , the 2D Fourier transform of ) , ( y x I : (7.9) By performing this transform, we obtain the 2D spatial frequency domain that can be used for a variety of image processing applications. In the following, we will discuss several examples in optics and image reconstruction in medical imaging. 7.2.1 Fourier Transform by Lenses In addition to ray optics most of us are familiar with, there is a significant part of optics known as Fourier optics. This part of optics is based on the Fourier transformation properties of optical devices such as a slit or a lens. The goal here is to present examples of the application of the 2D Fourier transform. Therefore we won’t go into the details governing the physics of Fourier optics here; such details can be found in Goodman (2005), or an introduction in Hecht (2016). An intuitive example of the Fourier transorm property of a lens is depicted in Figure 7.8. Here it can be seen that a spatially uniform wavefront, i.e. a constant input, is transformed into a dot at the focal distance (f, Fig. 7.8A), and a point light source placed at the focal distance of the lens is transformed into a uniform wavefront (Fig. 7.8B). Now recall from the examples in Section 6.2.1 and Equation (6.11) that the Fourier transform of a constant (or a DC electrical signal) is a weighted delta function , and vice versa. If we ignore the details of the weighting, this is exactly what the lens is showing in the examples of Figs. 7.8A, B. FIG 7.8 The Fourier transforming property of the lens can be employed to image objects and to optically process images. One well known and widely applied example is the so called 4f system (Fig. 7.8C). In this example we have an object placed at one focal
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VII. FOURIER TRANSFORM APPLICATIONS
p. 1 of 27 04/11/16
7.2 TWO DIMENSIONAL FOURIER TRANSFORM APPLICATIONS IN
IMAGING
The first aspect of application of Fourier transformation to imaging is that an
image has two dimensions, unlike the one dimension of a single recording of neural
activity. Therefore we need to extend the 1D Fourier transform in Equation (6.4) to a 2D
version. This can be done as follows. Let’s start with an image ),( yxI and create the
Fourier transform with respect to x and associated spatial frequency u. Then we take that
result and perform a Fourier transform with respect to y and its associated spatial
frequency v. These two steps give:
By combining the two integrals we get ),( yxIF , the 2D Fourier transform of ),( yxI :
(7.9)
By performing this transform, we obtain the 2D spatial frequency domain that can be
used for a variety of image processing applications. In the following, we will discuss
several examples in optics and image reconstruction in medical imaging.
7.2.1 Fourier Transform by Lenses
In addition to ray optics most of us are familiar with, there is a significant part of
optics known as Fourier optics. This part of optics is based on the Fourier transformation
properties of optical devices such as a slit or a lens. The goal here is to present examples
of the application of the 2D Fourier transform. Therefore we won’t go into the details
governing the physics of Fourier optics here; such details can be found in Goodman
(2005), or an introduction in Hecht (2016).
An intuitive example of the Fourier transorm property of a lens is depicted in
Figure 7.8. Here it can be seen that a spatially uniform wavefront, i.e. a constant input, is
transformed into a dot at the focal distance (f, Fig. 7.8A), and a point light source placed
at the focal distance of the lens is transformed into a uniform wavefront (Fig. 7.8B). Now
recall from the examples in Section 6.2.1 and Equation (6.11) that the Fourier transform
of a constant (or a DC electrical signal) is a weighted delta function , and vice versa. If
we ignore the details of the weighting, this is exactly what the lens is showing in the
examples of Figs. 7.8A, B.
FIG 7.8
The Fourier transforming property of the lens can be employed to image objects
and to optically process images. One well known and widely applied example is the so
called 4f system (Fig. 7.8C). In this example we have an object placed at one focal
VII. FOURIER TRANSFORM APPLICATIONS
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distance of the objective lens that creates a 2D Fourier transform of the object in the
Fourier plane at its focal distance. The Fourier plane is also placed at one focal distance
to a second lens, the collector lens that performs the second Fourier transform. Due to the
duality property of the Fourier transform and its inverse transform (Section 6.2.1), the
collector lens creates an inverted image of the object in the image plane, again at its focal
distance. The image is inverted because of the different signs in the exponentials of the
Fourier transform and its inverse (see Equations (6.4) and (6.8)). To show this, let’s
define the coordinates of the object plane as x,y and the ones of the image plane as X,Y. If
we now substitute X=-x and Y=-y (i.e. inverted coordinates) in the expression for the 2D
Fourier transform (Equation (7.9)), we indeed obtain the correct expression for the
inverse Fourier transform of the inverted object:
(7.10)
Since we deal with four focal distances in this procedure, this system is known as the 4f
system. One important property of this setup is that we now can process the image in the
frequency domain by placing masks in the Fourier plane. If we, for example place a
pinhole mask in the Fourier plane, we remove high spatial frequencies from the image
that is produced, i.e. we attenuate edges and keep contrast information of the image. If,
on the other hand, we place a mask blocking the center part in the Fourier plane, we
remove the low spatial frequencies and keep the edge information; we now have an edge
detector. An example of the latter procedure, albeit obtained via a digital filter procedure
and not via direct optical processing, is depicted in Figure 18.4.
7.2.2 Tomography
In this section we apply the Fourier transform to a problem in tomography, used
in medical imaging. We will approach tomography in a general fashion; the principles we
discuss apply both to scanning emission and absorption profiles. Consider emission of
activity and passive absorption as the same type of process. In the case of emission, each
little voxel (or pixel in the 2-D case) emits its own contribution to the total that is
measured outside the volume. The absorption model is slightly more complicated since
each pixel instead contributes to attenuation across the area. We can use Beer’s law to
express the intensity of the output Io of a beam as a function of input intensity Ii and
attenuation caused by absorption ak across k = 1 - N elements:
N
kka
io eII 1 . Using the
property that the absorption law is exponential, we can use the logarithm of the
absorption ratio A to obtain an additive effect for each element k, i.e.:
N
k
k
i
o aI
IA
1
ln
7.22..1 Measured Absorption – Radon Transform
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In the following, we develop the Radon transform, the Fourier slice theorem,
and filtered back projection as each applies to CT image reconstruction. These
techniques require reconstruction of a density function representing the internal structure
of an object from sensor readings taken from outside that object. This is typically
accomplished by calculating a series of 2-dimensional density functions (or slices)
through the object on a set of planes and reconstructing the 3-dimensional image from
those images. The task is to calculate the 2-dimensional density function with readings
from a sensor which typically rotates around the object in a given plane. The following
derivations use Fourier analysis to relate a filtered version of this measured signal to the
density function of the object within the measured region.
For ease of explanation we use polar coordinates to derive the theorem. Assume a
source and detector moving at an angle with respect to the X-axis around an object
enclosed in a circle with radius R (Fig. 7.9). The distance of the source-detector (SD) line
from the origin is t, and the detector measures absorption (or emission) of all the points
along the line. In polar coordinates, all points r,on the SD line relate to t as:
cosrt (7.11)
If the emitter/detector pair is rotating at a constant speed, t represents time and the
measurement at the detector becomes a time series.
Fig. 7.9
The value of varies between 0-180 degrees. The total absorption along SD is
represented by m(t,) and is determined by the contributions of arbitrarily small surfaces
ddrr (Fig. 7.9, inset). Denoting the absorption function inside the circle as a(r,),
which corresponds to the mass to be scanned, we obtain:
ddrrrtratmR
)cos(),(),( (7.12)
Think of ddrrraR
),( as the total absorption of the whole object inside the circle
with radius R. For a particular measurement m(t,) we are only interested in the
contributions along the line of response (LOR, Fig. 7.8). We pull these out by adding a
function that sifts for the values for and r on the LOR at a given t and . The delta
function that accomplishes this must evaluate to zero within the LOR, i.e. the argument
should be 0)cos( rt and )cos( rt in equation (7.12) accomplishes
sifting the points on the LOR (see CH 2 for the sifting property). Using integration limits
reflecting area R instead of - is appropriate because: 0),( ra for Rr .
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The 1-D continuous Fourier transform of m(t,) in the spatial domain is:
dtetmzM tzj
2),(),( (7.13)
where z represents the spatial frequency domain. Substituting (7.12) in (7.13) and
combining all terms related to t within the square brackets gives:
ddrrdtertrazMR
tzj
2)cos(),(),(
When using the sifting property of the function for the integration over t, this
expression evaluates to:
ddrrerazMR
rzj
)cos(2),(),( (7.14)
In the following section we will show that this expression is identical to the 2-D Fourier
transform of the absorption function a.
7.2.2.2 The Absorption Function in the Spatial Frequency Domain
According to equation (7.9), the 2-D Fourier transform of the absorption function
a(x,y) in Cartesian coordinates is:
dydxeyxavuA yvxuj
)(2),(),( (7.15)
Changing u and v into polar coordinates z and gives:
dydxeyxazzA yxzj
)sincos(2),()sin,cos( (7.16)
Similarly transforming x and y to polar coordinates r and gives:
ddrrerrazA rzj
)sinsincos(cos2)sin,cos(),( (7.17)
Using the trigonometric identity )cos(sinsincoscos and setting a(r,)=0
for all points outside the circle with radius R, equation (7.17) becomes:
VII. FOURIER TRANSFORM APPLICATIONS
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ddrrerazAR
rzj
)cos(2),(),( (7.18)
Thus the 2-D Fourier transform of the absorption function a evaluates to the same
expression as the 1-D transform of the measured Radon transform m (equations (7.14)
and (7.18)), i.e.:
),(),( zMzA (7.19)
Equation (7.19) is known as the Fourier slice theorem.
7.2.2.3 The Inverse Transform
The inverse transform of equation (7.18) returns A(z,) to the spatial domain:
ddzzezAra rzj )cos(2),(),(
(7.20)
Using (7.19) and defining ),(),( zMzzG equation (7.20) becomes:
ddzezGra rzj )cos(2),(),(
(7.21)
The seemingly arbitrary multiplication of M(z,) with z in the frequency domain equates
to convolution of m(t,) with a high pass filter characteristic in the spatial domain (CH
17). The inverse transform of G(z,) is g(t,) can therefore be considered a high pass
filtered (differentiated) version of m(t,).
In the above expression, the part dzezG rzj )cos(2)(
can be written as:
)cos()( 2
rwwithdzezG wzj (7.22)
Recognizing this as the inverse Fourier transform of ),( wg , and changing the
integration limits for to 0180 degrees (or 0radian), equation (7.21) evaluates to:
00
)),cos((),(),( drgdwgra (7.23)
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Because the function g(…) is a filtered/differentiated version of m(…), equation (7.23) is
the filtered backprojection equation.
7.2.2.4 Backprojection in Cartesian Coordinates
For ease of use, we can transform (7.23) from polar to Cartesian coordinates. We