# Wave Insert Development

The wave insert 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 (30°) angled ramps is embedded in a phantom and is configured to produce a waveform profile across the CT image.

A perfect CT image (with no loss of resolution) of the test object would produce a consistent geometric pattern of the intersection of a line with the pair of angled ramps. However, due to the finite resolution (x, y and z), the CT waveform profile will not yield the perfect profile; rather it will be influenced by slice thickness, and in-plane resolution (PSF, MTF), as well as noise limitations, and other sources of non-uniformity such as beam hardening etc.

Various characteristics of the waveform profile including, amplitude, frequency, and slope (rate of climb) of the peaks, can be studied using mathematical analysis such as the Fourier transform. These performance characteristics are encoded in the wave pattern.

The waveform profiles are visually examined and mathematically analyzed, to demonstrate the effect of Slice Thickness (z axis) and changes of in-plane (x,y) resolution; moreover, the harmonic analysis of the waveform can be used to predict, either the in-plane resolution (MTF), or the z-axis MTF when one of the two is already known.

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. Phantoms are desired that can not only produce a visual pattern to aid in the evaluation of the radial and 3-dimensional properties of the image; but also, are 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).

# Theoretical Aspects

Our approach as illustrated below, utilizes a periodic pattern embedded in a phantom module as a pair of angled ramps (30°), configured to produce a waveform profile of pixel values of the CT image of the angled ramps, taken across the image that results from scanning the pair of angled ramps. The trigonometric effect of the angle is accounted by the scaling for the magnified ramp size in the x,y plane and is applied, as appropriate, to the dimensions being used. A pair of ramps with opposed angles is used to provide two separate measurements, as well as 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 diminsh 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,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, 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 spatial resolution limitations caused by the finite z-axis slice thickness, and x,y in-plane resolution, as well as stochastic (noise) limitations and any other sources of non-uniformity such as beam hardening and faulty calibration of the CT scanner, as well as geometric mismatch (such as angulation) of the object with the scan plane.

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, and 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 circumstance, 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.

# 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 0.0mm to 4.6 mm.

The figure 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 and 𝜈 is the spatial frequency and the weighing factors on the contributing sine waves constitute the harmonics of the function and decreases as:1/1; 1/3; 1/5;…etc.

Figure Below - Left: Profiles. Right: Harmonics. Top to bottom: No smoothing, smoothing with a Gaussian kernal using signma = 0.5; 1; and 2. 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 Fig 3 if z-axis slice thickness had been kept constant, but in-plane resolution varied. For instance as illustrated in Fig. 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.

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