2013 b Me 210 Homework 3

7/29/2019 2013 b Me 210 Homework 3

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Homework 3: Medical Imaging

DUE MARCH 13, 2013

AT THE BEGINNING OF CLASS

In this homework assignment you will explore the computational problems required toreconstruct an image from projection data and implement some of the methods used to solve

these problems.

Part 1: Introduction and Report Outline (10 points)Prepare a one-page introduction (maximum) describing the concept of x-ray computed

tomography and image reconstruction problem. Provide an outline for your report.

Part 2: The Program LU decomposition (25 points)Write a MATLAB m-script file (name it hw3a.m) to obtain the LU decomposition of any 4*4

matrix using Crout’s algorithm. Test your program using the matrix below and display the

resulting L and U matrices. In this portion of the homework, include the program and the resultsobtained from running the code. Save the resulting L and U matrices to be used in Part 3.

A= [

]

Part 3: Forward and Backward substitution (25 points)Write a MATLAB m-script file (name it hw3b.m) to solve the equation A* x= B using forward

and backward substitution, where A is given in part 2 and

B = [] L and U matrices are calculated in Part 2. Include the program and matrix x in your report.

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Part 4: Image Reconstruction (40 points)In this part of the homework, you will need to generate a 2500 pixel (50 x 50) modified Shepp-

Logan phantom shown below in Figure 1. The Shepp-Logan phantom is formed from a series of eclipses with differing CT numbers made to approximate the human head.

Figure 1: A 50x 50 Shepp-Logan phantoms shown in grayscale

Figure 2: Arrangement of the pixels corresponding to the phantom

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A total of 3050 x-ray projections of this phantom, at 10 different angles from horizontal to

vertical (inclusive), have been recorded. A matrix W (dim 3050 x 2500) has been constructed

based on the projections. The elements of this matrix are between 0 and 1 depending on whether the pixel is included in the particular ray sum, with partial intersection through a portion of a

pixel given a partial weighting of less than one. The matrix W can be downloaded from

Blackboard. The vector S (dim 3050 x 1) contains the value of the ray sum for each projection.As discussed in class, the following equation relates the matrix W to the vector of ray sums Sthrough the vector of pixel CT numbers µ (dim 2500 x1).

(1)

In this part of the assignment you need to write a MATLAB m-file (name it hw3c.m) to perform

the tasks described below.

1. Load the file data.mat containing matrix W. You can load this file into your MATLAB program with the command load (‘data.mat’).

2. Generate of a 50 x 50 modified Shepp-Logan phantom using the Matlab function phantom. This is a 50 x50 matrix µ that contains pixel CT numbers. Display the phantomµ in as a gray scale image (hint: Matlab functions imagesc, image, or imshow andcolormap).

3. Ray sum construction: In this step, you will need to construct the vector S (dimension3050 x 1) that contains the values of the ray sum for each projection based on W and µ.

This task illustrates the process of obtaining ray sums in CT scan.

4. Image reconstruction: Based on the ray sum vector S and the matrix W, reconstruct theimage. In this step, you need to solve the system of equations (Eq. 1) for µ using the

linsolve function in MATLAB. This task illustrates the image reconstruction process in

CT scan, where the ray sum S and projection matrix W are known. Store the results intoa 50 x 50 matrix µ* so that the entries are arranged as shown in Figure 2 above. µ* is your estimate of the true matrix µ based on the projects recorded. Display the

reconstructed image µ* in gray scale.

In this part of your report include a copy of the MATLAB program, the run of the program andgray-scale displays of the original image µ and the reconstructed image µ*. Discuss the quality

of your reconstructed image relative to the original image.

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2013 b Me 210 Homework 3