# Technical Reports

## Technical Reports 2006

### 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

### 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

### 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

### 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).

### 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.

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

### 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

### 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

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.

### 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

### Grassmannians Of Two-Sided Vector Spaces

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.

### 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:

### 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:

### 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

### 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 CO2 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 CO2 with its aversion to H2O 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

### 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

### 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