Digital Curves Research Articles

Genetic algorithms for approximation of digital curves with line segments and circular arcs

A method for error‐bounded approximation of digital curves with line segments and circular arcs is proposed. In the proposed method, a chromosome is used to represent an approximation by a string, and the operations of genetic algorithms are designed especially to obtain the approximation for which an error is bounded by a given norm. Many experiments show that the convergence is guaranteed, and optimal or near‐optimal solutions can be obtained. The proposed method successfully reduced the total number of line segments and circular arcs required to approximate the digital curves under the error‐bound criterion.

High Detection Rates of Colorectal Neoplasia by Stool DNA Testing With a Novel Digital Melt Curve Assay

High Detection Rates of Colorectal Neoplasia by Stool DNA Testing With a Novel Digital Melt Curve Assay

Comparison and improvement of tangent estimators on digital curves

Many contour-based applications rely on the estimation of the geometry of the shape, such as pattern recognition or classification methods. This paper proposes a comprehensive evaluation on the problem of tangent estimators on digital curves. The methods taken into account use different paradigms: approximation and digital geometry. In the former paradigm, methods based on polynomial fitting, smoothing and filtering are reviewed. In the latter case of digital geometry, we consider two methods that mainly rely on digital straight line recognition [J.-O. Lachaud, A. Vialard, F. de Vieilleville, Fast, accurate and convergent tangent estimation on digital contours, Image and Vision Computing 25(10) (2007) 1572–1587] and optimization [B. Kerautret, J.-O. Lachaud, Robust estimation of curvature along digital contours with global optimization, in: Proceedings of Discrete Geometry for Computer Imagery, Lyon, France, Lecture Notes in Computer Science, vol. 4992, Springer, Berlin, 2008]. The comparison takes into account objective criteria such as multi-grid convergence, average error, maximum error, isotropy and length estimation. Experiments underline that adaptive methods based on digital straight line recognition often propose a good trade-off between time and precision and that if precision is to be sought, non-adaptive methods can be easily transformed into adaptive methods to get more accurate estimations.

ON THE LOCAL PROPERTIES OF DIGITAL CURVES

We propose a geometric approach to extract local properties of digital curves. This approach uses the notion of blurred segment 1. It is an extension of Reveilles' arithmetical approach 2 on discrete lines, more flexible, to take into account noise in digital images. This notion is also extended to the 3D space. Algorithms are presented to decompose a curve into a sequence of maximal blurred segments. A curvature estimator for 2D and 3D curves is proposed relying on this decomposition as well as a torsion estimator for 3D curves. All these estimators can naturally work with disconnected curves. They prove their efficiency in experiments and in comparisons with some well-known methods of the literature.

Tangential cover for thick digital curves

The recognition of digital shapes is a deeply studied problem. The arithmetical framework, initiated by Reveillès [Géométrie discrète, calcul en nombres entiers et algorithmique, Thèse d’Etat, 1991], provides a powerful theoretical basis, as well as many algorithms to deal with digital objects. The tangential cover, first presented in Feschet and Tougne [Optimal time computation of the tangent of a discrete curve: application to the curvature, in: G. Bertrand, M. Couprie, L. Perroton (Eds.), 8th Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, vol. 1568, Springer, Berlin, 1999, pp. 31–40] and Feschet [Canonical representations of discrete curves, Pattern Anal. Appl. 8(1–2) (2005) 84–94] is a useful tool for representing geometric digital primitives. It computes the set of all maximal segments of a digital curve and permits either to obtain minimal length polygonalization or asymptotic convergence of tangents estimations. Nevertheless, the arithmetical approach does not tolerate the introduction of irregularities, which are however inherent to the acquisition of digital shapes. The present paper is an extension of Faure and Feschet [Tangential cover for thick digital curves, in: D. Coeurjolly, I. Sivignon, L. Tougne, F. Dupont (Eds.), DGCI 2008, Lecture Notes in Computer Science, vol. 4992, Springer, Berlin, 2008, pp. 358–369], in which we propose a new definition for a class of the so-called “thick digital curves” that applies well to a large class of digital object boundaries. We then propose an extension of the tangential cover to thick digital curves and provide an algorithm with an O ( n log n ) time complexity, where n denotes the number of points of specific subparts of the thick digital curve. In order to keep up with this low complexity, some critical points must be taken into account. We describe all required implementation details in this paper.

A genetic algorithm with chromosome-repairing for min − # and min − ε polygonal approximation of digital curves

A genetic algorithm for solving min − ε polygonal approximation and min − # polygonal approximation is proposed in this paper. It combines traditional split-and-merge techniques with a novel chromosome-repairing scheme to cope with constraints. Due to this combination of techniques we call our new method SMCR. In this new scheme an infeasible solution cannot only be easily transformed into a feasible one, but also be optimized. The experimental results show that the proposed SMCR has higher performance than the other GA-based methods and some non-GA-based methods.

Formulas for the number of [formula omitted]-gaps of binary objects in arbitrary dimension

In this paper we define the notion of a gap in an arbitrary digital binary object S in a digital space of arbitrary dimension. Then we obtain an explicit formula for the number of gaps in S of maximal dimension, derive combinatorial relations for digital curves, and discuss possible applications to image analysis of digital surfaces (in particular planes) and curves.

A novel stochastic search method for polygonal approximation problem

A novel stochastic search method (NSSM) is proposed for the polygonal approximation problem. NSSM incorporates the ranking selection scheme, which is initially developed for solving the premature convergence of genetic algorithms (GAs), into the traditional split-and-merge technique. For avoiding getting trapped in a local optimum, NSSM randomly selects the splitting points and the merging points and determines the selection probability using the ranking selection scheme. Three groups of digital curves, including the synthesized benchmark curves and the real image curves, are used to test the performance of NSSM. The experimental results show the higher performance over the other methods including the GA-based methods and the local search methods.

Border and SurfaceTracing - Theoretical Foundations

In this paper we define and study digital manifolds of arbitrary dimension, and provide (in particular)a general theoretical basis for curve or surface tracing in picture analysis. The studies involve properties such as one-dimensionality of digital curves and (n-1)-dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in continuous topology. The presented approach is fully based on the concept of adjacency relation and complements the concept of dimension as common in combinatorial topology. This work appears to be the first one on digital manifolds based ona graph-theoretical definition of dimension. In particular, in the n-dimensional digital space, a digital curve is a one-dimensional object and a digital hypersurface is an (n-1)-dimensional object, as it is in the case of curves and hypersurfaces in the Euclidean space. Relying on the obtained properties of digital hypersurfaces, we propose a uniform approach for studying good pairs defined by separations and obtain a classification of good pairs in arbitrary dimension. We also discuss possible applications of the presented definitions and results.

753 Stool DNA Screening of Colorectal Neoplasia: Improved Adenoma Detection By Novel Digital Melt Curve Assay

753 Stool DNA Screening of Colorectal Neoplasia: Improved Adenoma Detection By Novel Digital Melt Curve Assay

Thinning-free Polygonal Approximation of Thick Digital Curves Using Cellular Envelope

Since the inception of successful rasterization of curves and objects in the digital space, several algorithms have been proposed for approximating a given digital curve. All these algorithms, however, resort to thinning as preprocessing before approximating a digital curve with changing thickness. Described in this paper is a novel thinning-free algorithm for polygonal approximation of an arbitrarily thick digital curve, using the concept of “cellular envelope”, which is newly introduced in this paper. The cellular envelope, defined as the smallest set of cells containing the given curve, and hence bounded by two tightest (inner and outer) isothetic polygons, is constructed using a combinatorial technique. This envelope, in turn, is analyzed to determine a polygonal approximation of the curve as a sequence of cells using certain attributes of digital straightness. Since a real-world curve=curve-shaped object with varying thickness, unexpected disconnectedness, noisy information, etc., is unsuitable for the existing algorithms on polygonal approximation, the curve is encapsulated by the cellular envelope to enable the polygonal approximation. Owing to the implicit Euclidean-free metrics and combinatorial properties prevailing in the cellular plane, implementation of the proposed algorithm involves primitive integer operations only, leading to fast execution of the algorithm. Experimental results that include output polygons for different values of the approximation parameter corresponding to several real-world digital curves, a couple of measures on the quality of approximation, comparative results related with two other well-referred algorithms, and CPU times, have been presented to demonstrate the elegance and efficacy of the proposed algorithm.

Piecewise Linear Digital Curve Representation and Compression Using Graph Theory and a Line Segment Alphabet

The use of an alphabet of line segments to compose a curve is a possible approach for curve data compression. Many approaches are developed with the drawback that they can process simple curves only. Curves having more sophisticated topology with self-intersections can be handled by methods considering recursive decomposition of the canvas containing the curve. In this paper, we propose a graph theory-based algorithm for tracing the curve directly to eliminate the decomposition needs. This approach obviously improves the compression performance, as longer line segments can be used. We tune our method further by selecting optimal turns at junctions during tracing the curve. We assign a polygon approximation to the curve which consists of letters coming from an alphabet of line segments. We also discuss how other application fields can take advantage of the provided curve description scheme.

Numerical calculation of digital curve length by using anchored discrete convolution

The paper presents a new method introducing an anchored discrete convolution for calculating the length of a digital curve. The method is based on discrete convolution by using convolution masks and point anchoring in the pixel. The use of ordinary convolution distorts the curve shape and gives large errors in length calculation. The advantage of anchoring is that it limits the point shifting into the pixel during the calculation of the curve length. The method is applied to an analytical arc and various calculations are performed. In addition different methods from the literature were compared and a real sample was tested.

Fast Polygonal Approximation of Digital Curves Using Relaxed Straightness Properties

Several existing DSS (digital straight line segment) recognition algorithms can be used to determine the digital straightness of a given one-pixel-thick digital curve. Because of the inherent geometric constraints of digital straightness, these algorithms often produce a large number of segments to cover a given digital curve representing a real-life object=image. Thus, a curve segment, which is not exactly digitally straight, but appears to be visually straight, is fragmented into multiple DSS when these algorithms are run. In this paper, a new concept of approximate straightness is introduced by relaxing certain conditions of DSS, and an algorithm is described to extract those segments from a digital curve. The number of such segments required to cover the curve is found to be significantly fewer than that of the exact DSS-cover. As a result, the data set required for representing a curve also reduces to a large extent. The extracted set of segments can further be combined to determine a compact polygonal approximation of a digital curve based on certain approximation criteria and a specified error tolerance. The proposed algorithm involves only primitive integer operations and thus runs very fast compared to those based on exact DSS. The overall time complexity becomes linear in the number of points present in the representative set. Experimental results on several digital curves demonstrate the speed, elegance and efficacy of the proposed method.

Adaptive geometry compression based on four-point interpolatory subdivision schemes with labels

We propose an adaptive geometry compression method with labels based on four-point interpolatory subdivision schemes. It can work on digital curves of arbitrary dimensions. With the geometry compression method, a digital curve is adaptively compressed into several segments with different compression levels. Each segment is a four-point subdivision curve with a subdivision step. Labels are recorded in data compression to facilitate merging the segments in data decompression. We provide high-speed four-point interpolatory subdivision curve generation methods for efficiently decompressing the compressed data. For an arbitrary positive integer k, formulae for the number of resultant control points of a four-point subdivision curve after k subdivision steps are provided. Some formulae for calculating points at the kth subdivision step are also presented. The time complexity of the new approaches is O(n), where n is the number of points in the given digital curve. Examples are provided to illustrate the efficiency of the proposed approaches.

Dominant point detection by reverse polygonization of digital curves

A polygonal approximation technique using reverse polygonization is presented in this paper. The reverse polygonization starts from an initial set of dominant points i.e. break points. In that, dominant points are deleted (one in each iteration) such that the maximal perpendicular distance of approximating straight line from original curve is minimal. A comparative study with some commonly referred algorithms is also presented, which shows that this technique can produce better approximation results. The algorithm has additional advantages like simplicity, polygonal approximation with any number of dominant points and up to any error value, and computational efficiency.

Cubic Bézier approximation of a digitized curve

In this paper we present an efficient technique for piecewise cubic Bézier approximation of digitized curve. An adaptive breakpoint detection method divides a digital curve into a number of segments and each segment is approximated by a cubic Bézier curve so that the approximation error is minimized. Initial approximated Bézier control points for each of the segments are obtained by interpolation technique i.e. by the reverse recursion of De Castaljau's algorithm. Two methods, two-dimensional logarithmic search algorithm (TDLSA) and an evolutionary search algorithm (ESA), are introduced to find the best-fit Bézier control points from the approximate interpolated control points. ESA based refinement is proved to be better experimentally. Experimental results show that Bézier approximation of a digitized curve is much more accurate and uses less number of points compared to other approximation techniques.

An Approach to Polygonal Approximation of Digital Curves Based on Discrete Particle Swarm Algorithm

An approach to polygonal approximation of regular digital curves based on PSO algorithm is presented. In this paper, each particle corresponds to a candidate solution to the polygonal approximation problem, which is represented as a binary vector. The offset error of centroid between the original curve and the approximation polygon, and the variance of distance error for each approximation segment are adopted in the fitness function to evaluate the feasibility degree of the candidate solution. The sigmoid function of iteration times is used as the acceleration factors instead of the constant factors to improve the global searching characteristics. Experimental results show that the proposed approach can get suitable approximation results for preserving the features of original curves.

A novel approach to polygonal approximation of digital curves

A new approach to polygonal approximation is presented in this paper. It starts from an initial set of dominant points (break points), where the integral square error from a given shape is zero. The proposed algorithm iteratively deletes most redundant dominant points till required approximation is achieved. Stabilization algorithm after elimination of each dominant point ensures high quality of approximation. Results of proposed algorithm are compared with classical algorithms. The proposed algorithm has additional benefits like polygonal approximation with any number of dominant points and up to any error value and robustness of results.

A simple algorithm for digital line recognition in the general case

This paper is concerned with the digital line recognition problem for lines of fixed thickness in the naive and general cases. Previous incremental algorithms from Debled-Rennesson and Reveillès [A linear algorithm for segmentation of digital curves, Int. J. Pattern Recognition Artif. Intell. 9(6) (1995)] and from Buzer [A linear incremental algorithm for naive and standard digital lines and planes recognition, Graphical Models 65(1–3) (2003) 61–76] deal with the 8-connected case or with sophisticated machinery coming from linear programming. We present the first elementary method that works with any set of points (not necessarily 8-connected) and we propose a linear time algorithm under some restrictions. This paper deals with implementation details giving pseudo-code of our method. We insist on linking the recognition problem to the intrinsic properties of convex hulls.