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Zooming in on A Transformed Image: an Undergraduate Project Caroline Haddad, SUNY Geneseo haddad@geneseo.edu Dawit Haile, Virginia State University dhaile@vsu.edu Helmut Knaust, University of Texas at El Paso hknaust@utep.edu

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Outline Background Wavelets Workshop 2006 Module Writing Workshop 2008 Project How to zoom in on region of interest How does the transform change the image? Solution Mathematica demonstration

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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. Thomas pjvanfleet@stthomas.edu Catherine Beneteau, co-PI, University of South Florida cbenetea@cas.usf.edu Caroline Haddad, co-PI, SUNY Geneseo haddad@geneseo.edu David Ruch, co-PI, Metropolitan State College of Denver ruch@mscd.edu pjvanfleet@stthomas.edu cbenetea@cas.usf.edu haddad@geneseo.edu ruch@mscd.edu

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Project Development Workshop Summer 2008, Madison, Wisconsin Our group: Caroline Haddad, SUNY Geneseo Dawit Haile, Virginia State University Helmut Knaust, University of Texas at El Paso

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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”.

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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)

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Possible Areas of Introduction Linear Algebra Wavelet Course Modeling Course (with a background in linear algebra) Imaging Course Signal Processing Course

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Learning Objectives or Outcomes 1. Students will gain better understanding of the inner workings of the DHWT.

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Learning Objectives or Outcomes 1. Students will gain better understanding of the inner workings of the DHWT. 2. Students will improve their programming skills

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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

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Instructor Deliverables Project Description, and Mathematica or Matlab Codes Student Deliverables Students will submit a module to do this and a report explaining the solution approach, and why any anomalies occur. Possible Future Work Future work: Try to get rid of edge effects for D4, D6. Can it be generalized for even D filters?

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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”.

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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.

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Depending on the Student and Time… 6.Repeat steps 2 - 5 for k = 2, and 3. 7. Repeat steps 2 - 5 for k arbitrary.

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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?

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Discrete Haar Wavelet Transform Matrix

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Applying W N to an Image

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Applying DHWT to Image

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Block Form of the Transform Matrix

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Region of Interest

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Region of Interest Zoom In

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Image after 2 iterations of DHWT

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Mathematica Code and Results http://baire.utep.edu/Zoom/

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Any questions? Thanks to Pat Van Fleet. Thanks to. Thanks to Contact Information: Caroline Haddad, SUNY Geneseo haddad@geneseo.edu Dawit Haile, Virginia State University dhaile@vsu.edu Helmut Knaust, University of Texas at El Paso hknaust@utep.edu

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