This article contains the following topics:
- Introduction
- Effect of Slice Thickness (z axis) on Ramp Profile
- Intensity Profile from Ramps
- Harmonics Plot from Ramps
Introduction
The wave insert, CTP721, was developed to sample the 3D resolution properties of a CT image, including in–plane (x,y) and z-axis information. The key development is to incorporate a z-axis aspect of a more traditional step (bar) phantom. The Phantom is amenable to mathematical analysis of the x, y, and z axis resolution properties, separately and combined. A periodic pattern of a pair of opposed (45°) angled aluminum ramps is embedded in a phantom and is configured to produce a waveform profile across the CT image. In an earlier version of this phantom, the ramps are at a 30° angle. The new 45° ramp angle equally balances the in-plane resolution and the z-axis resolution. A pair of ramps with opposed angles is used to provide a visual and quantitative sense of angle scan alignment by noting any differences in ramp pattern dimensions which will tend to expand or diminish as the relative ramp angle is changed by gantry and/or phantom angulation.
When the test object is imaged, the axial CT image of the test object will show a waveform profile that comprises a pattern of repeating signal values, seen above. A perfect image with near zero loss of resolution in the x, y, and z planes of a tomographic slice of the test object would produce a consistent pattern in the square wave waveform profile taken across a set of angled ramps in the image. This is due to the consistent parallel spacing of the angled ramps as well as the consistent thickness of each angled ramp. However, due to the finite resolution in the spatial performance of a tomographic imaging device, the CT waveform profile across the set of angled ramps will not yield the perfect waveform profile. The actual profile will be influenced by a number of factors, including spatial resolution limitations caused by the finite z-axis slice thickness and x,y in-plane resolution. In addition, stochastic (noise) limitations and any other sources of non-uniformity such as beam hardening and faulty calibration of the CT scanner will affect the profile. Finally, geometric mismatch (such as angulation) of the object with the scan plane will cause the actual waveform profile to deviate from the theoretical version.
Various characteristics of the waveform profile including, but not limited to; amplitude, frequency, and slope (rate of climb) of the peaks, as well as associated mathematical analyses of the waveform profile, such as the Fourier transform, can be analyzed to determine and evaluate imaging performance of the tomographic imaging device. Spatial performance includes the device’s ability to accurately image an object, including in-plane resolution characteristics, slice thickness, angular orientation of the slice plane, and uniformity of response across the scan field. If, for instance, there is a variation in slice thickness throughout the slice, or from one side of the slice to another, or if the mean of the profile is changing, due to non-uniformity of the scanner, these variations will be reflected in the image of the test object. The waveform profiles taken across the angled ramps in the image will encode these properties.
The waveform profiles from the tomographic images may be visually examined which may, in some circumstances, be sufficient to provide a general evaluation of the performance of the imaging device. Alternatively or additionally, the waveform profiles may be mathematically analyzed, for instance using automatic processing software.
The ability of a tomographic imaging device to produce accurate images, including 3-dimensional renditions of objects, may be vital in applications involving volume measurements and 3-dimensional planning of invasive medical procedures. The aim is to provide phantoms that can be used to simultaneously sample the radial and 3-dimensional extent of a tomographic image, rather than just local in–plane (x,y) or z-axis (thickness) information. Desired phantoms not only produce a visual pattern to aid in the evaluation of the radial and 3-dimensional properties of the image but are also amenable to mathematical analysis of the x, y, and z-axis image properties separately, as well as in a simultaneous fashion. The Wave Phantom is designed to integrate these capabilities in a single module test object, rather than requiring several separate test objects.
Effect of Slice Thickness (z axis) on Ramp Profile
To further illustrate some of the concepts of how a waveform profile can encode characteristics of a tomographic imaging device reference is made to the figure below. Section a, the side view of the interception of an ideal (uniform) 1mm tomographic slice with the test object is depicted; in b, the intercept pattern is shown; along with the corresponding wave profile as shown in c. The thickness of the slice is vertical in this illustration and would correspond to the z-axis in conventional CT scanning. A given slice intercepts portions of angled ramps separated by cast material. The resulting image represents only those portions that are present in this particular slice.
Likewise, the figure below depicts similar slice and waveform profile characteristics as those depicted above, but with ideal uniform slice thickness, varying from 0.5mm to 4.6 mm. It is important to note that the software will not perform wave ramp analysis for slice thicknesses greater than 2mm. The image below is included to help illustrate the theoretical characteristics of the wave ramp.
Figure 9.3 above further illustrates that as slice thickness decreases towards an infinitely thin slice, the shape of the corresponding waveform would approach a formal “square wave”, with infinitely steep slopes, and flat peaks.
It can be noted that an (infinite) square wave pattern is characterized by the well known Fourier series and resulting harmonics as provided from equation 1, and as the figure shown below.
Equation 1 𝑓𝑠𝑞𝑢𝑎𝑟𝑒(𝑥) = 4/𝜋 ∑∞𝑘=1 𝑠𝑖𝑛(2𝜋(2𝑘−1)𝜈𝑥) /2𝑘−1
= 4/𝜋 (𝑠𝑖𝑛(2𝜋𝜈𝑥)+1/3𝑠𝑖𝑛(6𝜋𝜈𝑥)+1/5𝑠𝑖𝑛(10𝜋𝜈𝑥)+…) ,
Where x is the spatial distance, 𝜈 is the spatial frequency, and the weighing factors on the contributing sine waves constitute the harmonics of the function and decrease as:1/1; 1/3; 1/5;…etc.
Figure 9.4: Left: Profiles, Right: Harmonics. Top to bottom: No smoothing, smoothing with a Gaussian kernel using sigma = 0.5; 1; and 2. The smoothing process with a Gaussian kernel removes noise or blurring from images by using a Gaussian distribution to average nearby points. In all cases pixel resolution is kept constant at 0.5mm.
However, a waveform plot extracted from images obtained by a tomographic imaging device will exhibit a rounding and/or blurring characteristic of the profile resulting in a repetitive more sine-wave like pattern. As mentioned previously, the two major reasons are (i) the influence of finite z-slice thickness as well as (ii) in-plane (x,y) point-spread function (psf), or blur resolution limitations in the x,y plane of the imaging device [3][4].
The effects of changes of in-plane (x,y) resolution will show similar changes in wave form plots of Figure 9.3 if z-axis slice thickness had been kept constant, but in-plane resolution varied. For instance as illustrated in Figure 9.4, as x, y resolution decreases, the psf increases in size (blurring increases) and the waveform plot will become more rounded and exhibit less extreme slopes in the peaks/valleys of the waveform - much like the effects of increased slice thickness but with continuous effects across the plane, not just at the intercepted region of the slice thickness with the angled ramps.
Intensity Profile from Ramps
This graph plots the pixel intensities through the wave insert. This profile gives rise to the harmonics through the Fourier transform. Below are the intensity profiles of wave ramps 1 and 2 from a scan with 1.25mm slice thickness. A scan with this slice thickness was chosen because it is a very common slice thickness that customers use.
Figure 9.5: A 1.25mm slice scan of Ramp 1 in the Wave Insert creates an oscillating pattern of pixel intensity [HU].
Figure 9.6: A 1.25mm slice scan of Ramp 2 in the Wave Insert creates an oscillating pattern of pixel intensity [HU].
Harmonics Plot from Ramps
This graph plots the measured harmonics and the theoretical ideal square wave harmonics. With thinner slices and better in-plane resolution, the measured values get closer to the ideal square wave. Below you will see a harmonics plot of the profile of wave ramps 1 and 2 from a scan with 1.25mm slice thickness.
Figure 9.7: The Fast Fourier transform is applied to the intensity profile of Ramp 1 to calculate the harmonic amplitudes of the critical frequencies.
Figure 9.8: The Fast Fourier transform is applied to the intensity profile of Ramp 2 to calculate the harmonic amplitudes of the critical frequencies.