∫
∞ 2( ) ( ) ifx
G f x g x eπ dx
[4-7]
where i is −1. If, for example, the input signal g(x) to the Fourier transform is a sine wave with a frequency , the function G(f ) will be a plot with spikes at frequen-cies 6, and it will be zero elsewhere. The function G(f ) will in general consist of complex numbers, with real and imaginary components.
The Fourier transform, FT[], converts a temporal or spatial signal into the fre-quency domain, while the inverse Fourier transform, FT21[], converts the frequency domain signal back to the temporal or spatial domain. The Fourier transform can be used to perform convolution. Referring back to Equation 4-1, the function G(x) can be alternatively computed as
{ }
( ) 1 [ ( )] [ ( )]
G x FT FT H x FT k x [4-8]
Equation 4-8 computes the same function G(x) as in Equation 4-1, but compared to the convolution procedure, in most cases the Fourier transform com-putation runs faster on a computer. Therefore, image processing methods that employ convolution filtration procedures such as in CT are often performed in the frequency domain for computational speed. Indeed, the kernel used in CT is often described in the frequency domain (descriptors such as ramp, Shepp-Logan, bone kernel, B41), so it is more common to discuss the shape of the kernel in the frequency domain (e.g., FT[k(x)]) rather than in the spatial domain (i.e., k(x)), in the parlance of clinical CT.
Fourier computations are a routine part of medical imaging systems. Fourier transforms are used to perform the filtering procedure in filtered back projection, for CT reconstruction. Inverse Fourier transforms are used in MRI to convert the measured time domain signal into a spatial signal. Fourier transforms are also used in ultrasound imaging and Doppler systems.
The Modulation Transfer Function, MTF(f)
Conceptual DescriptionImagine that it is possible to stimulate an imaging system spatially with a pure sinu-soidal wave form, as illustrated in Figure 4-13A. The system will detect the incoming sinusoidal signal at frequency f, and as long as the frequency does not exceed the Nyquist frequency (discussed later) of the imaging system (i.e., f , FN), it will pro-duce an image at that same frequency but in most cases with repro-duced contrast (Fig.
4-13A, right side). The reduction in contrast transfer is the result of resolution losses in the imaging system. For the input sinusoidal signals with frequencies of 1, 2, and 4 cycles/mm (shown in Fig 4-13A), the recorded contrast levels were 87%, 56%, and 13%, respectively, on the measured images. For any one of these three frequencies measured individually, if the Fourier transform was computed on the recorded signal, the result would be a peak at the corresponding frequency (Fig. 4-13A, right side).
Three such peaks are shown in Figure 4-13B, representing three sequentially and
1 cy / mm
A 2 cy / mm
4 cy / mm
1 mm
Input signal
87 %
56 %
13 % Measured signal
1.0
B 0.9 0.8 0.7 0.6
MTF (f) 0.5
Spatial Frequency (cy / mm) 0.4
0.3 0.2 0.1 0.0
0 1 2 3 4 5
FIGURE 4-13
■
■ A. Sinusoidal input signals are incident on a detector (intensity as a function of position), and three different frequencies are shown as input functions (left). The signals measured by the imaging system are shown on the right—the frequency is the same as the inputs in all cases, but the amplitude of the measured sig-nal is reduced compared to that of the input sigsig-nal. This reduction in amplitude is a result of resolution losses in the imaging system, which are greater with signals of higher frequencies. For the input at 1 cycle/mm, the original 100% amplitude was attenuated to 87%, and with the 2- and 4-cycle/mm input functions, the resulting signal amplitudes were reduced to 56% and 13%, respectively. B. This figure shows the amplitude reduction as a func-tion of spatial frequency shown in A. At 1 cycle/mm, the system reduced the contrast to 87% of the input. For 2 mm21 and 4 mm21, the signal was modulated as shown. This plot shows the MTF, which illustrates the spatial resolution of an imaging system as a function of the spatial frequency of the input signal.
individually acquired (and then Fourier transformed) signals. The amplitude of the peak at each frequency reflects the contrast transfer (retained) at that frequency, with contrast losses due to resolution limitations in the system. Interestingly, due to the characteristics of the Fourier transform, the three sinusoidal input waves shown in Figure 4-13A could be acquired simultaneously by the detector system, and the
1.0 0.9 0.8 0.7 0.6
MTF(f) 0.5
0.4 0.3 0.2 0.1 0.0
0.0 0.2 0.4 0.6 0.8 10 % MTF
limiting spatial resolution
1.0
Spatial Frequency (cycles / mm)
1.2 1.4 1.6 1.8 2.0
FIGURE 4-14
■
■ The limiting spatial resolution is the spatial frequency at which the amplitude of the MTF decreases to some agreed-upon level. Here the limiting spatial resolution is shown at 10% modulation, and the limiting spatial resolution is 1.6 cycles/mm.
Fourier Transform could separate the individual frequencies and produce the three peaks at F 5 1, 2, and 4 cycles/mm shown in Figure 4-13B. Indeed, if an input signal contained more numerous sinusoidal waves (10, 50, 100,….) than the three shown in Figure 4-13A, the Fourier transform would still be able to separate these fre-quencies and convey their respective amplitudes from the recorded signal, ultimately resulting in the full, smooth modulation transfer function (MTF) curve shown in Figure 4-13B.
Practical Measurement
It is not possible in general to stimulate a detector system with individual, spatial sinusoidal signals as described in the paragraph above. Rather, the LSF, discussed previously and illustrated in Figures 4-6 and 4-7, is used to determine the MTF in experimental settings. A perfect line source input (called a delta-function), it turns out, is represented in the frequency domain by an infinite number of sinusoidal functions spanning the frequency spectrum. Therefore, the Fourier transform of the LSF(x) computes the full MTF curve as shown in Figure 4-13B. Prior to the computa-tion, the LSF(x) is normalized to have unity area
∞
∫
x ∞LSF x dx( ) 1 [4-9]
Then the Fourier transform is computed, and the modulus (brackets) is taken, resulting in the MTF(f )
∞
=
∫
∞ 2( ) ( ) ifx
MTF f x LSF x eπ dx
[4-10]
Limiting Resolution
The MTF gives a rich description of spatial resolution, and is the accepted standard for the rigorous characterization of spatial resolution. However, it is often useful to have a single number value that characterizes the approximate resolution limit of an imaging system. The limiting spatial resolution is often considered to be the frequency at which the MTF crosses the 10% level (see Fig. 4-14), or some other agreed-upon and specified level.
Nyquist Frequency
Let’s look at an example of an imaging system where the center-to-center spacing between each detector element (dexel) is D in mm. In the corresponding image, it would take two adjacent pixels to display a full cycle of a sine wave (Fig. 4-15)—one pixel for the upward lobe of the sine wave, and the other for the downward lobe. This sine wave is the highest frequency that can be accurately measured on the imaging system. The period of this sine wave is 2D, and the corresponding spatial frequency is FN 5 1/2D. This frequency is called the Nyquist frequency (FN), and it sets the upper bound on the spatial frequency that can be detected for a digital detector system with detector pitch D. For example, for D 5 0.05 mm, FN 5 10 cycles/mm, and for D 5 0.25 mm, FN 5 2.0 cycles/mm.
If a sinusoidal signal greater than the Nyquist frequency were to be incident upon the detector system, its true frequency would not be recorded, but rather it would be aliased. Aliasing occurs when frequencies higher than the Nyquist frequency are imaged (Fig. 4-16). The frequency that is recorded is lower than the incident fre-quency, and indeed the recorded frequency wraps around the Nyquist frequency. For example, for a system with D 5 0.100 mm, and thus FN 5 5.0 cycles/mm, sinusoi-dal inputs of 2, 3, and 4 cycles/mm are recorded accurately because they obey the Nyquist Criterion. The Fourier transform of the detected signals would result in a
Sampling pitch Aperture width
∆ a
FIGURE 4-15
■
■ This figure illustrates a side view of detector elements in a hypothetical detector system.
Detector systems whose center-to-center spacing (pitch) is about equal to the detector width are very com-mon, and these represent contiguous detector elements. The sampling pitch affects aliasing in the image, while the aperture width of the detector element influences the spatial resolution (the LSF and MTF). A sine wave (shown) where each period just matches the width of two detector elements is the highest frequency sine wave that can be imaged with these detectors due to their sampling pitch.
measured signal
input signal
digital sampling FIGURE 4-16
■
■ The concept of aliasing is illustrated. The input sine wave is sampled with the sampling comb (arrows), but the Nyquist criterion is violated here because the input sine wave is higher than the Nyquist fre-quency. Thus, the sampled image data will be aliased and the lower frequency sine wave will be seen as the measured signal on the image.
histogram of the frequency distribution of the measured sinusoid frequencies, but since the input functions are single frequencies, the Fourier transform appears as a spike at that frequency (Fig. 4-17). For sinusoidal inputs with frequencies greater than the Nyquist frequency, the signal wraps around the Nyquist frequency—a fre-quency of 6 cycles/mm (FN 1 1) is recorded as 4 cycles/mm (FN21), and a frequency of 7 cycles/mm (FN 1 2) is recorded as 3 cycles/mm (FN22), and so on. This is seen on Figure 4-17 as well. Aliasing is visible when there is a periodic pattern that is imaged, such as an x-ray antiscatter grid, and aliasing appears visually in many cases as a Moiré pattern or wavy lines.
The Presampled MTF
Aliasing can pose limitations on the measurement of the MTF, and the finite size of the pixels in an image can cause sampling problems with the LSF that is measured to compute the MTF (Equation 4-10). Figure 4-6 shows the angled slit method for determining the presampled LSF, and this provides a methodology for computing the so-called presampled MTF. Using a single line perpendicular to the slit image (Fig. 4-6A), the sampling of the LSF measurements is D, and the maximum fre-quency that can be computed for the MTF is then 1/2D. However, for many medical imaging systems it is both possible and probable that the MTF has nonzero ampli-tude beyond the Nyquist limit of FN 5 1/2D. In order to measure the presampled MTF, the angled-slit method is used to synthesize the presampled LSF. By using a slight angle between the long axis of the slit and the columns of detector elements in the physical measurement of the LSF, different (nearly) normal lines can be sampled (Fig. 4-6B). The LSF computed from each individual line is limited by the D sam-pling pitch, but multiple lines of data can be used to synthesize an LSF, which has much better sampling than D. Indeed, oversampling can be by a factor of 5 or 10, depending on the measurement procedure and how long the slit is. The details of how the presampled LSF is computed are beyond the scope of the current discus-sion; however, by decreasing the sampling pitch from D to, for example, D/5, the
Fin = 2
Relative amplitude
Spatial Frequency (cycles/mm) Fin = 3
0 1 2 3 4 5
Fin = 4
Fin = 6
Fin = 7
0 1 2 3 4 5
Fin = 8
FIGURE 4-17
■
■ For a single-frequency sinusoidal input function to an imaging system (sinusoidally varying intensity versus position), the Fourier transform of the image results in a spike (delta function) indicating the measured frequency. The Nyquist frequency in this example is 5 cycles/mm. For the three input frequencies in the left panel, all are below the Nyquist frequency and obey the Nyquist criterion, and the measured (recorded) frequencies are exactly what was input into the imaging system. On the right panel, the input frequencies were higher than the Nyquist frequency, and the recorded frequencies in all cases were aliased—they wrapped around the Nyquist frequency—that is, the measured frequencies were lower than the Nyquist frequency by the same amount by which the input frequencies exceeded the Nyquist frequency.
Nyquist limit goes from FN to 5FN, which is sufficient to accurately measure the as flat panel detector systems (as a concrete example) have detector elements that are essentially contiguous (i.e., very little dead space between adjacent detector elements). Thus, for a detector where the detector pitch (the center-to-center dis-tance between adjacent detector elements) is D, the width of each detector element (a) is about the same (Fig. 4-16), that is, a D. This finite width of each detector element means that all of the signal which is incident on each dexel is essentially averaged together, and one number (the gray scale value) is produced for each detector element (in general). This means that on top of any blurring that may occur from other sources, the width of the dexel imposes a fundamental limit on spatial resolution. In a very literal sense, the detector width—idealized as a rectan-gular (RECT) response function—is the best the LSF can ever get on a digital imag-ing system. The RECT function is a rectangle-shaped function of width a, usually centered at x 5 0, such that the RECT function runs from 2a/2 to 1a/2. Let the LSF be a RECT function with width a, and the MTF is then computed as the Fourier transform of the LSF. This Fourier transform can be computed analytically, and the resulting MTF will be given by
∞
A plot of this MTF is illustrated in Figure 4-18. The function on the right side of Equation 4-11 is commonly called a sinc function. The sinc function goes to zero at F 5 1/a, but aliasing occurs above the Nyquist frequency at 1/2D. Thus, for a system which has an MTF defined by the sinc function, the possibility of aliasing is high
1.0 5 cycles/mm and the amplitude of the MTF goes to zero at 1/a = 10 cycles/mm. This MTF shows the SINC function—the analytical Fourier transform of the RECT function of width a. For an imaging system with a detector aperture of width a, this curve represents the best MTF possible for that detector.
since the MTF still has nonzero amplitude above FN. An imaging system that has an MTF defined by the sinc function is performing at its theoretical maximum in terms of spatial resolution.
Field Measurements of Resolution Using Resolution Templates Spatial resolution should be monitored on a routine basis for many imaging modali-ties. However, measuring the LSF or the MTF is more detailed than necessary for routine quality assurance purposes. For most clinical imaging systems, the evalu-ation of spatial resolution using resolution test phantoms is adequate for routine quality assurance purposes. The test phantoms are usually line pair phantoms or star patterns (Fig. 4-19). The test phantoms are imaged, and the images are viewed to estimate the limiting spatial resolution of the imaging system. There is a degree of subjectivity in such an evaluation, but in general viewers will agree within acceptable limits. These measurements are routinely performed on fluoroscopic equipment, radiographic systems, nuclear cameras, and in CT.
4.5
Contrast Resolution
Contrast resolution refers to the ability to detect very subtle changes in gray scale and distinguish them from the noise in the image. Contrast resolution is character-ized by measurements that pertain to the signal-to-noise ratio (SNR) in an image.
Contrast resolution is not a concept that is focused on physically small objects per se (that is the concept of spatial resolution); rather, contrast resolution relates more to anatomical structures that produce small changes in signal intensity (image gray scale), which make it difficult for the radiologist to pick out (detect) that struc-ture from a noisy background.