BIOENG 1340 Assignment 1 -Biomedical Imaging Solution

Instructions: Please show your solutions to each problem in full, explaining your approach clearly and including plots (as needed) to justify your answers.
For computer programs, please remember to turn in your code through the course's Blackboard website, in addition to your report containing plots / figures which are requested. If you have collaborated with another student on solving this homework assignment please state so (e.g. "I helped John with question 1").
NOTE: AS discussed in class, only Q3 in this assignment is Scored whereas it is recommended to ensure that you can solve QI and Q2 based on what we discussed in Class.
Working with Vectors & Matrices in Matlab.
Building and solving a basic set of linear equations in Matlab to estimate parameters of a linear system operating on a ID signal.
BASIC LINEAR ALGEBRA & APPLIED PROGRAMING
(50 points) Consider the arbitrary signal with k — 1 to N observations, and a linear signal model v[x] = ax + ß, with a & being constants to be determined so that v[x] fits f[x] in the least squares sense i.e. min I (v[x] — = O.
a.
b.
Please draw up what you understand is the Linear Time Invariant Systems Block-diagram Model view of this problem, wherein the System is defined as the v[x] = ax + linear signal model.
Write down the algebraic matrix formulation of this fitting problem, starting with a system of equations defining the fitting problem, given d. the linear signal model and the N observations. Note that this Linear Algebra / Matrix form of the system is the form which will help you solve for the parameters of the System i.e. a & B, and should something like A k — v, where k — [a, , elaborating on the contents of each term, assuming "N" discrete observations / recordings i.e. (Xk, f[Xk]) fork 1
to N.
Write a Matlab program to compute the least squares fit of the model to the data stored in the i.e. 'HW1_Q1.mati contains a signal (data stored in the variable 'f' such that the data constitutes, (Xk, f[xk]) data-points for fitting), assumlng that in Matlab solves the least-squares fitting problem. Plot the raw data as well as the best fit line, v[x] = ax + found (overlaid on top of each other). Remember to upload your code through Blackboard.
Compare your solution to the fitting problem using in Matlab to the solution for x = (ATA)-I AT b. Note that here indicates the inverse operator and "T" indicates the matrix transpose. Plot the best fit line, v[xJ = ax + B, and compare it against the solution obtained in part (a). Note that as per rules of matrix multiplication, (ATA)-L AT b must be evaluated by multiplying matrices from "right to left" i.e. first compute bl = AT b and then multiply the result with (ATA)-1
2.
(SO points) Repeat problem la to lc but now instead considering a quadratic ax2 + ßx + V, with a, ß and y being constants to be determined model v[x] =
3.
(50 points) For a linear fitting problem, show that (ATA) x = AT b has an algebraic equivalent which can derived from the sum of least-squares minimization problem, given N data points of (xo f[xk]) i.e. k —
1 to N, to minimize a functional Q[xJ to zero, where Q[xJ is given by the following:
(v[Xk] -
As discussed in class, please create a matrix 2x2 matrix system to solve for starting with ——0 and gg=O. prove that this algebraic equivalent turns out to be the following, showing all steps:
v[Xkl where, is the same as (ATA) and is the same as AT b.