INVITED SPEAKERS
Toni Bohnlein, Weizmann Institute of Science
Cécile Bordier, Centre Hospitalier Universitaire de Lille
Sara Brunetti, Università di Siena
Davide Coluzzi, Politecnico di Milano
Niccolò di Marco, Università di Firenze
Roberto Fedele, Politecnico di Milano
Gabriele Fici, Università di Palermo
JaquesOlivier Lachaud, Université Savoie Mont Blanc
Clément Lagisquet, Université Savoie Mont Blanc
Stefania Petra, Heidelberg University
Fabrizio Pizzagalli, Università di Torino
Csaba Vincze, University of Debrecen
Laurent Vuillon, Université Savoie Mont Blanc
ORGANIZING AND SCIENTIFIC COMMITTEE
Paolo Dulio  Politecnico di Milano
Paolo Finotelli – Université de Caen Normandie
Andrea Frosini  Università degli studi Firenze
Silvia Pagani  Università Cattolica del Sacro Cuore
Lama Tarsissi  Univ. Paris Est Marne La Vallée and Sorbonne University Abu Dhabi
SUPPORTED BY
The 16th edition of the Meeting is dedicated to Carla Peri (19572022), who used to be a member of the organizing and scientific committee since the first edition in 2007, and also did a very active research in discrete tomography.
The aim of the Meeting is to share interdisciplinary aspects between the experimental research concerning Xray tomography and the mathematical image reconstruction community. Properties and results coming from the various scenarios often overlap, even if different tools and strategies are employed in different areas.
The contact points can be identified with image reconstruction by means of Xrays, reconstruction algorithms, and reconstruction models under suitable assumptions.
A few special contributions coming from researches in neuroscientific imaging have been also included since the edition in 2017, later expanded, and now integral part of the event.
Participation and talks proposed by young researchers have been always encouraged
Generalized Microscopic Image Reconstruction Problems
Abstract We consider a generalization of the microscopic image reconstruction problem (MIR) where the task is to inspect a specimen represented as a collection of points typically organized on a grid in the plane. Assume each point x has an associated physical value l(x), which we would like to determine. However, it might be that obtaining these values precisely (by what we call a surgical probe) is difficult, risky, or impossible. The alternative is to employ aggregate measuring techniques (such as CT or MRI), whereby each measurement is taken over a larger window, and the exact values at each point are subsequently extracted by computational methods. We extend the MIR framework in a number of ways. First, we study a generalized setting where the inspected object is represented by an arbitrary graph G, and the vector l in Rn assigns a value l(v) to each node v. A probe P(v) centred at a node v will capture a window encompassing its entire neighbourhood N[v], i.e., P(v) sums l(u), for each u in N[v]. Additionally, we analyze a weighted variation in which a node v may have an aggregation coefficient w(v), namely P(v) sums l(u) and w(v) l(v), for each u in N[v]. We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector l from the collection of probes, P’ = P(v) for each v in V. We then consider cases where such reconstruction is impossible (namely, graphs G for which the probe vector P’ is inconclusive, in the sense that there may be more than one vector l yielding P. Assume that surgical probes (whose outcome at node v is the exact value of l(v)) are technically available, yet are expensive or risky, and must be used sparingly. We show that in such cases, it may still be possible to achieve reconstruction based on a combination of a collection of ordinary (aggregate) probes together with a suitable set of surgical probes. We aim at identifying the minimum number of surgical probes necessary for a unique reconstruction, depending on the graph topology. This is referred to as the Minimum Surgical Probing problem (MSP). Besides providing a solution for the above problems for arbitrary graphs, we also explore the range of possible behaviours of the (MSP) problem by determining the number of surgical probes necessary in certain specific graph families including grid graphs.
Brain connectivity in MRI: methods, interpretation and needs
Abstract After an introduction to MRI and its various acquisitions and the different school of thought that have led us to connectivity, I will present methods to perform connectivity analysis. From ICA to graph theory via statistical permutation methods, I will try to give you a general idea of the possible methods and the potential problems encountered. I will finish with concrete applications on patient populations before reviewing the current needs and the methods under development.
How to rebuild a binary image from its multilevel description based on generalized salient pixels
Abstract In Discrete Tomography, several classes of binary images presenting different kind of convexity have been considered for their reconstruction from projections.In Digital Image Analysis convexity estimators are among the most important shape descriptors. Shape feature extraction and representation plays an important role in many categories of applications like for example shape retrieval, shape recognition and classification, shape approximation and simplification, and so on. In this talk, we present a multilevel description of a binary image based on a special kind of convexity. In particular, the so called generalized salient pixels provides a decomposition of the image into Qconvex hulls at different levels and they are stored in a matrix called, GSmatrix (where GS stands for Generalized Salient). Therefore, there is a onetoone correspondence between the binary image and its GSmatrix. We show how to build the GSmatrix from the binary image and vice versa how to rebuild the binary image from its GSmatrix. Then, we play with GSmatrices to see how changes can modify the rebuild images. In Mathematical Morphology, a wide range of operators permits to process images for edge detection, noise removal, image enhancement and image segmentation, to mention same common usage. Among them, the two most basic operations are erosion and dilation. Virtually all other mathematical morphology operators can be defined in terms of combinations of erosion and dilation along with set operators such as intersection and union. We start by defining a new "erosion" operation based on the interaction of GSmatrix binary image and we investigate some consequences.
Brain Connectivity through Graph Theory: SPIDERNET a New Tool to Explore SubNetworks
Abstract Brain connectomics consists in the modelling of human brain as a network, mathematically represented as a numerical connectivity matrix. It is represented by collection of nodes (brain regions) and links (connections), describing any kind of relationship between pairs of nodes, at different scales.
The brain regions are strongly connected through neuroanatomical white matter (WM) pathways which can be quantified by the number of streamlines derived with deterministic WM tractography or other metrics. In parallel to structural connectivity (SC), synchronous and asynchronous activity of specific brain regions results in related complex cognitive functions, which can be investigated in terms of functional connectivity (FC). Conversely, FC represents the magnitude of temporal correlations between the blood oxygenation leveldependent (BOLD) time series produced by pairs of brain regions. In both cases, the resulting networks determine a complex system involving several interconnected nodes relevant to the parcellation of cortical areas and subcortical structures which result to be difficult to be interpreted. To overcome this limitation, circular representation of connectivity by "connectograms” is currently used via opensource tools, which, however, lack userfriendly interfaces and options to explore specific subnetworks.
A complete wholebrain network can be indeed made of thousands of links, and it is wellknown that this largescale network is associated with highlevel cognitive functions. However, the brain is composed of several interacting lowerscale subnetworks, which are characterized by distinct patterns of brain activation, identifying specific domains of behavior and cognition. Therefore, a common practice in explorative studies of brain connectivity is the extraction of valuable subnetworks referenced by the domain knowledge. Focusing on subnetworks, analyzed both qualitatively, by a reduced connectogram, and quantitively by local and global subgraph indexes, can lead to easier data interpretation driven by the addressed physiological and/or pathological problem.
In this context, we developed SPIDERNET (Software Package Ideal for Deriving Enhanced Representations of brain NETworks), an easytouse, flexible, and interactive tool for connectograms generation and subnetwork exploration.
The potential impact of the tool was tested on pilot cases.
The null label problem and its relation to the 2intersection graph
Abstract A 3uniform hypergraph H consists of a set V of vertices, and a subset of triples of V, called set of edges E. Let a null labeling be an assignment of +1 or 1 to the triples such that each vertex has a signed degree equal to zero. If a null labeling exists, we say that the hypergraph is null. Assumed as necessary condition the degree of every vertex of H to be even, the Null Labeling Problem consists in determining whether H has a null labeling. It is remarkable that null hypergraphs arise considering two hypergraphs with the same degree sequence. In particular, the symmetric differences of these two hypergraphs give a new hypergraph that is null. From a discrete tomography point of view, null hypergraphs arise from matrices with the same projections, i.e. solutions of the same reconstruction problem. Therefore they allow modeling of switching components, a very used notion in this field of research.
Although the problem is NPcomplete, the subclasses where the problem turns out to be polynomially solvable are of interest. We defined the notion of 2intersection graph related to a 3uniform hypergraph and we prove that if it is Hamiltonian then the related 3hypergraph has a null labeling. Then we aimed to deepen the knowledge of the structural properties of 2intersection graphs. Going into details, we studied when, given a graph G, it is possible to find a 3hypergraph such that its 2intersection graph is G. If it is possible, we say that G is reconstructable or equivalently, it has the 2intersection property. It’s easy to see that the question is relatively straightforward for some classes. However, using some suitable gadgets, we proved that the problem in its general form is NPComplete.
A furnace for in situ wettability experiments at high temperatures under Xray Microtomography
Abstract In this study, we analyzed the problem of a compact furnace, to be used for in situ experiments in a conebeam Xray microtomography commercial system. The design process was accomplished and outlined through its main steps, until the realization of a prototype. The furnace was conceived to carry out wettability experiments at temperatures up to 700 C and under inert atmosphere on sessile droplets of a molten metal alloy, with a few millimeters diameter, posed on a thin ceramic substrate. Xray imaging of the molten droplet is expected to permit an accurate threedimensional reconstruction of the droplet profile and a robust estimation of the related quantities (such as the contact angle and the surface tension) utilized for the assessment of metalceramic joints by brazing. The challenges faced during this project, mostly related to the constraints of the setup, and the novel solutions implemented were discussed also with the support of analytical and numerical tools, in terms of interaction of Xrays with matter, geometry and working principle, heat transfer and insulation, material selection
Minimal Forbidden Words and Digital Lines
Abstract Minimal Forbidden Words (a.k.a. Minimal Absent Words) are a powerful tool for dealing with words and languages (set of words). Given a language L that is closed under taking factors (i.e., a factorial language) the set of Minimal Forbidden Words of L is the set of words that do not belong to L but whose proper factors belong to L. There is a bijection between factorial languages and their sets of MFWs. Hence, one can study the combinatorial properties of a language by looking at its set of Minimal Forbidden Words. In the context of digital geometry, one is often interested in binary words approximating Euclidean segments in the plane (balanced words). Among them, Christoffel words play a central role. A Christoffel word is a balanced word with the Lyndon property, i.e., it is lexicographically smaller than all its nontrivial rotations. Another interesting class is that of digitally convex words, i.e., binary words approximating convex lines in the plane. These can be defined as those words whose Lyndon factorization consist of balanced (hence Christoffel) words. We discuss the set of MFWs of balanced words and of digitally convex words.
Multistage Deep Learning Denoising for Computed Tomography
Abstract Computed Tomography (CT) is a powerful tool in various areas. Recently, lowdose CT has drawn a lot of attention, because the high radiation dose in normal CT could be harmful to objects or patients. Reducing the projection numbers is also becoming a popular measure to improve the scanning speed. However, reconstructions from lowdose or/and limited projections typically suffer from high noise. Recently, deep learning has been proposed as an effective denoising technique for CT. However, when the noise level is high or when specific artifacts like ring or zinger artifacts occur, current deep learning approaches can yield suboptimal results. In this talk, we propose a deep learningbased multistage denoising algorithm. In our algorithm, the denoising takes place subsequently in the projection domain, the sinogram domain, and the reconstruction domain, in a supervised manner. In each stage, both the denoised output from the previous stages and the raw noisy data are utilized for denoising. Experiments show that our algorithm is effective in denoising highly noisy data with additional ring or zinger artifacts.
Alternative definitions for digital convexity Part 1
Alternative definitions for digital convexity Part 2
Abstract This talk proposes full convexity as an alternative definition of digital convexity, which is valid in arbitrary dimension. It solves many problems related to its usual definitions, like possible non connectedness or non simple connectedness, while encompassing its desirable features. Fully convex sets are digitally convex, but are also connected and simply connected. They have a morphological characterisation, which induces a simple convexity test algorithm. Arithmetic planes are fully convex too. Full convexity implies local full convexity, hence it enables local shape analysis, with an unambiguous definition of convex, concave and planar points. We propose also a natural definition of tangent subsets to a digital set. It gives rise to the tangential cover in 2D, and to consistent extensions in arbitrary dimension. We present two applications of tangency: the first one is a simple algorithm for building a polygonal mesh from a set of digital points, with reversibility property, the second one is the definition and computation of shortest paths within digital sets. In a second part of the talk, we study the problem of building a fully convex hull. We propose an iterative operator for this purpose, which computes a fully convex enveloppe in finite time. We can even build a fully convex enveloppe within another fully convex set (a kind of relative convex hull). We show how it induces several natural digital polyhedral models, whose cells of different dimensions are all fully convex sets. As perspective to this work, we study the problem of fully convex set intersection, which is the last step toward a full digital analogue of continuous convexity.
Graph theory: From simulation to information
Abstract This talk, which follows "From fibers to brain diseases", presents the techniques used to get meaningful informations from the data. First will be how is the graph (or multigraph) constructed, second is the link between clustering clustering (CC) and crowdfulness, third is the introduction of a new measure: the fluidity of the network, to understand how much the polymers flows in it.
Unet based deep neural networks for transmission tomography
Abstract The fusion of computer tomography and deep learning is an effective way of achieving improved image quality and artifact reduction in reconstructed images. The literature provides a variety of options for the combination of computer tomography and deep learning methods. Most of the time the researchers applied deep learning as a pre or postprocessing tools before or after the reconstruction, while others interested in performing the reconstruction with the neural network. In this presentation, we focus on the Unet based combination of the neural networks and computer tomography. We compared two of our own Unet based neural network architectures to three other Unet based approaches. In the case of our own architectures, the image reconstruction step is located inside the neural networks, which allows the network to be trained by taking the mathematical model of the projections into account. This strong connection enables us to enhance the projection data and the reconstructed image together. We tested the methods on two datasets. The datasets contain physically correct simulated data, and they show strong signs of beam hardening and electrical noise. We also performed a numerical evaluation of the neural networks on the reconstructed images according to three error measurements and provided a scoring system of the methods derived from the three measures. Our experimental results showed that the reconstruction step used in skip connections in deep neural networks improves the quality of the reconstructions.
A Geometric Approach to Coalition Resilient Outcomes in Social Graphs modeled by the Max kCut Game
Abstract We investigate strong Nash equilibria in the max kcut game, where we are given an undirected unweighted graph together with a set of k colors. Vertices represent players and edges their mutual interests. By selecting a color players induce a coloring. Each player v has the set of k colors as strategy set, and its payoff is the number of neighbors of v that has chosen a different color. The max kcut game has been extensively studied due to the great interest for its realworld applications with selfish agents. Very little is known about the existence of strong equilibria in such game. In this work we make some steps forward in the comprehension of it, proving that do not exist any subsets of nodes able to increase their own payoffs simultaneously. Instead of formulating a basic approach based only on the notions and tools of game theory, we split the nodes of the graph into three subsets (the coalition itself, the coalition boundary and the nodes without relationship with the coalition) and we use a novel approach based on discrete geometry and algorithms on graphs to study the properties of the adjacency matrix of the graph and obtain significant information on the coalition and its boundary.
Geometric Multilevel Optimization for Discrete Tomography
Abstract In this talk, I will present a geometric multilevel optimization approach to the reconstruction problem in discrete tomography based on undersampled projection data that is relevant to a range of applications and naturally leads to a hierarchy of models of varying discretization levels. Multilevel optimization is employed to take advantage of this hierarchy: while working at the fine level the search direction is computed based on a coarse model which speeds up tomographic reconstruction. We import few concepts of information geometry to the northotope and propose a smoothing operator that only uses firstorder information and incorporates constraints smoothly. We show that the proposed algorithm is well suited to the illposed reconstruction problem in discrete tomography and demonstrate its efficiency on several largescale examples.
Investigating genetic factors driving brain morphology through Neuroimaging studies
Abstract Imaging genetics aims to identify genetic variants associated with the structure and function of the human brain. Recent studies were able to identify and replicate common genetic variants influencing human brain structure as measured through magnetic resonance imaging (MRI). In this talk, the impact of imaging genetic studies will be presented as one of the methods for understanding the causal chain from genetic variant to increased risk for disorders.
Image reconstruction method for a discrete optical tomograph
Abstract Optical tomography belongs to the group of absorption tomography, for which the natural signal at the output of the sensor is a analog signal. In some applications it is advisable to use appropriate discretization systems. As a result of the operation of such systems, a discrete measurement signal is obtained. This issue is discussed on the example of a discrete optical tomograph for measuring moving gas bubbles. The presentation shows the structure of the tomograph. The prototype solution uses a light beam in the visible light range. To measure the intensity of light radiation, fiber optic detectors mounted in two perpendicular sensors were used. The small number of projections makes the reconstruction process more difficult. For this reason, a dedicated image reconstruction algorithm based on the Kohonen neural network has been proposed. Physical phenomena occurring on the border of the liquid and gas phases, which are the main cause of the change in the intensity of light reaching the detector, are discussed. Due to the fact that the shapes of the gas bubbles can differ significantly from each other and the number of projections is limited, the reconstruction of the image of the bubbles with the space was approximated by means of ellipses. Such an approximation is often sufficient to determine the parameters of mass transfer for the purposes of technological control of aeration and oxygenation processes.
Inpainting in discrete Sobolev spaces: the subtle bond between content and structure
Abstract In the literature, the so called “inpainting problem” consists in filling zones of “missing” information in an image, i.e., in a twodimensional discrete set ([9]). To have satisfactory results, the inpainted area has to be not “eyedistinguishable” from the rest of the given image by a human observer. Different solutions have been proposed to tackle the problem: on one hand, PDE based approaches ([2, 1, 5, 6, 8, 3]), on the other hand, exemplarbased ones ([7, 10]). Starting from this last kind of techniques we consider discrete Sobolev spaces to formulate a new functional from whose minimization good results in inpainting can be achieved. Moreover, we extend the work of [4] structuring a new priority index. Achieved results show an effective uncertainty reduction in the choice of the missing points, with a consequent improvement in the quality of the final reconstructions.
References
[1] M. Bertalmio, A. L. Bertozzi, G. Sapiro, NavierStokes, Fluid Dynamics and Image and Video Inpainting, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.CVPR 2001, (2001).
[2] M. Bertalmio, G. Sapiro, V. Caselles, C. Ballester, Image Inpainting, SIGGRAPH '00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, (2000), pp. 417424.
[3] M. Burger, L. He, C. B. Sch¨onlieb, CahnHilliard Inpainting and a Generalization for Grayvalue Image, SIAM J. Imaging Sci., 2 (4), (2009), pp. 11291167.
[4] A. Criminisi, P. Perez, K. Toyama, Region Filling and Object Removal by ExemplarBased Image Inpainting, IEEE Transactions on Image Processing, 13 (9), (2004), pp. 12001212.
[5] T. F. Chan, J. Shen, Nontexture Inpainting by CurvatureDriven Diffusions, Journal of Visual Communication and Image Representation, 12 (4), (2001), pp. 436449.
[6] T. F. Chan, J. Shen, Mathematical Models for Local Nontexture Inpaintings, SIAM J. Appl. Math., 62 (3), pp. 10191043.
[7] A. Efros, W. T. Freeman, Image Quilting for Texture Synthesis and Transfer, Proceedings of SIGGRAPH 2001, (2001), pp.341346.
[8] S. Esedoglu, J. Shen, Digital inpainting based on the MumfordShahEuler image model, European Journal of Applied Mathematics, 13, (2002), pp. 353370.
[9] S. Masnou, J. Morel, Level lines based disocclusion, Proceedings 1998 International Conference on Image Processing. ICIP98, 3, (1998), pp. 259263.
[10] L. Y. Wei, M. Levoy, Fast Texture Synthesis using Treestructured Vector Quantization, Proceedings of SIGGRAPH 2000, (2000), pp. 479488.
Women and mathematics. "Thank you Carla"
Abstract TO BE RECEIVED
Distancemean functions and their geometric applications
Abstract A distancemean function measures the average distance of points from the elements of a given subset K in the space. The level sets of a distancemean function are called generalized conics with elements in K as focal points. The most important discrete examples are polyellipses (polyellipsoids) as the level sets of a function measuring the arithmetic mean of distances from finitely many focal points, i.e. the distance sum must be constant for the points of a polyellipse (polyellipsoid). Polynomial lemniscates can also be given as the level sets of a function measuring the geometric mean of distances from finitely many focal points, i.e. the distance product must be constant for the points of a polynomial lemniscate. We have a lot of generalizations as well. Instead of formulating a basic definition to cover the wide range of possibilities we are going to present some applications in convex geometry (bisection of bodies by coordinate hyperplanes) and geometric tomography (reconstruction of planar bodies from the coordinate Xrays by the taxicab distancemean function and some related results).
Abstract In this talk, the first part presents how to construct biological fibers from proteins and how to build a model using tiling theory and discrete geometry. A second part is devoted to the description of the fiber formation in Parkinson’s and Alzheimer’s diseases. In a third part, we describe coarsegrain simulations where proteins are modelled as semiflexible polymers with punctate multifunctional binding sites in order to obtain a model of biomolecular condensates and to show some different behaviors (fluid versus rigid phase) according to various parameters (affinity, concentration, …). In fact, understanding the transition between fluid phase to rigid phase is important because this phenomenon is linked to certain neurodegenerative diseases. This is a joint work with Julian C. Shillcock (EPFL), Claire Lesieur (Ampère and IXXI), Clément Lagisquet (USMB), Jérémy Alexandre (EPFL) and John H. Ipsen (University of Souththern Denmark).
Conference room  7° floor
Building 14
Via Bonardi n. 9
20133 Milano  Italy
Subway: Line 2 (MM2 green line), stop: PIOLA. Then 50 meters on walk.
From Linate Airport: Bus n. 91, stop: ROMAGNAPIOLA, or n. 93, stop: PONZIOCELORIA. Then 5 minutes walking.
From Malpensa Airport Terminal 1: Train Malpensa Express to Milano Cadorna  Subway Line 2.
From Malpensa Airport Terminal 2: Shuttle service to Terminal 1  Train Malpensa Express to Milano Cadorna  Subway Line 2.
From Orio al Serio Airport: Orio Shuttle (or other companies) to Milano Centrale – Subway Line 2.
Monday 2  
10:15  10:30 
Opening 
10:30  11:25 
J.O. Lachaud 
11:30  12:25 
T. Bohnlein 
12:30  14:00 
Lunch Break 
14:00  14:55 
S. Petra 
15:00  15:25 
M. Rzasa 
15:30  16:00 
M. Seracini 
16:00  16:25 
Coffee break 
16:30  16:55 
C. Olasz 
17:00  17:30 
J. Shi 






Tuesday 3  
9:30  10:25 
G. Fici 
10:30  10:55 
R. Fedele 
11:00  11:30 
Coffee Break 
11:30  12:15 
D. Coluzzi 
12:20  12:40 
L. Tarsissi 
12:45  14:00 
Lunch Break 
14:00  14:55 
C. Bordier 
15:00  15:25 
F. Pizzagalli 
15:30  16:00 
Coffee Break 
16:00  16:25 
L. Vuillon 
16:30  16:55 
C. Lagisquet 


20:00 
Social Dinner 
Wednesday 4  
9:30  10:25 
C. Vincze 
10:30  10:55 
S. Brunetti 
11:00  11:30 
Coffee Break 
11:30  12:15 
N. Di Marco 
12:20  12:45 
G. Palma 
12:50 
Farewell Aperitif 














Red name = Online presentation
Building 14 “NAVE”  Mathematics Department, Conference Room, 7^{th} floor 
Building 14 “NAVE”  Mathematics Department, Seminar Room, 6^{th} floor 