Krzysztof Gdawiec
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Publikacje

Książki

Nauka programowania dla początkujących: podejście graficzne (okładka)
Domańska, D., Gdawiec, K.
Wydawnictwo UŚ, Katowice, (2017)
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Streszczenie. Książka jest łagodnym wprowadzeniem w świat programowania. Jak mówi chińskie przysłowie ,,Jeden obraz wart więcej niż tysiąc słów” dlatego w książce zastosowano podejście graficzne w nauce programowania. Czytelnik nauczy się pisania programów tworzących różne graficzne obiekty i wzory w języku Processing, który jest dialektem jednego z najpopularniejszych języków programowania — języka Java. Język Processing został stworzony na MIT (Massachusetts Institute of Technology) z myślą o artystach, dlatego nauka tego języka jest bardzo prosta i szybko można w nim tworzyć różne programy graficzne. W trakcie lektury książki Czytelnik pozna różne pojęcia i techniki programowania, np. zmienne, instrukcje warunkowe, pętle, tablice, funkcje, rekurencję. Na końcu każdego z rozdziałów znajdują się zadania do samodzielnego rozwiązania. Gdyby którekolwiek z zadań okazało się zbyt trudne, na końcu książki umieszczono przykładowe rozwiązania wszystkich zadań.
Książka skierowana jest zarówno do osób, które nigdy nie miały do czynienia z programowaniem i są zainteresowane poznaniem jego podstaw, jak i do osób, które programować umieją, ale chcą odświeżyć swoją wiedzę oraz chcą się zmierzyć z, prawdopodobnie nie poznanym przez nie dotychczas, językiem Processing. Jedynymi wymaganiami jakie postawiono na starcie przed Czytelnikiem to znajomość elementów szkolnej matematyki i chęć nauczenia się programowania.
Basics of Modelling and Visualization (cover)
Kotarski, W., Gdawiec, K., Machnik, G.T.
Basics of Modelling and Visualization
University of Silesia, Katowice, (2009)
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Abstract. This textbook presents basic concepts related to modelling and visualization tasks. Chapters 1-4 describe transformations in the plane and in the space, and geometrical forms of graphical objects such as curves, patches and fractals. Chapter 5 is about lights, materials, textures, colours that all are needed to enrich a severe appearance of pure geometrical objects leading to their photorealistic visualizations. In Chapter 6 freeware software such as POV Ray, MayaVi and Deep View are described. Using those software one can obtain photorealistic renderings and visualizations.
The textbook was prepared for students of the specialization ,,Modelling and Visualization in Bioinformatics'' but it should be helpful to anyone who is interested in computer graphics, modelling techniques, animation and visualization of data. Authors of this textbook believe that information presented in the book will be useful for students and will inspire their imagination in creation of photorealistic static 3D scenes and also will be helpful in creation of animations and visualization of data in an effective and professional way.

Czasopisma, materiały konferencyjne

Dynamics and Iterations
Gdawiec, K.
Procedural Generation of Aesthetic Patterns from Dynamics and Iteration Processes
International Journal of Applied Mathematics and Computer Science, (in press)
Abstract. Aesthetic patterns are widely used nowadays, e.g. in jewellery design, carpet design, as textures and patterns on wallpapers etc. Most of the work during the design stage is carried out by a designer manually. Therefore, it is highly useful to develop methods for aesthetic pattern generation. In this paper, we present methods for generating aesthetic patterns using the dynamics of a discrete dynamical system. The presented methods are based on the use of different iteration processes from fixed point theory (Mann, S, Noor etc.) and the use of an affine combination of these iterations. Moreover, we propose new convergence tests that enrich the obtained patterns. The proposed methods generate patterns in a procedural way and can be easily implemented on the GPU. The presented examples show that using the proposed methods we are able to obtain a variety of interesting patterns. Moreover, the numerical examples show that the use of the GPU implementation using shaders allows the generation of patterns in real time and the speed-up - compared to the CPU implementation - ranges from about 1000 to 2500 times.
Combined Root Finding
Gdawiec, K.
Nonlinear Dynamics 90(4), 2457-2479, (2017)
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Abstract. Fractal patterns generated in the complex plane by root finding methods are well known in the literature. In the generation methods of these fractals only one root finding method is used. In this paper, we propose the use of a combination of root finding methods in the generation of fractal patterns. We use three approaches to combine the methods: (1) the use of different combinations, e.g. affine and s-convex combination, (2) the use of iteration processes from fixed point theory, (3) multistep polynomiography. All the proposed approaches allow us to obtain new and diverse fractal patterns that can be used, for instance, as textile or ceramics patterns. Moreover, we study the proposed methods using five different measures: average number of iterations, convergence area index, generation time, fractal dimension and Wada measure. The computational experiments show that the dependence of the measures on the parameters used in the methods is in most cases a non-trivial, complex and non-monotonic function.
Polynomiography for MMP
Gdawiec, K., Kotarski, W.
Applied Mathematics and Computation 307, 17-30, (2017)
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Abstract. In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-Euler-Schroder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantari's recent results in finding the maximum modulus of a complex polynomial based on Newton's process with the Picard iteration to other MMP-processes with various non-standard iterations.
Shape Inversion with Iterations
Gdawiec, K.
Computer Graphics Forum 36(1), 35-45, (2017)
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Abstract. In this paper, we generalize the idea of star-shaped set inversion fractals using iterations known from fixed point theory. We also extend the iterations from real parameters to so-called q-system numbers and proposed the use of switching processes. All the proposed generalizations allowed us to obtain new and diverse fractal patterns that can be used, e.g., as textile and ceramics patterns. Moreover, we show that in the chaos game for iterated function systems - which is similar to the inversion fractals generation algorithm - the proposed generalizations do not give interesting results.
Switching in Polynomiography
Gdawiec, K.
Nonlinear Dynamics 87(4), 2235-2249, (2017)
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Abstract. Mandelbrot and Julia sets are examples of fractal patterns generated in the complex plane. In the literature we can find many generalizations of those sets. One of such generalizations is the use of switching process. In this paper we introduce some switching processes to another type of complex fractals, namely polynomiographs. Polynomiograph is an image presenting the visualization of the complex polynomial's root finding process. The proposed switching processes will be divided into four groups, i.e., switching of: the root finding methods, the iterations, the polynomials and the convergence tests. All the proposed switching processes change the dynamics of the root finding process and allowed us to obtain new and diverse fractal patterns.
Pseudoinversion fractals
Gdawiec, K.
Lecture Notes in Computer Science, vol. 9972, pp. 29-36, (2016)
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Abstract. In this paper, we present some modifications of inversion fractals. The first modification is based on the use of different metrics in the inversion transformation. Moreover, we propose a switching process between different metric spaces. All the proposed modifications allowed us to obtain new and diverse fractal patterns that differ from the original inversion fractals.
Polynomiography for Square Systems of Equations with Mann and Ishikawa Iterations
Gdawiec, K., Kotarski, W., Lisowska, A.
WSCG 2016 Short Papers Proceedings, pp. 1-5, (2016)
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Abstract. In this paper we propose to replace the standard Picard iteration in the Newton--Raphson method by Mann and Ishikawa iterations. This iteration's replacement influence the solution finding process that can be visualized as polynomiographs for the square systems of equations. Polynomiographs presented in the paper, in some sense, are generalization of Kalantari's polynomiography from a single polynomial equation to the square systems of equations. They are coloured based on two colouring methods: basins of attractions with different colours for every real root and colouring dependent on the number of iterations. Possible application of the presented method can be addressed to computer graphics where aesthetic patterns can be used in e.g. texture generation, animations, tapestry design.
Biomorphs via Modified Iterations
Gdawiec, K., Kotarski, W., Lisowska, A.
Journal of Nonlinear Science and Applications 9(5), 2305-2315, (2016)
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Abstract. The aim of this paper is to present some modifications of the biomorphs generation algorithm introduced by Pickover in 1986. A biomorph stands for biological morphologies. It is obtained by a modified Julia set generation algorithm. The biomorph algorithm can be used in the creation of diverse and complicated forms resembling invertebrate organisms. In this paper the modifications of the biomorph algorithm in two directions are proposed. The first one uses different types of iterations (Picard, Mann, Ishikawa). The second one uses a sequence of parameters instead of one fixed parameter used in the original biomorph algorithm. Biomorphs generated by the modified algorithm are essentially different in comparison to those obtained by the standard biomorph algorithm, i.e., the algorithm with Picard iteration and one fixed constant.
ICMMI2015
Gdawiec, K.
Advances in Intelligent Systems and Computing, vol. 391, pp. 499-506, (2015)
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Abstract. In the paper, a modification of rendering algorithm of polynomiograph is presented. Polynomiography is a method of visualization of complex polynomial root finding process and it has applications among other things in aesthetic pattern generation. The proposed modification is based on a perturbation mapping, which is added in the iteration process of the root finding method. The use of the perturbation mapping alters the shape of the polynomiograph, obtaining in this way new and diverse patterns. The results from the paper can further enrich the functionality of the existing polynomiography software.
Nonstandard Polynomiography
Gdawiec, K., Kotarski, W., Lisowska, A.
Abstract and Applied Analysis, vol. 2015, Article ID 797594, 19 pages, (2015)
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Abstract. In this paper a survey of some modifications based on the classic Newton's and the higher order Newton-like root finding methods for complex polynomials are presented. Instead of the standard Picard's iteration several different iteration processes, described in the literature, that we call as non-standard ones, are used. Kalantari's visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multi-parameter iterations do not destabilize the iteration process. Moreover, we obtain nicely looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.
SIO2014
Gdawiec, K., Kotarski, W., Lisowska, A.
Wielomianografia z niestandardową rodziną iteracji Eulera-Schrodera
Systemy Inteligencji Obliczeniowej. Uniwersytet Śląski, Katowice, pp. 75-85, (2014)
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Streszczenie. Wielomianografia, wprowadzona przez Kalantariego, jest to wizualizacja procesu rozwiązywania równań wielomianowych na płaszczyźnie zespolonej. Łączy w sobie dwa aspekty, matematyczny czyli rozwiązywanie równań z graficzną prezentacją tzw. wielomianografów. Z punktu widzenia grafiki komputerowej ten drugi aspekt wielomianografii jest ważniejszy, gdyż prowadzi do możliwości automatycznego generowania wzorów o walorach estetycznych. Celem pracy jest uogólnienie wielomianografii Kalantariego przez zastosowanie nowych wieloparametrowych schematów iteracyjnych w połączeniu z metodami wyższych rzędów z Rodziny Iteracji Eulera-Schrodera. Wielomianografy otrzymane w ten sposób istotnie poszerzają zbiór możliwych do wygenerowania stabilnych wzorów, które są znacząco różne od motywów generowanych za pomocą standardowej wielomianografii Kalantariego.
Shape Inversion
Gdawiec, K.
Fractals 22(4), 1450009, 7 pages, (2014)
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Abstract. In the paper, we generalized the idea of circle inversion to star-shaped sets and used the generalized inversion to replace the circle inversion transformation in the algorithm for the generation of the circle inversion fractals. In this way, we obtained the star-shaped set inversion fractals. The examples that we have presented show that we were able to obtain very diverse fractal patterns by using the proposed extension and that these patterns are different from those obtained with the circle inversion method. Moreover, because circles are star-shaped sets, the proposed generalization allows us to deform the circle inversion fractals in a very easy and intuitive way.
ICCVG2014
Gdawiec, K.
Lecture Notes in Computer Science, vol. 8671, pp. 25-32, (2014)
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Abstract. Polynomiography is a method of visualization of complex polynomial root finding process. One of the applications of polynomiography is generation of aesthetic patterns. In this paper, we present two new algorithms for polynomiograph rendering that allow to obtain new diverse patterns. The algorithms are based on the ideas used to render the well known Mandelbrot and Julia sets. The results obtained with the proposed algorithms can enrich the functionality of the existing polynomiography software.
WSCG2014
Gdawiec, K., Kotarski, W., Lisowska, A.
WSCG 2014 Poster Papers Proceedings, pp. 21-26, (2014)
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Abstract. In the paper visualizations of some modifications based on the Newton's root finding of complex polynomials are presented. Instead of the standard Picard iteration several different iterative processes described in the literature, that we call as non-standard ones, are used. Following Kalantari such visualizations are called polynomiographs. Polynomiographs are interesting from scientific, educational and artistic points of view. By the usage of different kinds of iterations we obtain quite new, comparing to the standard Picard iteration, polynomiographs that look aesthetically pleasing. As examples we present some polynomiographs for complex polynomial equation z^3 - 1 = 0. Polynomiographs graphically present dynamical behaviour of different iterative processes. But we are not interested in it. We are focused on polynomiographs from the artistic point of view. We believe that the new polynomiographs can be interesting as a source of aesthetic patterns created automatically. They also can be used to increase functionality of the existing polynomiography software.
Zakopane2013
Gdawiec, K., Kotarski, W., Lisowska, A.
Wielomianografia wyższych rzędów z iteracjami Manna i Ishikawy
Systemy Wspomagania Decyzji. Uniwersytet Śląski, Katowice, pp. 171-181, (2013)
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Streszczenie. Celem tego rozdziału jest przedstawienie modyfikacji wielomianografii wyższych rzędów uzyskanej poprzez zastąpienie standardowej iteracji Piccarda przez iterację Manna i Ishikawy. Wielomianografia, odkryta przez Kalantariego w 2000 roku, jest to wizualizacja procesu aproksymacji miejsc zerowych wielomianu zespolonego. Wielomianografia wiąże matematykę ze sztuką. Jest metodą, za pomocą której generuje się wzory o dużych walorach estetycznych. Zaproponowana w rozdziale modyfikacja wielomianografii prowadzi do istotnego poszerzenia zbioru możliwych do wygenerowania wzorów, które mogą być inspiracją dla grafików. Może ponadto rozszerzyć możliwości istniejącego oprogramowania do generowania wielomianografów.
CISIM2013
Gdawiec, K.
Lecture Notes in Computer Science, vol. 8104, pp. 358-366, (2013)
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Abstract. The aim of this paper is to present some modifications of the orbits generation algorithm of discrete dynamical systems. The first modification is based on introduction of a perturbation mapping in the standard Picard iteration used in the orbit generation algorithm. The perturbation mapping is used to alter the orbit during the iteration process. The second modification combines the standard Picard iteration with the iteration which uses the perturbation mapping. The obtained patterns have unrepeatable structure and aesthetic value. They can be used for instance as textile patterns, ceramics patters or can be used in jewellery design.
WSCG2013
Gdawiec, K.
WSCG 2013 Communication Proceedings, pp. 15-20, (2013)
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Abstract. The aim of this paper is to present a modification of the visualization process of finding the roots of a given complex polynomial which is called polynomiography. The name polynomiography was introduced by Kalantari. The polynomiographs are very interesting both from educational and artistic points of view. In this paper we are interested in the artistic values of the polynomiography. The proposed modification is based on the change of the usual convergence test used in the polynomiography, i.e. using the modulus of a difference between two successive elements obtained in an iteration process, with the tests based on distance and non-distance conditions. Presented examples show that using various convergence tests we are able to obtain very interesting and diverse patterns. We believe that the results of this paper can enrich the functionality of the existing polynomiography software.
Zakopane2012
Kotarski, W., Gdawiec, K., Lisowska, A.
Metody generowania estetycznych wzorów
Systemy Wspomagania Decyzji. Uniwersytet Śląski, Katowice, pp. 331-339, (2012)
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Streszczenie. W pracy przedstawiono wybrane metody generowania estetycznych wzorów za pomocą komputera. Do prezentacji wybrano trzy metody oparte na różnych podejściach: systemach dynamicznych, biomorfach oraz wielomianografii, które generują szerokie spektrum wzorów o dużych potencjalnych możliwościach ich praktycznego zastosowania. Wzory generowane automatycznie, na podstawie wybranych metod, mogą stanowić inspirację dla grafików komputerowych. Ponadto, metody te wzbogacone dodatkowo o formalne kryteria oceniające miarę estetyki generowanych wzorów takie jak: złożoność, symetrie, zwartość, spójność, wymiar fraktalny, mogą tworzyć podstawę systemu generującego automatycznie wzory o zadanych przez użytkownika parametrach estetycznych.
ISVC2012
Kotarski, W., Gdawiec, K., Lisowska, A.
Lecture Notes in Computer Science, vol. 7431, pp. 305-313, (2012)
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Abstract. The aim of this paper is to present some modifications of complex polynomial roots finding visualization process. In this paper Ishikawa or Mann iterations are used instead of the standard Picard iteration. Kalantari introduced the name polynomiography for that visualization process and the obtained images he called polynomiographs. Polynomiographs are interesting both from educational and artistic point of view. By the use of different iterations we obtain quite new polynomiographs that look aestheatically pleasing comparing to the ones from standard Picard iteration. As examples we present some polynomiographs for complex polynomial equation z^3 - 1 = 0, permutation and doubly stochastic matrices. We believe that the results of this paper can inspire those who may be interested in aesthetic patterns created automatically. They also can be used to increase functionality of the existing polynomiography software.
Gdawiec, K., Domańska, D.
Lecture Notes in Artificial Intelligence, vol. 7267, pp. 501-508, (2012)
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Abstract. The aim of this paper is to present a new method of two-dimensional shape recognition. The method is based on dependence vectors which are fractal features extracted from the partitioned iterated function system. The dependence vectors show the dependency between range blocks used in the fractal compression. The effectiveness of our method is shown on four test databases. The first database was created by the authors and the other ones are: MPEG7 CE-Shape-1PartB, Kimia-99, Kimia-216. Obtained results have shown that the proposed method is better than the other fractal recognition methods of two-dimensional shapes.
Gdawiec, K., Domańska, D.
International Journal of Applied Mathematics and Computer Science 21(4), 757-767, (2011)
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Abstract. One of the approaches in pattern recognition is the use of fractal geometry. The property of self-similarity of fractals has been used as a feature in several pattern recognition methods. All fractal recognition methods use global analysis of the shape. In this paper we present some drawbacks of these methods and propose fractal local analysis using partitioned iterated function systems with division. Moreover, we introduce a new fractal recognition method based on a dependence graph obtained from the partitioned iterated function system. The proposed method uses local analysis of the shape, which improves the recognition rate. The effectiveness of our method is shown on two test databases. The first one was created by the authors and the second one is the MPEG7 CE-Shape-1 PartB database. The obtained results show that the proposed methodology has led to a significant improvement in the recognition rate.
Zakopane2011
Gdawiec, K., Kotarski, W., Lisowska, A.
Automatyczne generowanie estetycznych wzorów za pomocą transformacji Gumowskiego-Miry
Systemy Wspomagania Decyzji. Uniwersytet Śląski, Katowice, pp. 219-226, (2011)
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Streszczenie. Celem niniejszej pracy jest przedstawienie sposobu użycia jednego z dyskretnych układów dynamicznych, tj. transformacji Gumowskiego-Miry, do automatycznego generowania estetycznych wzorów. Zaprezentowane zostaną również trzy algorytmy kolorowania otrzymanych wzorów. Przedstawione przykłady pokazują ogromne możliwości tworzenia niepowtarzalnych wzorów za pomocą zaprezentowanych algorytmów. Wygenerowane za pomocą zaproponowanego algorytmu wzory mogą zostać użyte jako wzory na tkaniny, ceramikę czy też jako podstawa do wykonania różnego rodzaju ozdób czy biżuterii.
ISVC2011
Gdawiec, K., Kotarski, W., Lisowska, A.
Lecture Notes in Computer Science, vol. 6939, pp. 691-700, (2011)
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Abstract. The aim of this paper is to present some modifications of the orbits generation algorithm of dynamical systems. The well-known Picard iteration is replaced by the more general one - Krasnosielskij iteration. Instead of one dynamical system, a set of them may be used. The orbits produced during the iteration process can be modified with the help of a probabilistic factor. By the use of aesthetic orbits generation of dynamical systems one can obtain unrepeatable collections of nicely looking patterns. Their geometry can be enriched by the use of the three colouring methods. The results of the paper can inspire graphic designers who may be interested in subtle aesthetic patterns created automatically.
Zakopane2010
Kotarski, W., Gdawiec, K., Lisowska, A.
Nieliniowe podziały i fraktale
Systemy Wspomagania Decyzji. Uniwersytet Śląski, Katowice, pp. 363-371, (2010)
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Streszczenie. W pracy przedstawia się uogólnienia techniki podziałów, których zastosowanie prowadzi do możliwości wygenerowania gładkich obiektów graficznych takich jak krzywe, płaty czy obiektów fraktalnych na podstawie początkowego zbioru punktów. Uogólnienia te idą w dwóch kierunkach. Pierwszy wprowadza parametr zespolony do podziału liniowego, zaś drugi nieliniowość za pomocą średnich innych niż średnia arytmetyczna. Omawiane uogólnienia podziałów w sposób istotny rozszerzają klasę obiektów graficznych, którą można wygenerować za pomocą liniowych podziałów. Podziały pozostają w ścisłym związku z metodami fraktalnymi, gdyż za ich pomocą i zadanego zbioru punktów kontrolnych definiuje się układy IFS stosowane do fraktalnego renderingu obiektów graficznych. W pracy jest również przedstawiony związek między podziałami i fraktalami. Wskazano ponadto na pewne zastosowania metody podziałów.
CACS2010
Kotarski, W., Gdawiec, K., Lisowska, A.
On Gumowski-Mira Aesthetic Superfractal Forms
Proceedings of The 2010 IRAST International Congress on Computer Applications and Computational Science, pp. 562-565, (2010)
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Abstract. Gumowski-Mira transform, in short GM, produces nice looking fractal forms that can be used to model ,,marine living creatures'' or aesthetic patterns useful for artistic design. Those original unrepeatable forms can inspire artistic design in jewellery such as pendants, necklaces, talismans. Moreover, GM can be a source of texture patterns for computer graphics and it suggests motives for fractal arts. We show that combination of GM with superfractals lead to enlarging a variety of fractal forms possible to create. Colours added to geometry enrich aesthetic appearance of superfractal forms generated with the help of GM.
Gdawiec, K.
Lecture Notes in Artficial Intelligence, vol. 6401, pp. 403-410, (2010)
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Abstract. From the beginning of fractal discovery they found a great number of applications. One of those applications is fractal recognition. In this paper we present some of the weaknesses of the fractal recognition methods and how to eliminate them using the pseudofractal approach. Moreover we introduce a new recognition method of 2D shapes which uses fractal dependence graph introduced by Domaszewicz and Vaishampayan in 1995. The effectiveness of our approach is shown on two test databases.
Zakopane2009
Kotarski, W., Gdawiec, K.
Proste i odwrotne schematy podziału
Systemy Wspomagania Decyzji. Uniwersytet Śląski, Katowice, pp. 229-238, (2009)
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CMS2009
Gdawiec, K., Kotarski, W., Lisowska, A.
Fractal Rendering of Arbitrary Catmull-Clark Surfaces
Computer Methods and Systems 2009, Kraków, pp. 401-406
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Abstract. In the paper we deal with the fractal rendering of arbitrary Catmull-Clark surfaces. To obtain Iterated Function System (IFS) needed for surface generation we use some facts about approximation of Catmull-Clark surface and fractal description of bicubic patches. First we approximate the given Catmull-Clark surface with bicubic B\'ezier patches and then for each patch we find corresponding IFS. In this way we obtain fractal description of the surface and therefore we can generate it fractally. Further, some examples of Catmull-Clark surfaces rendered fractally are also presented.
CMS2009
Kotarski, W., Gdawiec, K., Machnik, G.T.
Fractal Based Progressive Representation of 2D Contours
Computer Methods and Systems 2009, Kraków, pp. 407-412
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Abstract. In the paper we present a method, different from those presented in literature, for progressive representation of two dimensional contours. The method is based on fractal representation of a set of linear and quadratic curves that approximate a given contour. If one knows IFS for fractal rendering of every part of the contour, then using the set of all IFSs (the so-called PIFS) it is possible to generate that contour fractally. When starting iterations from a single points belonging to the segments of the contour in every iteration further points lying on the contour are generated. In every iteration number of points placed on the contour is doubling. So, the contour is presented progressively in higher and higher resolution showing gradually larger number of details.
Gdawiec, K.
Advances in Intelligent and Soft Computing, vol. 59, pp. 451-458, (2009)
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Abstract. One of approaches in pattern recognition is the use of fractal geometry. The property of the self-similarity of the fractals has been used as feature in several pattern recognition methods. In this paper we present a new fractal recognition method which we will use in recognition of 2D shapes. As fractal features we used Partitioned Iterated Function System (PIFS). From the PIFS code we extract mappings vectors and numbers of domain transformations used in fractal image compression. These vectors and numbers are later used as features in the recognition procedure using a normalized similarity measure. The effectiveness of our method is shown on two test databases. The first database was created by the author and the second one is MPEG7 CE-Shape-1PartB database.
Gdawiec, K.J.
IEEE Eurocon 2009, St. Petersburg, Russia, pp. 353-358
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Abstract. From the beginning of fractals discovery they found a great number of applications. One of those applications is fractal recognition. In this paper we introduce a fractal recognition method which is based on fractal description obtained from fractal image compression. Next, we present simple modification of this method and results of the tests.
IJPAM
Gdawiec, K.
International Journal of Pure and Applied Mathematics 50(3), 421-430, (2009)
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Abstract. The problem of fractal modeling is very simple when we know the mathematical description of a fractal. We just apply one of the well-known algorithms. The inverse problem of finding the mathematical description for given fractal is not so trivial and we do not know any general method to solve this problem. So there are several approaches to this problem e.g. via Bezier curves, fractal compression. In this paper we present automatic method for finding fractal description of 2D contours. Our algorithm uses fractal interpolation for this purpose. We also present some of practical examples.
Gdawiec, K., Kotarski, W.
Fraktalne rozpoznawanie obiektów dwuwymiarowych
Systemy Wspomagania Decyzji. Uniwersytet Śląski, Katowice, pp. 261-268 (2008)
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Streszczenie. Od momentu odkrycia fraktale znalazły szerokie zastosowania. Jednym z takich zastosowań jest wykorzystanie ich do rozpoznawania kształtów. W niniejszym artykule przedstawimy dwie metody fraktalnego rozpoznawania oparte o opis fraktalny powstający w wyniku fraktalnej kompresji obrazów. Następnie omówimy prostą modyfikację tych metod oraz wyniki przeprowadzonych badań.

Inne

SDJ
Domańska, D., Gdawiec, K.
Processing — a Language for Both Graphic Artists and Programmers
Software Developers Journal, (przyjęty)
Abstract. Nowadays, not only programmers need to create software. Artists also need to create programs because of their visions and the limitations of current software. In the article we will present a simple programming language which was developed especially for artists. The name of the language is Processing.
Curves as Fractals (example)
Gdawiec, K.
MathWorks, (2006)
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Fractals (example)
Gdawiec, K.
MathWorks, (2006)
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