A Tutorial on Well-Composedness

Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called well-composed images, with no topological issues. Furthermore, well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well defined. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. Believing in its prime importance in digital topology, we then propose this state of the art of well-composedness, summarizing its different flavors, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.     

Mathematical framework for topological relationships between ribbons and regions

In a map, there are different relationships between spatial objects, such as topological, projective, distance, etc. Regarding topological relations, if the scale of the map is changed and if some spatial objects are generalized, not only the shapes of those objects will change (for instance, a small area becomes a point and then disappears as the scale diminishes), but also their topological relations can vary according to scale. In addition, a mathematical framework which models the variety of this category of relationships does not exist. In the first part of this paper, a new topological model is presented based on ribbons which are defined through a transformation of a longish rectangle; so, a narrow ribbon will mutate to a line and then will disappear. Suppose a road is running along a lake, at some scales, they both appear disjointed whereas at some smaller scales, they meet. So, the topological relations mutate according to scale. In this paper, the different components of this mathematical framework are discussed. For each situation, some assertions are defined which formulate the mutation of the topological relationships into other ones when downscaling.     

Optimal morphological pattern restoration from noisy binary images

A theoretical analysis of morphological filters for the optimal restoration of noisy binary images is presented. The problem is formulated in a general form, and an optimal solution is obtained by using fundamental tools from mathematical morphology and decision theory. Consideration is given to the set-difference distance function as a measure of comparison between images. This function is used to introduce the mean-difference function as a quantitative measure of the degree of geometrical and topological distortion introduced by morphological filtering. It is proved that the class of alternating sequential filters is a set of parametric, smoothing morphological filters that best preserve the crucial structure of input images in the least-mean-difference sense     

Topological sensitivity analysis

The so-called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems. This derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain. However, the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study (because they have not the same topology). Therefore, some specific mathematical framework should be developed in order to obtain the derivatives. This work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts. The main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative. This idea leads to a more simple and constructive formulation than the ones found in the literature. Further, to point out the straightforward use of the proposed methodology, it is applied for solving some design problems in steady-state heat conduction.     

Some remarks on a Markov chain modelling cooperative biological systems

In this paper, we deal with a Markov chain introduced in our previous work, concerning a model for cooperative interactions in biological systems. Our aim is to obtain a mathematical explanation of some types of qualitative behaviour in cooperative systems. To this end, we point out further mathematical properties of the Markov chain and we find a diffusion continuous approximation of the chain, studying also some properties of the spectrum of both the discrete and the continuous associated operators.     

基于增量式学习的网络攻击检测数学建模仿真

针对现有网络攻击检测数学模型,存在网络攻击检测率较低、网络攻击误报率较高、检测时效性较差等问题,构建一种基于增量式学习的网络攻击检测数学模型.首先利用拓扑几何学原理构建一个网络信息之间的拓扑结构关系,再利用该关系进行网络信息去噪,去噪后对数据做零均值化处理和对时间序列进行拟合.最后利用拟合得到的时间序列进行网络攻击判定...     

Fibonacci序列构造广义M J混沌分形图谱周期性的研究

为更好的研究M-J混沌分形图谱的周期性,首先利用旋转逃逸时间算法绘制了正整数阶复映射的广义M-J混沌分形图谱,然后分析了广义Mandelbrot集(M-集)周期芽苞的分布规律,并验证了广义M-集周期芽苞存在Fi-bonacci序列拓扑不变性的规则;最后通过大量计算机数学实验,找出了M-集参数平面与动力平面上相应的Julia集图像结构之间的对应关系,同时给出了广义M-J集周期轨道的计算公式。     

Topology on Digital Label Images

In digital imaging, after several decades devoted to the study of topological properties of binary images, there is an increasing need of new methods enabling to take into (topological) consideration n-ary images (also called label images). Indeed, while binary images enable to handle one object of interest, label images authorise to simultaneously deal with a plurality of objects, which is a frequent requirement in several application fields. In this context, one of the main purposes is to propose topology-preserving transformation procedures for such label images, thus extending the ones (e.g., growing, reduction, skeletonisation) existing for binary images. In this article, we propose, for a wide range of digital images, a new approach that permits to locally modify a label image, while preserving not only the topology of each label set, but also the topology of any arrangement of the labels understood as the topology of any union of label sets. This approach enables in particular to unify and extend some previous attempts devoted to the same purpose.     

A robust projection plane selection strategy for UAV image stitching

Unmanned Aerial Vehicles (UAVs) are the most popular way to collect ground data today, thanks to their low cost and matchless convenience. However, UAVs are prone to unstable flight poses because they are so light in weight, which has resulted in a new challenge for UAV image stitching. In this paper, we propose a robust approach to stitch UAV images captured from approximately planar scenes without pose parameters assistance. The key idea of the proposed framework lies in an effective projection plane selection strategy, which is capable of resisting the perspective distortion from existing pose-perturbed images. To select a reasonable reference image as the projection plane, we first divide all the images into two groups (stable group and unstable group) according to their registration error under the affine model. Then, a specifically designed approach is used to define a weighted topological graph, which guarantees that the reference image is selected from the stable group while maintaining a global minimum accumulated registration error. Based on our cost topological graph, each unstable group image is locally attached to a stable group image via a homography. Finally, alignment parameters of all the stable group images are solved using affine model, after which global optimization is performed on the model of both groups. Comparing our results to those of the conventional approaches indicates that our proposed approach produced superior results in several challenging experiments involving both qualitative and quantitative evaluation.     

Combining topological and view-based features for 3D model retrieval

With the rapidly increasing of 3D models, the 3D model retrieval methods have been paid significant research attention. Most of the existing methods focus on taking advantage of one kind of feature. These methods can not achieve ideal retrieval results for different classes of 3D models. In this paper, we propose a novel 3D model retrieval algorithm by combining topological and view-based features. To preserve the topological structure of the 3D model, a multiresolutional reeb graph (MRG) is constructed according to the salient topological points. The view-based features are extracted from the images, which are rendered at each of the topological points. To preserve the spatial structure information of the images, we modify the bag-of-features (BOF) method by using the combined shell-sector model. We take the view-based features as the attribute information of the corresponding MRG nodes. The comparison between two 3D models is transformed to the problem of computing the similarity of the corresponding MRGs. Finally, we calculate the similarity between the query model and the models in the databases by adapting the earth mover distance method. Experimental results on two standard benchmarks show that our algorithm can achieve satisfactory retrieval performance.     

Preserving Topology by a Digitization Process

The main task of digital image processing is to recognize properties of real objects based on their digital images. These images are obtained by some sampling device, like a CCD camera, and represented as finite sets of points that are assigned some value in a gray-level or color scale. Based on technical properties of sampling devices, these points are usually assumed to form a square grid and are modeled as finite subsets of Z². Therefore, a fundamental question in digital image processing is which features in the digital image correspond, under certain conditions, to properties of the underlying objects. In practical applications this question is mostly answered by visually judging the obtained digital images. In this paper we present a comprehensive answer to this question with respect to topological properties. In particular, we derive conditions relating properties of real objects to the grid size of the sampling device which guarantee that a real object and its digital image are topologically equivalent. These conditions also imply that two digital images of a given object are topologically equivalent. This means, for example, that shifting or rotating an object or the camera cannot lead to topologically different images, i.e., topological properties of obtained digital images are invariant under shifting and rotation.     

Virtual double-sided image probing: A unifying framework for non-linear grayscale pattern matching

This paper focuses on non-linear pattern matching transforms based on mathematical morphology for gray level image processing. Our contribution is on two fronts. First, we unify the existing and a priori unconnected approaches to this problem by establishing their theoretical links with topology. Setting them within the same context allows to highlight their differences and similarities, and to derive new variants. Second, we develop the concept of virtual double-sided image probing (VDIP), a broad framework for non-linear pattern matching in grayscale images. VDIP extends our work on the multiple object matching using probing (MOMP) transform we previously defined to locate multiple grayscale patterns simultaneously. We show that available methods as well as the topological approach can be generalized within the VDIP framework. They can be formulated as particular variants of a general transform designed for virtual probing. Furthermore, a morphological metric, called SVDIP (single VDIP), is deduced from the VDIP concept. Some results are presented and compared with those obtained with classical methods.     

Vision-based sparse topological mapping

Most of the existing appearance-based topological mapping algorithms produce dense topological maps in which each image stands as a node in the topological graph. Sparser maps can be built by representing groups of visually similar images of a sequence as nodes of a topological graph. In this paper, we present a sparse/hierarchical topological mapping framework which uses Image Sequence Partitioning (ISP) to group visually similar images of a sequence as nodes which are then connected on the occurrence of loop closures to form a topological graph. An indexing data structure called Hierarchical Inverted File (HIF) is proposed to store the sparse maps so as to perform loop closure at the two different resolutions of the map namely the node level and image level. TFIDF weighting is combined with spatial and frequency constraints on the detected features for improved loop closure robustness. Our approach is compared with two other existing sparse mapping approaches which use ISP. Sparsity, efficiency and accuracy of the resulting maps are evaluated and compared to that of the other two techniques on publicly available outdoor omni-directional image sequences.     

Quantum walks on simplicial complexes

We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation suggests that localization of our quantum walks reflects not only topological but also geometric structures. On the other hand, our proposing quantum walk contains an intrinsic problem concerning exhibition of non-trivial behavior, which is not seen in typical quantum walks such as Grover walks on graphs.     

Topological landscapes: a terrain metaphor for scientific data

Scientific visualization and illustration tools are designed to help people understand the structure and complexity of scientific data with images that are as informative and intuitive as possible. In this context the use of metaphors plays an important role since they make complex information easily accessible by using commonly known concepts. In this paper we propose a new metaphor, called "Topological Landscapes," which facilitates understanding the topological structure of scalar functions. The basic idea is to construct a terrain with the same topology as a given dataset and to display the terrain as an easily understood representation of the actual input data. In this projection from an $n$-dimensional scalar function to a two-dimensional (2D) model we preserve function values of critical points, the persistence (function span) of topological features, and one possible additional metric property (in our examples volume). By displaying this topologically equivalent landscape together with the original data we harness the natural human proficiency in understanding terrain topography and make complex topological information easily accessible.     

Road network extraction from high resolution satellite images

In this paper, an approach of roads network extraction from high resolution satellite images is presented. First, the approach extracts road surface from satellite image using one-class support vector machine(SVM). Second, the road topology is built from the road surface. The last output of the approach is a series of road segments which is represented by a sequence of points as well as the topological relations among them. The approach includes four steps. In the first step one-class support vector machine is used for classifying pixel of the satellite images to road class or non-road class. In the second step filling holes and connecting gaps for the SVM's classification result is applied through mathematical morphology close operation. In the third step the road segment is extracted by a series of operations which include skeletonization, thin, branch pruning and road segmentation. In the last step a geometrical adjustment process is applied through analyzing the road segment curvature. The experiment results demonstrate its robustness and viability on extracting road network from high resolution satellite images.     

Hardwired polynomial evaluation

This paper is devoted to the evaluation of polynomials and elementary functions by special-purpose circuits. First we recall the basic results concerning the approximation of mathematical functions by polynomials (these results enable us to compute every continuous function if we are able to compute polynomials); then we describe a simple operator, suitable for VLSI implementation, which evaluates a polynomial in the range [0, 1]. Finally, we give some complexity results about the evaluation of the most usual elementary functions with our operator.     

General Adaptive Neighborhood-Based Pretopological Image Filtering

This paper introduces pretopological image filtering in the context of the General Adaptive Neighborhood Image Processing (GANIP) approach. Pretopological filters act on gray level image while satisfying some topological properties. The GANIP approach enables to get an image representation and mathematical structure for adaptive image processing and analysis. Then, the combination of pretopology and GANIP leads to efficient image operators. They enable to process images while preserving region structures without damaging image transitions. More precisely, GAN-based pretopological filters and GAN-based viscous pretopological filters are proposed in this paper. The viscous notion enables to adjust the filtering activity to the image gray levels. These adaptive filters are evaluated through several experiments highlighting their efficiency with respect to the classical operators. They are practically applied in both the biomedical and material application areas for image restoration, image background subtraction and image enhancement.