A (binary) image in the cubic grid is a finite set of cubes, conventionally considered as black. A known problem of the cubic grid is that the adjacency relation is not unique: face-, edge- and vertex-adjacency between the cubes lead to different topological properties of an image. Image repairing is the process of transforming a given image into another one, whose adjacency relations are not ambiguous. Starting from a work by Edelsbrunner and Kerber (2012), we propose to repair an image by deforming its cubes. In order to do this, we define an improved transformation, which maintains the centers of the voxels. Finally, we present a comparison of different repairing methods.

Fuzzy graphs equipped with aggregation functions offer a flexible mathematical framework for representing uncertainty, gradual transitions, and complex relational structures in medical image data. Unlike classical graph models that rely on the existence of sharp edges and binary similarity in images, fuzzy graphs encode variable degrees of connectivity and allow the integration of multiple image features through custom aggregation operators. Therefore, they may be especially relevant in segmentation tasks in images where boundaries are not precise, contrast can be at low level, or structures demonstrates very heterogeneous visual characteristics. In this paper, we examine the potential of possibilities applications fuzzy graph theory to digital image segmentation. Nodes and edges are defined as fuzzy entities, while aggregation functions use intensity, structure, and texture information, combining them to form unique similarity measures. Such a framework allows pixel regions to be represented with different degrees of membership, reflecting gradual transitions and local variations. By integrating multiple sources of information, fuzzy graphs provide a more flexible representation of local and global relationships within an image compared to traditional graph–based models. Although this study is at a conceptual stage, we outline how different aggregation strategies can influence the formation of fuzzy relations and their impact on segmentation outcomes. The goal of this approach is to offer a structured yet adaptable way of modelling uncertainty and heterogeneity in visual data. By investigating the fuzzy graph representation and aggregation methods in image processing, this paper highlights promising directions for future research in nuanced image segmentation and analysis, especially in contexts where classical methods may have difficulty detecting some very subtle variations and ambiguous boundaries.

We consider closed curves in the three regular and eight semiregular grids in the plane, in which each vertex and each edge can be repeated a limited number of times. We define the conditions for such curves to be self-avoiding, and we present a linear-time algorithm to check them. We define the orientation of such curves. We propose a classification of their vertices, and we give a unifying formula relating the number of different types of vertices, valid in the regular and semiregular grids. Our results can be used in the plane tiling applications. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We introduce an improved version of the discrete bijective reflection proposed in [Andres 2019] and we show that this new reflection is almost always identical to the reflection based on the quasi shear rotation for integer centers. The difference between the two reflections is restricted to a discrete set of angles of the form arctan (k+1/2) , where k∈ℕ . The quasi shear rotation (QSR) based reflection yields among the lowest errors compared to the continuous reflection. Our improved pivot reflection (IPR) provides the same error in most cases and is easier to compute. Our new approach therefore offers a promising direction for the computation of nD rotations.

We compute several vertex-degree-based topological indices of graphs defined by catacondensed coronoid systems. We do this by using our recently proposed formulae for graphs defined by finite sets of hexagons of arbitrary topology in the hexagonal grid, and the associated M-polynomial.

When the cubic grid is compressed in a diagonal direction, cubes become nonregular truncated octahedra and the cubic grid becomes combinatorially equivalent to the body centered cubic (BCC) grid. We propose a combinatorial coordinate system for this grid, which addresses all the cells in the grid through integer triplets and which reflects the grid geometry. This addressing scheme, together with the fact that each object in the BCC grid is manifold, simplifies topological analysis and processing of digital objects. For illustration purposes, we present a naive algorithm for finding the closure of a set of cells in the slanted cubic grid, which relies heavily on the use of combinatorial coordinates.

We introduce and investigate the concept of a discrete circle in the triangular grid, taking into account the presence of two distinct types of triangles, depending on their orientation. We propose an analytical definition of a discrete circle in this grid, incorporating a non-constant thickness that depends on both the type of the considered triangle and the position of the circle's center. This varying thickness ensures the edge connectivity of the resulting circle. Building on this analytical definition, we also propose an incremental generation algorithm that is linear in the number of triangles on the circle.

Each 3D rotation can be decomposed into three 2D rotations parametrized by three Euler angles. Each of the three 2D rotations can be expressed as a sequence of three 2D shears along coordinate axes, leading to a decomposition of a 3D rotation into nine (beam) shears in total. We define a 3D digitized rotation using nine digitized beam shears, i.e., we round the result of each shear before applying the next one. As digitized shears are bijective, our 3D digitized rotation inherits the same property. Experiments show that the average error of our digitized rotation compared to the continuous one is kept under 1 (around 0.8).

Vertex-degree-based topological indices have been widely investigated and used in chemical graph theory for describing, predicting and explaining physical, chemical and biological properties of molecules. As opposed to the large body of research on these indices for planar and molecular graphs, little has been done on their computation for 3D crystallographic structures. We fill this gap by computing some of the most widely used vertex-degree-based topological indices of rectangular blocks of unit cells in one of the basic crystallographic structures, the simple cubic grid.

Vertex-degree-based indices have been widely used for describing and predicting physical, chemical and biological properties of molecules. They are expressed through the numbers mi,j of the edges connecting the vertices of degrees i and j. We generalize some known formulae for mi,j from graphs defined by simply connected sets of hexagons to those defined by sets of arbitrary topology in the regular hexagonal grid. We compute the most commonly used indices for three families of graphs in this grid.

We propose an analytical definition of discrete circles in the hexagonal grid. Our approach is based on a non-constant thickness function. We determine the thickness using the (edge and vertex) flake model. Both types of circles are connected. We prove that edge flake circles are without simple points for integer radii. Incremental generation algorithms are deduced from the analytical characterization of both edge and vertex flake circles. We compare our approach with existing algorithms for the circle generation on the hexagonal grid. Our approach offers simpler algorithm and an analytical characterization that the other algorithms do not offer. The benefit of an analytical characterization is that it makes the question of the membership of a point to a primitive trivial.

We briefly review the algorithm for determining the orthogonal hull of a set of simple rectilinear polygons, proposed by Nicholl et al., based on determining their maximal vertices.

We propose a characterization of discrete analytical spheres, planes and lines in the body-centered cubic (BCC) grid, both in the Cartesian and in the recently proposed alternative compact coordinate system, in which each integer triplet addresses some voxel in the grid. We define spheres and planes through double Diophantine inequalities and investigate their relevant topological features, such as functionality or the interrelation between the thickness of the objects and their connectivity and separation properties. We define lines as the intersection of planes. The number of the planes (up to six) is equal to the number of the pairs of faces of a BCC voxel that are parallel to the line. & COPY; 2023 Elsevier Ltd. All rights reserved.

In this paper we consider the aggregation functions' application to problems of face recognition. A wide range of aggregation function classes can play a crucial role in multi-criteria decision-making theory. Some of the most well-known aggregation functions include generalized means, integrals based on non-additive measures, etc. Their usage also saves memory resources and can lead to an increase in the percentage of correctly classified objects (in our case, we deal with object recognition). Biometry and face recognition represent important modern problems in computer science, and some solutions are widely-adopted and applied in mobile devices and identity confirmation, border control, ID and other document verification, etc. Other than classic methods for face recognition, such as PCA, linear discriminant analysis, as well as various local descriptors and other methods, methods based on sparse data representation and deep learning methods have recently been used. Each part of the face can be compared individually, and a conclusion regarding the match-overlap of two faces should be drawn from all these comparisons. In our investigation, we analyze the quality of the decisions made in face recognition problems, with regards to the aggregation functions used, as well as the different methods used for recognizing certain parts of the face.

As opposed to the 3D cubic grid, the body-centered cubic (BCC) grid has some favorable topological properties: each set of voxels in the grid is a 3-manifold, with 2-manifold boundary. Thus, the Euler characteristic of an object O in this grid can be computed as half of the Euler characteristic of its boundary ∂ O . We propose three new algorithms to compute the Euler characteristic in the BCC grid with this surface-based approach: one based on (critical point) Morse theory and two based on the discrete Gauss–Bonnet theorem. We provide a comparison between the three new algorithms and the classic approach based on counting the number of cells, either of the 3D object or of its 2D boundary surface.

We consider paths in the 2D square grid, composed of grid edges, given as a sequence of moves in the four cardinal compass directions, without U-turns, but possibly passing several times through the same vertex or the same edge (if the path is open, it cannot pass twice through its starting vertex). We propose an algorithm which reports a self-crossing if there is one, or otherwise draws the path without self-crossings. The algorithm follows the intuitive idea naturally applied by humans to draw a curve: at each vertex that has already been visited, it tries to insert two new segments in such a way that they do not cross the existing ones. If this is not possible, a self-crossing is reported. This procedure is supported by a data structure combining a doubly-linked circular list and a skip list. The time and space complexity is linear in the length of the path.

A 0-tandem is a configuration of two voxels (n-cells) sharing exactly one vertex. We propose a formula connecting the number of 0-tandems in a simple nD digital open or closed 0-connected curve $$\gamma $$ with the number of cells in $$\gamma $$ . Our formula generalizes the formula by Brimkov et al. (2006) from closed to open curves, and the formula by Maimone and Nordo (2015) from open curves in 3D to open or closed curves in nD. We also propose an alternative formula valid for 3D curves.

We propose a new formula for computing discrete geometric moments on 2D binary images. The new formula is based on the inclusion-exclusion principle, and is especially tailored for images coming from computer art, characterized by a prevalence of horizontal and vertical lines. On the target class of images, our formula reduces the number of pixels where calculations are to be performed.

A 2D binary image is well-composed if it does not contain 2 × 2 blocks of two diagonal black and two diagonal white pixels, called critical configurations. Some image processing algorithms are simpler on well-composed images. The process of transforming an image into a well-composed one is called repairing. We propose a new topology-preserving approach, which produces two well-composed images starting from an image I depending on the chosen adjacency (vertex or edge adjacency), in the same original square grid space as I . The size of the repaired images depends on the number and distribution of the critical configurations. A well-composed image I is not changed, while in the worst case the size increases at most two times (or four times if we want to preserve the aspect ratio). The advantage of our approach is in the small size of the repaired images, with a positive impact on the execution time of processing tasks. We demonstrate this experimentally by considering two classical image processing tasks: contour extraction and shrinking.