Bioe 1330 / 2330 Biomedical Imaging Homework #3 (55 points) Jihang Wang and George Stetten Email: [email protected] Version: 1.00 Plagiarism Policy Discussion about homework is allowed, provided you include a list of the students with whom you worked on your homework. Each student MUST write up his/her actual results independently. Submission Instructions A complete homework submission consists of 2 parts: the answer sheet, and a hand-­‐in folder containing your Matlab code and images. Please hand in your answer sheet, and email me a folder containing your Matlab code snippets (I prefer the Matlab M-­‐files, but you can also copy and paste your code to Word, Text or PDF files) and the images. Both should be done by end of day, Tuesday 10/7/2014. Before emailing me your hand-­‐in folder, please check and make sure it contains the following images: projection0.jpg, projection90.jpg, sinogram.jpg, unfilteredImg.jpg and filteredImg.jpg. 1. (4 Points) The Greek letters 𝜉 (xi) and 𝜚 (“curly” rho) are often neglected, but essential in our course as the dummy variable for integration with x and the distance from the origin in the frequency domain (total frequency in both u and v). The script version of the lower case xi is basically an epsilon with a little tail added under it to the left: The curly rho has a little tail added under it: Draw each letter xi 20 times, to get it into your hand and cerebellum (where automatic muscular motions are stored). 2. (2 Points) The half-­‐value layer (HVL) of a tissue is dependent upon the type of tissue attenuating the X-­‐ray as well as the energy of the incident X-­‐ray. If the linear attenuation coefficient μ for bone is 10cm-­‐1 at the effective X-­‐ray energy of 68keV, what is the half-­‐value layer (HVL) of the bone for that effective energy? 3. (2 Points) By convention, radiation with energy greater than or equal to 13.6eV is considered ionizing radiation. What is the wavelength below which UV light is ionizing? 1 4. (3 Points) The inverse square law has very practical use in radiography. Suppose an acceptable chest radiography was taken using 67.5mA at 80kVp from 1.5m. Suppose that it was now requested that be taken at 1m at 80kVp. What mA setting should be used to yield the same exposure? 5. (6 Points) Do Problem 4.11 in the book (same in 1st and 2nd editions). 6. (8 Points) Consider the x-­‐ray image of a slab that consists of two different materials with different linear attenuation coefficients 𝜇! and 𝜇! , as illustrated below. Determine the intensity of detected photons along the y-­‐axis on the detector plane. Express your solution in terms of the y-­‐coordinate. You should consider the inverse square law and the oblique effect. Assume the x-­‐
ray source is an ideal point source with intensity 𝐼! . For simplicity, assume the slab is infinitely long in the y direction. (Hint: a number of separate cases need to be considered in this question) 7. The figure below shows a tissue slice that is imaged by a parallel beam x-­‐ray CT scanner. (1) (6 Points) Sketch the projection 𝑔(𝑙, 𝜃) as the function of 𝑙 for the cases of 𝜃 = 0, 90 and 135 degrees separately. Make sure to include the magnitudes of the projected values and the transition points along the 𝑙 axis in your sketch. 2 (2) (3 Points) Sketch the image obtained by backprojections from the superposition of just two projections: at 0 and 90 degrees. You should normalize the backprojection, spreading the measured attenuation evenly along each backprojection, using the dimensions of the imaged region as indicated on the figure. (3) (3 Points) Determine the Fourier transform of the original image along a line with orientation at 𝜃 = 0 degrees. Hint1: Consider using the Fourier transform of the rectangular function given by: 1 |𝑥| < 1/2
𝑔(𝑥) =
⟺ 𝐺 𝑢 = 𝑠𝑖𝑛𝑐(𝑢) 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Hint2: Consider using both the linearity and translation properties of Fourier transform. 8. In Matlab, create a 256 by 256 pixel “Shepp-­‐Logan” head phantom image using the command phantom(). 3 (1) (2 Points) In Matlab, separately draw the projections of the image at 𝜃 = 0 and 𝜃 = 90 as in figure 6.9 in the 1st edition of the textbook, or figure 6.8 in the 2nd edition. Save two the two images as “projection0.jpg” and “projection90.jpg” in your hand-­‐in folder. (2) (3 Points) Compute the Radon transform of the head phantom for three different sets of theta values using the function radon() where 𝑅1 has 18 projections, 𝑅2 has 90 projections, and 𝑅3 has 180 projections (Hint: Use the radon command format [R, xp] = radon(...) which returns a vector xp containing the radial coordinates corresponding to each row of R. ). (3) (2 Points) Display the sinogram image with the 180 projections using the command imagesc() with grayscale colormap. Save the image as “sinogram.jpg”. (a) (2 Points) What do the x and y axes represent? (b) (2 Points) Which projections do the first and center-­‐most columns correspond to? (c) (1 Points) Why does the centermost column have a wider profile than the first column? (4) In Matlab, you can use the command I = iradon(R, theta, interp, filter) to get the backprojection image. The parameter “interp” specifies the type of interpolation used in the back projection. In this question, we always use “linear”. The parameter “filter” specifies the filter used for frequency domain filtering. (a) (2 Points) Apply unfiltered backprojection to reconstruct the head phantom image from the projection data created in question (2) using the function above with “none” as the filter type. Display the 3 results using the subplot() function. Save the image as “unfilteredImg.jpg”. Describe what is wrong with the images created by unfiltered backprojection, and explain the cause. (b) (4 Points) Apply filtered backprojection in the same manner as question 4(a) with “Ram-­‐
Lak” as filter type (ramp filter). Display the 3 results using subplot() function. Save the image as “filteredImg.jpg”. Explain why the ramp filter is suitable for applying here and how it has corrected the problems in question 4(a). 4