# The nexus of conformal geometry, action principles and tau-functions: a pathway to novel constructive methods for shape analysis and imaging

Lead Research Organisation: Imperial College London
Department Name: Dept of Mathematics

### Abstract

The study of shapes is a central problem in medical imaging, computer vision and many engineering fields. Suppose we are given the silhouette of a human organ, such as the brain or colon, from an X-ray scan. Human organs change their shape because of disease progression, surgery and other treatments, and simply by natural growth. How are we to tell from such a silhouette, or 2D shape, whether the organ is diseased (as in the brain of an Alzheimer's patient for example) and, if so, the degree of progression of the disease?

One way is to compare, or "register", the shape of the scan against templates or averaged shape data from previous patients. This process is known as shape registration and it is a vibrant area of active research, along with shape classification and recognition. All this requires a way to compute how far one shape is from another. For mathematicians, the distance between entities is measured by defining a metric on a suitably defined "space" or set of entities; over the years, many such metrics over different spaces have been proposed for the purposes of shape registration in medical imaging.

Conformal geometry is the study of the conformal structure of a shape and, mathematically, it has many favourable attributes that render it particularly suitable for shape registration tasks. Mathematicians give the name "multiply connected'' to a shape that has holes in it. In medical imaging, it is common to identify on any given medical scan certain distinguishing features, attributes or "landmarks'' that have special significance. In the technologically important area of facial recognition, the 2D silhouette of a person's face might be punctuated by "holes'' around the eyes, nose and mouth which are considered in order that the image can impart more accurate information. The mathematical notion of multiple connectivity is therefore deeply relevant to these applications.

The conformal geometric approach to shape analysis has not been fully developed in the multiply connected setting, and this is an area where the PI has particular expertise acquired in past research on quite different application areas. The PI also has special interest in constructive methods, and his past work has included both identifying important mathematical tools and bringing these to implementational fruition by the development of computational methods and software.

This proposal develops a novel approach to shape analysis by building on surprising connections, found only in the last decade, between several different mathematical subfields.

Consider a simple closed curve in the plane. It has an inside and the outside. The Riemann mapping theorem of complex analysis says that there exists a "mapping" from a circular disc to the inside of the curve, and a different mapping from the same circular disc to the outside of the curve. Conformal geometers have considered a clever composition of these two mappings and have come up with an identifier known as a "fingerprint" of the original curve.

On the other hand, in a classic free boundary problem called the Hele-Shaw problem it has been recognized since 1972 that it is useful to consider the (Richardson) "moments" of the inside of a curve, as well as the moments of the outside of a curve. In the last decade, mathematical physicists have discovered that the interior moments can be generated by the exterior moments by means of a so-called tau-function. Surprisingly, this same tau-function has also been obtained by yet another community (conformal field theorists) using very different mathematical arguments.

Both the "fingerprint" of the curve and the "tau-function" of the curve come about by marrying data about the interior of the curve to data about the exterior of the curve. These objects must be related. But how? This project is about exploring those connections, and examining if novel constructive methods for shape analysis and medical imaging can be found as a result.

### Planned Impact

The academic impact of the work is likely to be wide-ranging owing to the preponderance of overlaps with diverse areas of mathematics, physics (e.g. Laplacian growth problems, random matrix theory, theory of orthogonal polynomials, conformal field theory, string theory) and, newly, to shape analysis and medical imaging. The PI has already co-organized two international workshops - focussing on some of these exciting new intersections - at the Pacific Institute for the Mathematical Sciences, Banff, Canada in 2007 and 2010 with an off-shoot workshop held at the Mittag-Leffler Institute in Sweden in Nov. 2011 (with a further workshop event scheduled for August 2012). This activity points to the current perceived international importance, in the scientific community, of the exciting mathematical intersections that have formed the seed of this proposal idea. To date, however, any ramifications of these ideas for the field of shape analysis and imaging technologies, as outlined in our Case for Support, has not been explored and this forms the unique crux of our vision. Any results will likely be of interest to all the disparate communities just listed and the PI is already well-placed in many of these for his results to be acknowledged by them. Furthermore, while he is new to the medical imaging communities, he has already been proactive in finding innovative ways to nurture connections with them, even before the project starts. The total academic impact of the proposal will therefore extend from physical scientists and engineers through to mathematical physicists and pure mathematicians with the imaging aspects being relevant to clinicians and medical imaging practitioners. The PI will continue in his efforts to enhance academic impact at the intersection of these communities via the route of workshop and conference organization as well as direct collaboration.

The applications of the results to shape analysis provide, in the longer term, broad potential for the outcomes to have both economic and societal impact. Any advances in shape analysis are likely to translate into enhanced visualisation capabilities for medical diagnosis, procedure planning, treatment, follow-ups and clinical research. The generality of the methods means that they are potentially amenable to the scanning of many organs (brain, colon, heart) and thereby have a broad relevance to public health. We propose to develop novel computer-aided methodologies for specific biomedical applications including, for example, brain imaging for tracking progression of the effects of drug addiction and Alzheimer's disease through to polyp detection for non-invasive colon cancer screening. Any results will have scope for commercialization in the event that they prove to be competitive with existing methods, or are ground-breaking in their efficacy. In this event, potential routes to successful commercialization have been identified. Imperial College is already affiliated (via Imperial Innovations) with other enterprises seeking to commercialize academic developments in imaging technology and the PI will espouse those connections once results reach the appropriate level of maturity. This will have positive impact in promoting economic growth, enhancing job creation and promoting the attractiveness of the UK for investment in imaging technologies.

The synergetic relationship with other workers in the department, the wider College and the city of London, interested in similar scientific challenges of imaging will enhance the impact by contributing to a vibrant research environment from which all works (the PI and PDRAs) will benefit in the form of rapid and effective training, knowledge transfer and assimilation. We envisage, by the end of the term of the fellowship, to have built a centre of excellence at Imperial College which will provide an academic UK hub for future research, training and investment.

### ORCID iD

Darren Gregory Crowdy (Principal Investigator / Fellow)

### Publications

10 25 50
Crowdy D (2015) Fourier-Mellin Transforms for Circular Domains in Computational Methods and Function Theory

Crowdy D (2016) Finite Gap Jacobi Matrices and the Schottky-Klein Prime Function in Computational Methods and Function Theory

Crowdy D (2016) Uniform flow past a periodic array of cylinders in European Journal of Mechanics - B/Fluids

Crowdy D (2015) A transform method for Laplace's equation in multiply connected circular domains in IMA Journal of Applied Mathematics

Description The study of shapes is a central problem in medical imaging, computer vision and many engineering fields. Suppose we
are given the silhouette of a human organ, such as the brain or colon, from an X-ray scan. Human organs change their
shape because of disease progression, surgery and other treatments, and simply by natural growth.

Conformal geometry is the study of the conformal structure of a shape and, mathematically, it has many favourable
attributes that render it particularly suitable for shape registration and categorisation tasks. Mathematicians give the name multiply connected to a shape that has holes in it.

We have made tangible and important theoretical insights into novel applications of conformal geometry in the description of shapes as they occur in different scientific contexts. This includes published work on surprising connections between random matrix theory and the so-called V-states in the study of vortex dynamics of the Euler equations of ideal fluids, as well as newly-found significance of a class of abstract mathematical shapes (recently dubbed double quadrature domains) in so-called hollow vortex systems. We have used perturbative methods based on a conformal geometric approach to study some very basic questions in compressible vortex flows, in particular with respect to how weak compressibility affects the speed of vortex streets. We believe this work to be of fundamental interest in fluid mechanics.

In addition, we have probed the underlying geometry of a new type of transform method aimed at solving (generally mixed) boundary value problems in complicated two-dimensional domains. The transform method is an essentially analytical construct, but it has proven to be very fruitful to examine its geometrical structure. A key finding is that we have unified and extended the notion of a Fourier transform beyond strips and polygons to more general circular-arc domains, whose boundaries comprise a mixture of straight lines and circular arcs. The theoretical developments were driven by the need for new methods in applications, and several publications showing how to use this new mathematical technology in a variety of applications have been prepared.

We have used the methods of applied complex analysis and conformal geometry to solve a variety of important mixed boundary value problems arising in the study of transport processes over superhydrophobic and other textured surfaces. This had led to a number of new interactions and collaborations with mechanical engineers (at Tufts University, Stevens University and The Technion in Israel).

In other work, we have used ideas associated the isomonodromic deformation theory and the Painleve equations to present a highly novel construction of the solution of the so-called "parameter problem" in applied conformal geometry (such as the construction of Schwarz-Christoffel mappings to polygons, and other mappings to so-called polycircular-arc domains). This is a highly interdisciplinary study involving the cross-over of many different mathematical ideas.

A new collaboration with Prof. A. Constantin (U. Vienna) builds on some earlier work of mine on new analytical solutions to problems in vortex dynamics involving the quasilinear elliptic Liouville equation. Some exciting new connections with those earlier ideas to integrable systems theory, operator theory and orthogonal polynomials has recently been identified and the work is currently being developed.
Exploitation Route The newly found connections between vortex dynamics and random normal matrices, is surprising and unexpected. It is likely to lead to a variety of new insights now that the connection has been made. Already, the published article has been selected by the editors as an IOP Select article, and, already within a few months of publication, it has been downloaded almost 600 times and it has been featured in the Highlights section of EuroPhysicsNews.

The new geometrical approach to transform methods has opened up a broad range of new possibilities for extensions. Generalising the approach to other partial differential equations (beyond the harmonic and biharmonic cases treated so far) is under examination. We have made progress on extending the approach to a class of problems in so-called "diffusive-wave field theory". This represents a new mathematical tool for use by engineering involved in this area.

A key advance during the course of the fellowship has been a new collaboration with an engineer Prof. Marc Hodes from Tufts University, MA, USA who visited Prof. Crowdy on a sabbatical visit at Imperial College during 2014. The two researchers engaged in discussion where the methods of conformal geometry in which Crowdy is an expert had potential application to some novel challenges involving the cooling of microchip circuitry used in everyday portable electronics. Many outstanding design and optimisation challenges exist within this field, in particular, with respect to the use of superhydrophobic surfaces to build effect heat sinks. This new interaction is growing, with Dr. Hodes working with both Crowdy and his PhD students. Crowdy has initiated the "Red Lotus Project" to bring together stakeholders and other academics interested in this general area. He organised the interdisciplinary workshop at Chicheley Hall in November 2016 and sponsored by the Royal Society. It was a key opportunity to identify novel challenge areas within this burgeoning field where conformal geometric methods can offer extraordinary advantage. In another advance, we have initiated contact with a Petroleum Engineering group at Texas A&M (led by Prof. R. Weijermars) who have been interested in exploiting the mathematical techniques we have developed for solving problems in multiply connected geometries for the purposes of design and optimisation challenges in onshore petroleum reservoir modelling.

The connection between isomonodromic deformations, the Painleve equations, and the associated tau functions to find constructive solutions to the accessory parameter problem in conformal geometry is a highly novel contribution that is likely to have high impact as the results become better known.
Sectors Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Energy,Manufacturing, including Industrial Biotechology

URL http://wwwf.imperial.ac.uk/~dgcrowdy/

Description The findings of our work on using conformal geometry to model the fabrication of microstructured optical fibres has led to a US-based spin-off company called LowReTech LLC, with CEO Dr. Peter Buchak (a former post-doc of the PI). The company was been awarded seed-fund start-up money from the National Science Foundations SBIR program. Prof. Crowdy is involved with the company in an advisory capacity, and as a consultant.
Sector Digital/Communication/Information Technologies (including Software)
Impact Types Economic

Description Role on IMA Research Committee
Geographic Reach National
Policy Influence Type Membership of a guideline committee
Impact This committee discusses the role of applied mathematics in broader societal, economic and policy perspectives.

Description Royal Society Pairing Scheme with MPs 2014
Geographic Reach National
Policy Influence Type Influenced training of practitioners or researchers
Impact I participated in the Royal Society Pairing Scheme with MPs in 2014. This enabled me to learn more about the policy space, and how education policy is shaped in Parliament and other branches of government. I was able to feed back what I learned to my colleagues, and this has been beneficial.

Description International Exchanges Award for collaboration with UFPE, Recife, Brazil
Amount £12,000 (GBP)
Organisation The Royal Society
Sector Charity/Non Profit
Country United Kingdom
Start 10/2016
End 10/2018

Description Pathways to Impact Award (internal award, funded by EPSRC)
Amount £76,000 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC)
Sector Public
Country United Kingdom
Start 10/2016
End 03/2017

Description Science Without Borders, Special Visiting Researcher Award
Amount R\$ 580,000 (BRL)
Organisation Government of Brazil
Department Coordination of Higher Education Personnel Training (CAPES)
Sector Public
Country Brazil
Start 01/2015
End 01/2018

Description Collaboration with Prof. Marc Hodes, Tufts University, MA, USA
Organisation Tufts University
Department Department of Mechanical Engineering
Country United States
PI Contribution We have initiated a multi-pronged collaboration with Prof. Marc Hodes and his group at Tufts. This involves frequent visits between the groups at Imperial College and Tufts.
Collaborator Contribution The partner group has contributed new research ideas, staff time and travel funds. They have also assisted in the organisation of workshops.
Impact Several joint papers are in preparation/submission. We have also submitted a joint grant proposal to the EPSRC/NSF scheme.
Start Year 2015

Description Collaboration with Prof. Ruud Weijermars, Texas A&M University
Organisation Texas A&M University
Country United States
PI Contribution We have initiated collaborations with a petroleum engineering group, led by Prof. Ruud Weijermars, at Texas A&M University, TX, USA. This has involved a joint publication, with the collaboration ongoing.
Collaborator Contribution They provided the problems, input data to the project, as well as staff time and other resources such as travel money for collaborative visits.
Impact Applying improved analytical methods for modelling flood displacement fronts in bounded reservoirs (Quitman field, east Texas), R Nelson, L Zuo, R Weijermars, D Crowdy, J. Petrol Sci. (2018). https://www.sciencedirect.com/science/article/pii/S0920410518301906
Start Year 2015

Description Mathematical modelling of microstructured optical fibre fabrication
Organisation University of Adelaide
Department School of Chemistry and Physics
Country Australia
PI Contribution We have used our expertise in the mathematics of slow viscous flows, and complex geometry, to provide mathematical models for the fabrication process of microstructured optical fibres.
Collaborator Contribution Our collaborators at the University of Adelaide are both in the Mathematics Department and Physics Department there (specifically, in IPAS: the Institute for Advanced Photonics and Sensing). Working closely with them we have performed experimental draws to test and confirm the predictions and scope of our mathematical models
Impact Several publications have appeared, or are in preparation, describing the mathematical model. Outputs in the form of software and user interfaces for the application of our modelling protocols are in preparation.
Start Year 2012

Title Improved method and apparatus for fabricating microstructured optical
Description The patent describes novel mathematical schemes, based on conformal geometry, to track the evolving shapes during the "fabrication draw" involved in making a breed of optical fibres called microstructured optical fibres. The basic idea rests on a so-called "geometrical pore model", or "GPM", which marries a conformal geometric description of the evolving shapes to the equations of slow viscous fluid motion.
IP Reference US10099955
Protection Patent application published
Year Protection Granted 2016
Licensed Commercial In Confidence
Impact The patent is currently being sponsored by Imperial Innovations, but discussions are currently active for this patent to be licensed exclusively to US-based company LowReTech LLC.

Title MOFSOFT
Description A front-end graphics user interface was developed (by Dr. Peter Buchak and Prof. Crowdy) that implemented mathematical models for the fabrication of microstructured optical fibres (devised by Crowdy and his collaborators).
Type Of Technology Software
Year Produced 2016
Impact This software is about to be licensed, via Imperial Innovations, to a US-based company LowReTech LLC.

Title Potential theory toolkit
Description With Dr. Everett Kropf, a post-doc supported by this grant, we developed a computational software tool, available for use by academics and industry stakeholders, for calculating potential theoretic results.
Type Of Technology Software
Year Produced 2016
Impact This software has underpinned a new collaboration between Prof. Crowdy and a petroleum engineer Dr. Ruud Weijermars as Texas A&M University who is interested in using it for onshore reservoir modelling. This has already led to a submitted joint publication.
URL https://github.com/ACCA-Imperial/PoTk

Title Software for computation of the Schottky-Klein prime function
Description A MATLAB software tool for the computation of an important special function called the Schottky-Klein prime function has been prepared. It can potentially be used in a variety of applied mathematical applications.
Type Of Technology Software
Year Produced 2015
Impact An invited paper to a special 50th anniversary issue of IMA J. Appl. Math will describe the background to the software, and its significance. This will aid with impact in the field.
URL https://github.com/ACCA-Imperial/SKPrime

Description ACCA Network
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I have led the institution of an international ACCA Network, which now has active nodes in the UK, Japan and Brazil. We have held several workshops, in the different countries, as well as smaller events such as Student Chapter Meetings. The idea is to provide positive brand recognition, and an outreach vehicle, for applied and computational complex analysis.
Year(s) Of Engagement Activity 2014,2015,2016
URL http://www3.imperial.ac.uk/ammp/research/acca

Description ACCA-UK/JP Second International Workshop, Kyoto University, Jan 18-19 2016
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact This was the second in an ongoing series of international workshops, under the auspices of the ACCA Network I founded, with an ACCA node in Japan.
Year(s) Of Engagement Activity 2016
URL https://www.math.kyoto-u.ac.jp/~karel/workshop-kyoto/index.html

Description ACCA-UK/JP Third International Workshop, Imperial College London, March 13/14 2017
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact This is the Third International ACCA (= "applied and computational complex analysis") workshop I have co-organized, jointly with my Japanese colleague Prof. Takashi Sakajo, as part of the international ACCA Network activities. The network is an ongoing initiative.
Year(s) Of Engagement Activity 2017
URL http://www.imperial.ac.uk/applied-computational-complex-analysis/acca-workshops/

Description Boeing Colloquium, University of Washington, Seattle USA
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Postgraduate students
Results and Impact I delivered an invited colloquium at University of Washington. The event is sponsored by Boeing Corporation and is a long-running and prestigious series.
Year(s) Of Engagement Activity 2015
URL https://depts.washington.edu/amath/news/boeing/

Description CMS Distinguished Lecture Series
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I delivered an invited series of 3 lectures including some on the outcomes of this project.
Year(s) Of Engagement Activity 2019
URL https://cms-math.net.technion.ac.il/prof-darren-crowdy/

Description Delivered invited Colloquium at University of Surrey, UK
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Public/other audiences
Results and Impact Invited Colloquium Speaker at Department of Mathematics, University of Surrey, UK on Feb 24 2017
Year(s) Of Engagement Activity 2017

Description Distinguished PIMS Colloquium Speaker
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Public/other audiences
Results and Impact I was invited, as Distinguished Colloquium Speaker, to deliver a lecture on my work on conformal geometry and special function theory to the Pacific Institute for Mathematical Sciences based at the University of British Columbia, Vancouver, Canada on Nov 28th 2016
Year(s) Of Engagement Activity 2016
URL https://www.math.ubc.ca/Dept/Events/index.shtml?period=future&series=all

Description ICMS Workshop on Applied Complex Analysis
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact The PI was an organiser of a specialised workshop, including public lectures, on the topic of applied complex analysis at the International Centre for Mathematical Sciences, Edinburgh.
Year(s) Of Engagement Activity 2017
URL http://icms.org.uk

Description Invited talk at workshop ("Mathematical Aspects of Physical Oceanography) at University of Vienna
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I was invited to give an invited talk at this workshop because of the burgeoning impact of some of my older work in the area of vortex dynamics dating back to the early 2000s. My talk summarised some of those earlier developments, and brought them to the attention of this new audience who have become interested in them.
Year(s) Of Engagement Activity 2018

Description NSF-CBMS Workshop: "Solving problems in multiply connected domains" (Principal Lecturer)
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I was invited to be Principal Lecturer at this special one-week workshop organised by the NSF and the Conference Board for Mathematical Sciences.
It took place June 18-22 2018 at University of California, Irvine, USA.
Year(s) Of Engagement Activity 2018
URL https://www.math.uci.edu/~ttrogdon/ACCA/index.html

Description Organiser of Newton Institute Program (4 months Sep-Dec 2019): "Complex analysis: techniques, applications and computations"
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact The Newton Institute is a world-renowned institute which holds long-term programs on areas of significant current interest in the mathematical sciences. I am one of 5 lead organisers who put forward a proposal for this workshop back in 2017. The event has been approved, is in organisation currently, and will take place in Sep-Dec 2019.
Year(s) Of Engagement Activity 2019
URL https://www.newton.ac.uk/event/cat

Description Principal Lecturer at NSF-CBMS Workshop, UC Irvine, California
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I was invited to deliver a series of 10 Lectures at this workshop as the "Principal Lecturer". I will also prepare a monograph based on the lectures to be published by the Society for Industrial and Applied Mathematics. These two related activities provide an opportunity for outreach, and for publicity of the research outcomes.
Year(s) Of Engagement Activity 2018
URL https://www.math.uci.edu/~ttrogdon/ACCA/index.html

Description Red Lotus Project Workshop at Chicheley Hall (Royal Society)
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact The PI was the lead organiser of this Chicheley Hall workshop (funded by the Royal Society: the PI was eligible to apply for this funding owing to his status as a Royal Society Research Fellow). The workshop involved 15 stakeholders from academia and industry.
Year(s) Of Engagement Activity 2016
URL http://www.imperial.ac.uk/applied-computational-complex-analysis/the-red-lotus-project/

Description Third International ACCA-UK/Japan Workshop, March 2017
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact An international 2-day workshop involving graduate students from both the UK and Japan, with invited plenary speakers. This is the third in an ongoing series of such workshops that promote international exchange and scientific diplomacy.
Year(s) Of Engagement Activity 2017
URL http://www.imperial.ac.uk/applied-computational-complex-analysis/acca-workshops/