Volume 29, issue 5, september 2009, pages 12511266. Madych, multiresolution analysis, haar bases, and self. The empire problem in penrose tilings computer science. Estimation of image statistics using generalized selfsimilar. Apr 01, 2014 read construction of wavelet sets using integral self affine multitiles, journal of fourier analysis and applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Abstract references similar articles additional information.
We investigate the connection between radix representations for and selfaffine tilings of. This generalises thurstons concept of a galoisdual tiling. In this paper, the properties of multiresolution analysis on the product of heisenberg group are investigated. Given a selfaffine periodic tiling of r n we construct an associated rregular multiresolution analysis and wavelet basis with the same lattice of translations and scaling matrix as the tiling. The use of generalized self similar tilings and nonseparable haar wavelets for characterization of image statistics over irregularly shaped regions is detailed.
Given a self affine periodic tiling of r n we construct an associated rregular multiresolution analysis and wavelet basis with the same lattice of translations and scaling matrix as the tiling. Wikimedia commons has media related to selfdual tilings pages in category self dual tilings the following pages are in this category, out of total. Self similar symmetric planar tilings 3 the right hand side. Introduction the mathematical literature provides several notions of selfsimilar tilings, which. Multiresolution analysis, haar bases and selfsimilar tilings. Construction of wavelet sets using integral selfaffine. Madych, multiresolution analysis, haar bases, and selfsimilar tilings, part 2. Haar bases and selfsimilar tilings orn, ieee trans. Taking into account the self similarity relation 2.
Wang, multiresolution analysis, selfsimilar tilings and haar wavelets on the heisenberg group, to appear in acta mathematica scientia series b. Selfsimilar lattice tilings and subdivision schemes. Wavelets, filter banks, and arbitrary tilings of the time. Multiresolution analysis, selfsimilar tilings and haar. The concept of starduality is described for selfsimilar cutandproject tilings in arbitrary dimensions. For example, each basis consists of appropriate dilates and translates of a finite collection of piecewise constant functions.
Abstract the purpose of this paper is to give the flavor of the subject of self similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings. We establish the calderon reproducing formula for functions in on the heisenberg group. Haarlike features with optimally weighted rectangles for. D 1 is the exceptional case in the list in that it generates a selfa ne tile but does not generate a number system, and this is the case we focus on. Madych, multiresolution analysis, haar bases and self similar tilings orn, ieee trans. Simple summation formulas for exponentials of the corresponding. Keywordstilings, fourier analysis, selfaffine tilings. Until now, all the operatoralgebraic machinery used to study tilings and tiling. Dynamics of selfsimilar tilings ergodic theory and. We illustrate the construction with examples having quadratic pisotvijayaraghavan units like. In particular, we will demonstrate the existence of such sequences for the koch snowflake fractal billiard table, a self similar sierpinski carpet billiard table and the tfractal billiard table. Tube formulas and complex dimensions of selfsimilar tilings. Regarding lattice tilings of t, it is known now 4 and quite hard to prove that, whenever d is a standard digit set, i. If the union of the latter ones is similar to the original tile, then the substitution is called self similar substitution.
In dimension two we further prove the existence of connected selfsimilar lattice tilings for parabolic and elliptic dilations. Multiresolution analysis, haar bases and wavelets on. Annairizi of arabias proof of the pythagorean theorem. Requirements for uniqueness of the image representation are presented and richness of the class of tiles is explored. In this paper, we generalize their results to the integral selfaffine multitiles setting. Author links open overlay panel heping liu a yu liu b haihui wang c. The dual tilings of the penrose tilings as well as the ammannbeenker tilings are calculated. We nd it easier to identify r2 with the complex plane c in order to simplify computations. By repetition we obtain a subdivision into arbitrary small triangles or rhombs. In this talk, we will present examples of sequences of compatible periodic orbits of prefractal billiard tables. Gabardo and yu first considered using integral selfaffine tiles in the fourier domain to construct wavelet sets and they produced a class of compact wavelet sets with certain selfsimilarity properties.
Multiresolution analysis, haar bases, selfsimilar tilings of r 551 the classical haar system defined by 1 is the simplest example of such a basis for l2r. For example, each basis consists of appropriate dila. An image representation based on a generalization of quad trees, related to the definition of haar wavelets and the resulting multiresolution decomposition of l 2 r n is presented. The construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of self similar tilings. Let us consider a similar puzzle, where the set of. The prescale function will satisfy a dilation equation and its lattice translates will. In this article, the properties of multiresolution analysis and self similar tilings on the heisenberg group are studied. A substitution at least here consists of rules how to enlarge a tile and replace the enlarged tile with other tiles.
Ams proceedings of the american mathematical society. The discrete part of the diffraction pattern of selfsimilar tilings, called the bragg spectrum. The ones marked may be different from the article in the profile. In this case m 1, i z, and a is the dyadic dilation ax 2x. Selfsimilar lattice tilings and subdivision schemes article in siam journal on mathematical analysis 331 june 2001 with 14 reads how we measure reads. None of the tilings are periodic, yet each is quasiperiodic. Andrew vince selfsimilar polygonal tilings a tiling t isselfsimilarif there is a similarity transformation r 2.
Tilings for pisot betanumeration mathematics tu graz. We apply our results to show that haarlike multivariable wavelets exist for. Each tiling is constructed by the same method from a single self similar polygon. Latticetiling properties of integral selfaffine functions. Figure 1 shows an example for n 4 using distinctly shaped. Ams transactions of the american mathematical society. The diffraction pattern of selfsimilar tilings centre. Moreover, we establish a theory to construct an orthonormal haar wavelet base in l2.
R 2 such that, for for every t 2t, the larger tile. Multiresolution analysis, selfsimilar tilings and haar wavelets on the heisenberg group author links open overlay panel heping liu a yu liu b haihui wang c show more. We investigate the connection between radix representations for and self affine tilings of. Moreover the properties of haar wavelet bases for are investigated.
We apply our results to show that haar like multivariable wavelets exist for all dilation matrices that are sufficiently large. That is, the n shapes can be assembled in n different ways so as to create larger copies of themselves, where the increase in scale is the same in each case. Let t be a selfsimilar tiling of the plane r2 for which the expansion map. A tiling based on sporadic generating pair a in figure. Walker, a primer on wavelets and their scientific applications, crc press, 1999 k. For example, each basis consists of appropriate dilates and translates of a finite collection of piecewise. From looking at the set of tiles and the region we wish to cover, it is not always clear whether such a task is even possible. Veerman, pitman research notes in mathematics 362, ed. If you are mathematically inclined, no doubt you see some mathematics in the tilings. On ir n, this problem was inten sively studied by grochenig and madych 7 see also 6. This cited by count includes citations to the following articles in scholar.
Abstractorthonormal bases for lz r are constructed that have properties that are similar to those enjoyed by the classical. Nonperiodic tilings of the plane exhibit no translational symmetry. Pisot substitutions and the hausdorff dimension of boundaries of atomic surfaces feng, dejun, furukado, maki, and ito, shunji, tsukuba. Orthonormal bases for l 2 r n are constructed that have properties that are similar to those enjoyed by the classical haar basis for l 2 r. The construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of selfsimilar tilings. On disklike selfaffine tiles arising from polyominoes gmainer, johannes and thuswaldner, jorg m.
Multiresolution analysis, selfsimilar tilings and haar wavelets on the heisenberg group. Each tiling is constructed by the same method from a single selfsimilar polygon. Multiresolution analysis and haar wavelets on the laguerre. Integral selfaffine functions let a e mdz be a d x d nonsingular integer matrix which is also expanding, i. Creating a cascade of haarlike classifiers step by step. A selftiling tile set, or setiset, of order n is a set of n shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of n shapes.
This geometric zeta function encodes scaling and curvature properties related to. Starting from the haar case, one could be attempted to explore less trivial ex. Moreover, we establish a theory to construct an orthonormal haar wavelet base in l 2. Corresponding tilings are determined by twoletter sturmian substitution sequences. The empire problem in penrose tilings by laura e ngerdean duane bailey, advisor a thesis submitted in partial ful llment of the requirements for the degree of bachelor of arts with honors in computer science williams college williamstown, massachusetts may 8, 2006. In the selfsimilar case for example when the cantor set is the transversal of a. Madych, multiresolution analysis, haar bases, and selfsimilar tilings of r n, ieee trans. One is through a multiresolution analysis of generated by a compactly supported prescale function. We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for s existence in terms of the weights wavelets in can be generated in several ways. Penrose tilings are a remarkable class of nonperiodic tilings for which the set of prototiles consists of just two shapes. Multifractal analysis of selfsimilar measures with overlap it is well known that the multifractal spectrum of a selfsimilar measure satisfying the open set condition is a closed interval. Wang, multiresolution analysis, selfsimilar tilings and haar wavelets on.
Multiresolution analysis, haar bases and selfsimilar tilings of r n grochenig, k. Transverse laplacians for substitution tilings iecl. Download pdf save cite this item table of contents. We characterize some analytic properties of integral selfaffine multitiles under certain. In the twodimensional plane, any selfaffine convex body is a polygon with at most five vertices 19. Selfsimilarity a selfsimilar system is fjgj j1, where each j is a contractive similarity mapping, i. In this article, the properties of multiresolution analysis and selfsimilar tilings on the heisenberg group are studied. Topological properties of a class of cubic rauzy fractals loridant, benoi t, osaka journal of mathematics, 2016. Toward discrete wavelets with irrational scaling factor.
The main focus is on spectral properties of such systems which are shown to be uniquely ergodic. A substitution tiling is called selfsimilar, if it can be generated by a selfsimilar substitution. The standard example in the plane is a square, or a parallelogram, or a triangle, with. The classical calderon reproducing formula reads where, and denotes the convolution on the calderon reproducing formula is a useful tool in pure and applied mathematics see 14, particularly in wavelet theory see 5, 6. Crystallographic haar wavelets, journal of fourier. The calderon reproducing formula associated with the. Haarlike features are shown with the default weights assigned to its rectangles. Selfsimilar polygonal tiling people university of florida. Selfsimilar tilings of nilpotent lie groups james rohal. The basic haar theory is presented, and then generalized to the case of multiple dilation matrices, allowing for the definition of almost arbitrary tilings.
Abstract the purpose of this paper is to give the flavor of the subject of selfsimilar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings. Recent work has made it clear that the design of multirate filter banks for signal. Affine transformations, haar wavelets, and image representation. They are also closely related to selfsimilar and selfaffine tilings of space 6, 7. This version of the notes was created on september 8, 2014. Moreover, we establish a theory to construct an orthonormal haar wavelet. We have already seen an example in section 1the penrose tiling with rhombic tiles and the. Pdf multiresolution analysis and haar wavelets on the laguerre. A self tiling tile set, or setiset, of order n is a set of n shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of n shapes. Haar bases and selfsimilar tilings of rn, ieee trans.