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It renders an image of a three dimensional scene of triangular primitives by reconstruction from a random sample of surface points which are chosen with a probability proportional to the projected area of the objects. The approach is independent of mesh connectivity and topology. It leads to a rendering time that grows only logarithmically with the numbers of triangles in the scene and to linear memory consumption, thus allowing walkthroughs of scenes of extreme complexity. We consider different methods for image reconstruction which aim at correctness, rendering speed and image quality and we develop an efficient data structure for sample extraction in output-sensitive time which allows for efficient dynamic updates of the scene. Experiments confirm that scenes consisting of some hundred billion triangles can be rendered within seconds with an image quality comparable to a conventional z-buffer rendering; in special cases, realtime performance can be achieved.","lang":"eng"}],"citation":{"apa":"Wand, M., Fischer, M., &#38; Meyer auf der Heide, F. (2000). <i>Randomized Point Sampling for Output-Sensitive Rendering of Complex Dynamic Scenes</i>. Universität Paderborn.","ieee":"M. Wand, M. Fischer, and F. Meyer auf der Heide, <i>Randomized Point Sampling for Output-Sensitive Rendering of Complex Dynamic Scenes</i>. Universität Paderborn, 2000.","short":"M. Wand, M. Fischer, F. Meyer auf der Heide, Randomized Point Sampling for Output-Sensitive Rendering of Complex Dynamic Scenes, Universität Paderborn, 2000.","chicago":"Wand, Michael, Matthias Fischer, and Friedhelm Meyer auf der Heide. <i>Randomized Point Sampling for Output-Sensitive Rendering of Complex Dynamic Scenes</i>. 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Thomas Braschs (und Jurek Beckers) filmische Reflexion über die Kunst nach Auschwitz: „Der Passagier – Welcome to Germany“.” In <i>Literatur und Demokratie</i>, edited by Alo Allkemper and Norbert Otto Eke, 285–300. Berlin: Erich Schmidt, 2000."},"publication":"Literatur und Demokratie","department":[{"_id":"465"}],"type":"book_chapter","place":"Berlin","date_created":"2020-09-15T05:46:14Z","date_updated":"2022-01-06T06:54:03Z","author":[{"last_name":"Eke","first_name":"Norbert Otto","full_name":"Eke, Norbert Otto","id":"104"}],"title":"Wahrnehmung im Augen-Schein. 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We show that many basic geometric properties have very efficient testing algorithms, whose running time is significantly smaller than the object description size.","lang":"eng"}],"publication":"Proceedings of the 8th Annual European Symposium on Algorithms (ESA'00)","doi":"10.1007/3-540-45253-2_15","series_title":"Lecture Notes in Computer Science","language":[{"iso":"eng"}],"date_updated":"2022-01-06T06:53:24Z","publication_status":"published","intvolume":"      4698","title":"Property Testing in Computational Geometry","year":"2000","publication_identifier":{"isbn":["9783540410041","9783540452539"],"issn":["0302-9743"]},"author":[{"full_name":"Czumaj, Artur","first_name":"Artur","last_name":"Czumaj"},{"last_name":"Sohler","first_name":"Christian","full_name":"Sohler, Christian"},{"full_name":"Ziegler, Martin","first_name":"Martin","last_name":"Ziegler"}],"place":"Berlin, Heidelberg","citation":{"ama":"Czumaj A, Sohler C, Ziegler M. Property Testing in Computational Geometry. In: <i>Proceedings of the 8th Annual European Symposium on Algorithms (ESA’00)</i>. Vol 4698. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer; 2000:155-166. doi:<a href=\"https://doi.org/10.1007/3-540-45253-2_15\">10.1007/3-540-45253-2_15</a>","bibtex":"@inproceedings{Czumaj_Sohler_Ziegler_2000, place={Berlin, Heidelberg}, series={Lecture Notes in Computer Science}, title={Property Testing in Computational Geometry}, volume={4698}, DOI={<a href=\"https://doi.org/10.1007/3-540-45253-2_15\">10.1007/3-540-45253-2_15</a>}, booktitle={Proceedings of the 8th Annual European Symposium on Algorithms (ESA’00)}, publisher={Springer}, author={Czumaj, Artur and Sohler, Christian and Ziegler, Martin}, year={2000}, pages={155–166}, collection={Lecture Notes in Computer Science} }","mla":"Czumaj, Artur, et al. “Property Testing in Computational Geometry.” <i>Proceedings of the 8th Annual European Symposium on Algorithms (ESA’00)</i>, vol. 4698, Springer, 2000, pp. 155–66, doi:<a href=\"https://doi.org/10.1007/3-540-45253-2_15\">10.1007/3-540-45253-2_15</a>.","short":"A. Czumaj, C. Sohler, M. Ziegler, in: Proceedings of the 8th Annual European Symposium on Algorithms (ESA’00), Springer, Berlin, Heidelberg, 2000, pp. 155–166.","chicago":"Czumaj, Artur, Christian Sohler, and Martin Ziegler. “Property Testing in Computational Geometry.” In <i>Proceedings of the 8th Annual European Symposium on Algorithms (ESA’00)</i>, 4698:155–66. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer, 2000. <a href=\"https://doi.org/10.1007/3-540-45253-2_15\">https://doi.org/10.1007/3-540-45253-2_15</a>.","apa":"Czumaj, A., Sohler, C., &#38; Ziegler, M. (2000). Property Testing in Computational Geometry. In <i>Proceedings of the 8th Annual European Symposium on Algorithms (ESA’00)</i> (Vol. 4698, pp. 155–166). Berlin, Heidelberg: Springer. <a href=\"https://doi.org/10.1007/3-540-45253-2_15\">https://doi.org/10.1007/3-540-45253-2_15</a>","ieee":"A. Czumaj, C. Sohler, and M. Ziegler, “Property Testing in Computational Geometry,” in <i>Proceedings of the 8th Annual European Symposium on Algorithms (ESA’00)</i>, 2000, vol. 4698, pp. 155–166."},"user_id":"15415","volume":4698,"page":"155-166","_id":"17990","publisher":"Springer","status":"public"},{"citation":{"short":"J. Fromme, S. Ganguin, S. Iske, D.M. Meister, U. Sander, eds., Medienbildung und Gesellschaft, VS Verlag, 2000.","chicago":"Fromme, J., S. Ganguin, S. Iske, Dorothee M. Meister, and U. Sander, eds. <i>Medienbildung und Gesellschaft</i>. VS Verlag, 2000.","apa":"Fromme, J., Ganguin, S., Iske, S., Meister, D. M., &#38; Sander, U. (Eds.). (2000). <i>Medienbildung und Gesellschaft</i>. VS Verlag.","ieee":"J. Fromme, S. Ganguin, S. Iske, D. M. Meister, and U. Sander, Eds., <i>Medienbildung und Gesellschaft</i>. VS Verlag, 2000.","ama":"Fromme J, Ganguin S, Iske S, Meister DM, Sander U, eds. <i>Medienbildung und Gesellschaft</i>. 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VS Verlag, 2000."},"date_created":"2020-08-21T13:20:57Z","department":[{"_id":"137"}],"type":"book_editor","publication_identifier":{"unknown":["ISSN: 2512-112X"]},"year":"2000","status":"public","title":"Medienbildung und Gesellschaft","publication_status":"published","date_updated":"2022-01-06T06:53:26Z","publisher":"VS Verlag","_id":"18121","language":[{"iso":"ger"}],"editor":[{"full_name":"Fromme, J.","last_name":"Fromme","first_name":"J."},{"first_name":"S.","last_name":"Ganguin","full_name":"Ganguin, S."},{"last_name":"Iske","first_name":"S.","full_name":"Iske, S."},{"id":"346","full_name":"Meister, Dorothee M.","first_name":"Dorothee M.","last_name":"Meister"},{"full_name":"Sander, U.","first_name":"U.","last_name":"Sander"}],"user_id":"21240"},{"user_id":"15415","volume":1963,"page":"450-458","_id":"18146","publisher":"Springer","status":"public","place":"Berlin, Heidelberg","citation":{"bibtex":"@inproceedings{Ziegler_Brattka_2000, place={Berlin, Heidelberg}, title={Computing the Dimension of Linear Subspaces}, volume={1963}, DOI={<a href=\"https://doi.org/10.1007/3-540-44411-4_34\">10.1007/3-540-44411-4_34</a>}, booktitle={SOFSEM 2000: Theory and Practice of Informatics}, publisher={Springer}, author={Ziegler, Martin and Brattka, Vasco}, year={2000}, pages={450–458} }","chicago":"Ziegler, Martin, and Vasco Brattka. “Computing the Dimension of Linear Subspaces.” In <i>SOFSEM 2000: Theory and Practice of Informatics</i>, 1963:450–58. Berlin, Heidelberg: Springer, 2000. <a href=\"https://doi.org/10.1007/3-540-44411-4_34\">https://doi.org/10.1007/3-540-44411-4_34</a>.","ama":"Ziegler M, Brattka V. Computing the Dimension of Linear Subspaces. In: <i>SOFSEM 2000: Theory and Practice of Informatics</i>. Vol 1963. Berlin, Heidelberg: Springer; 2000:450-458. doi:<a href=\"https://doi.org/10.1007/3-540-44411-4_34\">10.1007/3-540-44411-4_34</a>","short":"M. Ziegler, V. Brattka, in: SOFSEM 2000: Theory and Practice of Informatics, Springer, Berlin, Heidelberg, 2000, pp. 450–458.","ieee":"M. Ziegler and V. Brattka, “Computing the Dimension of Linear Subspaces,” in <i>SOFSEM 2000: Theory and Practice of Informatics</i>, 2000, vol. 1963, pp. 450–458.","mla":"Ziegler, Martin, and Vasco Brattka. “Computing the Dimension of Linear Subspaces.” <i>SOFSEM 2000: Theory and Practice of Informatics</i>, vol. 1963, Springer, 2000, pp. 450–58, doi:<a href=\"https://doi.org/10.1007/3-540-44411-4_34\">10.1007/3-540-44411-4_34</a>.","apa":"Ziegler, M., &#38; Brattka, V. (2000). Computing the Dimension of Linear Subspaces. In <i>SOFSEM 2000: Theory and Practice of Informatics</i> (Vol. 1963, pp. 450–458). Berlin, Heidelberg: Springer. <a href=\"https://doi.org/10.1007/3-540-44411-4_34\">https://doi.org/10.1007/3-540-44411-4_34</a>"},"doi":"10.1007/3-540-44411-4_34","language":[{"iso":"eng"}],"date_updated":"2022-01-06T06:53:26Z","publication_status":"published","intvolume":"      1963","year":"2000","title":"Computing the Dimension of Linear Subspaces","publication_identifier":{"isbn":["9783540413486","9783540444114"],"issn":["0302-9743"]},"author":[{"last_name":"Ziegler","first_name":"Martin","full_name":"Ziegler, Martin"},{"last_name":"Brattka","first_name":"Vasco","full_name":"Brattka, Vasco"}],"type":"conference","department":[{"_id":"63"}],"date_created":"2020-08-24T09:58:56Z","abstract":[{"text":"Since its very beginning, linear algebra is a highly algorithmic subject. Let us just mention the famous Gauss Algorithm which was invented before the theory of algorithms has been developed. The purpose of this paper is to link linear algebra explicitly to computable analysis, that is the theory of computable real number functions. Especially, we will investigate in which sense the dimension of a given linear subspace can be computed. The answer highly depends on how the linear subspace is given: if it is given by a finite number of vectors whose linear span represents the space, then the dimension does not depend continuously on these vectors and consequently it cannot be computed. If the linear subspace is represented via its distance function, which is a standard way to represent closed subspaces in computable analysis, then the dimension does computably depend on the distance function.","lang":"eng"}],"publication":"SOFSEM 2000: Theory and Practice of Informatics"},{"abstract":[{"lang":"eng","text":"What is the minimum number of hyperplanes that slice all edges of the d-dimensional hypercube? The answers have been known for d<=4.<br>This work settles the problem for d=5 and d=6. More precisely, a computer search implies that 4 hyperplanes do not suffice for this purpose (but 5 do).<br>We also develop computational approaches for attacking this extremal problem from combinatorial geometry in higher dimensions. They allow us to determine for example all maximal sliceable subsets of hypercube edges up to dimension 7."}],"citation":{"short":"M. Ziegler, C. Sohler, in: Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG’00), 2000, pp. 73–79.","chicago":"Ziegler, Martin, and Christian Sohler. “Computing Cut Numbers.” In <i>Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG’00)</i>, 73–79, 2000.","apa":"Ziegler, M., &#38; Sohler, C. (2000). Computing Cut Numbers. In <i>Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG’00)</i> (pp. 73–79).","ieee":"M. Ziegler and C. Sohler, “Computing Cut Numbers,” in <i>Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG’00)</i>, 2000, pp. 73–79.","ama":"Ziegler M, Sohler C. Computing Cut Numbers. In: <i>Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG’00)</i>. ; 2000:73-79.","bibtex":"@inproceedings{Ziegler_Sohler_2000, title={Computing Cut Numbers}, booktitle={Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG’00)}, author={Ziegler, Martin and Sohler, Christian}, year={2000}, pages={73–79} }","mla":"Ziegler, Martin, and Christian Sohler. “Computing Cut Numbers.” <i>Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG’00)</i>, 2000, pp. 73–79."},"publication":"Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG'00)","department":[{"_id":"63"}],"type":"conference","date_created":"2020-08-24T10:10:40Z","date_updated":"2022-01-06T06:53:26Z","author":[{"first_name":"Martin","last_name":"Ziegler","full_name":"Ziegler, Martin"},{"last_name":"Sohler","first_name":"Christian","full_name":"Sohler, Christian"}],"status":"public","title":"Computing Cut Numbers","year":"2000","user_id":"15415","language":[{"iso":"eng"}],"_id":"18150","page":"73-79"},{"citation":{"bibtex":"@article{Lorys_Wanka_Oesterdiekhoff_Kutylowski_2000, title={Periodification Scheme: Constructing Sorting Networks with Constant Period}, volume={45}, DOI={<a href=\"https://doi.org/10.1145/355483.355490\">10.1145/355483.355490</a>}, journal={Journal of the ACM}, author={Lorys, Krzysztof and Wanka, Rolf and Oesterdiekhoff, Brigitte and Kutylowski, Miroslaw}, year={2000}, pages={944–967} }","short":"K. Lorys, R. Wanka, B. Oesterdiekhoff, M. Kutylowski, Journal of the ACM 45 (2000) 944–967.","ama":"Lorys K, Wanka R, Oesterdiekhoff B, Kutylowski M. Periodification Scheme: Constructing Sorting Networks with Constant Period. <i>Journal of the ACM</i>. 2000;45:944-967. doi:<a href=\"https://doi.org/10.1145/355483.355490\">10.1145/355483.355490</a>","chicago":"Lorys, Krzysztof, Rolf Wanka, Brigitte Oesterdiekhoff, and Miroslaw Kutylowski. “Periodification Scheme: Constructing Sorting Networks with Constant Period.” <i>Journal of the ACM</i> 45 (2000): 944–67. <a href=\"https://doi.org/10.1145/355483.355490\">https://doi.org/10.1145/355483.355490</a>.","ieee":"K. Lorys, R. Wanka, B. Oesterdiekhoff, and M. Kutylowski, “Periodification Scheme: Constructing Sorting Networks with Constant Period,” <i>Journal of the ACM</i>, vol. 45, pp. 944–967, 2000, doi: <a href=\"https://doi.org/10.1145/355483.355490\">10.1145/355483.355490</a>.","mla":"Lorys, Krzysztof, et al. “Periodification Scheme: Constructing Sorting Networks with Constant Period.” <i>Journal of the ACM</i>, vol. 45, 2000, pp. 944–67, doi:<a href=\"https://doi.org/10.1145/355483.355490\">10.1145/355483.355490</a>.","apa":"Lorys, K., Wanka, R., Oesterdiekhoff, B., &#38; Kutylowski, M. (2000). Periodification Scheme: Constructing Sorting Networks with Constant Period. <i>Journal of the ACM</i>, <i>45</i>, 944–967. <a href=\"https://doi.org/10.1145/355483.355490\">https://doi.org/10.1145/355483.355490</a>"},"publication":"Journal of the ACM","abstract":[{"lang":"eng","text":"We consider comparator networks M that are used repeatedly: while the output produced by M is not sorted, it is fed again into M. Sorting algorithms working in this way are called periodic. The number of parallel steps performed during a single run of M is called its period, the sorting time of M is the total number of parallel steps that are necessary to sort in the worst case. Periodic sorting networks have the advantage that they need little hardware (control logic, wiring, area) and that they are adaptive. We are interested in comparator networks of a constant period, due to their potential applications in hardware design.\r\n\r\nPreviously, very little was known on such networks. The fastest solutions required time O(nε) where the depth was roughly 1/ε. We introduce a general method called periodification scheme that converts automatically an arbitrary sorting network that sorts n items in time T(n) and that has layout area A(n) into a sorting network that has period 5, sorts ***(n • T(n) items in time O(T(<n)• log n), and has layout area O(A(n)) • T(n)). In particular, applying this scheme to Batcher's algorithms, we get practical period 5 comparator networks that sort in time O(log3n). For theoretical interest, one may use the AKS netork resulting in a period 5 comparator network with runtime O(log2n)."}],"date_created":"2020-08-27T11:49:50Z","department":[{"_id":"63"}],"type":"journal_article","author":[{"first_name":"Krzysztof","last_name":"Lorys","full_name":"Lorys, Krzysztof"},{"last_name":"Wanka","first_name":"Rolf","full_name":"Wanka, Rolf"},{"full_name":"Oesterdiekhoff, Brigitte","last_name":"Oesterdiekhoff","first_name":"Brigitte"},{"last_name":"Kutylowski","first_name":"Miroslaw","full_name":"Kutylowski, Miroslaw"}],"title":"Periodification Scheme: Constructing Sorting Networks with Constant Period","year":"2000","status":"public","intvolume":"        45","date_updated":"2022-01-06T06:53:32Z","_id":"18446","language":[{"iso":"eng"}],"page":"944-967","volume":45,"doi":"10.1145/355483.355490","user_id":"15415"}]