In a digital image, color representation of a digital image sensor is limited to a narrow dynamic range. Especially, when extremely bright light is captured, the original color of a scene is saturated to the maximum value, up to which a digital image sensor can represent the color. This paper proposes an algorithm that corrects the color in a saturated region, where the original color is distorted and lost. For natural correction, i.e., to minimize the artifacts near the boundary of a saturated region, the proposed method uses the weighted sum of color value(s) in the saturated color channel(s) of neighborhood of saturated regions. In determining the weight of each pixel, saturation, hue, and color values are used with the certainty map. Using the certainty map, the proposed method can reliably distinguish the unsaturated and already desaturated neighboring pixels from the remaining pixels. Then, the proposed correction method computes the weight function using saturation, hue, and color values. Therefore, the proposed algorithm can get reliable corrected colors. Comparison of experimental results of the proposed and existing correction methods shows the effectiveness of the proposed saturated region correction method in the view of natural color restoration.
{"title":"CORRECTION OF SATURATED REGIONS IN RGB COLOR SPACE","authors":"Hae Jin Ju, Rae-Hong Park","doi":"10.5121/IJCGA.2016.6201","DOIUrl":"https://doi.org/10.5121/IJCGA.2016.6201","url":null,"abstract":"In a digital image, color representation of a digital image sensor is limited to a narrow dynamic range. Especially, when extremely bright light is captured, the original color of a scene is saturated to the maximum value, up to which a digital image sensor can represent the color. This paper proposes an algorithm that corrects the color in a saturated region, where the original color is distorted and lost. For natural correction, i.e., to minimize the artifacts near the boundary of a saturated region, the proposed method uses the weighted sum of color value(s) in the saturated color channel(s) of neighborhood of saturated regions. In determining the weight of each pixel, saturation, hue, and color values are used with the certainty map. Using the certainty map, the proposed method can reliably distinguish the unsaturated and already desaturated neighboring pixels from the remaining pixels. Then, the proposed correction method computes the weight function using saturation, hue, and color values. Therefore, the proposed algorithm can get reliable corrected colors. Comparison of experimental results of the proposed and existing correction methods shows the effectiveness of the proposed saturated region correction method in the view of natural color restoration.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"14 1","pages":"01-13"},"PeriodicalIF":0.0,"publicationDate":"2016-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.5121/IJCGA.2016.6201","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70617628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider random polytopes defined as the convex hull of a Poisson point process on a sphere in $R^3$ such that its average number of points is $n$. We show that the expectation over all such random polytopes of the maximum size of their silhouettes viewed from infinity is $Theta(sqrt{n})$.
{"title":"Silhouette of a random polytope","authors":"M. Glisse, S. Lazard, J. Michel, M. Pouget","doi":"10.20382/jocg.v7i1a5","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a5","url":null,"abstract":"We consider random polytopes defined as the convex hull of a Poisson point process on a sphere in $R^3$ such that its average number of points is $n$. We show that the expectation over all such random polytopes of the maximum size of their silhouettes viewed from infinity is $Theta(sqrt{n})$.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"1 1","pages":"86-99"},"PeriodicalIF":0.0,"publicationDate":"2016-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75632553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a simple technique for analyzing the size of geometric hypergraphs defined by random point sets. As an application we obtain upper and lower bounds on the smoothed number of faces of the convex hull under Euclidean and Gaussian noise and related results.
{"title":"Smoothed complexity of convex hulls by witnesses and collectors","authors":"O. Devillers, M. Glisse, X. Goaoc, Rémy Thomasse","doi":"10.20382/jocg.v7i2a6","DOIUrl":"https://doi.org/10.20382/jocg.v7i2a6","url":null,"abstract":"We present a simple technique for analyzing the size of geometric hypergraphs defined by random point sets. As an application we obtain upper and lower bounds on the smoothed number of faces of the convex hull under Euclidean and Gaussian noise and related results.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"66 1","pages":"101-144"},"PeriodicalIF":0.0,"publicationDate":"2016-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84028599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a set of $h$ pairwise disjoint polygonal obstacles with a total of $n$ vertices in the plane, the vertex-vertex visibility graph is an undirected graph whose nodes are vertices of the obstacles and whose edges are pairs of visible vertices. The vertex-edge and edge-edge visibility graphs are defined similarly. Ghosh and Mount gave a well-known output-sensitive $O(nlog n+k)$ time algorithm for computing these visibility graphs, where $k$ is the size of the corresponding graph. By developing new techniques based on an extended corridor structure, we augment Ghosh and Mount’s algorithm to build these visibility graphs in $O(n+hlog h+k)$ time, after the free space is triangulated. The new algorithm improves Ghosh and Mount’s algorithm by reducing its additive $O(nlog n)$ time factor to $O(n + hlog h)$. Like Ghosh and Mount’s algorithm, our algorithm can also compute several important structures such as the funnel structure and the enhanced visibility graph, which may have other applications.
{"title":"A new algorithm for computing visibility graphs of polygonal obstacles in the plane","authors":"D. Chen, Haitao Wang","doi":"10.20382/jocg.v6i1a14","DOIUrl":"https://doi.org/10.20382/jocg.v6i1a14","url":null,"abstract":"Given a set of $h$ pairwise disjoint polygonal obstacles with a total of $n$ vertices in the plane, the vertex-vertex visibility graph is an undirected graph whose nodes are vertices of the obstacles and whose edges are pairs of visible vertices. The vertex-edge and edge-edge visibility graphs are defined similarly. Ghosh and Mount gave a well-known output-sensitive $O(nlog n+k)$ time algorithm for computing these visibility graphs, where $k$ is the size of the corresponding graph. By developing new techniques based on an extended corridor structure, we augment Ghosh and Mount’s algorithm to build these visibility graphs in $O(n+hlog h+k)$ time, after the free space is triangulated. The new algorithm improves Ghosh and Mount’s algorithm by reducing its additive $O(nlog n)$ time factor to $O(n + hlog h)$. Like Ghosh and Mount’s algorithm, our algorithm can also compute several important structures such as the funnel structure and the enhanced visibility graph, which may have other applications.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"19 1","pages":"316-345"},"PeriodicalIF":0.0,"publicationDate":"2015-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81759845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-11-17DOI: 10.4230/LIPIcs.SoCG.2017.23
M. Buchet, T. Dey, Jiayuan Wang, Yusu Wang
In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth K in a metric space, but it got corrupted with noise so that some of the data points lie far away from K creating outliers also termed as ambient noise. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of K. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases. Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms are effective in practice.
{"title":"Declutter and Resample: Towards Parameter Free Denoising","authors":"M. Buchet, T. Dey, Jiayuan Wang, Yusu Wang","doi":"10.4230/LIPIcs.SoCG.2017.23","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2017.23","url":null,"abstract":"In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth K in a metric space, but it got corrupted with noise so that some of the data points lie far away from K creating outliers also termed as ambient noise. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of K. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases. \u0000 \u0000Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms \u0000are effective in practice.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"6 1","pages":"21-46"},"PeriodicalIF":0.0,"publicationDate":"2015-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79617117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Luis Barba, O. Cheong, M. G. Dobbins, R. Fleischer, A. Kawamura, Matias Korman, Y. Okamoto, J. Pach, Yuan Tang, T. Tokuyama, S. Verdonschot
Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, that is, whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any compact planar set) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional compact set containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) For any $d$-dimensional bounded convex polyhedron containing the origin, there exists a pair of antipodal points consisting of a point on a $lfloor d/2 rfloor$-face and a point on a $lceil d/2rceil$-face.
{"title":"Weight balancing on boundaries","authors":"Luis Barba, O. Cheong, M. G. Dobbins, R. Fleischer, A. Kawamura, Matias Korman, Y. Okamoto, J. Pach, Yuan Tang, T. Tokuyama, S. Verdonschot","doi":"10.20382/jocg.v13i1a1","DOIUrl":"https://doi.org/10.20382/jocg.v13i1a1","url":null,"abstract":"Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, that is, whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any compact planar set) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional compact set containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) For any $d$-dimensional bounded convex polyhedron containing the origin, there exists a pair of antipodal points consisting of a point on a $lfloor d/2 rfloor$-face and a point on a $lceil d/2rceil$-face.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"4 1","pages":"1-12"},"PeriodicalIF":0.0,"publicationDate":"2015-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90608643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-09-25DOI: 10.4230/LIPIcs.SWAT.2016.30
B. Aronov, Matias Korman, Simon Pratt, André van Renssen, Marcel Roeloffzen
An s-workspace algorithm is an algorithm that has read-only access to the values of the input, write-only access to the output, and only uses O(s) additional words of space. We give a randomized s-workspace algorithm for triangulating a simple polygon P of n vertices, for any s up to n. The algorithm runs in O(n^2/s+n(log s)log^5(n/s)) expected time using O(s) variables, for any s up to n. In particular, the algorithm runs in O(n^2/s) expected time for most values of s.
{"title":"Time-Space Trade-offs for Triangulating a Simple Polygon","authors":"B. Aronov, Matias Korman, Simon Pratt, André van Renssen, Marcel Roeloffzen","doi":"10.4230/LIPIcs.SWAT.2016.30","DOIUrl":"https://doi.org/10.4230/LIPIcs.SWAT.2016.30","url":null,"abstract":"An s-workspace algorithm is an algorithm that has read-only access to the values of the input, write-only access to the output, and only uses O(s) additional words of space. We give a randomized s-workspace algorithm for triangulating a simple polygon P of n vertices, for any s up to n. The algorithm runs in O(n^2/s+n(log s)log^5(n/s)) expected time using O(s) variables, for any s up to n. In particular, the algorithm runs in O(n^2/s) expected time for most values of s.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"10 1","pages":"105-124"},"PeriodicalIF":0.0,"publicationDate":"2015-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84409296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an algorithm that approximates the medial axis of a smooth manifold in $mathbb{R}^3$ which is given by a sufficiently dense point sample. The resulting, non-discrete approximation is shown to converge to the medial axis as the sampling density approaches infinity. While all previous algorithms guaranteeing convergence have a running time quadratic in the size $n$ of the point sample, we achieve a running time of at most $mathcal{O}(nlog^3 n)$. While there is no subquadratic upper bound on the output complexity of previous algorithms for non-discrete medial axis approximation, the output of our algorithm is guaranteed to be of linear size.
{"title":"Subquadratic medial-axis approximation in $mathbb{R}^3$","authors":"Christian Scheffer, J. Vahrenhold","doi":"10.20382/jocg.v6i1a11","DOIUrl":"https://doi.org/10.20382/jocg.v6i1a11","url":null,"abstract":"We present an algorithm that approximates the medial axis of a smooth manifold in $mathbb{R}^3$ which is given by a sufficiently dense point sample. The resulting, non-discrete approximation is shown to converge to the medial axis as the sampling density approaches infinity. While all previous algorithms guaranteeing convergence have a running time quadratic in the size $n$ of the point sample, we achieve a running time of at most $mathcal{O}(nlog^3 n)$. While there is no subquadratic upper bound on the output complexity of previous algorithms for non-discrete medial axis approximation, the output of our algorithm is guaranteed to be of linear size.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"78 1","pages":"249-287"},"PeriodicalIF":0.0,"publicationDate":"2015-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86103408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In arrangements of pseudocircles (i.e., Jordan curves) the weight of a vertex (i.e., an intersection point) is the number of pseudocircles that contain the vertex in its interior. We show that in complete arrangements (in which each two pseudocircles intersect) $2n-1$ vertices of weight 0 force an $alpha$ -subarrangement, a certain arrangement of three pseudocircles. Similarly, $4n-5$ vertices of weight 0 force an $alpha^4$-subarrangement (of four pseudocircles). These results on the one hand give improved bounds on the number of vertices of weight $k$ for complete, $alpha$-free and complete, $alpha^4$-free arrangements. On the other hand, interpreting $alpha$- and $alpha^4$-arrangements as complete graphs with three and four vertices, respectively, the bounds correspond to known results in extremal graph theory.
{"title":"Forcing subarrangements in complete arrangements of pseudocircles","authors":"R. Ortner","doi":"10.20382/jocg.v6i1a10","DOIUrl":"https://doi.org/10.20382/jocg.v6i1a10","url":null,"abstract":"In arrangements of pseudocircles (i.e., Jordan curves) the weight of a vertex (i.e., an intersection point) is the number of pseudocircles that contain the vertex in its interior. We show that in complete arrangements (in which each two pseudocircles intersect) $2n-1$ vertices of weight 0 force an $alpha$ -subarrangement, a certain arrangement of three pseudocircles. Similarly, $4n-5$ vertices of weight 0 force an $alpha^4$-subarrangement (of four pseudocircles). These results on the one hand give improved bounds on the number of vertices of weight $k$ for complete, $alpha$-free and complete, $alpha^4$-free arrangements. On the other hand, interpreting $alpha$- and $alpha^4$-arrangements as complete graphs with three and four vertices, respectively, the bounds correspond to known results in extremal graph theory.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"17 1","pages":"235-248"},"PeriodicalIF":0.0,"publicationDate":"2015-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74515393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $mathbb Z/ {n mathbb Z}$, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group $mathcal{W}$, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let $mathcal G$ be the group generated by $mathcal{W}$ as well as all integer translations in $mathbb Z^d$. We prove that if $P$ multi-tiles $mathbb R^d$ under the action of $mathcal G$, then we have the closed form $G_P(n) = text{vol}(P) G(n)^d$. Conversely, we also prove that if $P$ is a lattice tetrahedron in $mathbb R^3$, of volume $1/6$, such that $G_P(n) = text{vol}(P) G(n)^d$, for $n in { 1,2,3,4 }$, then there is an element $g$ in $mathcal G$ such that $g(P)$ is the fundamental tetrahedron with vertices $(0,0,0)$, $(1, 0, 0)$, $(1,1,0)$, $(1,1,1)$.
{"title":"Polyhedral Gauss sums, and polytopes with symmetry","authors":"R. Malikiosis, S. Robins, Yichi Zhang","doi":"10.20382/jocg.v7i1a8","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a8","url":null,"abstract":"We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $mathbb Z/ {n mathbb Z}$, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group $mathcal{W}$, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let $mathcal G$ be the group generated by $mathcal{W}$ as well as all integer translations in $mathbb Z^d$. We prove that if $P$ multi-tiles $mathbb R^d$ under the action of $mathcal G$, then we have the closed form $G_P(n) = text{vol}(P) G(n)^d$. Conversely, we also prove that if $P$ is a lattice tetrahedron in $mathbb R^3$, of volume $1/6$, such that $G_P(n) = text{vol}(P) G(n)^d$, for $n in { 1,2,3,4 }$, then there is an element $g$ in $mathcal G$ such that $g(P)$ is the fundamental tetrahedron with vertices $(0,0,0)$, $(1, 0, 0)$, $(1,1,0)$, $(1,1,1)$.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"16 1","pages":"149-170"},"PeriodicalIF":0.0,"publicationDate":"2015-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83333364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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