Zooming in on A Transformed Image: an Undergraduate Project

Caroline Haddad, SUNY Geneseohaddad@geneseo.edu

Dawit Haile, Virginia State Universitydhaile@vsu.edu

Helmut Knaust, University of Texas at El Pasohknaust@utep.edu

Outline

BackgroundWavelets Workshop 2006Module Writing Workshop 2008

ProjectHow to zoom in on region of interestHow does the transform change the image?

SolutionMathematica demonstration

Collaborative Research: A Phase II Expansion of the Development of a

Multidisciplinary Course on Wavelets and Applications

(DUE-0442684)September 2007 – June 2010

 Patrick Van Fleet, Project Director, University of St.

Thomaspjvanfleet@stthomas.edu

  Catherine Beneteau, co-PI, University of South Florida

cbenetea@cas.usf.edu  

Caroline Haddad, co-PI, SUNY Geneseohaddad@geneseo.edu

  David Ruch, co-PI, Metropolitan State College of Denver

ruch@mscd.edu

Project Development Workshop

Summer 2008, Madison, Wisconsin

Our group:Caroline Haddad, SUNY GeneseoDawit Haile, Virginia State UniversityHelmut Knaust, University of Texas at El Paso

The Project: Zooming in on a Region of Interest

Problem statement: Given an image transformed by the k iterations of a Wavelet Transform, write an algorithm that finds the inverse of a sub-matrix in the blur portion without inverting the entire matrix. This is known in the medical field as a "region of interest”.

Prerequisites

Basic Matrix Algebra, Haar Wavelet Transform as Applied to a Matrix or Vector, and its Inverse

Possibly another Wavelet Transform (such as one of Daubechies)

Possible Areas of Introduction

Linear Algebra

Wavelet Course

Modeling Course (with a background in linear algebra)

Imaging Course

Signal Processing Course

Learning Objectives or Outcomes

1. Students will gain better understanding of the inner workings of the DHWT.

Learning Objectives or Outcomes

1. Students will gain better understanding of the inner workings of the DHWT.

2. Students will improve their programming skills

Learning Objectives or Outcomes

1. Students will gain better understanding of the inner workings of the DHWT .

2. Students will improve their programming skills

3. Students will learn practical applications such as "region of interest" in medical imaging, e.g. in MRI's and CT scans, and “un-transforming” small portions of large audio files

Instructor DeliverablesProject Description, and Mathematica or Matlab

Codes

Student DeliverablesStudents will submit a module to do this and a report

explaining the solution approach, and why any anomalies occur.

Possible Future WorkFuture work: Try to get rid of edge effects for D4,

D6. Can it be generalized for even D filters?

One Possible Project Outline

1. Define and identify where in the transformed matrix to obtain "region of interest".

2. Identify any restrictions on the indices of the original matrix and indices of the sub-matrix (region of interest), if any.

3. Given a matrix transformed by one iteration (k = 1) of the HWT, decide what other portions of the transformed matrix will be needed to invert “the region of interest”.

One Possible Project Outline

4. Write a code to obtain the inverse.

5. Test it //This is done if the original picture document is known.

Depending on the Student and Time…

6. Repeat steps 2 - 5 for k = 2, and 3.

7. Repeat steps 2 - 5 for k arbitrary.

Depending on the Student and Time…

8. How about using D4? D6?

9. What do you expect to happen? What does happen when you attempt to invert the region of interest of the given transformed image?

10.Why do you think this happens?

11. With D6 you get weird edge effects on the top and left. For k = 1, 2, 3, the number of rows/columns affected for each is 4, 12, 28, respectively. How can you "fix" this?

Discrete Haar Wavelet Transform Matrix

Applying WN to an Image

Applying DHWT to Image

Block Form of the Transform Matrix

Region of Interest

Region of Interest

Region of Interest Zoom In

Image after 2 iterations of DHWT

Mathematica Code and Results

http://baire.utep.edu/Zoom/

Any questions?

Thanks to Pat Van Fleet. Thanks to . Thanks to

Contact Information:

Caroline Haddad, SUNY Geneseohaddad@geneseo.edu

Dawit Haile, Virginia State Universitydhaile@vsu.edu

Helmut Knaust, University of Texas at El Paso

hknaust@utep.edu