Kathryn Leonard, Ph.D. - US grants

This project explores shape models that unite human shape perception, computational tractability and mathematical rigor. In particular, it establishes geometry-based selection criteria for skeletal models, defining the best model to be the one that requires the fewest bits to approximate within a specified error tolerance. The goals of the project are to develop theoretical results establishing selection criteria for skeletal models and to apply those results to shape-dependent industrial projects.

Skeletal shape models are attractive for shape-based applications because they decompose shapes into salient parts that can be manipulated independently. Their primary downfall for practical applications, a lack of robustness to noises in the shape boundary, has only recently been addressed. In the classical definition of the skeletal model, each shape has a unique skeleton. That uniqueness creates a geometric rigidity that in turn leads to the lack of robustness. A recent generalized skeletal definition relaxes the uniqueness constraint, allowing multiple skeletal models for each shape. Multiple models provide the flexibility to accommodate noisy shape boundaries, but introduce a new problem in selecting the best skeletal model for a given shape.

The project engages capable but disadvantaged students who would otherwise be unaware of research as a career in exciting and relevant research. Broader impacts include extensive collaboration between research students and future teachers to develop learning activities for K-12 classrooms, development of course modules to incorporate concepts from digital image analysis into standard sophomore-level mathematics courses, and development of industrial applications in collaboration with students and industrial partners.

This summer Research Experience for Undergraduates (REU) in Mathematics at California State University Channel Islands will support 12 REU students each of three summers. Unique features include a focus on targeting talented students who might not otherwise have considered mathematics as a viable career choice, particularly those from traditionally under-represented groups, with a special emphasis on students who are native Spanish speakers or first-generation college students. Our group of experienced faculty mentors includes several bilingual (Spanish-English) faculty including one from a partner university in Mexico. We expect the emphasis on native Spanish speakers and students from under-represented groups will broaden the pipeline into graduate school and mathematical careers.

Students will spend the bulk of their time working on research in groups of size four per faculty mentor. Workshops, distinguished visitors and colloquia, exchanges with another regional REU, and social outings are also included in the activities. Our primary goal is for the student participants to conduct original mathematical research leading to publications. In so doing we wish to raise their levels of mathematical maturity and confidence while fostering an enthusiasm for mathematics. We will create and maintain a research community of mathematicians. We will improve the participants' abilities to communicate mathematics visually, orally, and in written form. We expect students to leave the program better equipped to pursue their mathematical studies as well as feeling excited and prepared to perform graduate-level mathematics in an academic or industrial setting.

This project will support the year-round Center for Undergraduate Research in Mathematics (CURM) which will promote academic year undergraduate research in mathematics and statistics based upon a unique model. CURM will support faculty members mentoring undergraduate students in research groups that consist of 2-5 students and 1 faculty mentor working together as a team on a research project at their own institution. CURM will accomplish this by administering mini-grants to approximately 11 faculty each year who will mentor approximately 35 undergraduate students conducting research in mathematics and statistics during the academic year at their home institutions across the U.S. Included will be workshops that will bring all of the participants together from around the U.S. to (1.) train the faculty mentors in mentoring skills, in helping their mentored students succeed in graduate school, and in setting up consistently funded undergraduate research groups at their own institution and (2.) allow students the opportunity to present their research to their peers, prepare for graduate school, and hone their technical communication skills.

This CURM project will target two critical transition phases: (a) undergraduate students making a successful transition to graduate school; and (b) newer mathematics faculty members making a successful transition to becoming successful and contributing members of their institution. We will target female participants and participants from undergraduate institutions. For the first transition point, undergraduate students will be mentored so that they develop the skills that will help them in graduate school and in their research. For the second transition point, we will provide new faculty members with the necessary skills to successfully set up and maintain long-term research programs that also involve mentoring undergraduates.

This grant supports travel for US women to the Workshop for Women in Shape Analysis (WiShT), to be held at the Nesin Mathematics Village in Istanbul, Turkey, in the week June 5-12, 2016. The workshop is designed to strengthen the shape modeling community by bringing together women researchers at various stages in their careers (from graduate student to senior researcher) and from across the world, to foster research collaboration and mentorship. Because shape modeling is interdisciplinary in nature, drawing from mathematics, computer science, and biotechnology, collaborations are especially crucial. Furthermore, since no single country has many women working in shape modeling, international collaborations are essential to forming robust networks of women researchers. Hosting the workshop in Turkey reflects a conscious effort to make the workshop more accessible for women from the Middle East and Africa. Participants will spend one week working together in small groups to solve one of a selection of open questions in shape modeling. Instead of the more typical workshop structure where participants watch presentations of established results, WiShT participants will begin generating new results in collaboration with other participants. Following the workshop, the research network will be maintained and strengthened by publishing a proceedings volume, establishing a website and listserv, and organizing follow-up conferences and reunions for participants. Mentoring and professional development will happen both formally and informally. For more information about the workshop see http://nesinkoyleri.org/eng/events/2016-wis2/index.php

Shape analysis is a critical component in many applications areas, including image recognition, medical imaging, biomedical engineering, graphics, and computer animation. The workshop will tackle three very different challenging problems in shape modeling and analysis in groups led by experienced researchers, and the broad range of participants will allow an interdisciplinary team of scholars to make substantial progress in each area. Group 1 will explore the many possible definitions of shape complexity and will study the relationships between them in the hope of deriving an overarching mathematical theory of shape complexity and an understanding of when each notion of complexity is most appropriate. The second group will focus on combinatorial optimization for shape-based segmentation models, both discrete and continuous. Each model presents challenges: continuous models may not converge to optimal solutions or may be much slower, while in the discrete setting, integrating shape information or targeting structures with a particular predefined prior is quite challenging. Group 2 will combine these approaches by exploring the mapping of variational formulations to the discrete domain and tackle the challenges associated with optimizing the resultant discrete models. Group 3 will focus on convolution skeletons in both two and three-dimensions. The goal is to develop a method to transform an input shape and associated convolution skeleton into a new skeleton that is robust, compact, and reproducible that gives a smooth parameterization of the original shape. Results from each group will be published in an AWM-Springer volume, as well as disseminated via conference and journal publications.