Summer 2022 Math 3607: Project 3
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Summer 2022 Math 3607: Project 3
2022
Instructions
• You are not allowed to use MATLAB commands and functions other than the ones discussed in lectures, accompanying live scripts, textbooks, and homework/practice problem solutions.
• You may be requested to explain your code to me, in which case a proper and satisfactory explanation must be provided to receive any credits on relevant parts.
• You are allowed to collaborate in a group of up to three students. In such a case, submit a single file to Gradescope. The uploader must select all group members at the time of upload. If the uploader does not select the group members, they will not receive credit.
• If any code is found to be plagiarized from the internet, you will receive a zero on the entire project and will be reported to the COAM.
• Do not carry out computations using Symbolic Math Toolbox. Any work done using sym, syms, vpa, and such will receive NO credit.
• Notation. Problems marked with r are to be done by hand; those marked with 旦 are to be solved using a computer.
• Answers to analytical questions (ones marked with r ) without supporting work or justifi- cation will not receive any credit.
1 Polynomial Evaluation of Matrices [25 points]
Let ppzq “ c1 ` c2 z ` ¨ ¨ ¨ ` cnzn ´1 . The value of p for a square matrix input is defined as ppXq “ c1I ` c2X ` ¨ ¨ ¨ ` cnXn´1 .
(a) r Show that if X = Rk ˆk has an EVD, then ppXq can be found using only evaluations of p at the eigenvalues and two matrix multiplications.
(b) 旦 Complete the following program which, given coefficients c “ pc1 , c2 , . . . , cnqT , evaluates the corresponding polynomial at x, which can be a number, a vector, or a square matrix. If x is a scalar or a vector, use Horner’s method1; if x is a square matrix, use the result from the previous part.
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function y = mypolyval (c , x) %MYPOLYVAL evaluates a polynomial at points x given its coeffs . % Input : % c coefficient vector (c_1 , c_2 , . . . , c_n)^T % x points of evaluation % - if x is a scalar or a vector , use Horner’s method % - if x is a square matrix , use the result from (a) % - otherwise , produce an error message . |
(c) 旦 Test your code by comparing against MATLAB’s built-in functions. Use polyval for cases where x is a scalar or a vector and polyvalm for cases where x is a square matrix. Recall that MATLAB uses a different convention in arranging polynomial coefficients, so you need to use flip accordingly.
2 旦 Embedding Watermark inside Image [25 points]
Let A and W be m-by-n matrices representing two grayscale images of the same pixel dimensions as shown below. The image W is embedded inside the image A using an SVD-based scheme described below.
Figure 1: Two original images. The image represented by A on the left and the image represented by W on the right.
Let A “ UΣVT be an SVD of A. For some small scaling factor α ą 0, construct a new matrix Σ ` αW and consider its SVD
Σ ` αW “ UwΣwVw(T) . (1)
Note that both Σ and Σw are diagonal matrices and, for a suitably chosen α , Σ « Σw . Thus, Aw obtained by
Aw “ UΣwVT (2)
is approximately equal to A and so represents an image which looks nearly identical to that of A, while containing information about the image W as well.
Your mission, should you choose to accept it, is to establish an algorithm to retrieve (an approxi- mation of) W embedded in Aw given Uw , Vw , Σ , and α .
Begin by loading the data with
>> load watermark % download and save ’watermark .mat’ first
This loads Aw, Uw, Vw, S, and alpha, which correspond to Aw , Uw , Vw , Σ , and α as described
above, respectively. Note that Aw is a matrix of double-precision floating point numbers whose elements range from 0 to 255.
(a) Write a MATLAB code to find a matrix W0, an approximation of W , out of Aw .
(b) Display the images represented by Aw and W0 side by side using subplot. Do they look
similar to their respective original images in Figure 2?
2022-07-19