Homepage for Patrick Van Fleet

Vita - Patrick J. Van Fleet

B-Splines

Contact Information

Address: Department of Mathematics, Mail #OSS201; University of St. Thomas; St. Paul, MN USA 55105
Phone Numbers: (651)962-5552 (Office); (651)962-5524 (CAM Lab); (651)962-5670 (FAX)
Electronic Communication: E-mail: pjvanfleet@stthomas.edu; World Wide Web: http://cam.mathlab.stthomas.edu/pvf/

Education

Southern Illinois University at Carbondale (SIU-C): Ph.D., Mathematics, 8/91, dissertation Numerical Evaluation of Multivariate Spline Functions directed by Dr. Edward G. Neuman.; M.S., Mathematics, 8/88.
Western Illinois University: B.S., Mathematics, 12/84.

Experience

Professor of Mathematics: September, 2003 - Present, University of St. Thomas.
Associate Professor of Mathematics: September, 1998 - August 2003, University of St. Thomas.
Director, Center for Applied Mathematics: September 1998 - Present, University of St. Thomas.
Associate Professor of Mathematics: September 1997 - August 1998, Sam Houston State University.
Assistant Professor of Mathematics: August 1992 - August 1997, Sam Houston State University.
Assistant Professor of Mathematics: August 1991 - August 1992, Vanderbilt University, visiting position.

Service and Memberships

Steering Committee, Midwest Numerical Analysis Conference, 2007-present.
Workshop Organizer, Ideal Data Representation, April 9-13, 2001, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN (part of the Special Year in Multimedia).
Co-organizer, Multiwavelets Conference: Theory and Application, March 20-22, 1997, Sam Houston State University, Huntsville, TX.
Associate Editor, Southwest Journal of Analysis and its Applications, 1995-2000.
Co-organizer, International Conference on Advances in Scientific Computing and Modeling , October 12-14, 1995, Eastern Illinois University, Charleston, IL.
Referee for Mathematical Reviews, various journals, proceedings, agencies, and publishers.
Corresponding member, Center for Approximation Theory, Texas A&M University, College Station, TX, 1992-present.
Member: American Mathematical Society, 1985-present; Mathematical Association of America, 1998-present; Society for Industrial and Applied Mathematics, 1998-present
Texas Project NExT Mentor, 1996-1998.

Books

Discrete Wavelet Transformations: An Elementary Approach with Applications, John J Wiley & Sons, Inc., Hoboken, NJ, January 2008.
Wavelet Theory: An Elementary Approach with Applications, with D. Ruch, in preparation, John J Wiley & Sons, Inc., Hoboken, NJ, 2009.

Journal Articles

Nonnegative Scaling Vectors on the Interval, with D. Ruch and Y. Yang, (in preparation).
Gibbs' Phenomenon for Nonnegative Compactly Supported Scaling Vectors, with D. Ruch, J. Math. Anal. Appl., No. 304, 2005, 370-382.: Abstract | PDF File
This paper considers Gibbs' phenomenon for scaling vectors in L²(R). We first show that a wide class of multiresolution analyses suffer from Gibbs' phenomenon.
To deal with this problem, in [11], Walter and Shen use an Abel summation technique to construct a positive scaling function P_r, 0 < r < 1, from an orthonormal scaling function $\phi$ that generates V₀. A reproducing kernel can in turn be constructed using P_r. This kernel is also positive, has unit integral, and approximations utilitizing it display no Gibbs' phenomenon. These results were extended to scaling vectors and multiwavelets in [9]. In both cases, orthogonality and compact support were lost in the construction process.
In this paper we modify the approach given in [9] to construct compactly supported positive scaling vectors. While the mapping into V₀ associated with this new positive scaling vector is not a projection, the scaling vector does produce a Riesz basis for V₀ and we conclude the paper by illustrating that expansions of functions via positive scaling vectors exhibit no Gibbs' phenomenon.
Multiwavelets and Integer Transforms, J. Comp. Anal. and Appl., No. 1, 5 , January 2003, 161-178.: Abstract | PDF File
In many applications of image processing, the given data are integer-valued. It is therefore desirable to construct transformations that map data of this type to an integer (or rational) ring. Calderbank, Daubechies, Sweldens, and Yeo [1] devised two methods for modifying orthogonal and biorthonal wavelets so that they map integers to integers.
The first method involves appropriately scaling the transform so that data that has been transformed and truncated can be recovered via the inverse wavelet transform. In developing this method, the authors of [1] created a useful factorization of the 4-tap Daubechies orthogonal wavelet transform [2]. We have observed that this factorization can be extended to 4-tap multiwavelets of arbitrary size.
In this paper we will discuss this generalization and illustrate the factorization on two multiwavelets. In particular, the well-known Donovan, Geronimo, Hardin, and Massopust (DGHM) [3] multiwavelet transform can be scaled so that it maps integers to integers. Since this transform is (anti)symmetric in addition to orthogonal, regular, and compactly supported, the ability to modify it so that it maps integers to integers should be useful in image processing applications.
Some Recurrence Formulas for Box Splines and Cone Splines, Approx. Th. and its App. , No. 1,18(2002),81-89.: Abstract | PDF File
A degree elevation formula for multivariate simplex splines was given by Micchelli [6] and extended to hold for multivariate Dirichlet splines in [8]. We report similar formulae for multivariate cone splines and box splines. To this end, we utilize a relation due to Dahmen and Micchelli [5] that connects box splines and cone splines and a degree reduction formula given by Cohen, Lyche, and Riesenfeld in [2].
On the Construction of Positive Scaling Vectors, with D. Ruch, Proceedings of Wavelet Analysis and Multiresolution Methods , T.X. He, ed., Marcel Dekker, New York, 2000, 317-339.: Abstract | PDF File
Let $\phi$ be a compactly supported, orthonormal scaling function that generates the linear space V₀ in L²(R). We are motivated by the work of Walter and Shen [7]. In this paper, the authors use an Abel summation technique to build a positive scaling function P_r, 0 < r < 1, for the space V₀. A reproducing kernel can in turn be constructed using P_r. This kernel is also positive, has unit integral, and approximations utilitizing it display no Gibbs' phenomenon.
A natural question and thus the purpose of this paper is to ask whether the work in [7] can be generalized to the scaling vector setting; and, if so, what advantages are there to be gained. In this paper, we show that such a generalization does indeed exist and an anologous kernel can be constructed. The construction requires that the sum of the integer translates of each component of the scaling vector be nonnegative - a requirement not necessarily satisfied by a general scaling vector. We obtain our results by assuming this property is satisfied and then conclude the paper by exhibiting an invertible linear transformation that takes a scaling vector satisfying modest conditions and maps it to one suitable for our construction.
Moment Computation in Principal Shift Invariant Spaces, with D. Eubanks and J. Wang, J. App. Math. and Stoch. Anal., No. 4, 11(1998), 465-479.: Abstract | PDF File
An algorithm is given for the computation of moments of f in S, where S is either a principal h-shift invariant space or S is a finitely generated h-shift invariant space. An error estimate for the rate of convergence of our scheme is also presented. In so doing, we obtain a result for computing inner products in these spaces. As corollaries, we derive Marsden-type identities for principal h-shift invariant spaces and finitely generated h-shift invariant spaces. Applications to wavelet/multiwavelet spaces are presented.
On the Evaluation of Simplicial Splines, Proceedings of CAM*97, University of Central Oklahoma, 1997, 314-328.: Abstract | PDF File
We consider the problem of large-scale evaluation of multivariate simplicial splines. These splines arise naturally as the multivariate analog of B-splines and are useful in applications such as finite element methods and computer-aided geometric design (CAGD). Simplicial splines have much in common with their univariate analog. They are piecewise polynomial functions whose smoothness and support can be controlled by knot placement. In addition, these splines obey a recurrence formula. Unlike the univariate case, implementation of this formula poses some problems. If x in R^s lies on an (s-1)- dimensional hyperplane H connecting two or more knot points, then the recurrence formula cannot be utilized.
Results on the evaluation of multivariate simplicial splines are documented in [7,10]. In both cases, the authors use perturbation techniques when faced with evaluating the spline at x in H. In [7], the author suggests a stable means for evaluating simplicial splines at points on the interior of regions formed by (s-1) -dimensional hyperplanes and we augment this method with a scheme for evaluating points on any such hyperplane. Our method is especially suited to large-scale evaluation of the spline and it is numerically stable when the evaluation is performed near or on (s-1)-dimensional hyperplanes. Examples are given to demonstrate the algorithm.
On the Support Properties of Scaling Vectors, with P. Massopust and D. Ruch, J. App. and Comp. Harmonic Anal., 3(1996), 229-238.: Abstract | PDF File
In Chui and Wang [3], support properties are derived for a scaling function generating a function space V₀ in L²(R). Motivated by this work, we consider support properties for scaling vectors. In [9], Goodman and Lee derive necessary and sufficient conditions for the scaling vector {\phi_1,...., \phi_r}, r > 0, to form a Riesz basis for V₀ and develop a general theory for spline wavelets of multiplicity r > 1. We consider conditions under which linear combinations of scaling functions generate V₀. These conditions also characterize all other scaling vectors that generate the same V₀. In addition we describe the scaling vectors of minimal support for V₀.
Next, we give sufficient conditions on the two-scale symbol for scaling vectors under which a given matrix refinement equation can be solved. A spline-wavelet example illustrates these results.
For the single scaling function \phi, the support of \phi is characterized by the degree of the two-scale symbol. The situation is more complicated in the scaling vector case. We prove a result that gives the support of the scaling vector under certain conditions on the coefficient matrices. This result is illustrated by an example of fractal wavelets derived in Geronimo, Hardin, and Massopust [8].
On Multipower Equations: Some Iterative Solutions and Applications, with D. Ruch, Zeitschrift für Analysis und ihre Anwendungen (J. of Anal. and App.), 15(1996), 201- 222.: Abstract | PDF File
A generalization of McFarland's iterative scheme for solving quadratic equations in Banach spaces is reported. The notion of a uniformly contractive system is introduced and subsequently employed to investigate the convergence of a new iterative method for approximating solutions to this wider class of multipower equations. Existence and uniqueness of solutions are addressed within the framework of a uniformly contractive system. To illustrate the use of the new iterative scheme, we employ it when approximating solutions to a Hammerstein equation and a Chandrashekar equation. Due to the nature of the examples, we have found that wavelet/scaling function bases are a natural choice for the implementation of our iterative method.
Recursion Formulas for the Moments of Box Splines, in: Proceedings of Advances in Scientific Computing and Modeling , S.K. Dey and J. Ziebarth, eds., Eastern Illinois University, 1996, 143- 149.: Abstract
In 1983, deBoor and DeVore [3] introduced multivariate box splines. These splines are analogs of univariate B- splines (see [1]). In addition they can be viewed as probability distribution functions. We consider the numerical computation of the moments of multivariate box splines with arbitrary knot sets. To this end, we derive recurrence formulas for the moments of box splines. An algorithm for the implementation of our results is constructed.
An Algorithm for the Evaluation of Box Splines, Proceedings of the CAM*95, University of Central Oklahoma, 1995, 269-282.: Abstract
We report an extension of a result due to M. Daehlen [8] whereby an s-variate box spline can be evaluated using -1)-variate simplex splines. In order to derive his algorithm, Daehlen employed a result of Cohen, Lyche, and Riesenfeld [7] that illustrates how an s-variate cone spline can be expressed as a combination of (s-1)-variate simplex splines. The result of [7] defines a special linear map \pi : R^s to R that depends on a nonsingular matrix V. In order for the algorithm to succeed, Daehlen requires the condition that V must be chosen so that \pi(xⁱ) > 0 for each of the box spline knot points xⁱ, i=1,...,n. We report a method for constructing V that satisfies the condition \pi(xⁱ) > 0. We then discuss the implementation of our result Daehlen's algorithm.
Moments of Dirichlet Splines and Their Applications to Hypergeometric Functions, with E. Neuman, J. Comp. App. Math., 53(1994), 225-241.: Abstract
Dirichlet averages of multivariate functions are employed for a derivation of basic recurrence formulas for the moments of multivariate Dirichlet splines. An algorithm for computing the moments of multivariate simplex splines is presented. Applications to hypergeometric functions of several variables are discussed.

Grants and Research Awards

Principal Investigator (with coPIs C. Beneteau, C. Haddad, and D. Ruch), Collaborative Research: A Phase II Expansion of the Development of a Multidisciplinary Course on Wavelets and Applications, National Science Foundation CCLI Phase II Expansion Program, $383848 award for September 2007 - January 2010.
Principal Investigator, Wavelets and Applications: A Multi-Disciplinary Undergraduate Course With an Emphasis on Scientific Computing, National Science Foundation EMD Program, $74505 award for June 2005 - May 2007.
Principal Investigator, A Coder for Integer Multiwavelet Transforms, University of St. Thomas Research Assistance Grant,$3500 for Summer, 2000.
Principal Investigator, Multiwavelet Integer-to-Integer Transforms, University of St. Thomas Research Assistance Grant, $3500 for Summer-Fall 1999.
Principal Investigator (with J. Herman and C. Shakiban), University of St. Thomas WebCampus Grant, $3500 for Summer 1999.
Co-investigator (with Project Director D. Eubanks), Project FOURIER: A Browser-Based Program for Enhancing an Interdisciplinary Undergraduate Fourier Analysis Course, National Science Foundation, $135678 award for July 1999-June 2001.
Principal Investigator, Multiwavelet Edge Detection, Sam Houston State University Research Enhancement Award, $5000 award for Summer, 1997.
Project Director (with J. Carroll, M. Carpenter, L. Graham, C. Hallum, J. Hebert, H. Konen, and W. So): Enhancement of Image Assessment Capabilities for Natural Resource Characterization, $740000 contract from the Strategic Environmental Research and Development Program, September 19, 1996 -- September 30, 1997.
Co-investigator (with P. Massopust, D. Ruch, W. So, Project Director: J.Wang): Wavelets Based on Several Scaling Functions and Related Applications, National Science Foundation, $120000 award for August 1, 1995 -- July 31, 1997.
Principal Investigator: Degree Elevation for Splines, Sam Houston State University Research Enhancement Award, $5900 award for Summer, 1995.
Co-investigator (with M. Carpenter, J Hebert, and W. So, project director C. Hallum): A Change Analysis Strategy in Support of Environmental Monitoring Using Multiple Sensor Platforms, Texas Regional Institute for Environmental Studies, $250000 award for August 15, 1994 - August 15, 1996.
Co-principal investigator (with D. Ruch): An Iterative Solution to Hammerstein Equations Using Spline Wavelets, Sam Houston State University Research Enhancement Award, $7500 award for Summer, 1994.
Co-investigator (with M. Carpenter, project director D. Ruch): Wavelet Techniques and Statistical Analysis for Energy Savings Analysis, O&M Diagnostics, and Data Verification, Texas Engineering Experimental Station, $24900 award for Summer, 1994.
Principal investigator: The Theory and Application of a New Family of Wavelet Functions , Sam Houston State University Research Enhancement Award, $6000 award for Summer, 1993.

Sponsored Workshops/Minicourses

Wavelets Workshop, Wavelets and Applications: A Multi-Disciplinary Undergraduate Course With an Emphasis on Scientific Computing, June 4-7, 2008, Center for Applied Mathematics, University of St. Thomas, St. Paul, MN (offered by the Mathematical Association of America's Professional Experiences (PREP) Program and supported by the National Science Foundation).
Minicourse, with D. Ruch, Wavelets and Applications: A Multi-Disciplinary Undergraduate Course With an Emphasis on Scientific Computing, January 7 & 9, 2008, Joint Mathematical Meetings, San Diego, CA.
Wavelets Workshop, Wavelets and Applications: A Multi-Disciplinary Undergraduate Course With an Emphasis on Scientific Computing, June 6-9, 2007, Center for Applied Mathematics, University of St. Thomas, St. Paul, MN (offered by the Mathematical Association of America's Professional Experiences (PREP) Program and supported by the National Science Foundation).
Minicourse, Wavelets and Applications: A Multi-Disciplinary Undergraduate Course With an Emphasis on Scientific Computing, January 6 & 8, 2007, Joint Mathematical Meetings, New Orleans, LA.
Wavelets Workshop, Wavelets and Applications: A Multi-Disciplinary Undergraduate Course With an Emphasis on Scientific Computing, June 7-10, 2006, Center for Applied Mathematics, University of St. Thomas, St. Paul, MN (offered by the Mathematical Association of America's Professional Experiences (PREP) Program and supported by the National Science Foundation).

Invited Lectures

Panelist, MathFest, Madison, WI, July 30, 2008: "Teaching an Interdisciplinary Mathematics Course".
Colloquium, University of Minnesota Duluth, Duluth, MN, April 10, 2008: "Basic Image Processing with Wavelets".
Colloquium, SUNY Geneseo, Geneseo, NY, November 8, 2007: "Basic Image Processing with Wavelets".
Colloquium, SUNY Geneseo, Geneseo, NY, November 9, 2007: "Wavelets and Lossless JPEG2000 Compression".
Research Symposium, SUNY Geneseo, Geneseo, NY, November 10, 2007: "Wavelets and Lifting".
Colloquium, Western Illinois University, Macomb, IL, March 23, 2007: "Wavelets and Lossless Compression in the JPEG2000 Image Compression Standard".
Colloquium, The University of Texas at Tyler, Tyler, TX, February 9, 2007 : "Wavelets - From Theory to Practical and Into the Classroom".
Colloquium, University of South Florida, Tampa, FL, December 8, 2006: "Multiwavelets on the Interval".
Seminar, Macalester College, St. Paul, MN, November 28, 2006: "Wavelets and Lossless Compression Using the JPEG2000 Image Compression Standard".
Colloquium, CAM Colloquium Series, St. Paul, MN, November 7, 2005: "A Simple Introduction to Digital Image Compression".
Colloquium, Metropolitan State College of Denver, Denver, CO, April 22, 2005: "An Introduction to Wavelets and an Application to Image Processing"
Colloquium, Sam Houston State University, Huntsville, TX, April 18, 2005: "Wavelets and Signal Denoising"
Colloquium, Gustavus Adolphus College, St. Peter, MN, April 12, 2005: "Wavelets and Applications to Digital Imaging".
Seminar, Iowa State University, April 16, 2004: "Nonnegative Scaling Vectors on the Interval".
Seminar, Illinois Institute of Technology, April 26, 2004: "Factoring Orthogonal Multiwavelet Transforms".
Coughlin Lecture, North Hennepin Community College, March 4, 2004: "Wavelets - A New Mathematical Tool with Applications in Image and Audio Processing".
Colloquium, University of Wisconsin-LaCrosse, October 23, 2003: "An Introduction to Wavelets via Filtering."
Conference, C*A*M (Conference on Applied Mathematics), Edmond, OK, October 25-27, 2002: "Positive Scaling Vectors on the Interval."
Colloquium, Western Illinois University, March 27, 2002: "Multiwavelets and Integer Transforms".
AMS Meeting, Joint Meetings of the AMS, MAA, and SIAM, Special Session on Wavelets and Education, San Diego, CA, January 6-9, 2002: "Mathematical Programming in an Undergraduate Wavelets Course".
Colloquium, Undergraduate Student Seminar, University of Minnesota at Duluth, October 25, 2001: "Factoring a 4-term Orthogonal Multiwavelet Transformation".
AMS Meeting, American Mathematical Society Southeastern Section Meeting, Special Session on Numerical Analysis and Approximation Theory, Chattanooga, TN, October 5-6, 2001: "Using Positive Multiscaling Functions on the Interval to Solve a Certain Class of Multipower Equations".
Conference, International Symposium on Analysis, Combinatorics and Computing, Dalian Univerity of Technology, Dalian, People's Republic of China, August 5-8, 2000: "Multiwavelets and Integer Transformations".
Conference International Conference on Scientific Computing and Mathematical Modeling, Special session to honor Gilbert Walker on the occasion of his retirement, Milwaukee, WI, May 25-27, 2000: "Positive Scaling Vectors on the Interval".
Seminar, IMA Multimedia Seminar, University of Minnesota, Minneapolis, MN, October 18, 1999: "Integer to Integer Transforms Using Scaling Vectors".
AMS Meeting, American Mathematical Society Western Section Meeting, Special Session on Wavelets and Approximation Theory, Austin, TX, October 8-10, 1999: "Integer to Integer Transforms Using Scaling Vectors".
Colloquium, CAM Colloquium Series, St. Paul, MN, May 12, 1999: "Wavelets and Imaging".
AMS Meeting, American Mathematical Society Central Section Meeting, Special Session on Wavelets, Champaign, IL, March 20-22, 1999: "On the Construction of Positive Scaling Vectors".
Invited Speaker, Texas NExT Meeting, Dallas, TX, November 22, 1997: "Successfully Completing NSF Research Grants".
AMS Meeting, American Mathematical Society Western Section Meeting, Special Session on (Multi)wavelets and Numerical PDEs, Albuquerque, NM, November 8-9, 1997: "Extensions and Applications of Positive Wavelet Expansions".
Panelist, Texas NExT Meeting, Tyler, TX, October 25, 1996: ``You're the Professor, What's Next?''
Colloquium, Sigma Xi Lecture Series, Sam Houston State University, October 18, 1995, Huntsville, TX: ``Wavelets: A New Splash in Mathematics''.
Colloquium, Southern Illinois University, October 10, 1995, Carbondale, IL: Multiwavelets and Properties of Scaling Vectors".
Seminar, Eidgenössische Technische Hochschule, June 22, 1995, Zürich, Switzerland: "Wavelets and Remote Sensing".
Seminar, Center for Approximation Theory, Texas A&M University, November 29, 1994, College Station, TX: "Support Properties for Scaling Vectors and Related Results".
Colloquium, University of Southwestern Louisiana, October 7, 1993, Lafayette, LA: "Wavelets and Their Applications".
Colloquium, Eidgenössische Technische Hochschule, November 24, 1993, Zürich, Switzerland: "Wavelets and Their Applications".
Seminar, Center for Approximation Theory, Texas A&M University, September 29, 1992, College Station, TX: "On the Moments of Dirichlet Splines and Their Applications to Hypergeometric Functions".
Colloquium, Dept. of Mathematics, University of Toledo, February 24, 1992, Toledo, OH: "Multivariate Splines and Their Applications".

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Vita - Patrick J. Van Fleet

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