May 4, 2021 – May 5, 2021


Researches based on Tomography touch on problems related to areas of pure mathematics as well as to applications. The contact point can be identified with image reconstruction by means of X-rays. The usual related topics base, on one side on improvements of reconstruction algorithms, on the other side on providing new uniqueness reconstruction models under suitable assumptions. Properties and results coming from the various scenarios often overlap, even if different tools and strategies are employed in different areas.


The aim of the Meeting is to share interdisciplinary aspects between the experimental research concerning X-ray tomography and the mathematical image reconstruction community.

Due to the backdrop of the pandemic, we have been forced to consider alternatives to our usual appointment, so we opted to turn it into an online event. It will be held on Zoom, the link will be sent out a few days before the Meeting to all registered participants.



Paolo Dulio - Politecnico di Milano
Paolo Finotelli - Politecnico di Milano
Andrea Frosini - Universita' degli studi Firenze
Silvia Pagani - Universita' Cattolica del Sacro Cuore
Carla Peri - Universita' Cattolica del Sacro Cuore
Lama Tarsissi - Sorbonne University Abu Dhabi / LIGM-UMR8049



Andreas Alpers, University of Liverpool Link to Video
Monia Cabinio, Fondazione Don Carlo Gnocchi, Milano Link to Video
Peter Gritzmann, Technische Universität München Link to Video
Robert Tijdeman, Leiden University Link to Video
Petra E. Vértes, University of Cambridge
Laurent Vuillon, Université de Savoie Link to Video


Peter Balazs, University of Szeged Link to Video
Joost Batenburg,
CWI Amsterdam Link to Video
Yan Gerard, Université Clermont Auvergne Link to Video

Imants Svalbe, Monash University Link to Video



Andreas Alpers - University of Liverpool

Title: On Reconstructing Polytopes from Refraction Data
Waves, such as electromagnetic waves or acoustic waves, can interact with matter in multiple ways. While classical computerized tomography is based on absorption, and imaging of polycrystalline materials is typically based on diffraction, we focus here on refraction. Refraction, governed mathematically by Snell's Law, is the change of direction of a wave that occurs when the wave travels from one medium to another.

In this talk we consider the inverse problem of reconstructing a polytope (representing a homogeneous object with given refractive index) from such refraction data. Focusing mainly on the two-dimensional case, we will discuss how such polytopes can be reconstructed, how many directions are needed, and which objects will be invisible (ghosts). A central role plays a classic result of H. Minkowski.

(Joint work with Peter Gritzmann and Stefan König)


Monia Cabinio - Fondazione Don Carlo Gnocchi, Milano

Title: Advanced MRI techniques in brain research.

With the advent of Magnetic Resonance Imaging (MRI), the possibility to study in vivo brain morphology and functioning has become widely used. The continuous innovations in the field allowed the development of advanced MRI sequences and techniques, used to qualitatively and quantitatively assess several brain tissue properties. Morphometrical indices of the gray matter (e.g. cortical thickness, volumes of subcortical structures…), microstructural properties of the white matter (e.g. Fractional anisotropy, Mean diffusivity), functional cortical activation associated (or not) to specific cognitive/motor tasks are just some of the data that can be extracted from MRI acquisitions, and their utility is currently of great importance in both clinical settings (diagnosis/rehabilitation) and in research. The present talk will be focused on the role of MRI in the brain research field, especially considering advanced MRI techniques and different experimental designs.


Peter Gritzmann – Technische Universität München

TitleOn Polyatomic Tomography over Abelian Groups: Some Remarks on Consistency, Tree Packings and Complexity
We deal with an inverse problem of reconstructing matrices from their marginal sums. More precisely, we are interested in the existence of r×s matrices for which only the following information is available: The entries belong to known subsets of c distinguishable abelian groups, and the row and column sums of all entries from each group are given. This generalizes Ryser’s classical problem of characterizing the set of all 0–1-matrices with given row and column sums and is a basic problem in (polyatomic) discrete tomography. We show that the problem is closely related to packings of trees in bipartite graphs, prove consistency results, give algorithms and determine its complexity. In particular, we find a somewhat unusual complexity behavior: the problem is hard for “small” but easy for “large” matrices.

(Joint work with Barbara Langfeld)


Robert Tijdeman - Leiden University

Title: Linear time reconstruction by discrete tomography.
The goal of discrete tomography is to reconstruct an unknown function f defined on a finite grid A via the given set of line sums in finitely many directions. This is complicated by the presence of ghosts (or switching functions) which allow many solutions to exist in general. Under the assumptions that A is two-dimensional, the line sums are exact, and the range of f is a field or unique factorization domain Ceko, Pagani and I succeeded in developing a method to construct a function f* which has the same line sums as f has in time linear in the product of the number of directions and the size of A. (Here linear is expressed in the number of elementary operations.) Ceko, Petersen, Svalbe and I studied so-called boundary ghosts which have a relatively small support and surround a big central area. The mentioned result of CPT makes it possible to reconstruct f in the central area in linear time. Ceko and I extended the boundary ghosts to three dimensions. Ceko, Pagani and I showed that a multiple application of their two-dimensional algorithm suffices for these three-dimensional boundary ghosts.


Petra Vertes - Lecturer at the University of Cambridge Fellow of the Alan Turing Institute

Title: Bridging the gap: From macro-scale neuroimaging to micro-scale transcriptomics.

The last 20 years have witnessed extraordinarily rapid progress in neuroscience, however the translation of this progress into improved understanding and treatment of mental health symptoms has been comparatively slow. One central challenge has been to reconcile different scales of investigation, from genes and molecules to cells, circuits, tissue, whole-brain and ultimately behaviour. In this talk I will describe my work on linking macro-scale neuroimaging data to the micro-scale Allen Human Brain Atlas (a brain-wide, whole-genome map of gene expression) and how we can apply these tools in the nascent field of imaging transcriptomics to further our understanding of schizophrenia and other neuropsychiatric disorders.


Laurent Vuillon, Université de Savoie

Title: Numeration systems and discrete geometry in DT and related problems
In this talk, we will focus first on the classical tomographical problem of the reconstruction of a binary matrix from projections in presence of absorption. We will explain the numeration system in base Fibonacci and the discrete geometry techniques behind the results on this topic. In a second time, we will present the Markoff numbers and the 100 years Uniqueness Frobenius' Conjecture.

In fact, we find for this topic a 1D reconstruction problem using a generalized numeration systems in 2 bases, namely in base Fibonacci and in base Pell. In a third time, we will present some discrete geometry techniques to tackle 3 conjectures of the book of Aigner.





Is the Era of Feature-Based Discrete Tomographic Reconstruction Over?

Coordinator: Peter BalazsUniversity of Szeged

With Joost Batenburg, Antal Nagy, Daniel Pelt and Laszlo Varga

Tomographic reconstruction can be formulated as the problem of solving a system of linear equations. Being this problem in general severely ill-posed, it is reformulated to an optimization task complemented by some prior information which is incorporated in a term of the objective function. Up to know, various such priors have been investigated how they could facilitate the discrete reconstruction: smothness, convexity, orientation, centroid, boundary, etc. Even though optimization may be challenging, this approach proved to be useful in many cases of limited projection data access.

Deep learning is recently an extremely hot topic in image processing. This new paradigm seems to overwrite everything, and it has bigger and bigger impact on the field of image reconstruction, too. Does it mean that the era of classical feature-based reconstruction is over? The aim of this round-table is to share opinions and discuss ideas on this topic.


•  Explorative tomography

Coordinator: Joost Batenburg CWI Amsterdam

With Tom Bultreys, Tristan van Leeuwen  and Nicola Viganò

Current tomographic imaging devices usually work according to a pre-set acquisition scheme, which governs the tomographic projection angles, X-ray dose, and other parameters. This approach is suitable for repetitive tasks, but for exploratory research, where one does not know what structures will be found inside the object, the acquisition can – and should – be adapted with respect to the observations made during the scanning process.
Mathematically, the standard workflow pre-specifies the set of equations (acquisition scheme) of the inverse problem and the prior function (object knowledge). In contrast, an exploratory workflow allows the inverse problem to be formed dynamically during the scan (dynamic forward operator) and the cost function to be adapted to the specific observations.

In this discussion we will talk about imaging experiments, models, and algorithms for exploratory tomographic imaging. A guiding topic will be how to setup a common set of problems and data-driven benchmarks to facilitate research and objective comparison between methods.


Reconstructing "Convex" Lattice Sets

Coordinator: Yan GerardUniversité Clermont Auvergne

With Sara Brunetti, Andrea Frosini and Lama Tarsissi

Discrete Tomography yields a number of results of complexity which have mostly been obtained from 1980 to 2000. The first milestone dates back to 1957 with a polynomial time algorithm for reconstructing lattice sets from 2 X-rays [Gale][Ryser]. Later on, many extensions of the initial problem have been proved to be NP-complete, for instance by increasing the number of directions of X-rays or by searching for a solution in a prescribed class of lattice sets.
A natural extension is to consider classes of convex lattice sets where the convexity of a lattice set might be understood in different ways. This direction of research has leaded in the nineties to two new milestones of the field:
- reconstructing a digital convex set from X-rays can be done in polynomial time [Brunetti,Daurat] under the assumption that it contains at least four directions with ordered slopes whose cross-ratio  is not in 4 3 2 3/2 or 4/3 [Gardner,Gritzmann]. These works do not provide any result with only 2 or 3 directions...
- reconstructing an HV-convex polyomino from horizontal and vertical X-rays can be done in polynomial time...
The two positive results are somehow restricted. And they are deeply different since the first one is built on a uniqueness result from R. Gardner and P. Gritzmann while the second one uses a 2-SAT reduction in order to deal with multiple solutions.
Since these two results, there have been notable works for extending our knowledge about the reconstruction of "convex" lattice sets according to several X-rays...
The topic of the round table is to do a survey of these results - a kind of state of the art around the reconstruction of convex lattice sets - and to exchange ideas around the most challenging and promising open problems that we have.


Compact and connected binary ghosts

Coordinator: Imants Svalbe - Monash University

With Matthew Ceko, Tim Petersen and Robert Tijdeman

Diffuse discrete compact arrays of signed element values that project as zero sums are called ghosts. These ghosts impose strict theoretical and practical constraints on our capacity to tomographically reconstruct any shape from its projected views. Here we review the construction of compact 'primitive' binary ghosts, spread over N discrete view directions, that are composed using either 1) the minimal number (2N) of elements, 2) boundary- or surface-only elements, or 3) that are comprised of the maximal number (2^N) of elements. The total number of elements in a ghost and the connectivity between their locations controls the degree and ease with which a shape's details can be reconstructed from its projected views. We use maximal ghosts (whose elements tile or fully fill areas or volumes) to induce the largest possible ambiguity in reconstructed images, and propose boundary ghosts (whose elements form connected boundaries or surfaces) to promote low reconstruction ambiguity. The location of a ghost that has been embedded into some image space remains invisible for those N projected views. We also review here the construction of related types of diffuse discrete arrays that, for several directions, project as zero sums, except for a delta-like central ray. The ghost centroid can be located as the intersection of the back-projection of those projected central rays. These arrays also exhibit a delta-like response for their autocorrelation. Such arrays may serve as unobtrusive n-dimensional digital fiducial markers or security/watermark signatures.


This conference is organized by


Politecnico di MilanoUniversità Cattolica del Sacro CuoreUniversità degli Studi di Firenze

Conference Information