Emily C.

Finite Mathematics and Calculus with Applications

10 months, 3 weeks ago

The digital images are represented with matrices. Each entry of the matrix is a
number that shows the level of brightness (intensity) of the corresponding region in the
image. The above figure Cameraman is represented with a matrix of size 256 by 256,
i.e., 256 × 256 = 262144 real numbers. Each of these numbers stores about 1 byte in
the computer, so this image will require considerable space in the machine. However, it
is possible to reduce the storage required for images by decomposing them into smaller
blocks. For instance, we can reproduce a matrix in size [512,512] by the production of
two matrices in size [512,1] and [1,512]. Hence, instead of keeping 512 × 512 numbers,
it would be enough to keep only 512 × 2. Unfortunately, it is not possible to have
direct decomposition for most of the natural images. Yet, we can approximate them
with reasonable errors. In this project, you are supposed to compress the given image
Cameraman with a minimum error and a reasonable strategy.
The images can be compressed in many different techniques. Here, you are going
to use Singular Value Decomposition (SVD) for image compression. SVD decomposes
any matrix A with size [m, n] as A = USV t where S is a diagonal matrix (with size
[m, n]), U and V are orthogonal matrices (with size [m, m] and [n, n] respectively). SVD
gives a decomposition of matrix A with a similar application of diagonalization. Diagonal
matrix S stores the square roots of eigenvalues of AtA in reducing order (please note that
eigenvalues store the energy/information and larger eigenvalues store larger information.
So the larger eigenvalues and corresponding eigenvectors are more informative than
others.). Columns of V are the eigenvectors of AtA. The order of these eigenvectors are
same with the corresponding eigenvalues. Hence, first column of V is the eigenvector of
AtA correspond to largest eigenvalue of AtA, etc.
Remember from Linear Algebra that d linearly independent eigenvectors span d-
dimensional vector space. Since columns of V store n orthogonal (hence linearly in-
dependent) eigenvectors, columns of V form a basis for any n-dimensional space. In
particular, they form a basis for the row space of A (it makes sense when we consider
that AtA store similarity information of columns of A). More importantly this basis
is in a special form: basis elements are ordered according to how informative they are
about the matrix A. Thus, when we use only first k columns of V, and corresponding
columns/rows of matrices S and U, and remove the remaining columns/rows we get an
approximation to matrix A. Lets call this approximation as ”k-rank approximation of
A” (please note that k-rank approximation of A is supposed to be a matrix in the same
size of A).
Follow the following steps and answer the questions clearly. Please include all the
figures and plots you generated in your report.
1. Load the attached grayscale image cameraman.tif. To use MATLAB functions
without an error I recommend you convert the image cameraman to double.
2. How many entries do you need to store if compression is not applied to the given
image (what is the total number of pixels in the image)?
3. Find U,S,V matrices as a result of SVD. What are the size of each of these matrices?
4. Calculate 3, 50, 70, 100-rank approximations through SVD. For each case calculate
the approximation error as the mean-squared error (an error will be a scalar.), and
also calculate the total number of entries the machine needs to store. Plot the
errors with respect to rank as a line graph.
5. Plot the figures for each of the approximations you generated in the previous step.
In the title of the figures state the rank of the approximation, calculated mean-
squared error and how many entries need to be stored. Include these figures in
your report.
6. Visually which approximations are good enough in your opinion? So by considering
the number of entries how much can you compress the image without large error.
Please note that you are allowed to use built-in MATLAB functions for the mean-
squared error calculation and SVD. Hint: Useful MATLAB functions would be: svd(.),
imshow(.,[]), mse(.,.) . Please check the details from mathworks.com.