Objectives To Understand;

1. Sources of resolution loss

2. Point spread function, line spread function, edge response function

3. Convolutions and convolution theorem

4. Use of the modulation transfer function to describe the spatial frequency dependent resolution of imaging systems.

5. Calculation of the modulation transfer function

6. Magnification radiography

7. Image sampling and its effects on resolution

The spatial resolution of a system is a measure of the system’s ability to see details of various sizes. In this section we will describe a method for quantitatively describing and measuring the spatial resolution of a system.

Resolution can be limited by any of the elements in the imaging chain or, as in conventional photography, by motion of the subject.

Let’s look at a few examples of effects which can limit resolution.

Figure 1 illustrates the effect of x-ray focal spot size on spatial resolution.

Due to the finite size of the focal spot, the image of a single point in the image can be no smaller than the image of the focal spot itself. The focal spot image is called the focal spot point spread function. As magnification, defined as (d1 + d2) /d1, is increased, focal spot blurring increases.

Figure 2 illustrates the degradation of spatial resolution due to the finite thickness of a radiographic intensifying screen.

filmX-ray

Intensifying screen

t

When an x-ray is absorbed in the screen it produces visible light. This light then diverges before reaching the film producing a point spread function due to the screen. This point spread function increases with screen thickness leading to a tradeoff between screen detection efficiency and spatial resolution.

Another source of blurring is patient motion, shown below. In this case motion during the exposure leads to another point spread function ( 3A ) equivalent to that produced in the case of a stationary object and a finite focal spot ( 3B ) .

Blurredimage

AB

Moving object

v

Detector Detector

MotionEquivalentFocal spot

Blurredimage

A further example of resolution loss occurs when an image is digitized and represented by a picture element ( pixel ) matrix. This is illustrated below, which shows images of a chest phantom represented by pixel matrices of various sizes.

256 x 256 128 x 128

32 x 3264 x 64

1024x

1024

512 x 512

Studies have been done to show that for diagnosis of clinical chest films information must be digitized to 4096 x 4096 to maintain diagnostic accuracy for the most demanding tasks. However, most diagnostic tasks are done acceptably well with 2048 x 2048 matrices. The smaller matrix is more convenient in terms of data storage and retrieval. There is also evidence that digital enhancement of contrast can compensate to some extent for slight losses in resolution.

In order to quantitatively describe the effects of various system elements on the overall resolution of an imaging system the concept of modulation transfer function has been developed. In this description of the resolution properties of an imaging system components are described in terms of their ability to produce images of sinusoidally varying test objects of various spatial frequencies.

where k is the angular spatial frequency, related to the spatial frequency fx by

The concept of spatial frequency is illustrated in Figure 5 which illustrates the variation in x-ray transmission through an object with a sinusoidally varying transmission in the x-direction of the form

(2)

The contrast associated with the sinusoidal waveform is given by

Note that many discussions of MTF use for the angular spatial frequency. We will use k to be consistent with our later discussions of magnetic resonance imaging where k is used for spatial angular frequency and is used for temporal angular frequency. Spatial frequency fx is usually given in units of line pairs per mm, where one line pair or one cycle refers to a bright and dark band in the sinusoidal object transmission.

Suppose the waveform of Figure 5 is sent into an arbitrary imaging system as shown in Figure 6.

Nin Imaging system Nout

The effect of the imaging system will in general be to multiply the signal variation by a complex system transfer function M(k) which reduces contrast and in general produces a phase shift giving

The factor MTF(k) modifying the magnitude of output signal variation is called the modulation transfer function and is given by the ratio of the output to input contrast for spatial frequency k. We will refer to both fx and k as spatial frequency. The distinction between linear and angular frequency should be clear from the context of the discussion.

Physical objects can be represented as a weighted sum of spatial frequency components. Because M(k) varies with k, the imaging system will generally generate a distorted version of the actual x-ray transmission, typically failing to represent the higher spatial frequency content of the transmission.

Fourier Series Representation of the Transmitted ImageLet’s consider the case of a one dimensional transmitted fluence N(x) representing a slit of width L as shown in Figure 7a. The image profile is shown in Figure 7b.

Detector

slit

x

a

b

N(x)

xL/2-L/2

Since the transmitted image is symmetric about x=0, it can be represented as a cosine series(in general sines and cosines are required) of the form,

where

and, for any n,

(7)

(6)

(8)( dx

multiply both sides of Eq. 6 by cos(kn’x) and integrate from-L/2 to L/2

1n

2/L

2/Lnnn

2/L

2/L

n0

2/L

2/L

n

dx)x'kcos()xkcos(a

dx)x'kcos(2/adx)x'kcos()x(N

0

=Lan/2 if k= k’

= 0 k not = k’

Orthogonality

This leads to

Assuming that N(x) has a value of unity in some appropriate unit and doing the integrals, we obtain

and

giving

The graph of an which will be compared later to the Fourier transform of the transmitted image is shown below.

Note that the points go through zero at k = 2/L or fx = 1/L. (See equations 2 and 7)

See Bracewell-Chapter 2

Usually, it is more convenient to express the image as an integral over the various spatial frequency components required to represent the image. Any distribution in x can be written as a sum of integrals of the form

Since

and

(12)

This is just a sum over spatial frequencies similar to the discrete example above, but now with a continuous distribution of frequency components. The quantities Ñ+ and Ñ- are in general complex numbers and are basically the weighting coefficients for the various frequencies analogous to the discrete points in Figure 8.

For convenience, equation 11 is usually written in compact form as

(13)

The weighting function for the various sinusoidal frequencies Ñ(k) is called the Fourier transform of N(x), or FT(N(x)).

It should be realized that the appearance of negative spatial frequencies in equation 13 is a reflection of the fact that there must, in general, be terms of the form e-ikx. These terms are of course associated with positive, physically realizable spatial frequencies. The artificial concept of negative spatial frequencies just arises in association with writing the integral in compact form.

The Fourier transform or expansion coefficient of the image is related to the image through the Fourier transform relationship,

(14)

Equation (13) expressing N(x) in terms of Ñ(k) is called the inverse Fourier transform. N(x) and Ñ(k) are called a Fourier transform pair. It is interesting to mention at this point that in magnetic resonance imaging, data is obtained in the form of Ñ(k) in “k space”. The image is then obtained through an two dimensional transform equation analogous to equation 13.

Following a brief introduction to the Dirac delta function, which will be presented below, the path from equation 13 to 14 may become more clear. However ,at this point lets continue our discussion of the single slit experiment of Figure 7.

slit

x

a

b

N(x)

xL/2

The Fourier transform of the detected image is given by equation 14 as

where we have used the definition of the sinc function, namely

(16)

This sinc function is shown in Figure 9 and, aside from the normalization, has the same shape as the expansion coefficient distribution shown in Figure 8. Note that in the integral representation the weighting of the complex exponentials is symmetric about fx =o which is necessary to reconstruct the cosine behavior corresponding to the expansion of equation 6. You can think of a point at each positive frequency having an equally weighted point at negative frequency in order to construct a cosine function in accordance with the last of equations 12.

Note that the weighting function (Fourier transform) goes to zero at fx = 1/L and 2/L as in the discrete case.

The frequency space representation of images will be helpful in the understanding of spatial resolution as described by the modulation transfer function.

A physical realization of this representation is to consider the fact that it would be possible in the laboratory to construct an image on film of any one dimensional image by making x-ray exposures through a series of carefully positioned sinusoidal bar patterns of appropriately selected transmissions and spatial frequencies.

In fact the idea can be generalized to two dimensions by using bar patterns rotated by ninety degrees. In thisway it would be possible, in principle, to make an image of the Mona Lisa on film, at least a black and white version.

Jean Baptiste Joseph Fourier1768-1830

ky

kx

The Mona Lisa in k-space

Low frequency Mona k - space

ky

kx

k - space

ky

kx

High frequency Mona

Spock in k-space

The frequency content of a given image is related to its size and shape. In general, for example, the Fourier transform a broad slit will contain mostly low spatial frequencies, while a narrow slit will require much higher frequencies.

This comparison is shown in Figure 10 for slits of width L and L/4. The curve for the slit of width L corresponds to the sinc function of Figure 9. The first zero of the Fourier transform of the slit of width L/4 occurs at fx = 1/(L/4) = 4/L.

Figure 10

Note regarding area of integrals in x-space and k-space:

Since

so the signal at the origin of k-space is equal to the integral over all image space signal.

It can also be seen that since

i.e. the value at the center of x-space is the integral of the k-space signal.

dk)k(N~21

)0(N

In Figure 10 it is assumed that the intensity has been kept the same as the slit width has been decreased. The Fourier transform goes from

Since N(0) is the same in each case (fixed intensity through the slit), the integrals in k-space are the same.

One further example will serve to introduce a mathematical function which will be useful in several subsequent sections. Figure 11 shows the Dirac delta function and the magnitude of its Fourier transform.

Figure 11

(x-a)

a x

Magnitude of Fourier transform of delta function

0 k

The delta function, located for example at x=a, is an infinitely narrow, infinitely intense signal distribution defined by the following properties

The integral property of the delta function whereby it selects out the value of the integrand at the point where the argument of the delta function is zero is very convenient in a number of applications. For example in calculating the Fourier transform of x-a) we obtain

(18)

For points away from x=o the complex Fourier transform is modulated by the phase factor e-ika which shuffles signal between the real and imaginary parts. However, the magnitude of the Fourier transform is constant at all frequencies.

The MTF of each imaging element multiplies the expansion coefficient (Fourier transform) of the image entering that element at each spatial frequency. For example consider the fluence N(x) incident on the imaging system element which records or further transmits a modified representation of the fluence NS(x) as shown in Figure 12.

Ns(x)

N(x)

If we represent the input fluence by equation 13,

then the system element will represent the image as

The system has degraded the information at each spatial frequency by the value of the system transfer function at that frequency.

The MTF of a given system element can be measured if the Fourier transform of the signal incident on that element is known. For example, by using a thin slit which simulates a delta function input distribution, the MTF of a detector can be found.

The geometry for this experiment is shown in Figure 13.

Figure 13

Because of the narrow slit and near unit magnification, the rays from the finite focal do not diverge before hitting the detector and the input intensity distribution may be considered to be a delta function at x=0, (0). The signal recorded by the detector ND will be given by

where MD(k) is the system transfer function associated with the detector.

Since from equation 18, for a delta function at x=o,

we can write

Since the right side is in the form of a Fourier transform, we can solve for MD(k) by doing the inverse transform, giving

(22)

In general the response of a system element, in this case ND(x), to a line image input is called the Line Spread Function (LSF) for that element. A general formula for a normalized system transfer function may be obtained by normalizing to the zero frequency value.

(23)

The MTF is usually defined as the magnitude of this transfer function as a function of positive spatial frequencies. This is a general recipe for finding the MTF of a system element providing that elements LSF is known.

III

III

Pick the one you think has the best MTF

Students usually select the image with the highest MTF values at low spatial frequencies.

Although the curve with a higher spatial frequency cutoff displays objects of finer detail, the contrast at low spatial frequencies is inferior and presents a less appealing image.

Which transfer function is more suitable depends on whether the imaging task is to find large or small objects.

As a further example, we will now consider the measurement of the MTF associated with the focal spot. In this case the LSF of the focal spot is found by imaging a narrow slit as before but this time with arbitrary magnification m= (d1+d2)/d1 as shown below.

If we use non-screen film as a detector, we may assume that the MTF of the film is unity for all spatial frequencies where the focal spot MTF is non zero. The LSF is a magnified version of the focal spot intensity distribution F(y),

(24)

Let us calculate the focal spot MTF assuming that the focal spot distribution is a rectangle function defined by

Then

Using equation 23 we can calculate the focal spot transfer function Mf as

This MTF associated with this function is shown in Figure 15. The MTF has zeros at detector plane frequencies of fx = n / [a(m-1)].For a 1mm focal spot, Table 1 shows the spatial frequency at the first zero of the MTF for various magnifications. Clearly, the effect of focal spot blurring increases quickly with magnification.

10 lp/mm

2 lp/mm

1 lp/mm

Magnification Spatial Frequency

An extension of this example allows us to calculate the MTF associated with uniform motion. Figure 16 shows the equivalent focal spot as seen from the moving object.

Focal Spot

object

v∆t

Equivalent Focalspot

•

object

detector

d1 d2

Figure 16

In this case the focal spot LSF has a width of mVt and the equivalent focal spot width am is given by

or

If we go back to equation 27

Eq. 27

we obtain the transfer function by substituting the equivalent focal spot for a

As V increases, the first zero of the transfer function moves to lower and lower spatial frequencies.

Mf

We have already seen that due to various factors such as finite focal spot size, motion or limitations in the detector, the image of a line object such as a slit will instead be recorded as a line spread function. Consider a one dimensional imaging system which is otherwise perfect except for the detector as shown in Figure 17.No

N(x’)

Detector

NR(x)

The correct signal N(x’) at each point in x’ will be spread to remote points x in the detector in accordance with the line spread function LSF(x-x’) which indicates how much of the signal aimed at x’ will show up at x as the recorded signal NR(x). This is shown for a delta function N(x’) distribution in Figure 18.

NR(x)

x

x’

x-x’

N(x’) = (x-x’)

Figure 18

In the case of a more general image distribution N(x’) the recorded signal will be given by the sum of all of the signal contributions spread from all points within the image.

The integral in equation 28 is called the CONVOLUTION of N(x) and LSF(x) and is indicated by the sign.

Such a representation of the blurred image assumes that the system is STATIONARY meaning that the LSF is the same at all points and that the system is LINEAR meaning that contributions from all coordinates sum at a distant coordinate linearly.

For a two dimensional system, equation 28 can be generalized in terms of a POINT SPREAD FUNCTION PSF(x-x’, y-y’).

(28)

See Bracewell - Chapter 3

The convolution C(x) of two functions A(x) and B(x) is given by the integral

(29)

where the limits in general go to + or - infinity but more often are determined by the range of the functions involved. Some of the treatments of convolution are confusing in terms of visualizing what is going on. Perhaps the easiest way is to pretend that one of the functions is made up of a continuous distribution of delta functions. Lets try to illustrate this.

Suppose we want to convolve the two functions in figure 19.

x

x

A(x)

B(x)

a

Substituting into equation 1, the convolution is given by

(30)

The delta function picks out the value of B at the point x’ = X, giving

(31)

That is a shifted version of B(x) with its new origin at the location of the delta function as shown in Figure 20.

x

C(x)

x

Now let us consider a quantity analogous to the line spread function, namely the edge response function, ERF .

The edge response function is the convolution of the edge with the system line spread function which describes how the signal from each of the continuous set of delta functions making up the edge distributes its signal to distant locations.

We will see that the edge response function is related to the line spread function and provides an alternate means of measuring system MTF.

Suppose we convolve the image of a sharp edge with the line spread function of the imaging element. The functions to be convolved are shown in Figure 20.

We have represented the edge as a continuum of delta functions, in this case all equally weighted. The convolution can then be represented as a set of displaced line spread functions centered at each of the delta functions making up the edge function.

Figure 20

Figure 21

A few of these are shown in Figure 21. It can be seen that addition of the contributions of all of these displaced line spread functions will produce an edge response function which is gradually rolled off at the edge and which comes to a constant value at large distances from the edge.

This way of thinking about the convolution process works for more general functions. For example you should convince yourself that the convolution of two rectangle functions produces a triangle function.

Once again, represent one of the rectangle functions as a continuum of delta functions and add up the displaced versions of the other rectangle function.

The edge response function provides a more convenient way of measuring MTF than the line spread function, simply because it is easier to make an edge than a slit.

The mathematical relationship between the two is illustrated below. The ERF is the convolution of the edge E(x) with the LSF. Assuming E(x) =0 for x<0 and 1 otherwise,

if we make the substitution that u=x-x’ and du = -dx’ we get

(33)

To understand the meaning of this relationship it is helpful to draw the integration over the LSF as shown in Figure 22.

du

LSF(u)

ERF

integralunder curve

The incremental increase in the ERF integral is given by

(34)

In other words, by the fundamental theorem of calculus, we can obtain line spread function as the derivative of the edge response function.

The recipe of equation 23

can then be used to calculate the system transfer function and MTF. The edge response method is commonly used to evaluate radiographic detector resolution.

(23)

Suppose that NR(x) is the convolution of N(x) and LSF(x) ie,

(35)

Then, in the Fourier transform convention we have chosen ( equations 13 and 14) the k space transforms are related by (See Figure 28 in Math Appendix)

This can be proved by writing out the explicit integral forms of these relationships as shown in the appendix to this section.

This result can be extended to a series of imaging elements, each of which degrade the system resolution by imposing an additional convolution of its input image with its LSF.

For example, in a chain with two image elements and an input image N(x), we would have

Applying the convolution theorem in two steps we obtain

In other words, recalling that the MTF is basically the FT of the LSF, normalized to 1 at zero spatial frequency, the MTF of the eventually recorded image is the product of the original image transform multiplied by MTFs of all of the serial imaging elements leading to the overall system MTFS for a series of N imaging elements of the form

An illustration of equation 37 occurs in the case of magnification radiography.

When radiographic magnification is used, the overall system MTF is the product of the focal spot MTF and the MTF of the detector. The geometry is shown in Figure 23.

From equation 27, the MTF of the focal spot is given by,

where fx is the spatial frequency in the detector plane. The relationship between the detector frequency fx and the patient frequency fpatient is

(38)

If we model the detector line spread function as a rectangle function of width d, the detector MTF is given by

The product MTF MfsMDet is plotted in Figure 24 for magnifications of 1.0, 1.33 and1.6 for the case of a =0.75 mm, and d = 0.25 mm.

At small magnification, the detector resolution is the dominant degrading factor. At a(m-1)/m = d/m which in this case occurs at m= 1.33 the line spread functions of the focal spot and detector are matched and the resolution is optimal. At this magnification, the component MTF values each go through zero at a spatial frequency of 5.3 lp/mm in the patient plane. At larger magnifications the degradation due to the focal spot overcomes any further gain due to magnification of the image relative to the detector resolution element.

Optimal magnification obviously will vary with the actual focal spot size and detector resolution. In general, smaller focal spots permit greater magnification. Better detectors require less magnification.

See Bracewell - Chapter 10, Hasegawa - Chapter 6

We have seen in a previous chapter how digitization of the image has noise consequences. Sampling an image for the purpose of eventually representing it as a pixel matrix can also affect the fidelity of the representation of the image if certain criteria are not met.

For example, if the sample spacing is not sufficiently small, high spatial frequencies can be interpreted as lower spatial frequencies. This phenomenon, illustrated in Figure 25 is called aliasing. The sample points, which in this case are too far apart to adequately characterize the spatial frequency shown, represent it instead as a low spatial frequency.

We will describe the basic Sampling Theorem which states that if a function (e.g. one dimensional image) to be sampled has a maximum frequency (is band limited ), then there is a maximum sampling distance which will faithfully represent the function.

Figure 25

The maximum frequency is called the Nyquist Frequency and is given by

(39)

wherex is the distance between samples, the eventual pixel size.

Figure 26 illustrates the basic considerations involved in sampling.

Figure 26 A shows the image to be sampled. B and C indicate sampling at two different spacingsx1 and x2.

The sampling is accomplished by a sequence of delta functions collectively called the shah function represented by III(x).

The Fourier transform of a shah function III(x /x) (sometimes called a comb funtion) with spacing x is another shah function xIII(xx) with spacing in fx space of 1/x.

The Fourier transforms of B and C are shown in I and J

Figure 26

Sampling A with B or C is just a multiplication by the shah function resulting in the sampled images shown in D and E.

The Fourier transform of the original image shown in H has been band-limited to the Nyquist frequency, usually by means of an analog or digital filter.

According to the convolution theorem, the frequency space representation of this image is given by the convolution of the Fourier transform of the image (H) with the Fourier transforms of the shah functions I and J, resulting in the frequency space representations of the image shown in F and G.

In the case of F, the sampling space x1 has been chosen to be such that there are two samples per cycle at the Nyquist frequency. In this case the replicated Fourier transforms of the original image do no overlap and there is no image distortion.

In G the sampling has been too sparse, resulting in overlap of the replicated frequency space images. In this way, large negative spatial frequencies from the first replicated distribution are aliased back into the positive spatial frequencies of the primary image leading to distortions and artifacts.

One of the most striking manifestations of aliasing of this sort occurs in magnetic resonance imaging where there is a direct relationship between spatial position and temporal frequency. Inability to band limit in one of the image directions leads to the appearance of objects at temporal frequencies above the Nyquist frequency, wrapping from the right side of the image to a lower frequency at the left side of the image. This leads to the often seen “nose in the back of the head” artifact.

When the properly sampled image represented by Figure 26 D and F are used to form a pixel representation of the image, Figure 26 D is basically convolved with a rectangle function Rect( x /x1), sometimes represented by II( x /x1) which is a rectangle with width x1. The digital image is then given by

(40)

This is shown in Figure 27A. The Fourier transform of this, by the convolution theorem is given by

(41)

and is shown in Figure 27B.

Note that the replicated frequency spectrum shown in Figure 26F is now modulated by the pixel frequency spectrum. Note that the pixel frequency spectrum has its first zero at twice the Nyquist frequency and therefore contributes only minor degradation within the frequency range up to the Nyquist frequency.

.

Integral Form of the delta function

The delta function may be represented in the form of an integral. This representation of the delta function has the previous properties stated above, but is a convenient form to recognize whenever it occurs in the course of manipulating integrals. The integral form of the delta function is given by.

(42)

This form is useful in showing the relationship between a function and its Fourier transform as expressed by equations 13 and 14. From equation 13, for example, we have

To solve for the Fourier transform we can multiply by e-ik’x and integrate over x, giving

Reversing the order of integration this equals

since the term in brackets is just 2(k-k’). Therefore, we obtain equation 14,

Suppose that NR(x) is the convolution of N(x) and LSF(x, .ie,

Taking the Fourier transform we obtain

Performing the x integral first we get,

letting u=x-x’, du = dx, we get

This convolution theorem is illustrated in Figure 28 for the case of the two rectangle functions by an example taken from “THE FAST FOURIER TRANSFORM” by E. Oran Brigham, Prentice Hill Inc.

In this example the Fourier transform of each of the rectangle functions is a sinc function. The convolution of the two rectangles is a triangle function as stated above. When the two sinc functions are multiplied in frequency space they form a sinc2 function which is the Fourier transform of the triangle as stated by the convolution theorem.

Tables of Fourier transform pairs may be found in Bracewell, page 100 or Hasegawa, Table 5.1 and Figure 5.1. Note that Bracewell’s s corresponds to our k, while Hasegawa’s u corresponds to fx.