Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

Occluded iris classification and segmentation using self-customized artificial intelligence models and iterative randomized Hough transform

A fast and accurate iris recognition system is presented for noisy iris images, mainly the noises due to eye occlusion and from specular reflection. The proposed recognition system will adopt a self-customized support vector machine (SVM) and convolution neural network (CNN) classification models, where the models are built according to the iris texture GLCM and automated deep features datasets that are extracted exclusively from each subject individually. The image processing techniques used were optimized, whether the processing of iris region segmentation using iterative randomized Hough transform (IRHT), or the processing of the classification, where few significant features are considered, based on singular value decomposition (SVD) analysis, for testing the moving window matrix class if it is iris or non-iris. The iris segments matching techniques are optimized by extracting, first, the largest parallel-axis rectangle inscribed in the classified occluded-iris binary image, where its corresponding iris region is crosscorrelated with the same subject’s iris reference image for obtaining the most correlated iris segments in the two eye images. Finally, calculating the iriscode Hamming distance of the two most correlated segments to identify the subject’s unique iris pattern with high accuracy, security, and reliability

Finding planar regions

Preiskovali smo problem iskanja ravnin na trianguliranem terenu. Za množico točk v prostoru zgradimo Delaunayjevo triangulacijo in z dvema različnima metodama poiščemo ravnino na terenu. V prvi metodi uporabimo algoritem za iskanje največjega konveksnega poligona. Algoritem se dobro obnese na manjši množici točk, na večji množici točk, pa zaradi svoje kvadratične časovne zahtevnosti ne pride v poštev. V drugi metodi uporabimo aproksimacijski algoritem. Ta se bolje obnese tudi na večji množici podatkov, kot tudi na realnih geografskih podatkih, ki jih lahko dobimo na spletnem portalu LIDAR. Implementiran vmesnik nam pomaga, da na enostaven način testiramo obe metodi in vizualiziramo rezultate.We studied the problem of finding planar regions in a triangulated terrain. For a set of points in 3-space, we construct the Delaunay triangulation. Then, with two different methods we look for a region which is flat. The first method uses an algorithm to find the largest convex polygon. The algorithm works well on smaller sets of points, but on larger sets of points it performs poorly due to its quadratic time complexity. In the second method, we use an approximation algorithm. It performs better on larger data sets, as well as on real geographical data, which can be obtained on the web portal LIDAR. The implemented interface helps us to test both methods in an easy way and visualize the results

ROC curves in cost space

The final publication is available at Springer via http://dx.doi.org/10.1007/s10994-013-5328-9ROC curves and cost curves are two popular ways of visualising classifier performance, finding appropriate thresholds according to the operating condition, and deriving useful aggregated measures such as the area under the ROC curve (AUC) or the area under the optimal cost curve. In this paper we present new findings and connections between ROC space and cost space. In particular, we show that ROC curves can be transferred to cost space by means of a very natural threshold choice method, which sets the decision threshold such that the proportion of positive predictions equals the operating condition. We call these new curves rate-driven curves, and we demonstrate that the expected loss as measured by the area under these curves is linearly related to AUC. We show that the rate-driven curves are the genuine equivalent of ROC curves in cost space, establishing a point-point rather than a point-line correspondence. Furthermore, a decomposition of the rate-driven curves is introduced which separates the loss due to the threshold choice method from the ranking loss (Kendall τ distance). We also derive the corresponding curve to the ROC convex hull in cost space; this curve is different from the lower envelope of the cost lines, as the latter assumes only optimal thresholds are chosen.We would like to thank the anonymous referees for their helpful comments. This work was supported by the MEC/MINECO projects CONSOLIDER-INGENIO CSD2007-00022 and TIN 2010-21062-C02-02, GVA project PROMETEO/2008/051, the COST-European Cooperation in the field of Scientific and Technical Research IC0801 AT, and the REFRAME project granted by the European Coordinated Research on Long-term Challenges in Information and Communication Sciences & Technologies ERA-Net (CHIST-ERA), and funded by the Engineering and Physical Sciences Research Council in the UK and the Ministerio de Economia y Competitividad in Spain.Hernández Orallo, J.; Flach ., P.; Ferri Ramírez, C. (2013). ROC curves in cost space. Machine Learning. 93(1):71-91. https://doi.org/10.1007/s10994-013-5328-9S7191931Adams, N., & Hand, D. (1999). Comparing classifiers when the misallocation costs are uncertain. Pattern Recognition, 32(7), 1139–1147.Chang, J., & Yap, C. (1986). A polynomial solution for the potato-peeling problem. Discrete & Computational Geometry, 1(1), 155–182.Drummond, C., & Holte, R. (2000). Explicitly representing expected cost: an alternative to ROC representation. In Knowl. discovery & data mining (pp. 198–207).Drummond, C., & Holte, R. (2006). Cost curves: an improved method for visualizing classifier performance. Machine Learning, 65, 95–130.Elkan, C. (2001). The foundations of cost-sensitive learning. In B. Nebel (Ed.), Proc. of the 17th intl. conf. on artificial intelligence (IJCAI-01) (pp. 973–978).Fawcett, T. (2006). An introduction to ROC analysis. Pattern Recognition Letters, 27(8), 861–874.Fawcett, T., & Niculescu-Mizil, A. (2007). PAV and the ROC convex hull. Machine Learning, 68(1), 97–106.Flach, P. (2003). The geometry of ROC space: understanding machine learning metrics through ROC isometrics. In Machine learning, proceedings of the twentieth international conference (ICML 2003) (pp. 194–201).Flach, P., Hernández-Orallo, J., & Ferri, C. (2011). A coherent interpretation of AUC as a measure of aggregated classification performance. In Proc. of the 28th intl. conference on machine learning, ICML2011.Frank, A., & Asuncion, A. (2010). UCI machine learning repository. http://archive.ics.uci.edu/ml .Hand, D. (2009). Measuring classifier performance: a coherent alternative to the area under the ROC curve. Machine Learning, 77(1), 103–123.Hernández-Orallo, J., Flach, P., & Ferri, C. (2011). Brier curves: a new cost-based visualisation of classifier performance. In Proceedings of the 28th international conference on machine learning, ICML2011.Hernández-Orallo, J., Flach, P., & Ferri, C. (2012). A unified view of performance metrics: translating threshold choice into expected classification loss. Journal of Machine Learning Research, 13, 2813–2869.Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, 30(1/2), 81–93. doi: 10.2307/2332226 .Swets, J., Dawes, R., & Monahan, J. (2000). Better decisions through science. Scientific American, 283(4), 82–87