Four Color Theorem

We give a pictorial, and absurdly simple, proof that transparently illustrates why four colours suffice to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal planar map. We show, moreover, why the proof cannot be expressed within classical graph theory.

We give a pictorial, and absurdly simple, proof that transparently illustrates why four colours suffice to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal planar map. We show, moreover, why the proof cannot be expressed within classical graph theory.

The harmony search algorithm is a recently developed meta heuristic capable of solving discrete and continuous valued optimization problems. However, the nature of pre-defined constant parameters limits the exploitation of the algorithm. In this paper a number of deterministic parameter control rules are used in harmony search algorithm to fine-tune these parameters individually and dynamically, making harmony search a more dynamic algorithm which is able to achieve better results in solving the four color map problem. In this problem, it is said one can color the regions of a map using a maximum of four colors only with the constraint that no neighboring regions should share the same color. Neighboring regions mean two regions having a common boundary, not just a common point. Experimental results reveal that the new approach can find better solutions when compared to the original harmony search and several other heuristics, making harmony search a strong mechanism to perform optimization tasks.

The Four-Color Mapping Problem has been solved using different optimization algorithms. Harmony Search (HS) is one of those algorithms, which is based on the imitation of the behavior of musicians when composing their music. The HS algorithm can be summarized in three steps... initialization, improvisation, and selection. We introduced in this paper an approach to enhance the performance of HS algorithm, in solving the Four-Color Mapping Problem. A modification has been applied to the initialization section of the HS algorithm, which affects the improvisation process, resulting in a boost in the performance of the improvisation process, and consequently, reducing the time and number of cycles taken to solve the Four-Color Mapping Problem compared to the HS algorithm. In this paper, tests have been carried out on maps with different numbers of regions, using both HS and Modified Harmony Search (MHS) algorithms. The obtained results of the MHS algorithm are better than those of the original HS one.

The present work describes a new methodology for the automatic detection of the glottal space from laryngeal images based on active contour models (snakes). In order to obtain an appropriate image for the use of snakes based techniques, the proposed algorithm combines a pre-processing stage including some traditional techniques (thresholding and median filter) with more sophisticated ones such as anisotropic filtering. The value selected for the thresholding was fixed to the 85% of the maximum peak of the image histogram, and the anisotropic filter permits to distinguish two intensity levels, one corresponding to the background and the other one to the foreground (glottis). The initialization carried out is based on the magnitude obtained using the Gradient Vector Flow field, ensuring an automatic process for the selection of the initial contour. The performance of the algorithm is tested using the Pratt coefficient and compared against a manual segmentation. The results obtained su...

Curve evolution implementations of the Mumford-Shah functional are of broad interests in image segmentation. These implementations, however, have initialization problems. A mathematical analysis of the initialization problem for the Chan-Vese implementations is provided in this paper. The initialization problem is shown to result from the non-convexity of the Mumford-Shah functional and the top-down hierarchy of the model’s use of global region information in the image. Based on the analysis, efficient implementation methods are proposed for the Chan-Vese models. The proposed methods based on region competition and the Mumford Shah functional. This algorithm is able to automatically and efficiently segment objects in complicated images. Using a bottom-up hierarchy, the method avoids the initialization problem in the Chan-Vese models and works for images with multiple junctions and color images. It is then extended to textured images using Gabor filters and fractal methods. Experimen...

In this paper a marker-controlled and regularized watershed segmentation is proposed. Only a few previous studies address the task of regularizing the obtained watershed lines from the traditional markercontrolled watershed segmentation. In the present formulation, the topographical distance function is applied in a level set formulation to perform the segmentation, and the regularization is easily accomplished by regularizing the level set functions. Based on the well-known Four-Color theorem, a mathematical model is developed for the proposed ideas. With this model, it is possible to segment any 2D image with arbitrary number of phases with as few as one or two level set functions. The algorithm has been tested on real 2D fluorescence microscopy images displaying rat cancer cells, and the algorithm has also been compared to a standard watershed segmentation as it is implemented in MATLAB. For a fixed set of markers and a fixed set of challenging images, the comparison of these two methods shows that the present level set formulation performs better than a standard watershed segmentation.

This paper presents a segmentation method combining an active contour approach with a watershed pre-segmentation. Segmentation is performed on two modalities: one being the color of the regions, the other one being the change due to motion. On the one hand, watershed methods are very efficient to provide an oversegmentation of homogeneous color regions. On the other hand, active contours methods are efficient to obtain a smooth segmentation. We apply in this paper an active contour method to segment the change detection mask. The incertitude on the motion induces artifacts on the resulting contours. Our original contribution consists in using the intra-image watershed segmentation to correct the contours. To reach such a goal, we construct a distance map from the oversegmentation of the homogeneous color region. The active contour is then corrected to better fit to the real object by introducing a specific term based on this distance map in the evolving equation of the active contour.

The four color map theorem states that no more than four colors are needed to color the regions of any map so that no two adjacent regions have the same color. More specifically, the theorem states that no more than four colors are required to color the vertices of any planar graph where any two adjacent vertices don't have the same color. This paper shows how the theorem arises from a broken correlation between the number of colors and the number of vertices (dots), and, how repairing this broken correlation leads to maps requiring more than four colors. However, if three dimensions are not invoked on a on planer graph, it can be shown that no more than four colors are needed to color it. Premise: All maps, regardless of their shapes and sizes, can be represented by planer dot-and-line diagrams. Such drawings make analysis easier and more universal. Each dot (vertex) represents a country and each line represents a border. The rules are any two dots connected by a line cannot be the same color and lines cannot intersect.

The proof of the four-color theorem is led from its definition to the notion of a "complete triangular network" CTN(n) to a description of its construction to see if it is possible to construct a CTN(n) enforcing the 5th color. A negative answer then allows us to claim that the four-color theorem is thereby proved. 1 MSC 54C99

L. Rueda, D. Mery, and J. Kittler (Eds.): CIARP 2007, LNCS 4756, pp. 281–290, 2007. © Springer-Verlag Berlin Heidelberg 2007 ... A Method for Segmentation of Local Illumination ... Eduardo Garea Llano, Jose Luís Gil Rodríguez, and Sandro Vega

In this paper we present a method for the automatic localization of local light variations and its photometric normalization in face images affected by different angles of illumination causing the appearance of specular light. The proposed approach is faster and more efficient that if the same one was carried out on the whole image through the traditional photometric normalization methods (homomorphic filtering, anisotropic smoothing, etc.). The process consists in using an algorithm for unsupervised image segmentation based on the active contour without edges approach with level set representation model for localization of regions affected by specular reflection combined with a normalization method based on the local normalization that considers the local mean and variance into the located region. The performance of the proposed approach is compared through two experimental schemes to measure how the similarity is affected by illumination changes and how the proposed approach improves the effect caused by these changes.

Efficient implementation methods are proposed for Chan-Vese models [3] [16]. The proposed methods do not require solutions of PDEs and are therefore fast. The advantages of level set methods, such as automatic handling of topological changes, are preserved. These methods utilize region information to guide the evolution of initial curves. Gaussian smoothing is applied to regularize the evolving curves. These algorithms are able to automatically and efficiently segment objects in complicated images. Experimental results show that the proposed methods work efficiently for images without strong noise. However, they still have initialization problems, as do the Chan-Vese models.

This paper generalizes the methods in a previous paper [10] in two ways. First, a more comprehensive analysis of the initialization problem of the Chan-Vese models is given. Second, the image segmentation method proposed in [10] is improved by applying bimodal curve evolution with region competition. The improved method maintains the advantages of the previous method. It is efficient, stable in the presence of strong noise and able to handle complicated images. It outperforms the previous method for images with weak edges. Experimental results in this paper demonstrate these improvements.

Noninvasive imaging of unstained living cells allows to study living specimens without altering them, and is a widely used technique in biotechnology for determining biological and biochemical roles of proteins. Fluorescence and contrast images are both used complementarily for better outcomes. However, segmentation of contrast images is particularly difficult due to the presence of lighting/shade-off artifacts, defocused scans, or overlapping. In this work, we make use of the optical properties intervening during the image formation process for cell segmentation. We propose the shear oriented polar snakes, an active contour model that implicitly involves phase information. Experimental results confirms the method suitability for cell images segmentation.

Рассматривается понятие математического доказательства в связи с применением компьютеров в математике. Исследуются особенности этого понятия, а также связанных эпистемологических проблем. Особое внимание уделяется проблеме обо- зримости на примере доказательства теоремы о четырех красках. Настоящая ра-бота представляет собой вторую статью в серии о понятии теоретико-типового доказательства.

The problem of how many colors are required for a planar map has been used as a focal point for discussions of the limits of human direct understanding vs. automated methods. It is important to continue to investigate until it is convincingly proved map coloration is an exemplary irre-ducible problem or until it is reduced. Meanwhile a new way of thinking about surfaces which hide N-dimensional volumes has arisen in physics employing entropy and the holographic principle. In this paper we define coloration entropy or flexibility as a count of the possible distinct colorations of a map (planar graph), and show how a guaranteed minimum colora-tion flexibility changes based on additions at a boundary of the map. The map is 4-colorable as long as the flexibility is positive , even though the proof method does not construct a col-oration. This demonstration is successful, resulting in a compact and easily comprehended proof of the four color theorem. The use of an entropy-like method suggests comparisons and applications to issues in physics such as black holes. Therefore in conclusion some comments are offered on the relation to physics and the relation of plane-section color ability to higher dimensional spaces. Future directions of research are suggested which may connect the concepts to not only time and distance and thus entropic gravity but also momentum.

This paper addresses the shape-based segmentation problem using level sets. In particular, we propose a fast algorithm to solve the piece-wise constant Chan-Vese segmentation model with shape priors and labeling functions. Instead of directly solving the underlying PDE, we calculate the energy and check how it changes when we move image points from inside the region enclosed by the evolving interface to the outside region and vice-vera. This algorithm is then extended to the case of multi-phase Chan-Vese model, with multiple selective shape priors and a corresponding labeling function for each prior. This makes our algorithm different from that in [1] and other similar works in different aspects. On one hand, our algorithm is not restricted to two regions, but allows segmentation into several regions. On the other hand, more than one shape prior can be taken into account in our implementation. In addition, the proposed algorithm improves dramatically the computational speed. Experimental results, on both synthetic and real images, demonstrate the performance of our algorithm and the computational improvements it offers.

This paper addresses the shape-based segmentation problem using level sets. In particular, we propose a fast algorithm to solve the piece-wise constant Chan-Vese segmentation model with shape priors and labeling functions. Instead of directly solving the underlying PDE, we calculate the energy and check how it changes when we move image points from inside the region enclosed by the evolving interface to the outside region and vice-vera. This algorithm is then extended to the case of multi-phase Chan-Vese model, with multiple selective shape priors and a corresponding labeling function for each prior. This makes our algorithm different from that in [1] and other similar works in different aspects. On one hand, our algorithm is not restricted to two regions, but allows segmentation into several regions. On the other hand, more than one shape prior can be taken into account in our implementation. In addition, the proposed algorithm improves dramatically the computational speed. Experimental results, on both synthetic and real images, demonstrate the performance of our algorithm and the computational improvements it offers.

In this paper we introduce novel regularization techniques for level set segmentation that target specifically the problem of multiphase segmentation. When the multiphase model is used to obtain a partitioning of the image in more than two regions, a new set of issues arise with respect to the single phase case in terms of regularization strategies. For example, if smoothing or shrinking each contour individually could be a good model in the single phase case, this is not necessarily true in the multiphase scenario. In this paper, we address these issues designing enhanced length and area regularization terms, whose minimization yields evolution equations in which each level set function involved in the multiphase segmentation can "sense" the presence of the other level set functions and evolve accordingly. In other words, the coupling of the level set function, which before was limited to the data term (i.e. the proper segmentation driving force), is extended in a mathematically principled way to the regularization terms as well. The resulting regularization technique is more suitable to eliminate spurious regions and other kind of artifacts. An extensive experimental evaluation supports the model we introduce in this paper, showing improved segmentation performance with respect to traditional regularization techniques.

The Mumford-Shah model has been one of the most influential models in image segmentation and denoising. The optimization of the multiphase Mumford-Shah energy functional has been performed using level sets methods that optimize the Mumford-Shah energy by evolving the level sets via the gradient descent. These methods are very slow and prone to getting stuck in local optima due to the use of gradient descent. After the reformulation of the 2-phase Mumford-Shah functional on a graph, several groups investigated the hierarchical extension of the graph representation to multi class. The discrete hierarchical approaches are more effective than hierarchical (or direct) multiphase formulation using level sets. However, they provide approximate solutions and can diverge away from the optimal solution. In this paper, we present a discrete alternating optimization for the discretized Vese-Chan approximation of the piecewise constant multiphase Mumford-Shah functional that directly minimizes the multiphase functional without recursive bisection on the labels. Our approach handles the nonsubmodularity of the multiphase energy function and provides a global optimum if the image estimation data term is known apriori.

This work presents an algorithm for the segmentation of myocardial regions in echocardiography imaging based on the statistics of the radiofrequency image. We formulate the problem of segmentation in a Maximum Likelihood framework using the Generalized Gaussian as an a priori distribution. We minimize the resulting functional using a Radial Basis Functions-based multiphase level set model. Numerical results obtained on both simulation and in vivo data demonstrate the capacity of our approach to segment myocardial regions in echocardiography imaging.

We propose a game theory approach to simultaneously restore and segment noisy images. We define two players: one is restoration, with the image intensity as strategy, and the other is segmentation with contours as strategy. Cost functions are the classical relevant ones for restoration and segmentation, respectively. The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium. For the computation of this equilibrium we present an iterative method with relaxation. The results of numerical experiments performed on some real images show the relevance and efficiency of the proposed algorithm.

In this paper, we propose a novel segmentation algorithm for color images. This method is a combination of edge information with region information and a geometric active contour without re-initialization, called distance regularized level set evolution. The information given by a new edge detector using morphological gradient is more accurate than normal gradient computing methods for color images. And the information of the region containing objects is relied on Chan-Vese minimal variance criterion. With both of these information, the model can have its initial contour that is more flexible to construct anywhere, fast to evolve and quite exact to stop at the boundary of objects. The suggested algorithm has been applied on natural color images with good performance. Some experimental results have shown to compare our model with others with respect to accuracy and computational efficiency.

A marker-controlled and regularized watershed segmentation is proposed for cell segmentation. Only a few previous studies address the task of regularizing the obtained watershed lines from the traditional marker-controlled watershed segmentation. In the present formulation, the topographical distance function is applied in a level set formulation to perform the segmentation, and the regularization is easily accomplished by regularizing the level set functions. Based on the well-known Four-Color theorem, a mathematical model is developed for the proposed ideas. With this model, it is possible to segment any 2D image with arbitrary number of phases with as few as one or two level set functions. The algorithm has been tested on real 2D fluorescence microscopy images displaying rat cancer cells, and the algorithm has also been compared to a standard watershed segmentation as it is implemented in MATLAB. For a fixed set of markers and a set of challenging images, the comparison of these two methods shows that the present level set formulation performs better than a standard watershed segmentation.

In this work a marker-controlled and regularized watershed segmentation is proposed. Only a few previous studies address the task of regularizing the obtained watershed lines from the traditional marker-controlled watershed segmentation. In the present formulation, the topographical distance function is applied in a level set formulation to perform the segmentation, and the regularization is easily accomplished by regularizing the level set functions. Based on the well-known Four-Color theorem, a mathematical model is developed for the proposed ideas. With this model, it is possible to segment any 2D image with arbitrary number of phases with as few as one or two level set functions. The algorithm has been tested on real 2D fluorescence microscopy images displaying rat cancer cells, and the algorithm has also been compared to a standard watershed segmentation as it is implemented in MATLAB. For a fixed set of markers and a fixed set of challenging images, the comparison of these two methods shows that the present level set formulation performs better than a standard watershed segmentation.

Abstract We propose a game theory approach to simultaneously restore and segment noisy images. We define two players: one is restoration, with the image intensity as strategy, and the other is segmentation with contours as strategy. Cost functions are the classical relevant ones for restoration and segmentation, respectively. The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium.