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BME 210 Midterm2Solution

BME 210 Midterm2Solution - BME 210 Spring 2008...

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BME 210 Spring 2008 Name: ____________Solution____________ MIDTERM EXAMINATION #2 (1 hr 15 min) (15 points) 1. Consider the 9 element pixel grid shown below with the 10 projections indicted. Write down the W matrix for these projects, such that W s μ = ( , s μ are the vectors of CT numbers and projections). When a ray sum goes through a pixel, assume it goes through that pixel’s center and therefore use a value of 1 for the weight for that pixel. S 1 S 2 S 3 S 4 S 5 μ 1 μ 2 μ 3 μ 4 μ 5 μ 6 μ 7 μ 8 μ 9 S 6 S 7 S 8 S 9 S 10 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 W = 1
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(15 points) 2. Solve the following system of equations for 1 2 3 , , and μ μ μ using forward and backward substitution. Follow the process of the forward and backward substitution algorithms. 1 2 3 1 0 0 3 2 1 4 5 1 11 1 0 0 10 3 3 3 3 1 45 5 1 0 0 3 3 � � � � - μ � � � � � � � � � � � � - μ = � � � � � � � � � � μ - � � � � � � � � - - � � Let 1 1 2 2 3 3 3 2 1 1 11 0 3 3 45 0 0 3 y y y μ μ μ - � � � � � � � � - = � � � � � � � � � � � � Therefore, 1 2 3 1 0 0 4 5 1 0 10 3 3 1 5 1 3 y y y � � � � � � � � = � � � � � � � � -
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