Two linear-time algorithms for computing the minimum length polygon of a digital contour

AbstractThe Minimum Length Polygon (MLP) is an interesting first order approximation of a digital contour. For instance, the convexity of the MLP is characteristic of the digital convexity of the shape, its perimeter is a good estimate of the perimeter of the digitized shape. We present here two novel equivalent definitions of MLP, one arithmetic, one combinatorial, and both definitions lead to two different linear time algorithms to compute them. This paper extends the work presented in Provençal and Lachaud (2009) [26], by detailing the algorithms and providing full proofs. It includes also a comparative experimental evaluation of both algorithms showing that the combinatorial algorithm is about 5 times faster than the other. We also checked the multigrid convergence of the length estimator based on the MLP

Relative Convex Hull Determination from Convex Hulls in the Plane

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.Comment: 15 pages, 4 figures, Conference paper published. We corrected two typing errors in Definition 2: $I_S$ has to be defined based on $O_S$, and $I_E$ has to be defined based on $O_E$ (not just using $O$). These errors appeared in the text of the original conference paper, which also contained the pseudocode of an algorithm where $I_S$ and $I_E$ appeared as correctly define

On the tiling by translation problem

On square or hexagonal lattices tiles or polyominoes are coded by words. The polyominoes that tile the plane by translation are characterized by the Beauquier-Nivat condition. By using the constant time algorithms for computing the longest common extensions in two words, we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon. For pseudo-hexagon polyominoes not containing arbitrarily large square factors we also have a linear algorithm. The results are extended to more general tiles. Key words: Tiling polyominoes, plane tesselation, longest common extensions

Lyndon + Christoffel = Digitally Convex

Discrete geometry redefines notions borrowed from Euclidean geometry creating a need for new algorithmical tools. The notion of convexity does not translate trivially, and detecting if a discrete region of the plane is convex requires a deeper analysis. To the many different approaches of digital convexity, we propose the combinatorics on words point of view, unnoticed until recently in the pattern recognition community. In this paper we provide first a fast optimal algorithm checking digital convexity of polyominoes coded by their contour word. The result is based on linear time algorithms for both computing the Lyndon factorization of the contour word, and the recognition of Christoffel factors that are approximations of digital lines. By avoiding arithmetical computations the algorithm is much simpler to implement and much faster in practice. We also consider the convex hull computation and relate previous work in combinatorics on words with the classical Melkman algorithm

Randomized phase III trial in elderly patients comparing LV5FU2 with or without irinotecan for first-line treatment of metastatic colorectal cancer (FFCD 2001–02)

International audienceBACKGROUND:Metastatic colorectal cancer (mCRC) frequently occurs in elderly patients. However, data from a geriatric tailored randomized trial about tolerance to and the efficacy of doublet chemotherapy (CT) with irinotecan in the elderly are lacking. The benefit of first-line CT intensification remains an issue in elderly patients.PATIENTS AND METHODS:Elderly patients (75+) with previously untreated mCRC were randomly assigned in a 2 × 2 factorial design (four arms) to receive 5-FU (5-fluorouracil)-based CT, either alone (FU: LV5FU2 or simplified LV5FU2) or in combination with irinotecan [IRI: LV5FU2-irinotecan or simplified LV5FU2-irinotecan (FOLFIRI)]. The CLASSIC arm was defined as LV5FU2 or LV5FU2-irinotecan and the SIMPLIFIED arm as simplified LV5FU2 or FOLFIRI. The primary end point was progression-free survival (PFS). Secondary end points were overall survival (OS), safety and objective response rate (ORR).RESULTS:From June 2003 to May 2010, 71 patients were randomly assigned to LV5FU2, 71 to simplified LV5FU2, 70 to LV5FU2-irinotecan and 70 to FOLFIRI. The median age was 80 years (range 75-92 years). No significant difference was observed for the median PFS: FU 5.2 months versus IRI 7.3 months, hazard ratio (HR) = 0.84 (0.66-1.07), P = 0.15 and CLASSIC 6.5 months versus SIMPLIFIED 6.0 months, HR = 0.85 (0.67-1.09), P = 0.19. The ORR was superior in IRI (P = 0.0003): FU 21.1% versus IRI 41.7% and in CLASSIC (P = 0.04): CLASSIC 37.1% versus SIMPLIFIED 25.6%. Median OS was 14.2 months in FU versus 13.3 months in IRI, HR = 0.96 (0.75-1.24) and 15.2 months in CLASSIC versus 11.4 months in SIMPLIFIED, HR = 0.71 (0.55-0.92). More patients presented grade 3-4 toxicities in IRI (52.2% versus 76.3%).CONCLUSION:In this elderly population, adding irinotecan to an infusional 5-FU-based CT did not significantly increase either PFS or OS. Classic LV5FU2 was associated with an improved OS compared with simplified LV5FU2.CLINICALTRIALSGOV:NCT00303771