# Technical Reports

## Technical Reports 2006

### #16/2006: Johnathan M. Bardsley and N'djekornom Laobeul

### Tikhonov Regularization for Ill-Posed Poisson Likelihood Estimation: Analysis and Computation

**Johnathan M. Bardsley and N'djekornom Laobeul**

Department of Mathematical Sciences

The University of Montana (USA)

**Abstract**

The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative logarithm of the Poisson likelihood in place of the ubiquitous least squares fit-to-data. However if the underlying mathematical model is assumed to have the form *z*=*Au*, where *A* is a linear, compact operator, Poisson likelihood estimation is ill-posed, and hence some form of regularization is required. In a recent paper by the first author, a numerical method is presented and analyzed for Tikhonov regularized Poisson likelihood estimation, but no theoretical justification of the approach is given. Our primary objective in this paper is to provide such a theoretical justification. We then briefly present a computational method of that is very effective and computationally efficient for this problem. The practical validity of the approach is then demonstrated on a synthetic example from astronomical imaging.

**Keywords:** regularization, ill-posed problems, maximum likelihood estimation, image reconstruction, nonnegatively constrained minimization

**AMS Subject Classification:** 65J22, 65K10, 65F22

**Download Technical Report:** Pdf (237 KB)

### #15/2006: Johnathan M. Bardsley

### An Efficient Estimation Scheme for Phase-Diversity Time Series Data

**Johnathan M. Bardsley**

Department of Mathematical Sciences

The University of Montana (USA)

**Abstract**

We present a two-stage method for obtaining both phase and object estimates from phase-diversity time series data. In the first stage, the phases are estimated for each time frame using the limited memory BFGS method. In the second stage, an algorithm that incorporates a nonnegativity constraint as well prior knowledge of data noise statistics is used to obtain an estimate of the object being observed. The approach is tested on real phase-diversity data with 32 time frames, and a comparison is made between it and a previously developed approach. Also, the image deblurring algorithm in stage two is tested against other standard methods and is shown to be the best for our problem.

**Keywords:** phase-diversity, image deblurring, nonlinear and nonnegatively constrained optimization

**AMS Subject Classification:** 65F20, 65F30

**Download Technical Report:** Pdf (3188 KB)

### #14/2006: Nicholas Mousoulides, Constantinos Christou, Bharath Sriraman

### From Problem Solving to Modelling- A meta-analysis

**Nicholas Mousoulides, Constantinos Christou**

University of Cyprus, Cyprus

**Bharath Sriraman**

The University of Montana, USA

**Abstract**

Mathematical modelling is a complex mathematical activity, and the teaching and learning of modelling and applications involves many aspects of mathematical thinking and learning (Burkhardt & Pollak, 2006; Niss, 1987; Kaiser, Blomhøj & Sriraman, 2006). An increasing number of mathematics education researchers have begun focusing their research efforts on mathematical modelling, especially at the school level. It is not simply a case of researchers changing their agenda, as much as a growing awareness among the mathematics education community of the need for change (Lesh, Kaput & Hamilton, 2007; Sriraman & Lesh, 2006). More than 25 years ago, a NSF funded project investigated the question: what is needed, beyond having a mathematical idea that enables students to use it in everyday problem solving situations? (Lesh, Landau & Hamilton, 1983). The answer to this question has begun to emerge after 25 years of systemic work in the domain of modelling. In this paper, we chronicle and meta-analyze the emergence of modelling perspectives around the world from the genre of problem solving research and synthesize major strands in the extant literature.

**Keywords:** design research; learning; modelling; modelling design; modelling research review; problem solving; teaching

**AMS Subject Classification:** 97

**Download Technical Report:**Pdf (172 KB)

### #13/2006: Günter Törner, Bharath Sriraman

### The Steiner "Theories of Mathematics Education"-program of 1987: Where are we today?

**Günter Törner**

University of Duisburg-Essen, Germany

**Bharath Sriraman**

The University of Montana, USA

**Abstract**

In this contribution we discuss the six theses presented by Hans-Georg Steiner (1987), which were instrumental in the community becoming interested in theories and philosophies of mathematics education. We discuss overlooked aspects of this seminal paper particularly in light of recent developments in the field of mathematics education. Nearly twenty years later, we reflect on the development of Steiner’s program for theory development and examine if any progress has been made at all on the open questions that Steiner (1987) posed to the community.

**Keywords:** Epistemology; Hans-Georg Steiner; Philosophy of mathematics education; Relativism; Theory development

**AMS Subject Classification:** 97

**Pre-print of:** Törner, G. and Sriraman, B. (2007).The Steiner TME-program of 1987: Where are we today? Invited paper for special issue of the *Zentralblatt für Didaktik der Mathematik* (In memoriam Hans-Georg Steiner).

**Download Technical Report:**Pdf (81 KB)

### #12/2006: Luis Moreno-Armella, Bharath Sriraman, Guillermina Waldegg

### Mathematical Objects and the Evolution of Rigor

**Luis Moreno-Armella**

Cinvestav, Mexico

**Bharath Sriraman**

The University of Montana, USA

**Guillermina Waldegg (Posthumously)**

Cinvestav, Mexico

**Abstract**

In this paper we discuss the origins and the evolution of rigor in mathematics in relation to the creation of mathematical objects. We provide examples of key moments in the development of mathematics that support our thesis that the nature of mathematical objects is co-substantia1 with the operational inventions that accompany them and that determine the normativity to which they are subjected.

**Keywords:** Arithmetic; Calculus; Geometry; History of mathematics; Mathematics foundations; Normativity; Non-Euclidean Geometry; Operationality; Proof

**AMS Subject Classification:** 01, 03, 97

**Pre-print of:** Moreno-Armella, L., Sriraman, B.,& Waldegg, G . Mathematical Objects and the evolution of rigor. In press in the *Mediterranean Journal for Research in Mathematics in Mathematics Education.*

**Download Technical Report:**Pdf (95 KB)

### #11/2006: Gabriele Kaiser, Bharath Sriraman

A global survey of international perspectives on modeling in mathematics education

**Gabriele Kaiser**

University of Hamburg, Germany

**Bharath Sriraman**

The University of Montana, USA

**Abstract**

In this article we survey the current debate in mathematical modeling and describe different perspectives on this debate. We relate these perspectives with earlier perspectives and show similarities and differences between these different approaches.

**Keywords:** Hans Freudenthal; international trends; mathematical modeling; pragmatism; teaching and learning; scientific-humanism; theoretical frameworks in modeling

**AMS Subject Classification:** 97

**Pre-print of:** Pre-print of Kaiser, G., and Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. *Zentralblatt für Didaktik der Mathematik*, 38(3), pp.xxx-xxx

**Download Technical Report:**Pdf (117 KB)

### #10/2006: Nicholas Mousoulides, Bharath Sriraman, Constantinos Christou

### An Empirical Investigation of Sixth grade Students’ Modelling Processes in Cyprus

**Nicholas Mousoulides**

University of Cyprus, Cyprus

**Bharath Sriraman**

The University of Montana, USA

**Constantinos Christou**

University of Cyprus, Cyprus

**Abstract**

Results from the recent PISA 2003 study documented three types of problem solving activities, namely decision making, system analysis and design and trouble shooting (OECD, 2004). The present study conducted in Cyprus provides a sound theoretical foundation for modelling processes involved in mathematical problem solving of young learners by elaborating on PISA’s results and considering previous innovative contributions. Within this framework, we develop a test for measuring the modelling processes involved in different types of problems in mathematical problem solving. We investigate sixth grade students’ modelling processes in problem solving in Cyprus and propose an empirical model that operationalizes and encompasses most of the previous research in the area. We aim to generate a categorization of modelling processes students need to master in succeeding in different problem solving situations which require engaging in modelling processes.

**Keywords:** Problem solving, modelling processes, mathematics teaching, structural equation modelling.

**AMS Subject Classification:** 97

**Download Technical Report:**Pdf (260 KB)

### #9/2006: Athanasios Gagatsis, Bharath Sriraman, Iliada Elia & Modestina Modestou

### Exploring Young Children's Geometrical Strategies through Dynamic Transformation of Polygons and Geometrical Models

**Athanasios Gagatsis**

University of Cyprus, Cyprus

**Bharath Sriraman**

The University of Montana, USA

**Iliada Elia & Modestina Modestou**

University of Cyprus, Cyprus

**Abstract**

This study explores young children's strategies while transforming polygons, through the use of geometrical models. Data were collected from 291 children ranging from 4 to 8 years of age in Cyprus. Children were asked to draw a stairway of specific polygons, with each shape being bigger or smaller than its preceding one. Relationships between children's responses in the transformation tasks, their ability to recognize geometric shapes and their IQ level were investigated. Results showed that children used three alternative strategies in the transformation tasks. Children's IQ score was directly associated with their transformation strategies, while only a low recognition ability was associated with the use of a defective strategy.

**Keywords:** geometric models, geometric transformation, dynamic intuition, polygons, van Hiele levels, geometric shapes' recognition, IQ, implicative and similarity diagram

**AMS Subject Classification:** 97

**Pre-print of:** Gagatsis, A., Sriraman, B ., Elia,I., and Modestou, M (2006) Exploring Young Children’s Geometrical Strategies through Dynamic Transformation of Polygons and Geometrical Models. In press in *Nordic Studies in Mathematics Education*.

**Download Technical Report:**Pdf (235 KB)

### #8/2006: John Bardsley, Aaron Luttman

### Total Variation-Penalized Poisson Likelihood Estimation for Ill-Posed Problems

**J. Bardsley**

Department of Mathematical Sciences

The University of Montana (USA)

**Aaron Luttmann**

Division of Science and Mathematics

Bethany Lutheran College (USA)

**Abstract**

The noise contained in data measured by imaging instruments is often primarily of Poisson type. This motivates, in many cases, the use of the Poisson likelihood functional in place of the ubiquitous least squares data fidelity when solving image deblurring problems. We assume that the underlying blurring operator is compact, so that, as in the least squares case, the resulting minimization problem is ill-posed and must be regularized. In this paper, we focus on total variation regularization and show that the problem of computing the minimizer of the resulting total variation-penalized Poisson likelihood functional is well-posed. We then prove that, as the errors in the data and in the blurring operator tend to zero, the resulting minimizers converge to the minimizer of the exact likelihood function. Finally, the practical effectiveness of the approach is demonstrated on synthetically generated data, and a nonnegatively constrained, projected quasi-Newton method is introduced.

**Keywords:** total variation regularization, ill-posed problems, maximum likelihood estimation, image deblurring, nonnegatively constrained minimization

**PACS numbers:** 02.30.Zz, 02.50.-r, 07.05.Pj

**Download Technical Report:**Pdf (238 KB)

### #7/2006: Nyman, Adam

### Grassmannians Of Two-Sided Vector Spaces

**Adam Nyman**

Department of Mathematical Sciences

University of Montana

**Abstract**

Let \(k\subset K\)be an extension of fields, and let \(A\subset M_{n}(K)\) be a *k*-algebra. We study parameter spaces of *m*-dimensional subspaces of \(K^n\) which are invariant under \(A\). The space \(\mathbb{F}_{A}(m,n)\), whose *R*-rational points are *A*-invariant, free rank \(m\) summands of \(R^n\), is well known. We construct a distinct parameter space, \(\mathbb{G}_{A}(m,n)\), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection \(\mathbb{F}_{A}(m,n)\cap\mathbb{G}_{A}(m,n)\), which we denote by \(\mathbb{H}_{A}(m,n)\). Under suitable hypotheses on \(A\), we construct affine open subschemes of \(\mathbb{F}_{A}(m,n)\) and \(\mathbb{H}_{A}(m,n)\) which cover their *K*-rational points. We conclude by using \(\mathbb{F}_{A}(m,n)\), \(\mathbb{G}_{A}(m,n)\), and \(\mathbb{H}_{A}(m,n)\) to construct parameter spaces of two-sided subspaces of two-sided vector spaces.

**Keywords:** Grassmannian, two-sided vector space, noncommutative vector bundle, bimodule.

**AMS Subject Classification:** Primary 15A03, 14M15, 16D20; Secondary 14A22.

**Download Technical Report:**Pdf (259 KB)

### #6/2006: Heikki Haario, Leonid Kalachev and Esko Tirronen

### Optimal experimental protocol for identification of dissolution and kinetics parameters in the presence of fast reaction

**Heikki Haario**

Lappeenranta University of Technology

Lappeenranta, Finland

**Leonid Kalachev**

University of Montana

Missoula, MT, USA

and

**Esko Tirronen**

Kemira Oyj

Espoo, Finland

**Abstract**

To model the reactive solid phase contribution, knowledge of the solubility and mass transfer should be extracted from empirical measurements. Often unknown factors may complicate the interpretation of the data. In this paper we consider an example and present a practical procedure that allows one to avoid the difficulties. A simplified model is derived for the mass transfer and kinetics. This model provides a straightforward way for estimation of the parameters of interest.

**Keywords:** Parameter estimation, mass transfer and kinetics, asymptotic methods

**AMS Subject Classification:**

**Download Technical Report:**Pdf (140 KB)

### #5/2006: Emily Stone, John Goldes and Martha Garlick

### A Two Stage Model for Quantitative PCR

**Emily Stone and John Goldes**

Department of Mathematical Sciences

The University of Montana

Missoula, MT 59801

and

**Martha Garlick**

Department of Mathematics and Statistics

Utah State University

Logan, UT 84322-3900

**Abstract**

In this paper we develop a suite of deterministic models for the reactions of quantitative PCR based on the law of mass action. The models are created by adding more reaction species to a base model (the logistic equation). Qualitative analysis is preformed at each stage and parameters are estimated by fitting each model to data from Roche LightCycler (TM) runs..

**Keywords:**

**AMS Subject Classification:**

**Download Technical Report:**Pdf (313 KB)

### #4/2006: J. Bardsley, J.K. Merikoski, and R. Vio

### The Stabilizing Properties of Nonnegativity Constraints in Image Deblurring Problems

**J. Bardsley**

Department of Mathematical Sciences

The University of Montana (USA)

**J.K. Merikoski**

Department of Mathematics, Statistics and Philosophy

University of Tampere (Finland)

and

**R. Vio**

Chip Computers Consulting s.r.l. (Venice, Italy)

**Abstract**

*Aims.* It is well known from practice that incorporating nonnegativity constraints in image deblurring algorithms often yields solutions that are much more stable with respect to errors in the data. In the current literature, no formal explanation of the stabilizing effects of nonnegativity constraints has been given. In this paper, we present both theoretical and computational results in support of this empirical finding.

*Methods.* Our arguments are developed in the context of the least-squares approach. For our analysis, we express the solution of a nonnegatively constrained least squares-problem as a pseudo-solution of a linear system. The conditioning of the corresponding coefficient matrix is then compared with the conditioning of the coefficient matrix of the linear system without constraints.

*Results.* In general, the matrix corresponding to the nonnegatively constrained problem is better conditioned than that of the unconstrained problem, and as a result, the corresponding solutions are typically more stable with respect to errors in the data.

*Conclusions.* In astronomical imaging, some form of regularization is either implicitly or explicitly used in all algorithms. The most standard regularization techniques, e.g., Tikhonov and iterative regularization, bias solutions in a way that is not compatible with the true solution. The incorporation of nonnegativity constraints, on the other hand, provides stability in a way that is fully compatible with the true solution. When incorporated into the least-squares framework, the resulting algorithms often yield reconstructions that are either on par, or are of a higher quality, than those obtained with more sophisticated approaches. This suggests that incorporating prior information about the object has a greater impact on results than does the sophistication of the algorithm that is used. We emphasize that this is not an academic result. In fact, simple algorithms such as those based on linear least-squares approaches are easy to implement, more flexible regarding the incorporation of constraints and are available in the most popular packages. Consequently, we believe that in many situations they should be the first choice in deblurring problems.

**Keywords:** Methods: data analysis – Methods: statistical – Techniques: Image processing

**AMS Subject Classification:** 65F20, 65F30

**Download Technical Report:**Pdf (287 KB)

### #3/2006: Aaron Luttman, John Bardsley

### A Variational Approach to Video Segmentation for Botanical Data

**Aaron Luttman**

University of Montana

and

**John Bardsley**

University of Montana

**Abstract**

In order to engage in photosynthesis, plant leaves absorb CO_{2} via the opening of pores in their surfaces called *stomata*. Water evaporates through open stomata, however, which is a detriment to plant function. Thus a plant will seek a stomatal aperture that balances its need for CO_{2} with its aversion to H_{2}O loss. In order to visualize a particular leaf's stomatal aperture, an experimentalist injects the leaf with dye so that it fluoresces. The regions with a higher relative intensity then correspond to areas in which the stomata are closed and the darker regions where the stomata are open. A camera is used to collect the emitted light, and a fluorescence pattern is measured. Images are continually recorded as these patterns change with time, resulting in a video sequence. Our task in this paper is to segment one such video sequence into fluorescing and non-fluorescing regions. After preprocessing the video, we take a variational approach to the segmentation problem. The associated three-dimensional evolution equation is solved using a semi-implicit numerical scheme. Results of the segmentation for actual leaf data are presented.

**Keywords:** Image Segmentation, Shape Reconstruction, Variational Methods

**AMS Subject Classification:** 68U10

**Download Technical Report:**Pdf (548 KB)

### #2/2006: John Bardsley, Stuart Jefferies, James Nagy, Robert Plemmons

### A Computational Method for the Restoration of Images with an Unknown, Spatially-Varying Blur

**John Bardsley**

University of Montana

**Stuart Jefferies**

Maui Scientific Research Center

**James Nagy**

Emory University

and

**Robert Plemmons**

Wake Forest University

**Abstract**

In this paper, we present an algorithm for the restoration of images with an unknown, spatially-varying blur. Existing computational methods for image restoration require the assumption that the blur is known and/or spatially-invariant. Our algorithm uses a combination of techniques. First, we section the image, and then treat the sections as a sequence of frames whose unknown PSFs are correlated and approximately spatially-invariant. To estimate the PSFs in each section, phase diversity is used. With the PSF estimates in hand, we then use a technique by Nagy and O'Leary for the restoration of images with a known, spatially-varying blur to restore the image globally. Test results on star cluster data are presented.

**Keywords:** Image reconstruction-restoration, Atmospheric turbulence, Inverse problems, Phase retrieval

**AMS Subject Classification:** 65F20, 65F30

**Download Technical Report:**Pdf (369 KB)

### #1/2006: Johnathan M. Bardsley, James G. Nagy

### Covariance-Preconditioned Iterative Methods For Nonnegatively Constrained Astronomical Imaging

**John Bardsley**

Department of Mathematical Sciences

University of Montana

and

**James Nagy**

Department of Mathematics and Computer Science

Emory University

**Abstract**

We consider the problem of solving ill-conditioned linear systems \(A\mathbf{x}=\mathbf{b}\) subject to the nonnegativity constraint \(\mathbf{x}\geq0\), and in which the vector \(\mathbf{b}\) is a realization of a random vector \(\mathbf{\hat{b}}\), i.e. \(\mathbf{b}\) is noisy. We explore what the statistical literature tells us about solving noisy linear systems; we discuss the effect that a substantial black background in the astronomical object being viewed has on the underlying mathematical and statistical models; and, finally, we present several covariance-based preconditioned iterative methods that incorporate this information. Each of the methods presented can be viewed as an implementation of a preconditioned modified residual-norm steepest descent algorithm with a specific preconditioner, and we show that, in fact, the well-known and often used Richardson-Lucy algorithm is one such method. Ill-conditioning can inhibit the ability to take advantage of *a priori* statistical knowledge, in which case a more traditional preconditioning approach may be appropriate. We briefly discuss this traditional approach as well. Examples from astronomical imaging are used to illustrate concepts and to test and compare algorithms.

**Keywords:**image restoration, linear models, preconditioning, statistical methods, weighted least squares

**AMS Subject Classification:** 65F20, 65F30

**Download Technical Report:**Pdf (310 KB)