Assignment 2: Math Basics
Methods In Medical Image Analysis (BioE 2630 : 16-725 : 18-791 : 42-735) - Spring 2014

Creative Commons License ITK Math Basics Assignment by Prahlad G Menon and John Galeotti, © 2012-2013 Carnegie Mellon University, is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond the scope of this license may be available by sending email to itk ATgaleotti.net.

Please contact Dr. Menon (and CC Dr. Galeotti) with any questions: pgmenon+miia ATandrew.cmu.edu , CC to galeotti+miia ATcs.cmu.edu

10 points total

Due Date: Email your submission to Dr. Menon by midnight (~11:59 PM EST) on Monday night, Jan. 27.

Problem 1  (7 points)

Consider the signal, s[k], with k = 1…N, and the signal model f[k] which is the Fourier series expansion of s[k], such that:

     f[k] = (1/N) Σn=0N-1 c[n] Φn[k]          (1)

where, c[k] are coefficients and Φn[k] are terms of the Fourier series with N total terms.

  1. Write down the formulation of this fitting problem expressed in the matrix form, assuming that the Fourier terms,Φn[k] form a matrix, A.
  2. Provide a means of solving this problem for the coefficients of the Fourier series expansion in the least squares sense and prove that:
         c = (ATA)-1 AT f          (2)
    where, c is a vector, A is a symmetric matrix and f is a vector.
  3. In this specific case of A representing Fourier terms, given properties of being orthonormal, how can you further simplify equation (2) ..?

Problem 2  (3 points)

Consider the 2x2 matrix, A = [ 1, 3; 3, 1]

  1. Calculate the eigenvalues and eigenvectors (v1 and v2, say) for A.
  2. What is the vector norm of the eigen vectors of A ..?
    Hint: what are |v1| and |v2|..?