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UTEP Summer Research Experience for Undergraduates: Applications of Discrete Wavelet Transformations

June 10 - July 19, 2013


Program Description

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The National Research Experience for Undergraduates Program (NREUP) is a program of the Mathematical Association of America, funded by the National Security Agency (H98230-13-1-0270) and the National Science Foundation (DMS-1156582).

The target audience for the program are primarily sophomores and juniors majoring in Mathematics. Successful completion of Calculus II is required; successful completion of Matrix Algebra and/or Principles of Mathematics is desirable.

  • Project Director. Helmut Knaust, Department of Mathematical Sciences, The University of Texas at El Paso, El Paso TX 79968.
  • Project Description. Five undergraduate students majoring in Mathematics will participate in an intensive six-week summer research program on wavelets.
    • Fingerprint.png
      The first group of students will investigate whether the FBI fingerprint algorithm [1] can be adapted to other classes of similar images such as facial portraits. Historically the adoption by the FBI of a compression algorithm for storing fingerprints digitally was the first major "applied" success of the wavelet research community. The algorithm is derived from the general JPEG2000 compression algorithm, but cleverly exploits the special character of the images to be processed by selecting a particular wavelet package and quantization scheme, thereby achieving superior results for fingerprint compression compared to Fourier or more general-purpose discrete wavelet transformation techniques.
    • The second student group will research the topic of image fusion. Fusing high resolution gray-scale satellite images with lower resolution multispectral images is of major interest to geographers using remote sensing, in particular to study food production. Some earlier fusion techniques did not use wavelet techniques, but recently discrete wavelet transform methods have been successfully employed [2,3]. The major idea is to replace certain portions of the transformed gray-scale image by corresponding portions of each channel of the transformed color image. The results are superior to those simply using resampling techniques for multi-spectral images. The students will study various fusion techniques, try to find other possible replacement and implementation schemes, and investigate methods to compare the quality of fusion results.
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  • Stipends. Students will receive a stipend of $3,700 for participation in the program; additionally they will receive a UTEP food allowance. Textbooks and other materials will be provided. Participants need to be U.S. citizens or permanent residents.
  • Expectations. This is a full-time job: Students will devote at least 40 hours per week to the project. Participants may sign up for an independent study course during the summer if they wish to receive academic credit for the summer research experience, but will not take other summer courses.
  • Week-by-week. The summer research experience will start on June 10, and will last six weeks until July 19.
    • Weeks 1 and 2. Students will become familiar with the basics of discrete wavelet transformations by daily lectures and self-study. A suitable resource is van Fleet's undergraduate textbook [5]. Additionally, students will learn the basics of Mathematica by working on topic-related programming exercises.
    • Week 3. Students will use Mathematica to implement the FBI fingerprint algorithm [4], and several fusion algorithms (e.g. [2] and [3]), respectively.
    • Week 4 and 5. Students will engage in the research portion of the project extending the known results obtained in Week 3.
    • Week 6. Students will prepare a poster of their research results, and write up their findings in a research report.
  • Learning Outcomes. The following learning outcomes will result from the summer research experience. Participants will:
    • get a taste of doing research in Mathematics. This will motivate and prepare them for participation in a graduate program in the Mathematical Sciences.
    • get exposed to the methods and issues involved in applying mathematics to a relevant engineering problem.
    • learn how to use a computer algebra system for mathematical investigations, as a computational and visualization aid, and for the implementation of mathematical algorithms.
    • be able to give and defend a mathematical presentation to a group of their peers and the professional community.
    • develop an understanding of the theoretical underpinnings of wavelet transforms and their applications.
  • References.
  1. J.N. Bradley, C.M. Brislawn, T. Hopper. The FBI Wavelet/Scalar Quantization Standard for gray-scale fingerprint image compression. SPIE, Vol. 1961, Visual Information Processing II (1993), pp. 293-304.
  2. C.M. Brislawn, J.N. Bradley, et al. The FBI Compression Standard for Digitized Fingerprint Images. Preprint, 1996.
  3. Shutao Li, James T. Kwok, Yaonan Wang. Using the discrete wavelet frame transform to merge Landsat TM and SPOT panchromatic images. Information Fusion 3 (2002) pp. 17-23.
  4. Zhenhua Li, Zhongliang Jing, Xuhong Yang, Shaoyuan Sun. Color transfer based remote sensing image fusion using non-separable wavelet frame transform. Pattern Recognition Letters 26 (2005), pp. 2006-14.
  5. David K. Ruch & Patrick J. Van Fleet. Wavelet Theory: An Elementary Approach with Applications. Wiley (2010).
  6. Patrick J. Van Fleet. Discrete Wavelet Transformations. Wiley (2008).

Miscellaneous Information


Student Reports and Posters

Image Fusion Team


Fingerprint Team


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