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2D neighborhood motion maps allowed us to study local alterations of discrete spaces under digitized rigid motions. Nevertheless, computation of 3D neighborhood motions maps is a challenging problem.

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Euclidean rotations in \(\mathbb{R}^n\) are bijective and isometric maps. Nevertheless, they lose these properties when digitized in \(\mathbb{Z}^n\). For \(n=2\), the subset of bijective digitized rotations has been described explicitly by Nouvel and Rémila and more recently by Roussillon and Cœurjolly. In the case of 3D digitized rotations, the same characterization has remained an open problem. We have been studied the problem and up-to-date, we have proposed an algorithm for certifying the bijectivity of 3D digitized rational rotations using the arithmetic properties of the Lipschitz quaternions.

Published in *Linux+ Magazine (Polish edition)*, 2010

**Author(s):** K. Pluta

**Abstract (PL):** Po ostaniach dość głośnych atakach na użytkowników Linuksa za pośrednictwem popularnego portalu z różnymi motywami dla środowiska GNOME, wiele osób zaczęło sobie zadawać pytanie: czy na pewno użytkownicy Linuksa są tacy bezpieczni jak zawsze twierdzili?

**File(s)**: **PDF**

Published in *Linux+ Magazine (Polish edition)*, 2010

**Author(s):** K. Pluta

**Abstract:** In the article we discuss eyeOS software.

**File(s)**: **PDF**

Published in *Linux+ Magazine (Polish edition)*, 2010

**Author(s):** K. Pluta

**Abstract:** In the article we discuss Nano blogger CMS.

**File(s)**: **PDF**

Published in *SMiSKT*, 2012

**Author(s):** K. Pluta, M. Postolski and M. Janaszewski

**Abstract:** The article presents a new conception of 3D human bronchial tree model which is useful to test algorithms for quantitative analysis of bronchial tubes based on tomographic images. The proposed model has been developed as an extension of the algorithm to generate the human bronchial tree by Hiroko Kitaoka, Ryuji Takaki and Bela Suki, The new model has been extended with geometrical deformations of branches and procedure which iteratively add noise and smooth a tree in voxel space. The presented conception has been implemented in the form of computer algorithms which generate 3D images of bronchial trees in voxel space. The article presents results of the implemented algorithms which are more like the segmented, real, bronchial trees than model Kitaoka, Takaki and Suki. Moreover the authors present influence of the algorithm parameters on the results and usefulness of the generated models for testing procedures of quantitative analysis of bronchial trees.

Recommended citation: Pluta K., Postolski M., & Janaszewski M. (2012). Algorytmy modelowania drzewa oskrzelowego. Zeszyty Naukowe Wyższej Szkoły Informatyki, 11, 152-170.

Published in *WSINF*, 2013

**Author(s):** K. Pluta

**Abstract:** The thesis presents new conception of 3D model of human bronchial tubes, which represents bronchial tubes extracted from CT images of the chest. The new algorithm which generates new model is an extension of the algorithm (basic algorithm) proposed by Hiroko Kitaoka, Ryuji Takaki and Bela Suki. The basic model has been extended by geometric deformations of branches and noise which occur in bronchial trees extracted from CT images. The manuscript presents comparison of results obtained with the use of the new algorithm and the basic one. Moreover, the discussion of usefulness of generated new models for testing of algorithms for quantitative analysis of bronchial tubes based on CT images is also included.

**File(s):** **PDF**, **Source code**

Recommended citation: Pluta K.: Algorytmy Modelowania Geometrii Drzew Oskrzelowych w Przestrzeni 3D. Bachelor Thesis, University of Computer Science in Łódź, 2013

Published in *IPC*, 2013

**Author(s):** K. Pluta, M. Janaszewski and M. Postolski

**Abstract:** The article presents new conception of 3D model of human bronchial tubes, which represents bronchial tubes extracted from CT images of the chest. The new algorithm which generates new model is an extension of the algorithm (basic algorithm) proposed by Hiroko Kitaoka, Ryuji Takaki and Bela Suki. The basic model has been extended by geometric deformations of branches and noise which occur in bronchial trees extracted from CT images. The article presents comparison of results obtained with the use of the new algorithm and the basic one. Moreover, the discussion of usefulness of generated new models for testing of algorithms for quantitative analysis of bronchial tubes based on CT images is also included.

Recommended citation: Pluta K., Janaszewski M., & Postolski M. (2012). New Algorithm for Modeling of Bronchial Trees. Image Processing & Communications, 17(4), 179-190. doi:10.2478/v10248-012-0045-8

Published in *HAL (research report)*, 2014

**Author(s):** Pluta K., Kenmochi Y., Passat N., Talbot H. and Romon P.

**Abstract:** Rigid transformations in \(\mathbb{R}^n\) are known to preserve the shape, and are often applied to digital images. However, digitized rigid transformations, defined as digital functions from \(\mathbb{Z}^n\) to \(\mathbb{Z}^n\) do not preserve shapes in general; indeed, they are almost never bijective and thus alter the topology. In order to understand the causes of such topological alterations, we first study the possible loss of voxel information and modification of voxel adjacencies induced by applications of digitized rigid transformations to 3D digital images. We then show that even very simple structured images such as digital half-spaces may not preserve their topology under these transformations. This signifies that a simple extension of the two-dimensional solution for topology preservation cannot be made in three dimensions.

Recommended citation: Pluta K., Kenmochi Y., Passat N., Talbot H., Romon P. (2016) Topological alterations of 3D digital images under rigid transformations. HAL, https://hal.archives-ouvertes.fr/hal-01333586

Published in *DGCI*, 2016

**Author(s):** K. Pluta, P. Romon, Y. Kenmochi, N. Passat

**Abstract:** Rigid motions are fundamental operations in image processing. While they are bijective and isometric in \(\mathbb{R}^2\), they lose these properties when digitized in \(\mathbb{Z}^2\). To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on \(\mathbb{Z}^2\), initially proposed by Nouvel and Rémila for rotations on \(\mathbb{Z}^2\). This allows us to study bijective rigid motions on \(\mathbb{Z}^2\), and to propose two algorithms for verifying whether a given rigid motion restricted to a given finite subset of \(\mathbb{Z}^2\) is bijective.

**File(s)**: **Pre-print (PDF)**, **BibTeX**

**Errata is not provided but several typos and mistakes were corrected in the journal version of this paper** (see **Bijective Digitized Rigid Motions on Subsets of the Plane**)

Recommended citation: Pluta K., Romon P., Kenmochi Y., Passat N. (2016) Bijective Rigid Motions of the 2D Cartesian Grid. In: Normand N., Guédon J., Autrusseau F. (eds) Discrete Geometry for Computer Imagery. DGCI 2016. Lecture Notes in Computer Science, vol 9647. Springer, pp 359-371, doi:10.1007/978-3-319-32360-2_28

Published in *CTIC*, 2016

**Author(s):** K. Pluta, P. Romon, Y. Kenmochi, N. Passat

**Abstract:** Euclidean rotations in \(\mathbb{R}^n\) are bijective and isometric maps. Nevertheless, they lose these properties when digitized in \(\mathbb{Z}^n\). For \(n=2\), the subset of bijective digitized rotations has been described explicitly by Nouvel and Rémila and more recently by Roussillon and Cœurjolly. In the case of 3D digitized rotations, the same characterization has remained an open problem. In this article, we propose an algorithm for certifying the bijectivity of 3D digitized rational rotations using the arithmetic properties of the Lipschitz quaternions.

**File(s)**: **Pre-print (PDF)**, **BibTeX**, **Errata (2017-06-20)**

Recommended citation: Pluta K., Romon P., Kenmochi Y., Passat N. (2016) Bijectivity Certification of 3D Digitized Rotations. In: Bac A., Mari JL. (eds) Computational Topology in Image Context. CTIC 2016. Lecture Notes in Computer Science, vol 9667. Springer, pp 30-41, doi:10.1007/978-3-319-39441-1_4

Published in *CASC*, 2016

**Author(s):** K. Pluta, G. Moroz, Y. Kenmochi, P. Romon

**Abstract:** Rigid motions are fundamental operations in image processing. While bijective and isometric in \(\mathbb{R}^3\), they lose these properties when digitized in \(\mathbb{Z}^3\). To understand how the digitization of 3D rigid motions affects the topology and geometry of a chosen image patch, we classify the rigid motions according to their effect on the image patch. This classification can be described by an arrangement of hypersurfaces in the parameter space of 3D rigid motions of dimension six. However, its high dimensionality and the existence of degenerate cases make a direct application of classical techniques, such as cylindrical algebraic decomposition or critical point method, difficult. We show that this problem can be first reduced to computing sample points in an arrangement of quadrics in the 3D parameter space of rotations. Then we recover information about remaining three parameters of translation. We implemented an ad-hoc variant of state-of-the-art algorithms and applied it to an image patch of cardinality 7. This leads to an arrangement of 81 quadrics and we recovered the classification in less than one hour on a machine equipped with 40 cores.

**File(s)**: **Pre-print (PDF)**, **BibTeX**, **Errata (2018-06-25)**

Recommended citation: Pluta K., Moroz G., Kenmochi Y., Romon P. (2016) Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image. In: Gerdt V., Koepf W., Seiler W., Vorozhtsov E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science, vol 9890. Springer, doi:10.1007/978-3-319-45641-6_27

Published in *JMIV*, 2017

**Author(s):** K. Pluta, P. Romon, Y. Kenmochi, N. Passat

**Abstract:** Rigid motions in \(\mathbb{R}^2\) are fundamental operations in 2D image processing. They satisfy many properties: in particular, they are isometric and therefore bijective. Digitized rigid motions, however, lose these two properties. To investigate the lack of injectivity or surjectivity and more generally their local behavior, we extend the framework initially proposed by Nouvel and Rémila to the case of digitized rigid motions. Yet, for practical applications, the relevant information is not global bijectivity, which is seldom achieved, but bijectivity of the motion restricted to a given finite subset of \(\mathbb{R}^2\). We propose two algorithms testing that condition. Finally, because rotation angles are rarely given with infinite precision, we propose a third algorithm providing optimal angle intervals that preserve this restricted bijectivity.

**File(s)**: **Pre-print (PDF)**, **BibTeX**, **Errata (2017-06-03)**

Recommended citation: Pluta K., Romon P., Kenmochi Y., Passat N. Journal of Mathematical Imaging and Vision (2017), doi:10.1007/s10851-017-0706-8

Published in *EGU 2017*, 2017

**Author(s):** **G. Domej**, C. Bourdeau, L. Lenti, K. Pluta

**A part of the abstract:** “Landsliding is a worldwide common phenomenon. Every year, and ranging in size from very small to enormous, landslides cause all too often loss of life and disastrous damage to infrastructure, property and the environment. One main reason for more frequent catastrophes is the growth of population on the Earth which entails extending urbanization to areas at risk…” (see PDF)

**File(s)**: **Abstract (PDF)**

Published in *DGCI*, 2017

**Author(s):** K. Pluta, P. Romon, Y. Kenmochi, N. Passat

**Abstract:** Euclidean rotations in \(\mathbb{R}^2\) are bijective and isometric maps, but they lose generally these properties when digitized in discrete spaces. In particular, the topological and geometrical defects of digitized rigid motions on the square grid have been studied. In this context, the main problem is related to the incompatibility between the square grid and rotations; in general, one has to accept either relatively high loss of information or non-exactness of the applied digitized rigid motion. Motivated by these considerations, we study digitized rigid motions on the hexagonal grid. We establish a framework for studying digitized rigid motions in the hexagonal grid—previously proposed for the square grid and known as neighborhood motion maps. This allows us to study non-injective digitized rigid motions on the hexagonal grid and to compare the loss of information between digitized rigid motions defined on the two grids.

**File(s)**: **Preprint (PDF)**, **BibTeX**

Recommended citation: Pluta K., Romon P., Kenmochi Y., Passat N. (2017) Honeycomb geometry: Rigid Motions on the Hexagonal Grid. In: Discrete Geometry for Computer Imagery. DGCI 2017. Lecture Notes in Computer Science, vol 10502. Springer, pp 33-45, doi:10.1007/978-3-319-66272-5_4

Published in *To appear in Journal of Mathematical Imaging and Vision*, 2017

**Author(s):** K. Pluta, T. Roussillon, D. Cœurjolly, P. Romon, Y. Kenmochi, V. Ostromoukhov

**Abstract:** Digitized rotations on discrete spaces are usually defined as the composition of a Euclidean rotation and a rounding operator; they are in general not bijective. Nevertheless, it is well known that digitized rotations defined on the square grid are bijective for some specific angles. This infinite family of angles has been characterized by Nouvel and Rémila and more recently by Roussillon and Cœurjolly. In this article, we characterize bijective digitized rotations on the hexagonal grid using arithmetical properties of the Eisenstein integers.

**File(s)**: **Preprint (PDF)**, **BibTeX**

Published in *UPE*, 2017

**Author(s):** K. Pluta

**Abstract:** In digital geometry, Euclidean objects are represented by their discrete approximations, e.g. subsets of the lattice of integers. Rigid motions of such sets have to be defined as maps from and onto a given discrete space. One way to design such motions is to combine continuous rigid motions defined on Euclidean space with a digitization operator. However, digitized rigid motions often no longer satisfy properties of their continuous siblings. Indeed, due to digitization, such transformations do not preserve distances, while bijectivity and point connectivity are generally lost. In the context of 2D discrete spaces, we study digitized rigid motions on the lattices of Gaussian and Eisenstein integers. We characterize bijective digitized rigid motions on the integer lattice, and bijective digitized rotations on the regular hexagonal lattice. Also, we compare the information loss induced by non-bijective digitized rigid motions defined on both lattices. Yet, for practical applications, the relevant information is not global bijectivity, but bijectivity of a digitized rigid motion restricted to a given finite subset of a lattice. We propose two algorithms testing that condition for subsets of the integer lattice, and a third algorithm providing optimal angle intervals that preserve this restricted bijectivity. We then focus on digitized rigid motions on the 3D integer lattice. First, we study at a local scale geometric and topological defects induced by digitized rigid motions. Such an analysis consists of generating all the images of a finite digital set under digitized rigid motions. This problem amounts to computing an arrangement of hypersurfaces in a 6D parameter space. The dimensionality and degenerate cases make the problem practically unsolvable for state-of-the-art techniques. We propose an ad hoc solution, which mainly relies on parameter uncoupling, and an algorithm for computing sample points of 3D connected components in an arrangement of second degree polynomials. Finally, we focus on the open problem of determining whether a 3D digitized rotation is bijective. In our approach, we explore arithmetic properties of Lipschitz quaternions. This leads to an algorithm which answers whether a given digitized rotation—related to a Lipschitz quaternion—is bijective.

**File:** **TEL public repository**

Recommended citation: Pluta K.: Discrete Spaces on Discrete Spaces. PhD Thesis, University Paris-Est, 2017

Published in *Italian Journal of Engineering Geology and Environment*, 2017

**Author(s):** G. Domej, C. Bourdeau, L. Lenti, S. Martino, K. Pluta

**Abstract:** Ranging in size from very small to tremendous, landslides often cause loss of life and damage to infrastructure, property and the environment. They are triggered by a variety and combinations of causes among which the role of water and seismic shaking have the most serious consequences. In this regard, seismic wave amplification due to topography as well as to the impedance contrast between the landslide mass and its underlying bedrock are of particular interest. Therefore, high resolution reconstruction of the lateral confinement of the landslide mass and the exact measurement of the mechanical properties are a necessity. A global chronological database was created to study and compare 2D and 3D geometries of landslides, i.e. of landslides properly sliding on a rupture surface. It contains 277 seismically and non-seismically induced landslides whose rupture masses were measured in all available details allowing for statistical analyses of their shapes and to create numerical models thereupon based. Detailed studies reveal that values of distinct geometrical parameters have different statistical behaviors. As for dimension related parameters, occurrence frequencies follow decreasing exponential distributions and mean values progressively increase with landslide magnitude. In contrast, occurrence frequencies of shape-related parameters follow normal distributions and mean values are constant throughout different landslide magnitudes. Dimensions and shapes of landslides are thus to be regarded in a precise and distinctive manner when analyzing seismically induced slope displacements.

**File(s)**: **Article (PDF)**, **BibTeX**

The collaborative project DGtal aims at developing generic, efficient and reliable digital geometry data structures, algorithms and tools. It takes the form of an open-source C++ library DGtal and a set of tools and binaries DGtalTools.

Educational and simple software to study flows over 3D meshes.

A plugin for MeshLab which allows to apply a mean curvature flow to a mesh.

RigidMotionsMapleTools is a set of Maple modules for which the main objective is to generate unique neighborhood motion maps (NMM, for short). These neighborhood motion maps are the images of an image patch which is a finite set of integer points. The tool was introduced in CASC16

Wolfram Language implementation of Backward algorithm introduced in DGCI16.

Wolfram Language implementation of Forward algorithm introduced in DGCI16.

Wolfram Language implementation of quaternion certification algorithm introduced in CTIC16.

A set of tools written in C++, Qt and DGtal, in order to study digitized rigid motions on regular grids.

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Undergraduate course, *University Paris-Est Marne-la-Vallée, IMAC*, 2014

**Topic:** Introduction to algorithms and programming in C.

Master 1, *ESIEE Paris*, 2015

**Topic:** Design of graphical user interface in Qt and C++.

Workshop/Master 1, *ESIEE Paris*, 2015

**Topic:** Implementation of parallel directional collapse.

Master 1, *University Paris-Est Marne-la-Vallée, IMAC*, 2016

**Topic:** Introduction to object oriented programming in C++.

Master 2, *University Paris-Est Marne-la-Vallée, SIS*, 2016

**Topic:** Introduction to digital geometry. We asked students to use **DGtal** to implemented several notions in the framework of digital geometry.

Master 1, *ESIEE Paris*, 2017

**Topic:** Introduction to computational geometry. But we, mostly, discussed polyhedral meshes and operations defined on them like: simplification schemas, subdivision schemas (Cutmull-Clark), etc.