An iterative point elimination technique to retain significant vertices on digital planar curves

A dynamic programming approach for fitting digital planar curves with line segments and circular arcs

Determination of optimal polygon from digital curve using L1 norm

In this chapter, we introduce basic 2D digital geometry. The main topic in 2D geometry is curves. A 2D digital curve is a simple path in \(\Sigma_{2}\). A simple closed digital curve is usually the boundary of a connected component. We first discuss how we precisely define a curve in a graph and Euclidean space, then we discuss how we represent digital curves. Digital curves have two important applications in computer graphics and computer vision: (a) Construction of a digital line when two end points are given, and (b) Determination of a closed digital curve to identify a connected region in computer vision. At the end of this chapter, we present two classic theorems related to 2D digital geometry: Pick’s theorem and Minkowski’s theorem. In addition, we discuss the basic concept of image segmentation, one of the major applications of 2D digital planes.

This letter presents a novel computing paradigm for polygonal approximation of digital planar curves. While the existing heuristic algorithms, such as genetic algorithm (GA) and particle swarm optimization (PSO), have achieved considerable success in solving the two types of polygonal approximation problems, more efficient optimization schemes are still desirable for practical applications. We propose to embed the split-and-merge local search in the Monte Carlo sampling framework, to combine strength of the local optimization and the global sampling. The proposed algorithm is essentially a well-designed basin hopping scheme that performs stochastic exploration in the reduced potential energy space. Experimental results on several benchmarks indicate that the proposed algorithm can achieve high approximation accuracy and is highly competitive to the state-of-the-art alternative algorithms with less computational cost.

This paper applies reverse engineering on the Bresenham's line drawing algorithm [J. E. Bresenham, IBM System Journal, 4, 106–111 (1965)] for polygonal approximation of digital curve. The proposed method has a number of features, namely, it is sequential and runs in linear time, produces symmetric approximation from symmetric digital curve, is an automatic algorithm and the approximating polygon has the least non-zero approximation error as compared to other algorithms.

In this investigation, we propose a scheme which integrates the two complementary approaches, i.e., the raster approach and the vector approach, to minimize the number of feature points but still preserve the shape of digital curves with high fidelity. The raster form of the digital curve is first encoded in the Freeman chain code space. Then the second order derivative operator combined with a Gaussian filter is applied to detect zero crossings. The turning points on the curve detected as zero crossings in Freeman chain code space are then transformed to the image space. The second stage is then applying dynamic strip algorithm to further eliminate unnecessary points remaining in the first stage. In order to avoid over- elimination of the points, the maximum length of the strip may also be assigned in the scheme. To quantitatively analyze the performance of the proposed scheme, the perpendicular distance of each point on the original curve to the fitted line segment characterized by the dominant points is calculated. The applicability of the proposed scheme in surface modeling from a contour map is also tested.

Moment-based methods for polygonal approximation of digitized curves

A method for polygonal approximation of digital planar curves is proposed. This method is relied on the supression of dominant points. Initially, the method uses the set of all breakpoints as dominant points. The method eliminates dominant points until a final condition is satisfied. The residuary dominant points are the polygonal approximation. This aproximation is compared with other classical algorithms. The experimental results show that this method works well for digital planar curves with features of several sizes.KeywordsComputer visiondigital planar curvespolygonal approximationdominant points

This paper presents an algorithm for approximating planar curves with conjugated circular arcs and straight-line segments. Approximation of digital planar curves using line segments and analytic curve pieces is a problem of fundamental importance in image analysis and pattern recognition. A specific variation of this problem is approximation by conjugated arcs and line segments, which also arises in robotics, particularly in motion planning. We propose an algorithmic solution to this problem, designed for the automatic generation of programs for robotic manipulators in the AS language. The originality of the method lies in employing a special representation of planar curves that is invariant under translation, rotation, and scaling. Such a representation enables the development of an optimal algorithm that automatically determines the number and placement of conjugated arcs and line segments approximating a given curve with a prescribed accuracy. The simplicity, flexibility with respect to input data, and low computational requirements of the proposed method make it suitable not only for image recognition tasks but also for robotic trajectory planning.

We propose an optimal split algorithm for multiresolution polygonal approximation of digital curves. Instead of using a sequence of heuristic split steps as in previous methods, we apply optimal approximation to obtain the next (higher) resolution levels using the previous (lower) resolution level as a starting point. We compare the proposed approach against a fast heuristic multiresolution algorithm based on a merge strategy under the L/sub 2/ error measure.

The paper presents a new technique for efficient polygon approximation of digitized planar curves. The polygon approximation algorithms based on sequential scan, split-and-merge and iterative techniques have some drawbacks: they shift the corner points of the given curve, they distort the original symmetry of the curve, polygon approximation is dependent on starting point and the starting points are taken as break points, they cannot preserve the identity of the segments whose lengths lie between (epsilon) and 2 (epsilon) , where (epsilon) is allowed maximum absolute deviation error. The proposed technique grows the edges of polygon approximation which is based on principle of merging. The edge/s are grown at point/s where the minimum merging error is produced. This simultaneous growing of edges overcomes the drawbacks present in the sequential scan, the split-and-merge, and the iterative techniques of polygon approximation. Merging is done on the sides of the initial polygon approximation obtained by template matching. The technique provides a scope for parallel implementation of the total task.

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r − 1 \leq r-1 , we show that the proposed method exhibits an error of the order of 4 r 4r for eigenvalue approximation and of the order of 3 r 3r for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order 4 r 4r can be obtained. This improves upon the order 2 r 2r for eigenvalue approximation in the collocation/iterated collocation method and the orders r r and 2 r 2r for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.

Scanning accuracy of nondental structured light extraoral scanners compared with that of a dental-specific scanner

Summary With the constantly increasing quantity of data recorded in the oil and gas industry, data analytics and data-driven algorithms are gaining popularity. Meanwhile, they are highly sensitive to employed data management methods. Using drilling data as an example, the methods for data quality improvement play an ever increasing role in the data preparation phase. One of the most common data issues, especially in real-time sensing systems, is missing data (gaps between measured points). Various data imputation approaches (forward filling, interpolation, regression modeling, etc.) have been used to fill gaps to complete data sets as standard data processing procedures. The metrics, such as coefficient of determination (R2), mean absolute error, root mean square error, and mean absolute percentage error, are common ways to evaluate the imputation approaches, assuming that the ground truth data are at hand. In reality, the ground truth of missing data is not available for quantitative method evaluation. Especially when data are received occasionally, the ground truth is impossibly or inaccurately estimated, leading to tricky situations. For instance, how to evaluate infilling methods quantitatively or how to compare the methods’ behaviors? To the best of the authors’ knowledge, there are so far few existing methods to quantitatively estimate the accuracy of data imputation without the ground truth. To some extent, one may lack proven confidence in using imputed data for following up data analysis and data-driven modeling. In this study, a novel approach has been developed to quantitatively evaluate the data imputation quality. The presented method is built on an analytical estimation of the R2 bounds in the context of linear interpolation. The reasons for choosing linear regression as a benchmarking method are twofold. First, the bounds evaluation of R2 can be determined analytically for data imputation in the context of linear regression. Second, linear regression is a relatively simple imputation method, and we assume that more sophisticated methods, such as random forest (RF), k-nearest neighbors (KNN), and other iterative imputation techniques, can achieve higher accuracy with proper parameter tuning. The evaluation method is demonstrated by examples using real-time drilling data logs. Moreover, the application to drilling logs also enables informed decision-making in relation to gap filling and data resampling workflows.

Numerical proper reparametrization of parametric plane curves