The spatial resolution of an x-ray or CT system is a measure of how the ability of a system to differentiate small structures. If you imagine imaging a very small point like object an image of that object is called the Point Spread Function (PSF). When this function is radially averaged the Line Spread Function (LSF) is generated. Taking the Fourier Transform of the LSF yields the Modulation Transfer Function (MTF) which measures how spatial frequencies pass through the system. The way that we measure how different frequencies of noise propagate through the system is the Noise Power Spectrum (NPS). Finally a metric of how well the system is using the information in the incoming x-rays is the Detective Quantum Efficiency (DQE). In this article we will be going through these concepts so that you can understand the basics of what is called linear systems theory.

# Image Sharpness and Image Spatial Resolution

The spatial resolution or the image sharpness is something that everyone is familiar with in their camera or television. As radiologic technologist or radiographers it is important to understand the spatial resolution of the system and methods that can be used to measure the spatial resolution.

With higher image resolution we can differentiate smaller objects. For example, small fine cracks in bone can be well visualized in x-ray imaging, or very small calcifications in breast mammography imaging.

Typically, when we discuss the image resolution it is for high contrast imaging such as bone or contrast agent. We use other metrics to denote the ability of the imaging system to observe low contrast features.

X-ray imaging has higher spatial resolution than other modalities such as CT, MRI, SPECT, PET or ultrasound. In this post we present the general framework for understanding and measuring spatial resolution.

**R****a****d Take-home Point**: Image resolution is an important component of x-ray system properties and determines how well very small structures are visualised.

## Direction Observation of Spatial Resolution

The most direct way to observe the spatial resolution of a system is make an image with objects that vary in size. In this way you can look for the smallest objects that you can differentiate.

The easiest way to design a phantom to assess the resolution is to have alternating bar patterns of air and metal (e.g. lead). Other methods include having bars that start off far apart and then get closer together.

In any case the direct observation techniques all have one thing in common that an image is taken, and then a person reads the image to determine what is the smallest feature that can be seen.

If we look at bar patterns, there are alternating sequence of light and dark areas in the image. The thickness of these bars defines frequency of white and dark bars. The bars that are larger correspond to lower spatial frequencies and the bars that are smaller correspond to the higher spatial frequencies.

As the frequency of the bars increases, it becomes more and more difficult to differentiate the separate light and dark bars. In the figure you can see a sample line pair phantom for several different systems. The systems toward the bottom of the image have the lowest resolution and the systems toward the top have the highest resolution. The same bar pattern is imaged on each system.

On the systems with higher resolution it is easier to differentiate more bar patterns. Thus, the resolution of the system can be determined by imaging an alternating phantom and specifying the last set of bar patterns that can be seen as separate. The frequency of these bars is measured in line pairs per millimeter (lp/mm).

The advantages of direct measurement based on a line pair phantom is that it is direct and can be very easily explained. The primary disadvantage of the direct measurements is that it is subjective and there is can be variation from one user to another (e.g. the other techs may not see the small line patterns as clearly as you do).

**Rad Take-Home Point**: Image resolution can be directly visualized with images of a bar pattern where the limiting resolution can be determined by the smallest set of line pairs that can be seen.

## Linear Systems for X-ray Technologists

When images are taken on a real x-ray system there will always be some blurring that takes place in the system due to the size of the focal spot and the size of the detector. The idea of linear systems modeling is that we can model the blurring in our x-ray system.

The blurring of the image that occurs when an image is made on an x-ray system can be thought of as starting with a perfect, â€˜ideal imageâ€™ and then blurring it out more. As the size of the detector elements gets larger the blurring will be increased.

We can think about each point in an ideal image and what happens in reality is that in the measured image each point in the ideal image is blurred by adding contributions from its neighbors. That blurring is called a point spread function (PSF). Each point in the image gets spread to the neighboring points in the image.

As the PSF gets larger there is more blurring into neighboring regions and as the PSF gets smaller there is less blurring.

In the figure you can see that if you start with an ideal image then apply the PSF to the whole image by blurring each point in the image, that the output image is generated. In this way we have a model for how the system behaves.

The blurring is usually modeled in 2D on the images (both up and down and left to right). The blurred shape we see is function of the system and it is called Point Spread Function.

In the figure for the perfect system, we can easily see all of the bar patterns. But in a real system the bar patterns will be blurred and the neighboring objects become hard to tell apart (visually differentiate).

In the next section we will describe how this linear system can be used to make measurements of spatial resolution.

**Rad Take-home Point**: The point spread function (PSF) is a description of the system blurring in image space.

## Modulation Transfer Function (MTF) (Frequency Space)

In the figure in this section you can see an ideal bar pattern and a realistic (blurred) bar pattern. If we draw a line down the middle of the bar patterns an make a plot of the values in the image we see that they go up and down (bright and dark in the image).

In the ideal image case they go up and down the same amount for all bar patterns. The amplitude separation between the bright and dark bars is the contrast for those bars.

In the real system the contrast is reduced for the higher resolution bars (seen on the bottom of the figure). Next, we will introduce a method to describe the resolution directly in frequency space.

Each of the bar patterns has a given width and this is measured in units of line pairs per mm (lp/mm).

The wider bar patterns have a lower frequency (fewer line pairs per mm) and the narrow bar patterns have a higher frequency (more line pairs per mm).

On the bottom of the figure we have a curve that goes down for higher frequencies. This curve is called the Modulation Transfer Function (MTF). The MTF is used to describe how quickly the ability to see the finer bar patterns goes down (i.e. how does the system effect the higher spatial frequencies).

**Rad Take-home Point**:

- The modulation transfer function (MTF) describes the frequency behavior of the system and is a curve that has lower values for high frequencies which represent the small image structures.
- Small image structures correspond to high frequencies, and large image structures correspond to low frequencies.

## Link between PSF and the MTF

We described the point spread function (PSF) and the modulation transfer function (MTF). In this section we will describe the link between the PSF and the MTF.

The point spread function (PSF) is a blurring function that is measured in the image space. The PSF is often assumed to be symmetric. If we take the Fourier Transform of that point spread function, we get the MTF or the Modulation Transfer Function.

This is the same Fourier transform that is used in MRI imaging where the data is collected in Frequency space and the Fourier transform is used to convert to image space (image reconstruction).

The advantage of using the MTF compared with direct visual observation is that MTF provides a very quantifiable number, that is not dependent on a subjective reader.

The different points on the MTF curve describe the system resolution can be compared from system to system. It is standard process to report the frequency (x-axis value) when the MTF curve is at 50% of the maximum value (MTF_{50}), and when the MTF curve is at 10% of the maximum value (MTF_{10}).

We can do this over and over on different systems and get a quantifiable number and all we have to do is scan a wire. If the wire is much smaller than the size of the detector, then all we can ignore the size of the wire.

So all we need to do is to scan a wire and take the Fourier Transform to get the MTF.

That relates to how well you can see the small structures in your image and we can do this in a quantifiable and repeatable way. This gives the same type of information as the direct observation of bar patterns but is less dependent on the reader of the images.

**Rad Take-home Point**: The modulation transfer function (MTF) is the Fourier transform (frequency space representation) of the point spread function (PSF).

# Detective Quantum Efficiency (DQE) and Noise Power Spectrum and more on linear systems

When you go to buy a new car one of the important things that you usually check is how efficient the car is and how many miles you will be able to travel on average on a gallon of gas (mpg, or kpl-kilometers per liter). Likewise, if your water heater breaks down as you are franticly scrolling through the options for a replacement you will often look at the efficiency of the water heater.

The same concept applies in x-ray imaging and we would like to have a metric that can be compared across imaging systems so that we can compare the efficiency of the imaging system at converting from incoming x-rays into a diagnostic medical image.

This is where the Detective Quantum Efficiency (DQE) metric comes in and can be used to compare among different imaging systems in order to determine which system is the most efficient for each imaging task that may be performed on the system.

The first person to demonstrate the basic relationship between contrast, object size and the ability of human to visualize objects was Albert Rose in 1948. He also introduced concept of Detective Quantum Efficiency (although he used a slightly different name, and the details of the theory came later). At the heart of the DQE methodology is Linear Systems theory, so we will briefly introduce this topic again here.

## Linear Systems Theory

Linear systems theory is a general methodology which is applied to x-ray imaging to study the process of converting the input signal generated by x-rays as they are converted into the final image.

Linear systems theory gets its name from the assumption that the system will behave linearly, i.e. no funny behavior where small changes in the input can lead to large changes in the output.

A powerful component of linear systems theory is the ability to decompose the image into different spatial frequencies. Just as piece of music can be made up of different notes added together (i.e. different spatial frequencies), we may form an image by breaking it down into different spatial frequencies.

The image signal and the spatial frequencies are related by the Fourier transform. So if we have an image and we take the Fourier transform of that image we will separate the image into the different spatial frequencies. The lower frequencies provide more of the contrast in an image, and the higher frequencies provide more of the edge details.

The Fourier transform gives us the ability to think about the x-ray image as a combination of sinusoids (i.e. sine waves at different angles). Linear systems theory allows us to track how different spatial frequencies as they pass through the imaging system?

In the image we have different sinusoids with different frequencies. This means sinusoids with lower frequency oscillate less frequently in spatial domain (i.e. large regions) and sinusoids with higher frequency oscillate much more frequently (i.e. edge regions). Lower frequency sinusoids depict large parts of body while small features like a crack in the skull or microcalcifications are a result of higher frequencies waves passing through the imaging system.

**R****a****d Take-home Point**: Linear systems theory can be used to track how different spatial frequencies are passed through the system, and example components of linear systems theory are discussed below: MTF, NPS and DQE.

## Modulation Transfer Function

The MTF measures the amplitude (ie. signal brightness) change of different spatial frequencies as they pass through the imaging system.

We can make an analogy to the resolution bars, higher spatial frequencies correspond to more bars (i.e. more line pairs) in the same distance. We discussed this in the image resolution section.

So if we look at a lower frequency signal, i.e. the top sine wave â€“ say this lower frequency signal had an amplitude A. Then after it goes through our system, it has an amplitude of 0.9A. Then we can mark one point on our MTF graph, so the MTF has a value of 0.9 for spatial frequency f.

And then for something thatâ€™s a higher frequency that oscillates twice as often, the question is if we pass it through our imaging system then whatâ€™s the amplitude when it comes out? For instance, in this case, itâ€™s 0.75. This provides us with another point on our MTF curve. The MTF has a value of 0.75 for a spatial frequency of 2f.

Then if we have a sine wave with a frequency of 4f and we pass it through our system, then that amplitude is 0.5 in this made up example. That gives us another point on our MTF curve.

Then finally, we can think about another waver here with a frequency that is 8f and, for instance, say that that output is 0.1 times the original amplitude A. So that would give us another point on our MTF curve. And then you can think about a curve that passes through these points. That is the concept behind the MTF curve.

So, the concept of MTF is aimed at measuring reduction of the signal amplitude different spatial frequencies (i.e. more reduction at higher spatial frequencies since the detectors have a finite size). The MTF measurements are typically made for high contrast objects, which is the best case for measuring the spatial resolution.

**R****a****d Take-home Point**: The MTF measures the amplitude (ie. signal brightness) change of different spatial frequencies as they pass through the imaging system.

## Noise Power Spectrum

As we have discussed in other posts, x-ray images are inherently noisy and the relative noise can be tracked by using metric such as signal to noise and contrast to noise level.

However, in the section we would like to describe a method analogous to the MTF except for measuring noise. The concept being that we would like to understand how noise at different spatial frequencies passes through an imaging system as well.

Different frequencies of an image have different noise. So, we can talk about lower frequency noise or higher frequency noise. As in the case of MTF, we can put noisy signal into the imaging system and then measure the Noise Power Spectrum. In this case by examining the variations in the noise at the output rather than the signal amplitude.

If we look at the plot, we can also fit a curve here in a similar manner as MTF. This is referred to as the Noise Power Spectrum (NPS).

In practice, what we do to measure the noise power spectrum is to take the Fourier transform of noisy image (e.g. from a water phantom). These local Fourier transforms are taken on overlapping image patches and scanned throughout the noisy image. is weâ€™ll take noisy images that are taken on the medical imaging systems and weâ€™ll use the Fourier Transform again â€“ the Fourier at different overlapping regions in order to actuallyTransforms calculate this.

**R****a****d Take-home Point**: Noise power spectrum measures noise variations as a function of spatial frequency.

## Detective Quantum Efficiency (DQE)

As we mentioned above, the motivation for the DQE is to provide a metric of the efficiency of the x-ray system.

One common example is the fact that we know for all automobiles how far on average they can travel with one gallon of gasoline. Letâ€™s consider an exercise like that here, except instead of how much mileage we can get we want to know what type of an image we can make.

If we imagine that we have initial set of ideal photons (with quantum noise). These photons have traveled through the patient and they have initial noise, they are still ideal because they didnâ€™t yet get measured by an actual system. The signal to noise ratio of these ideal photons is referred to here as SNR_{IN}.

The photons then go through the system itself and there is system blurring and noise associated with the detector itself. The signal to noise ratio of the actual measured signal is referred to as SNR_{OUT}.

DQE is simply a ratio of SNR_{OUT }divided by SNR_{IN}, where (f) means that these are all functions of spatial frequency.

Sometimes (f_{x,} f_{y}) are used to represent the spatial frequencies in the two different directions in the image x and y, whereas we just write f for simplicity here assuming that the behavior is the same between the two directions on the detector.

Also, the DQE is proportional to the ratio of the MTF^{2} divided by the NPS. If the system has less blurring as a function of the different spatial frequencies, then the DQE will be higher. When comparing systems for a given task we are looking for the system with the higher DQE at a representative spatial frequency for that task.

In the full expression of DQE(f) there are some other quantities as well such as system gain. However, here we want to highlight the relationship between DQE, MTF and NPS. From this relationship you can see that the DQE is very strongly dependent on the DQE since the

**R****a****d Take-home Point**: The DQE is the ratio of the SNR_{OUT }divided by SNR_{IN}, where DQE is proportional to MTF squared and inversely proportional to NPS.

## A Sample DQE from x-ray detectors

Finally, weâ€™ll show one sample case where DQE can be used to compare different x-ray technologies. In the figure below we see the comparison of DQE curves for different x-ray image acquisition technologies. This provides a concrete use case of the DQE methodology.

The triangles here represent the standard screen film case. So they have a given DQE and that DQE is decreasing as function of spatial frequency. Then we want to compare that against other technologies to convert our x-ray into this signal. One of them is computed radiography (CR). Thatâ€™s shown in these boxes in this figure. Remember that CR is a system that does not require film development.

In this plot the CR is very comparable to the standard screen film.

Then for the more recent detector technologies show improved DQE in this example. For instance, cesium iodide (CsI), you can see that case here in the dark shaded squares. In that case, the x-ray photons are converted to light photons and using a columnar structure. This column like structure leads to less blurring than traditional scintillators. Since there is less spreading of light the MTF will be higher and this leads to significantly higher DQE.

Finally, among the detectors compared here the one that has the highest DQE at high spatial frequencies is this amorphous selenium detector. When the x-rays come in, rather than being converted to light photons first, the x-rays are directly measured (i.e. converted to electrons).

For this direct conversion detector there is a higher MTF than the CsI detector since the signal is converted directly from x-rays to electrons. So, thereâ€™s less blurring of the image signal. Again, this one does best at the high spatial frequencies because of the MTF benefit from having a detector with very little detector blur (remember DQE goes as MTF squared so improvements in MTF significantly improve the DQE).

**R****a****d Take-home Point**: The DQE can be used to compare different imaging systems such as different detector technology for the efficiency of the conversion from ideal photons to image signal taking into account both resolution and noise.