Complexity and Algorithms for the Discrete Fr\'echet Distance Upper Bound with Imprecise Input

We study the problem of computing the upper bound of the discrete Fr\'{e}chet distance for imprecise input, and prove that the problem is NP-hard. This solves an open problem posed in 2010 by Ahn \emph{et al}. If shortcuts are allowed, we show that the upper bound of the discrete Fr\'{e}chet distance with shortcuts for imprecise input can be computed in polynomial time and we present several efficient algorithms.Comment: 15 pages, 8 figure

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Combining region-based and imprecise boundary-based cues for interactive medical image segmentation

We present an approach combining both region selection and user point selection for user- assisted segmentation as either an enclosed object or an open curve, investigate the method of image segmentation in specific medical applications (user-assisted segmentation of the media–adventitia border in intravascular ultrasound images, and lumen border in optical coherence tomography images), and then demonstrate the method with generic images to show how it could be utilized in other types of medical image and is not limited to the applications described. The proposed method combines point-based soft con- straint on object boundary and stroke-based regional constraint. The user points act as attraction points and are treated as soft constraints rather than hard constraints that the segmented boundary has to pass through. The user can also use strokes to specify region of interest. The probabilities of region of interest for each pixel are then calculated, and their discontinuity is used to indicate object boundary. The combinations of different types of user constraints and image features allow flexible and robust segmentation, which is formulated as an energy minimization problem on a multilayered graph and is solved using a shortest path search algorithm. We show that this combinatorial approach allows efficient and effective interactive segmentation, which can be used with both open and closed curves to segment a variety of images in different ways. The proposed method is demonstrated in the two medical applications, that is, intravascular ultrasound and optical coherence tomography images, where image artefacts such as acoustic shadow and calcification are commonplace and thus user guidance is desirable. We carried out both qualitative and quantitative analysis of the results for the medical data; comparing the proposed method against a number of interactive segmentation techniques

Bounding and Estimating the Hausdorff distance between real space algebraic curves

This is the author’s version of a work that was accepted for publication in Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Rueda S.L., Sendra J., Sendra J.R., (2014). "Bounding and Estimating the Hausdorff distance\ud between real space algebraic curves ". Computer Aided Geometric Design. vol 31 (2014)\ud 182-198; DOI 10.1016/j.cagd.2014.02.005In this paper, given two real space algebraic curves, not necessarily bounded,\ud whose Hausdor distance is nite, we provide bounds of their distance. These\ud bounds are related to the distance between the projections of the space curves onto\ud a plane (say, z = 0), and the distance between the z-coordinates of points in the\ud original curves. Using these bounds we provide an estimation method for a bound\ud of the Hausdor distance between two such curves and we check in applications that\ud the method is accurate and fas