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Modulation transfer function (MTF) is a method of describing “the extent to which a piece of equipment degrades the images created in it or passing through it.”  Therefore it can be said that MTF is a measure of the ability of the imaging system to handle contrast as a function of spatial frequency , the greater the MTF of a system , at high spatial frequencies , the more adept it is at capturing fine detail.
MTF= modulation in image at a particular spatial frequency
modulation in object at the same spatial frequency
MTF is only defined for a linear system where the output is linearly proportional to the input. Additionally “an object whose brightness (I) varies sinusoidally with a frequency of (L) lines/mm is imaged through the optical system. The modulation of the object (Mo) is defined as Mo = (IMAX – Imam)/ (IMAX + Imam), where IMAX = maximum intensity, Imam = minimum intensity. The image can be measured using the same procedure, where modulation of the image (Mi) is defined as Mi = (IMAX – Imin)/ (IMAX + Imin).” 
Figure 1: The top image displays an object whose brightness (I) varies sinusoidally with a frequency of (L) lines/mm. The bottom image is a plot of the modulation transfer factors at various spatial frequencies
The following describes a method to determine the MTF in an x-ray system. “An x-ray beam that varies sinusoidally in space is created by means of a spatially modulated attenuator, and falls on the screen. The resulting pattern of light is mapped by scanning the screen with a light meter or by exposing the film. The MTF at this frequency is the ratio of the output and input modulations. After the MTF for this spatial frequency is obtained the process is repeated at other frequencies.” 
(ii) Typical values for the MTF for:
Type of system
Definition of contrast
“Radiographic contrast refers to the extent to which the various different tissue structures within the body are displayed as different shades of grey in the image.”  It is determined by the properties of the tissues, thickness, density and chemical composition.
Factors which determine contrast
X-ray tube output: High energy beams from the x-ray tube result in radiation scatter in the body. This results in a low contrast image. To reduce the affects of scatter the beam energy is low. This produces an image of high contrast but results in a higher radiation dose to the patient.
The patient: As the x-ray beam passes through the patient’s body, it undergoes Compton scattering. This scattering reduces the contrast of the image. Also the thinner the patient the less dose required when compared to a thicker patient which requires an increased dose in order to obtain images.
Image receptor : is ” the ability of a film or screen-film combination to capture and depict contrast is determined by the rate at which the optical density changes with the logarithm of the exposure or relative exposure , or of air kerma, by the slope of the characteristic curve , d(optical density) / d(log_10 kerma).” 
Method to indicate the contrast in an image
To obtain an indication of the contrast the simplest method uses a special bar phantom. The bar phantom contains holes of different diameters and which have been preset at certain contrast level. The test measures the systems ability to resolve low contrast objects.
Figure 2: This image shows a bar phantom with holes of different diameters and at preset at certain contrast level.
Show how the concepts of Fourier analysis can be used to determine the size of the focal spot of an x-ray system.
Convolution is used to mathematically describe the blurring of an image under certain assumptions. In turn this can be used to determine the focal spot size. The assumption is made that “the focal spot of the X-ray source is not point-like. It is extended in a plane which is parallel to the surface of the detector system.” 
Figure 3: shows the setup assumed and relates variables to the corresponding objects.
Convolution characterizes image blurring and this blurring can be removed by deconvolution. Using the above equation can find that the “deconvolution of t with gd yields an estimate of f.” 
Describe in some detail an experimental procedure to arrive at this
The focal spot size can be determined by using a pinhole camera which takes an image of the focal spot. Using a gold/platinum plate, a small hole is drilled into it, about 75Î¼m in diameter, this is the pinhole camera. This pinhole is then inserted into lead, so that the x-rays which do not pass through the hole are absorbed. The image of the focal spot is projected onto film. To get the effective focal spot, the x-ray tube must be horizontal, when the pinhole is directly below the focal spot and in a plane parallel to the axis. In order to minimize errors the image of the focal spot is magnified (M=d2/d1). The image below shows the geometry for determining the size of the focal spot.
Figure 4: Geometry for the determination of focal-spot sizes using a pinhole
Finite -size focal spot
Image of focal spot
Why do you think that such a phantom can give a wrong estimate of the true focal spot size?
A bar phantom indirectly measures the sharpness of an image. To achieve an accurate focal spot size, a bar phantom which contains a series of line pairs of diminishing separation is required. The lines pairs should run parallel and perpendicular to each other; this allows the width and length of the focal spot to be estimated. The following image shows a standard bar phantom.
Figure 5: This image shows a standard bar phantom with equally spaced lines.
The bar phantom given in the question does not contain line pairs of equal separation. Line pairs in this phantom are running in only one direction. If this bar phantom was used, it would result in a wrong estimate of the focal spot size as the correct length and width of the focal spot size would be unable to be determined.
Figure 6: Image given in the question .
Describe the basic construction of a modern CT scanner and provide a table with the most important performance specifications of such a device
The CT scanner comprises of 3 main sections; x-ray tube, gantry and the detector. CT imaging requires the use of high energy x-rays in order to obtain accurate images. For this reason the x-ray tube of a CT scanner must be able to handle extreme cooling and heating. The x-ray tube of helical CT scanner (“technique that involves continuous movement of the patient through the scanner with the ability to scan faster and with higher definition of internal structures.”  ) experiences large volumes of heat, for this reason an anode of high heat capacity and rapid cooling is necessary. Whereas translate -rotate CT scanners(“uses both translation and rotation of a tube detector assembly to collect the projection data for each slice.”  ) have an oiled-cooled stationary x-ray tube.
Figure 7: This image shows the main components of a CT scanner
In recent years CT x-ray tube designs have incorporated copper and graphite blocks in the tungsten anode. In order to minimize sampling time, a rotating anode x-ray tube with an rpm of 10,000 and pulsed x-ray beams are used to reach higher x-ray outputs. The gantry houses the x-ray tube and the detectors which are placed opposite each other. An aperture in the gantry allows a patient to be moved through the scanner. To rotate the gantry slip rings are used. “The slip ring allows electric power to be transferred from a stationary power source onto the continuously rotating gantry. State of the art CT scanners with slip rings can now rotate continuously and do not have to slow down to start and stop.”  The CT scanner uses solid-state detectors, which are arranged in multiple rows, with the numbers in each row varying with different manufactures. The detectors used require certain properties and are “chosen for their detection efficiency, short response time, and stability of operation” 
The Following is a table of the most important performance specifications for a Siemens SOMATOM AR, STAR 
Heat storage, hu (X-ray tube anode)
Tube cooling (X-ray tube anode):
Heat dissipation rate, hu/min (X-RAY TUBE)
Number and type of detectors (GANTRY)
Continuous rotate-rotate, low- voltage slip ring
X-ray fan beam angle, Â° (GANTRY)
Gantry opening, cm (GANTRY)
Provide a description of the mathematics of image reconstruction in CT
Reconstruction Algorithm is necessary for image reconstruction . There are a number of mathematical techniques used to calculate the slice of the image.
Fourier transform(FT); the x-ray attenuation at each angular position is divided into frequency components at varying amplitudes. Using the frequency component the image is created in frequency space, into a spatially correct image and can then be reconstructed using an inverse Fourier transform.
Iterative techniques; x-ray attenuation at one angular orientation is compared to similar attenuation at a different position. The difference in x-ray attenuation at the varying positions is added. This is then repeated at all angular directions. To ensure the convergence of the reconstructed data decreasing amounts of the attenuation difference is added each time. This technique is slow and has been replaced by other more efficient methods of image reconstruction.
Central slice theorem; is a fundamental concept in image reconstruction. “This theorem states that the 1-D FT of the projection of an object is the same as the values of the 2-D FT of the object along a line drawn through the center of the 2-D FT plane.” 
Filtered back projection; usually referred to as the convolution method. It uses a one dimensional integral equation for the reconstruction of the image. “In the convolution method of using integral equations, a de-blurring function is combined (convolved) with the x-ray transmission data to remove most of the blurring before the data are back projected.”  The de-blurring function is a filter that removes the frequency that causes blurring in the composite image. The image can be reconstructed while the data is being collected.
Figure 8: This image, gives image reconstruction process, can be expressed compactly in the above equation, where the terms have been grouped to reflect the “filtered-back-projection” approach
Standard tests a medical physicist would perform on a CT scanner.
The following are some of the standard tests performed on a CT scanner.
CT number; accuracy is measured by scanning a water filled phantom. If the CT number varies slightly it can be adjusted by using a correction factor for the pixel value.
Noise; evaluated by taking the standard deviation of CT numbers. The standard deviation can be determined by using the CT numbers obtained by scanning the water filled phantom.
Resolution; if scanning phantoms returns a low contrast resolution; it can indicate changes in component performance as it affects noise.
Patient dose; this is measured using ionization chambers. These chambers determine the dose by calculating the exposure conditions such as the x-ray beam and slice thickness used in the CT scan.
The following table shows the standard tests and the frequency at which they are performed. 
CT number , Accuracy
CT number ,Constancy
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