## Abstract

This article gives a brief overview of the research in microlocal analysis, tomography, and integral geometry of Professor Eric Todd Quinto, Robinson Professor of Mathematics at Tufts University, along with the collaborators and colleagues who influenced his work.

The research of Eric Todd Quinto, Robinson Professor of Mathematics at Tufts University, spans Radon transforms, microlocal analysis, and their application to a broad range of problems in tomography. His career would not be as rich without the support of important mentors and friends over the years, and this article outlines his work through his collaborations.

Professor Quinto was lucky to meet Nobel Laureate and Tufts Physics Professor Allan Cormack, a pioneer in tomography, when he started at Tufts in 1977. Cormack introduced him to the field of tomography, and they wrote several articles together. In 1980, he arranged for Professor Quinto to attend the first Oberwolfach tomography workshop, where Quinto met founders of the field, including Frank Natterer and Alfred Louis. They discussed important tomography problems and an applied way of thinking and became lifelong friends. Many of Louis’ students, including Bernadette Hahn, Peter Maaß, Andreas Rieder, Gaël Rigaud, and Thomas Schuster became mathematical collaborators, as well as good friends, sharing mathematics and life.

Professor Quinto has been lucky to work with talented and creative younger researchers, including Gaik Ambartsoumian, Raluca Felea, Jürgen Frikel, Christine Grathwohl, Jakob Jørgensen, Esther Klann, Venky Krishnan, Peer Kunstmann, Cliff Nolan, Ronny Ramlau, James Webber, and others, as well as Tufts colleagues Christoph Börgers and Fulton Gonzalez.

Sharing mathematics through teaching and mentoring his students has always been satisfying to Professor Quinto, and he has enjoyed doing research with many talented undergraduates and his Ph.D. students Yiying Zhou, Aleksei Beltukov, Anuj Abhishek, and Alejandro Coyoli.

## Tomography

The transform that models X-ray tomography is a special case of the generalized Radon transform that Professor Quinto studied in graduate school. When he received his Ph.D. from MIT in 1978, the field of tomography was just beginning. Mathematicians such as Larry Shepp and Kennan Smith introduced the field to mathematicians by presenting the theory and practical applications in mathematical language. It was a perfect time to start in the field. Intrigued by Allan Cormack’s work in exterior tomography, Quinto developed several reconstruction algorithms, which he tested on industrial data [14, 16].

Later, Jan Boman introduced Professor Quinto to his former student Ozan Öktem who was working on single particle electron microscopy. Öktem, Quinto, and biologist Ulf Skoglund developed simple and effective algorithms for this highly ill-posed limited data problem and tested them on real data. Professor Quinto enjoyed learning about the biological and practical challenges of this application [17, 18].

Some of Professor Quinto’s most important work involves the melding of his expertise in tomography with his expertise in microlocal analysis. His first article joining these two themes, [15], was among the articles that introduced microlocal analysis to applied tomographers. In it, he described visible and invisible singularities in limited-data tomography. Scientists found the article readable and useful for their work, and Quinto has written other introductory articles, such as [11]. He continued to use these ideas to explain visible and invisible features as well as artifacts in a range of tomographic problems (e.g., [6, 5, 20]. See also [12, 10]. Now, microlocal analysis has permeated tomography.

To describe the results, we introduce the wavefront set. If *f* is a
distribution, its wavefront set,
*f* is not smooth *and* directions in which it is not smooth at
those points. For example, if *f* is the characteristic function of a
set *A* with smooth boundary, then
*f* is not smooth at all points on

For
*f* is

the transform

the integral of *g* over parameters for all lines
containing

Much of Professor Quinto’s research focuses on limited data X-ray CT, in
particular which singularities of a function *f* are visible and which
are invisible in tomographic reconstructions. In addition, he analyzes
when there can be artifacts or singularities added to the
reconstruction that are not in the original object. If limited data
are given over an open set
*limited data Radon transform (for A)* to be

*A*. Thus,

*Rf*for

*A*, suppressing both some visible singularities and artifacts.

## Theorem 1 ([15, 5]).

*Let D be an elliptic
pseudodifferential operator in the p-variable, and let f be an integrable
function or a distribution of compact support.
Let A be an open symmetric subset of
*

*If
*

*For complete data, all singularities of f are visible:
*

Therefore, if
*f* under

A precise analysis of these added artifacts including geometric
descriptions is in [5]; sometimes the artifacts can be
entirely along lines

## Integral geometry

Professor Quinto was fortunate to have mentors including his Ph.D. advisor, Victor Guillemin, and informal advisor Sigurdur Helgason. Guillemin developed seminal ideas in microlocal analysis, and he proved that Radon transforms are Fourier integral operators in the highly influential book with Sternberg [8]. Professor Quinto’s first article [13] uses their framework to analyze the microlocal properties of generalized Radon transforms satisfying the Bolker condition, and this was also used in his later work in both tomography and integral geometry.

Sigurdur Helgason proved fundamental properties of Radon transforms
and taught Quinto their properties, including elegant support
theorems, which Quinto generalized. To describe the first of these
support theorems, we consider the hyperplane transform. Let
*generalized Radon hyperplane
transform*

where

Professor Quinto’s work with Jan Boman combined their perspectives and expertise
and was the start of a rich collaboration. Jan Boman introduced Quinto
to a powerful microlocal analytic regularity theorem of Kawai,
Kashiwara, and Hörmander [9, Theorem 8.5.6] that
allows one to infer support restrictions on a function at the boundary
of its support if a normal direction is not in its analytic wavefront
set. Many Radon transforms are real analytic Fourier integral
operators, and they used this theorem and the analytic microlocal
regularity of these FIO to eat away at

## Theorem 2 ([3, Theorem 2.2]).

*Let
*

*W*be a connected symmetric open set in

*f*is zero in a neighborhood of the hyperplane

*f*is disjoint from the union of hyperplanes

To our knowledge, this was the first result to use analytic microlocal analysis to prove support theorems for generalized Radon transforms. Important results for analytic Radon transforms on hyperfunctions and distributions have subsequently been proven using other microlocal techniques.

The proof of Theorem 2 is simple enough to describe.
One starts with a hyperplane
*W* that is disjoint
from
*W* to a
hyperplane
*and*
*f*. This contradiction shows that
*W* meets

Inspired by seminal work of Gelfand on admissible complexes, and a
fundamental article by Greenleaf and Uhlmann [7], Boman and
Quinto proved support theorems for Radon transforms on line complexes
in

Professor Quinto’s work in the 1990s and 2000s with Mark Agranovsky on stationary sets for the wave equation was rewarding because Professor Agranovsky’s deep knowledge of harmonic analysis complemented his expertise in microlocal analysis. They had fun bouncing ideas off of each other by e-mail and creating new mathematics.

Let
*f* is defined as

where
*f* is
trivially recoverable from data over all
*Rf*. Let

## Definition 3.

The set
*set of injectivity*
for *R* on compactly supported functions if, whenever

Agranovsky and Professor Quinto developed theorems to characterize sets of
injectivity for the spherical transform in a range of papers starting
for the plane [1]. Let
*Coxeter system* of lines.

## Theorem 4 ([1]).

*The set
*

Therefore, if *S* consists of curves, any one of which is not a line, then
*S* is a set of injectivity. In fact, *S* can be any infinite number of
points that is not contained in any Coxeter system.

The proof outline is as follows. Assume *f* is a nontrivial compactly
supported function in the plane and

are all zero for
*k* such that
*S* is contained in the zero set of a harmonic
polynomial. The authors then use subtle properties of harmonic
polynomials in the plane to show that the zero sets of such
polynomials are either in a Coxeter system or have at least two
connected components that are smooth curves that are bounded away from
each other. Then, they use analytic microlocal analysis to show that
if
*f* must be zero.
This final argument is easiest to see when the two curves are parallel
lines,
*R* with centers
restricted to
*f* integrates
to zero over all circles centered on both

Agranovsky and Quinto also partially characterized stationary sets for
the spherical transform in

For more information about Professor Quinto’s research, see https://sites.tufts.edu/tquinto/. Articles in the bibliography provide more complete references to work in the field.

Professor Quinto is not only an outstanding organizer of various international conferences, including numerous SIAM conferences and the Inverse Problems: Modeling and Simulation (IPMS) conference series, thus playing a leading role in the inverse problems community, but also above all a good-natured and friendly person.

Apart from his notable contributions to the field of inverse problems, Professor Quinto is also known for his warm personality, soft-spoken way and spirit of camaraderie and of course, for his booming laughter, that brings instant cheer to everyone around. Professor Quinto is a special human being loved by everyone in the inverse problems community and not only! For decades, Professor Quinto has been a friend to hundreds of young patients as a volunteer at Boston Children’s Hospital, bringing smiles and good cheer, rocking the babies, talking and playing with the kids and even helping them with Maths homework. In 2003 in recognition of his leadership as a volunteer, Professor Quinto received the Bob Groden Distinguished Service Award, a Children’s Hospital humanitarian honor. Professor Quinto’s compassion and earnestness obviously came also through to his students who have been praising him for being caring and excited about teaching [21]. A truly brilliant mathematician with a big heart!

On behalf of his colleagues, students, and friends from all over the world, we would like to wish Todd every success in his scientific work and much happiness with his wife, Judy Larsen, and daughter Laura Quinto.

## References

[1] M. L. Agranovsky and E. T. Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal. 139 (1996), no. 2, 383–414. 10.1006/jfan.1996.0090Search in Google Scholar

[2] M. L. Agranovsky and E. T. Quinto, Stationary sets for the wave equation in crystallographic domains, Trans. Amer. Math. Soc. 355 (2003), no. 6, 2439–2451. 10.1090/S0002-9947-03-03228-8Search in Google Scholar

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[14] E. T. Quinto, Tomographic reconstructions from incomplete data—numerical inversion of the exterior Radon transform, Inverse Problems 4 (1988), no. 3, 867–876. 10.1088/0266-5611/4/3/019Search in Google Scholar

[15]
E. T. Quinto,
Singularities of the X-ray transform and limited data tomography in

[16] E. T. Quinto, Local algorithms in exterior tomography, J. Comput. Appl. Math. 199 (2007), no. 1, 141–148. 10.1016/j.cam.2004.11.055Search in Google Scholar

[17] E. T. Quinto and O. Öktem, Local tomography in electron microscopy, SIAM J. Appl. Math. 68 (2008), no. 5, 1282–1303. 10.1137/07068326XSearch in Google Scholar

[18] E. T. Quinto, U. Skoglund and O. Öktem, Electron lambda-tomography, Proc. Natl. Acad. Sci. USA 106 (2009), no. 51, 21842–21847. 10.1073/pnas.0906391106Search in Google Scholar PubMed PubMed Central

[19] J. W. Webber and E. T. Quinto, Microlocal properties of seven-dimensional lemon and apple radon transforms with applications in compton scattering tomography, Inverse Problems 38 (2022), no. 6, Paper No. 064001. 10.1088/1361-6420/ac65adSearch in Google Scholar

[20] J. W. Webber, E. T. Quinto and E. L. Miller, A joint reconstruction and lambda tomography regularization technique for energy-resolved x-ray imaging, Inverse Problems 36 (2020), no. 7, Paper No. 074002. 10.1088/1361-6420/ab8f82Search in Google Scholar

[21]
*Angel Among Us*, Tufts Journal, November 19 (2008), available at https://tuftsjournal.tufts.edu/2008/11_2/people/01/.
Search in Google Scholar

**Received:**2022-04-20

**Accepted:**2022-04-20

**Published Online:**2022-06-25

**Published in Print:**2022-08-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston