Suppose an image has 9 pixels, pixel size 1 1, and each pixel has constant attenuation coefficient f The vector of attenuation coefficients can be written as 1 J2 J3 J4 J5 J6 J7 J8 J9 Suppose we send 6 lines through the object and get 6 measurements, mi. These projec- tions were taken at 0 and 90 The vector of measurements can be written as Recall that the goal of the inverse problem is given the measurements m, compute the attenuation coefficients, f. (a) Fill in the measurement matrix A corresponding to the two projections (at 0 and 90) (b) Compute a basis for the column space of A: R(A). What is the rank of A? (c) Compute a basis for the null space: N(A). What is the dimension of the null space? (d) Show that the rank-nullity theorem is satisfied (e) Compute a basis for the row space of A: R(A) (f) Use your computed vectors to write down a basis for R9 (g) The next question investigates how many projections we need until Af m admits a unique solution, assuming that a solution exists. To answer this question, find the measurement matrices corresponding to more projection images and use Matlab to compute the ranks of these matrices Hint: Start with 2 projection images (matrix from question 1) and compute the rank. Add a projection at 45 (see class notes) to get 3 projection images. Continue to add more projections until A has full column rank Suppose an image has 9 pixels, pixel size 1 1, and each pixel has constant attenuation coefficient f The vector of attenuation coefficients can be written as 1 J2 J3 J4 J5 J6 J7 J8 J9 Suppose we send 6 lines through the object and get 6 measurements, mi. These projec- tions were taken at 0 and 90 The vector of measurements can be written as Recall that the goal of the inverse problem is given the measurements m, compute the attenuation coefficients, f. (a) Fill in the measurement matrix A corresponding to the two projections (at 0 and 90) (b) Compute a basis for the column space of A: R(A). What is the rank of A? (c) Compute a basis for the null space: N(A). What is the dimension of the null space? (d) Show that the rank-nullity theorem is satisfied (e) Compute a basis for the row space of A: R(A) (f) Use your computed vectors to write down a basis for R9 (g) The next question investigates how many projections we need until Af m admits a unique solution, assuming that a solution exists. To answer this question, find the measurement matrices corresponding to more projection images and use Matlab to compute the ranks of these matrices Hint: Start with 2 projection images (matrix from question 1) and compute the rank. Add a projection at 45 (see class notes) to get 3 projection images. Continue to add more projections until A has full column rank