Course: CIS451/651 Data Compression in Multimedia
Professor: Paul D. Amer
Semester: Spring 2013
Title:  Homework - Chapter 14 - Subband Encoding

Tasks

  1. (5 pts)  Perform a 3-level 2-dimensional Haar subband transform as presented in class on the following 8X8 "image". (Submit work showing all intermediate transforms.)
  2. (5 pt) Quantize the transformed image from Problem 1 in the following two ways to form two quantized images:
  3. (5 pts) Perform an inverse 3-level 2-dimensional Haar subband transfrom as presented in class on the following 8X8 matrix.  You may use Matlab or the Excel spreadsheet developed for Problem 2. Show your work for all intermediate steps.  Assume the forward transform transformed each row in all three levels, and then transformed each column in all three levels.  Hence your inverse should invert each column in all three levels (first invert level 3, then invert level 2, then invert level 1), and then invert each resulting row in all three levels (first invert level3, then 2, then 1).

8  6  4  3  6  0  7 -1
1 -2  0  4  3  8  5 -3
0 -3  9  4  2  5  7  0
4  1 -3  4  3  0  0  0
1  1 -1 -1 -1  1  1  1
2  2  2  2  2  2  2  2
0  1  0  1  0  1  1  0
3  2  0  1  7  2  2 -4

  1. (5 pts) Derive the 3-level 1-dimensional  inverse Haar Transform basis set for N=8. Show all work, not just the final answer. (The answer consists of 8 - 1X8 vectors.) (In class, we showed the forward 3-level 1-D Haar Transform). Draw an analogous picture to that shown in Figure 13.3 for your basis set. Using Matlab, apply your inverse to the matrix in the previous problem (twice; once in each dimension) to verify the your answer to Problem 3. Show your work.
  1. (4 pts) Does the order of performing transforms matter? Consider the 4X4 image used in the class presentation, and repeated here:

    18  8  9  1
    20  6 11 15
    16  8  6  6
     6  6 -2  2

    • In the class exercise, we first transformed level 1 and 2 on the rows, and then transformed level 1 and 2 on the columns. For this example, transform levels 1 and 2 on the columns and then levels 1 and 2 on the rows. Show all intermediate results. Do you get the same final result as in class?
    • For the original example, now transform level 1 on the rows, then level 1 on the columns, then level 2 on the rows, and finally level 2 on the columns. Show all intermediate results. Do you get the same result as in class and/or the previous part of this question?

    For all of the following problems, in addition to submitting your hardcopy answer in class, please email your answer to the Professor.

  2. (Grad students: required; Undergrads: Extra Credit: 2 pts) Derive the 3-level 2-dimensional Haar Transform basis matrices for N=8. (The answer consists of 64 - 8X8 matrices.)
  3. (Grad students: required; Undergrads: Extra Credit: 2 pts) Derive the 3-level 2-dimensional inverse Haar Transform basis matrices for N=8. (The answer consists of 64 - 8X8 matrices.)
  4. (All students: Extra Credit: 2 pts) Derive the 3-level 1-dimensional normalized Haar Transform basis set for N=8. (The normalized transform uses (a+b)/(2^.5) and (a-b)/(2^.5) rather than (a+b)/2 and (a-b)/2). Draw an analogous picture to that shown in Figure 13.3 for your basis set.
  5. (All students: Extra Credit: 2 pts) Derive the 3-level 1-dimensional inverse normalized Haar Transform basis set for N=8. Draw an analogous picture to that shown in Figure 13.3 for your basis set. How does your answer relate to the previous question's answer?
  6. (All students: Extra Credit: 2 pts) Derive the 3-level 2-dimensional normalized Haar Transform basis matrices for N=8. (The answer consists of 64 - 8X8 matrices.)
  7. (All students: Extra Credit: 2 pts) Derive the 3-level 2-dimensional inverse normalized Haar Transform basis matrices for N=8. (The answer consists of 64 - 8X8 matrices.)

Notes

  1. Graduate students must do all assignments individually. Undergraduate students may collaborate in groups of 2 for assignments. Only one submission with both names should be turned in from a group.
  2. Clearly label your answers, and please submit answers in the order assigned.
  3. (repeated from course syllabus) Academic Honesty: Unless explicitly stated otherwise, students are not permitted to access or compare any homework, or program-project answers with those of any other student or group past or present, alive or dead, or any Internet web site prior to submitting the assignment. Comparing answers, or getting answers off the Internet before submitting one's work is considered cheating. If you do not have time to complete an assignment, it is better to submit partial solutions than to get answers from someone else. While it is obviously difficult to enforce this policy, students who do not follow this policy should be keenly aware that in this class, they a re cheating, and if caught, will be prosecuted according to University guidelines. This applies both to the student (or group) who gets answers and the student (or group) who gives answers.
  4. (repeated from course syllabus)  Lateness Policy: Assignments are due at the beginning of class. Unexcused late assignments will be penalized up to 10% per school day (weekends do not count) up to a 2-day maximum penalty of 20%. Without prior discussion with the professor, assignments will not be accepted more than two school days late without a university approved excuse.