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ELG7173 - Final Exam Winter 2005

You have 3 hours to complete this exam. The exam has four questions; you are required to answer any three of them. Each question is worth equal marks. This is a closed book exam; however, you are permitted to bring one 8.5"×14" sheet of notes into the exam. You are permitted to use a calculator. You may not communicate with anyone during the exam except the instructor.

You may make assumptions to simplify the problems as long as they don't change the calculations by more than 10%. You may use the following conditions and equations for your calculations:

  • E = mc², c= 3×108 m/s
  • E = , h = 6.63×10−34 m²kg/s
  • eV = 1.6×10−19 J
  • F = ma
  • electron mass = 9.11×10−31 kg
  • Hydrogen Gyromagnetic ratio, γ = 42.58 MHz/T
  • V = IR
  • Accoustic impedance Z = ρc
  • MRI signal for 90°−FID: kρ*[1−exp(−TR/T1)]
  • MRI signal for Spin-Echo: kρ*exp(−TE/T2) [1−exp(−TR/T1)]
  • MRI signal for Inversion-Recovery: kρ*[1−2exp(−TI/T1)+exp(−TR/T1)]

1. X-ray / Nuclear Medicine

    1A. Sketch the mechanical layout of one technology to detect high energy photons from X-ray or medical radioisotope sources. (ie. scintillating film, semiconductors, pulse ionization chambers, etc.) Discuss at least two advantages or disadvantages (or one advantage and one disadvantage) of the technology you sketched.

    1B. Describe one physical effect related to the detector technology you described, which limits the spatial resolution of X-ray CT imaging. Using backprojection image reconstruction, describe how this effect impacts image resolution.

    1C. How is the Anger camera (in SPECT) able offer better resolution than the spacing of detectors? Can this approach be used for X-ray systems?

    1D. A SPECT camera images a small concentration of radioisotope (ie. a point source) in tissue. The detector is placed 50 cm from the source. The collimator is directly in front of the detector; it is 10 cm long, and forms a rectangular 2mm×2mm grid. Sketch the system and representative photon pathways. What is the detection efficiency of this SPECT system?

    1E. A PET system is now used to image the same source (using a different radioisotope with the same biological affinity, but which releases positrons). The PET system has a single 1 m diameter ring of detectors. Each detector in the ring is 1 cm square, and there is no space between detectors. Sketch the system and representative photon pathways. What is the detection efficiency of this PET system?

2. Magnetic Resonance Imaging

    2A. Compare the MRI pulse sequences: A) 90°−FID and B) Spin-Echo. Sketch each pulse sequence, showing RF pulse, the gradient pulses and the output magnetization in the z- and xy-directions (Mz, Mxy).

    2B. How is the Spin-Echo pulse sequence able to measure T2 while 90°−FID is not?

    2C. We are concerned that a patient may have a type of brain cancer. This cancer type has a T1 of 1.5 s and a density of 1.1 g/cm³, while the surrounding brain tissue has a T1 of 1.2 s and a density of 1.05 g/cm³. T2 is identical for both the cancerous and non-cancerous tissues. Is there an advantage of Spin-Echo over 90°−FID in terms of contrast for this application. What is best choice of TR to optimize contrast between that cancerous region and the normal brain tissue? For simplicity, define contrast as the difference between the signals from each tissue type.

    2D. In order to improve MRI image quality, we choose to use a Wiener filter. Show the Fourier domain representation of the Wiener filter, and describe a protocol to measure the value of each parameter.

    2E. Representing the Wiener filter in the Fourier domain implies an assumption that the underlying system is linear and space invariant. Discuss the validity of modelling MRI as linear and space invariant. Is the MRI signal a linear function of density? Is the MRI signal a linear function of T1? Is the MRI signal a linear function of RF pulse strength?

3. Image Enhancement

    3A. Unsharp masking (USM) subtracts a low-pass filtered version of an image (multiplied by α) from the original image. For this question, use a USM convolution kernel based on the following low-pass filter, FLP, and α=½, Given the image f, calculate the output image of f processed by this unsharp mark. Show only the top pixel row of the pixels values which do not depend on padding of the image.
    FLP = 0.2×
    0 1 0
    1 1 1
    0 1 0
    4 2 4 9 9 5
    3 1 2 8 8 7
    3 0 4 7 6 7
    0 4 0 8 6 7

    3B. The USM amplifies high frequencies but differs from a classic high-pass filter. Sketch the spatial frequency response of both the USM and a high-pass filter. Describe how they differ.

    3C. Typically, the unsharp mask is considered to be an heuristic image enhancement technique. However, under certain circumstances (ie. for a special case of image degradation), the USM would be model-based image enhancement. Describe a situation in which the USM would be a model-based image enhancement technique.

    3D. Some applications use a directional low-pass filter as part of the USM, instead of a non-directional low-pass filter. Describe how this will change the behaviour of the USM.

    3E. Assume the 1-dimesional fast Fourier transform (FFT) has a complexity of O( N log2N ) operations, for a signal of length N. For this question, ignore the computational cost of addition, and assume the constant is 1.0, so that the 1D FFT requires exactly N log2N multiplications. Filtering can be implemented by either spatial domain convolution, or in the Fourier domain. For a 512×512 pixel image, for what size of convolution kernel is convolution more expensive (in terms of number of multiplications) than frequency domain filtering?

4. Image Transformations

Consider a 7×7 image with each pixel represented using 3-bits, giving pixels values from 0 to 7. Pixels have the following values:
0 4 0 4 4 2 2
3 5 7 3 5 7 2
3 5 5 3 5 7 4
4 3 3 3 3 4 3
3 5 4 3 4 5 4
3 3 5 5 7 4 4
3 0 3 3 3 3 0

    4A. Calculate the image histogram. Implement grey-scale windowing to stretch the histogram over the 8-bit range (0 to 255), such that the lower and upper thresholds include 10% of image pixels. Show the histogram after grey-scale windowing. If necessary, round grey level values to the nearest integer.

    4B. Implement histogram equalization over the 8-bit range for the original image Show the equalized histogram. If necessary, round grey level values to the nearest integer.

    4C. Implement thresholding on the original image, using a threshold value of 4.5. Implement the morphological operator erosion on the thresholded image using a 3×3 square structuring element. Sketch the output image from erosion.

    4D. Describe the Hough transform for line detection. Briefly discuss one advantage and disadvantage of the Hough transform in terms of its sensitivity to features and noise.

    4E. A simplification of the Hough transform can be made to only detect horizontal lines. Describe how this would be implemented, and what output you would expect from applying this transform to the original image.

Last Updated: $Date: 2007-11-24 01:39:40 -0500 (Sat, 24 Nov 2007) $