DOCSLIB.ORG

Wavelets for Computer Graphics: a Primer Part 2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Wavelets for Computer Graphics: a Primer Part 2 Wavelets for Computer Graphics: A Primer Part 2 Eric J. Stollnitz Tony D. DeRose David H. Salesin University of Washington j 1 Introduction It is often convenient to put the different scaling functions § i(x) for a given level j together into a single row matrix, Wavelets are a mathematical tool for hierarchically decomposing ¨ j j ¢¥¢¥¢ j § (x) := [ § (x) (x)], functions. They allow a function to be described in terms of a coarse 0 mj © 1 overall shape, plus details that range from broad to narrow. Regard- less of whether the function of interest is an image, a curve, or a where mj is the dimension of Vj. We can do the same for the surface, wavelets provide an elegant technique for representing the wavelets: levels of detail present. j j ¢¥¢ ¢ j (x) := [ (x) © (x)], In Part 1 of this primer we discussed the simple case of Haar 0 nj 1 wavelets in one and two dimensions, and showed how they can be where nj is the dimension of Wj. Because Wj is the orthogonal com- used for image compression. In Part 2, we present the mathematical plement of Vj in Vj+1, the dimensions of these spaces satisfy mj+1 = theory of multiresolution analysis, then develop bounded-interval mj + nj. spline wavelets and describe their use in multiresolution curve and surface editing. The condition that the subspaces Vj be nested is equivalent to requir- ing that the scaling functions be re®nable. That is, for all j = 1, 2, ¤ there must exist a matrix of constants Pj such that 2 Multiresolution analysis ¨ ¨ j © 1(x) = j(x) Pj. (1) The Haar wavelets we discussed in Part 1 are just one of many bases that can be used to treat functions in a hierarchical fashion. In this In other words, each scaling function at level j 1 must be express- section, we develop a mathematical framework known as multires- ible as a linear combination of ª®nerº scaling functions at level j. © j j © 1 j j 1 olution analysis for studying wavelets [2, 11]. Our examples will Note that since V and V have dimensions m and m , respec- © j j j 1 continue to focus on the Haar basis, but the more general mathe- tively, P is an m m matrix (taller than it is wide). matical notation used here will come in handy for discussing other Since the wavelet space Wj © 1 is by de®nition also a subspace of Vj, wavelet bases in later sections. we can write the wavelets j © 1(x) as linear combinations of the scal- ¨ © ing functions j(x). This means there is an mj nj 1 matrix of con- Multiresolution analysis relies on many results from linear algebra. j Some readers may wish to consult the appendix in Part 1 for a brief stants Q satisfying ¨ review. © j 1(x) = j(x) Qj. (2) As discussed in Part 1, the starting point for multiresolution analysis is a nested set of vector spaces Example: In the Haar basis, at a particular level j there are mj = 2j scaling functions and nj = 2j wavelets. Thus, ¡ ¡£¢¤¢¥¢ 0 ¡ 1 2 V V V there must be re®nement matrices describing how the two scaling functions in V1 and the two wavelets in W1 can be As j increases, the resolution of functions in Vj increases. The basis made from the four scaling functions in V2: functions for the space Vj are known as scaling functions. 1 0 1 0 The next step in multiresolution analysis is to de®ne wavelet spaces. j j j+1 2 1 0 2 1 0 For each j, we de®ne W as the orthogonal complement of V in V . P = and Q = 0 1 0 1 This means that Wj includes all the functions in Vj+1 that are orthog- 0 1 0 1 onal to all those in Vj under some chosen inner product. The func- tions we choose as a basis for Wj are called wavelets. Remark: In the case of wavelets constructed on the un- bounded real line, the columns of Pj are shifted versions 2.1 A matrix formulation for re®nement of one another, as are the columns of Qj. One column therefore characterizes each matrix, so Pj and Qj are com- The rest of our discussion of multiresolution analysis will focus on © ¥ ¥ pletely determined by sequences ( ¥ ¤ , p 1, p0, p1, ) and wavelets de®ned on a bounded domain, although we will also refer © ¥ ¥ ( ¥ , q 1, q0, q1, ), which also do not depend on j. Equa- to wavelets on the unbounded real line wherever appropriate. In the j tions (1) and (2) therefore often appear in the literature as ex- bounded case, each space V has a ®nite basis, allowing us to use ma- pressions of the form trix notation in much of what follows, as did Lounsbery et al. [10] and Quak and Weyrich [13]. § § (x) = pi (2x i) ¦ Eric J. Stollnitz, Tony D. DeRose, and David H. Salesin. Wavelets for com- i § puter graphics: A primer, part 2. IEEE Computer Graphics and Applica- (x) = qi (2x i). tions, 15(4):75±85, July 1995. i These equations are referred to as two-scale relations for notation can also be used for the decomposition process outlined in scaling functions and wavelets, respectively. Section 2.1 of Part 1. Consider a function in some approximation space Vj. Let's assume Note that equations (1) and (2) can be expressed as a single equation we have the coef®cients of this function in terms of some scaling using block-matrix notation: function basis. We can write these coef®cients as a column matrix T ¢¥¢¥¢ j j j j ¨ ¨ © © ¡ ¡ j of values C = [c c © ] . The coef®cients c could, for ex- j 1 j 1 = Pj Qj . (3) 0 mj 1 i ample, be thought of as pixel colors, or alternatively, as the x- or 2 Example: Substituting the matrices from the previous ex- y-coordinates of a curve's control points in IR . ample into Equation (3) along with the appropriate basis Suppose we wish to create a low-resolution version Cj © 1 of Cj with functions gives a smaller number of coef®cients mj © 1. The standard approach for © j © 1 j 1 creating the m values of C is to use some form of linear ®lter- 1 0 1 0 ing and down-sampling on the mj entries of Cj. This process can be 1 1 1 1 2 2 2 2 1 0 1 0 expressed as a matrix equation § § § § § § [ ] = [ ] 0 1 0 1 0 1 2 3 0 1 0 1 j © 1 j j 0 1 0 1 C = A C (6) © where Aj is an mj 1 mj matrix of constants (wider than it is tall). It is important to realize that once we have chosen scaling func- j © 1 j tions and their re®nement matrices Pj, the wavelet matrices Qj are Since C contains fewer entries than C , this ®ltering process clearly loses some amount of detail. For many choices of Aj, it is somewhat constrained (though not completely determined). In fact, j © 1 ¨ since all functions in j © 1(x) must be orthogonal to all functions possible to capture the lost detail as another column matrix D , © © j © 1 j 1 j 1 computed by ¤¦¥ § § in (x), we know ¢ = 0 for all k and . k £ j © 1 j j To deal with all these inner products simultaneously, let's de®ne D = B C (7) © some new notation for a matrix of inner products. We will denote where Bj is an nj 1 mj matrix of constants related to Aj. The pair of ¨ © © © j © 1 j 1 j 1 j 1 ¥ ¥ j j ¤ § ¢ § by [ ¢ ] the matrix whose (k, ) entry is . £ £ k matrices A and B are called analysis ®lters. The process of splitting © Armed with this notation, we can rewrite the orthogonality condi- the coef®cients Cj into a low-resolution version Cj © 1 and detail Dj 1 tion on the wavelets as is called analysis or decomposition. ¨ © j © 1 j 1 j j ¥ [ ¢ ] = 0. (4) If A and B are chosen appropriately, then the original coef®cients £ © Cj can be recovered from Cj © 1 and Dj 1 by using the matrices Pj Substituting Equation (2) into Equation (4) yields and Qj from the previous section: © © ¨ ¨ j © 1 j j j j j 1 j j 1 ¥ [ ¢ ] Q = 0. (5) C = P C + Q D . (8) £ © Recovering Cj from Cj © 1 and Dj 1 is called synthesis or reconstruc- A matrix equation with a right-hand side of zero like this one is j j known as a homogeneous system of equations. The set of all pos- tion. In this context, P and Q are called synthesis ®lters. ¨ ¨ j © 1 j ¥ sible solutions is called the null space of [ ¢ ], and the £ 2 j Example: In the unnormalized Haar basis, the matrices A columns of Q must form a basis for this space. There are a multitude 2 of bases for the null space of a matrix, implying that there are many and B are given by: j ¨ different wavelet bases for a given wavelet space W . Ordinarily, we 1 1 1 0 0 uniquely determine the Qj matrices by imposing further constraints A2 = 2 0 0 1 1 © in addition to the orthogonality requirement given above. For exam- ¨ ple, the Haar wavelet matrices can be found by requiring the least 1 1 1 0 0 B2 = © number of consecutive nonzero entries in each column.
Load more

Recommended publications
  • Splines and Wavelets
    6 Splines and wavelets A fundamental aspect of our formulation is that the whitening operator L is nat- urally tied to some underlying B-spline function, which will play a crucial role in the sequel. The spline connection also provides a strong link with wavelets [UB03]. In this chapter, we review the foundations of spline theory and show how one can construct B-spline basis functions and wavelets that are tied to some specific oper- ator L. The chapter starts with a gentle introduction to wavelets that exploits the analogy with Legos blocks. This naturally leads to the formulation of a multiresolu- tion analysis of L2(R) using piecewise-constant functions and a de visu identification of Haar wavelets. We then proceed in Section 6.2 with a formal definition of our gen- eralized brand of splines—the cardinal L-splines—followed by a detailed discussion of the fundamental notion of Riesz basis. In Section 6.3, we systematically cover the first-order operators with the construction of exponential B-splines and wavelets, which have the convenient property of being orthogonal. We then address the the- ory in its full generality and present two generic methods for constructing B-spline basis functions (Section 6.4) and semi-orthogonal wavelets (Section 6.5). The pleas- ing aspect is that these results apply to the whole class of shift-invariant differential operators L whose null space is finite-dimensional (possibly trivial), which are pre- cisely those that can be safely inverted to specify sparse stochastic processes. 6.1 From Legos to wavelets It is instructive to get back to our introductory example of piecewise-contant splines in Chapter 1 (§1.3) and to show how these are naturally connected to wavelets.
    [Show full text]
  • A Suite on Wavelet Differentiation Algorithms
    Algorithm 929: A Suite on Wavelet Differentiation Algorithms MANI MEHRA and KAVITA GOYAL, Indian Institute of Technology Delhi A collection of the Matlab routines that compute the values of the scaling and wavelet functions (φ(x)and ψ(x) respectively) and the derivative of an arbitrary function (periodic or non periodic) using wavelet bases is presented. Initially, the case of Daubechies wavelets is taken and the procedure is explained for both collocation and Galerkin approaches. For each case a Matlab routine is provided to compute the differenti- ation matrix and the derivative of the function f (d) = D(d) f . Moreover, the convergence of the derivative is shown graphically as a function of different parameters (the wavelet genus, D and the scale, J) for two test functions. We then consider the use of spline wavelets. Categories and Subject Descriptors: G.1.2 [Numerical Analysis]: Approximation—Wavelets and fractals; G.1.8 [Numerical Analysis]: Partial Differential Equations—Multigrid and multilevel methods General Terms: Algorithms, Theory Additional Key Words and Phrases: Differentiation matrices, wavelets, wavelet methods (Galerkin and collocation), splines. ACM Reference Format: Mehra, M. and Goyal, K. 2013. Algorithm 929: A suite on wavelet differentiation algorithms ACM Trans. Math. Softw. 39, 4, Article 27 (July 2013), 28 pages. DOI: http://dx.doi.org/10.1145/2491491.2491497 1. INTRODUCTION The term differentiation matrix D(d) denotes the transformation between grid point values of a function and its d-th order derivative. In the case of the Galerkin ap- proach [Canuto et al. 1988] this matrix is the product of three matrices, that is, D(d) = C−1 D(d)C.
    [Show full text]
  • On Spline Wavelets
    On Spline Wavelets Dedicated to Professor Charles Chui on the occasion of his 65th birthday Jianzhong Wang Abstract. Spline wavelet is an important aspect of the constructive theory of wavelets. This paper consists of three parts. The ¯rst part surveys the joint work of Charles Chui and me on the construction of spline wavelets based on the duality principle. It also includes a discussion of the computational and algorithmic aspects of spline wavelets. The second part reviews our study of asymptotically time- frequency localization of spline wavelets. The third part introduces the application of Shannon wavelet packet in the sub-band decompo- sition in multiple-channel synchronized transmission of signals. x1. Introduction In this paper, I give a brief survey of my joint work with Charles Chui on spline wavelet. In the theory of wavelet analysis, an important aspect is to construct wavelet bases of a function space, say of the Hilbert space L2(R) ([32], [64], [65]). Recall that a (standard) wavelet basis of L2(R) is the basis generated by the dilations and translates of a \wavelet" function Ã: More precisely, if the set of functions j=2 j fÃj;kgj;k2Z ;Ãj;k = 2 Ã(2 x ¡ k); (1) forms an unconditional basis (also called a Riesz basis) of L2(R); then we call à a wavelet and call the system (1) a wavelet basis. Three of most basic properties of a wavelet are its regularity, its decay speed, and the order of its vanishing moments. Therefore, a wavelets is usually chosen to be compactly supported (or exponentially decay) and to have certain degree of regularity so that they are \local" in both x-domain (time domain) and !-domain (frequency domain).
    [Show full text]
  • Construction of Biorthogonal B-Spline Type Wavelet Sequences with Certain Regularities
    Construction of Biorthogonal B-spline Type Wavelet Sequences with Certain Regularities Tian-Xiao He∗ Department of Mathematics and Computer Science Illinois Wesleyan University, Bloomington, IL 61702-2900, USA [email protected] Abstract In this paper we study a refinement equation of the form φ(x) = P n∈Z hnφ(2x − n), where {hn} is a finitely supported sequence. 1 P n Let the symbol of the sequnce m(z) := 2 n hnz take the form 1+z N k0 Pk j P P j m(z) = ( 2 ) z j=0 ajz where j aj = 1 and j(−1) aj 6= P 2 P 2 0. Under the assumption 0≤odd j≤k |aj| + 0≤even j≤k |aj| 2N−1 < 2 we show that the corresponding φ is in L2. Then the B-spline type wavelet sequences that possess the largest possible regularities and required vanishing moments are characterized. AMS Subject Classification: 39A70, 41A80, 65B10. Key Words and Phrases: symbolic summation operator, power series, generating function, Euler’s series transform. 1 Introduction We start by setting some notations. We define a low-pass filter as ∗The author would like to thank the Illinois Wesleyan University for a sabbatical leave during which the research in this paper was carried out. 1 2 Tian-Xiao He −1 X −inξ m0(ξ) = 2 cne . (1.1) n Here, we assume that only finitely many cn are nonzero. However, some of our results can be extended to infinite sequences that have sufficient decay for |n| → ∞. Next, we define φ by ˆ ∞ −j φ(ξ) = Πj=1m0(2 ξ).
    [Show full text]
  • A BRIEF STUDY on CARDINAL B- SPLINE WAVELETS Rupali Gupta1, Gagan Batra2 1,2Mathematics Department, Khalsa College for Women, Ludhiana, Punjab (India)
    A BRIEF STUDY ON CARDINAL B- SPLINE WAVELETS Rupali Gupta1, Gagan Batra2 1,2Mathematics Department, Khalsa College For Women, Ludhiana, Punjab (India) ABSTRACT This paper focuses on the concepts of Cardinal B-Spline wavelets, their Properties and description of their relations. Spline wavelets are extremely regular and usually symmetric or anti-symmetric. They can be designed to have compact support and to achieve optimal time-frequency localization (B-spline wavelets). The underlying scaling functions are the B-splines, which are the shortest and most regular scaling functions of order L which will be further classified into linear, quadratic or biquadratic spline wavelets. Their graphical representations are enclosed herein. Keywords: Compact Structures, Control points, Interpolation,Orthonormal, Spline wavelets. I. INTRODUCTION In this paper, we try to visualize the basic concepts of B- spline cardinal wavelets and their properties[1]. A spline wavelet is a wavelet constructed using a spline functionwhich is basically orthogonal, but do not have any compact supports. The interpolatory spline wavelets were introduced by C.K. Chui and J.Z. Wang. There is a certain class of wavelets, unique in some sense, having compact supports with some special properties. These special wavelets are called B-spline wavelets or cardinal B-spline wavelets. A Basis spline, often called as B-spline is a spline function that has the minimum support with respect to some given degree, sleekness, and domain partition. All spline functions of a given degree can be expressed as Linear combination[6] of B-splines of that degree. Due to this reason, B-splines of order ‘n’ are basis functions for spline functions of the same order defined over the same knots.
    [Show full text]
  • 4.4 Basic Spline Wavelet Transform
    Basic Spline Wavelet Transform and Pitch Detection of a Speech Signal by Ken Wenchian Lee Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degrees of Bachelor of Science and Master of Science at the Massachusetts Institute of Technology May 1994 © 1994 Ken Wenchian Lee. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part. // Signature of Author . .... ... .. o d ;... .. .. V1~ 4'.7 o. Department of Electrical Engineering and Computer Science Certified by ...................... r...L. Yr......... .................................- Professor Peter Elias Department of Electrical Engineering and Computer Science Thesis Supervisor (Academic) Certified by .......................................; ..... ......... ... ......................... Dr. Forrest Tzeng o-\ C Supervisor(COMSAT Laboratories) I . Accepted by ............... ·J.......... ............. deric R. Morgenthaler on Graduate Students Basic Spline Wavelet Transform and Pitch Detection of a Speech Signal by Ken Wenchian Lee Submitted to the Department of Electrical Engineering and Computer Science on January 28, 1994, in Partial Fulfillment of the Requirements for the Degrees of Bachelor of Science and Master of Science Abstract In recent years the wavelet transform and its applications have gained tremendous interest and have come under intense investigation. Successful applications of the transform are inherently dependent on the wavelets themselves, whose properties necessary for a specific application are not well known. In this thesis, besides examining wavelet theory, we study the wavelet issues important in the accurate pitch detection of a speech signal in no noise, low noise, and high noise environments. To have control over the properties of the wavelets, we employ a transform based on wavelets built from a sum of shifted B-spline functions.
    [Show full text]
  • Non-Equispaced B-Spline Wavelets Is Concluded by the Short Section 3.5 on the Non-Decimated B-Spline Wavelet Transform
    Non-equispaced B-spline wavelets Version with detailed proofs Maarten Jansen Universit´elibre de Bruxelles Departments of Mathematics and Computer Science October 20, 2016 Abstract This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non- equispaced knots. The new construction extends the equispaced, biorthog- onal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the con- text of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolatingwavelets on non-equidistant data. Finally, the third con- tribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non- orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet trans- forms on irregular point sets with a focus on their use for smoothing or other applications in statistics. Keywords arXiv:1502.01533v2 [math.ST] 19 Oct 2016 B-spline; wavelet; Cohen-Daubechies-Feauveau; penalised splines; non-equispaced; non-equidistant; lifting AMS classification 42C40; 65T60; 65D07; 41A15; 65D10 1 1 Introduction Ever since the early days of wavelet research, spline wavelets have enjoyed special attention in the community.
    [Show full text]
  • Wavelet Bases
    VII Wavelet Bases One can construct wavelets ψ such that the dilated and translated family 1 t 2jn ψj,n(t)= ψ − j j √2 2 (j,n) Z2 ! " #$ ∈ is an orthonormal basis of L2(R). Behind this simple statement lie very different point of views which open a fruitful exchange between harmonic analysis and discrete signal processing. j j Orthogonal wavelets dilated by 2 carry signal variations at the resolution 2− . The construction of these bases can thus be related to multiresolution signal approximations. Following this link leads us to an unexpected equivalence between wavelet bases and conjugate mirror filters used in discrete multirate filter banks. These filter banks implement a fast orthogonal wavelet transform that requires only O(N) operations for signals of size N. The design of conjugate mirror filters also gives new classes of wavelet orthogonal bases including regular wavelets of compact support. In several dimensions, wavelet bases of L2(Rd) are constructed with separable products of functions of one variable. Wavelet bases are also adapted to bounded domains and surfaces with lifting algorithms. 7.1 Orthogonal Wavelet Bases Our search for orthogonal wavelets begins with multiresolution approximations. For f L2(R), the + ∈ partial sum of wavelet coefficients ∞ f,ψ ψ can indeed be interpreted as the difference n= $ j,n% j,n between two approximations of f at the−∞ resolutions 2 j+1 and 2 j. Multiresolution approximations % − − compute the approximation of signals at various resolutions with orthogonal projections on different spaces Vj j Z. Section 7.1.3 proves that multiresolution approximations are entirely characterized by a particular{ } ∈ discrete filter that governs the loss of information across resolutions.
    [Show full text]
  • Spline Wavelets in Numerical Resolution of Partial Differential
    Spline Wavelets in Numerical Resolution of Partial Differential Equations Jianzhong Wang1 Department of mathematics, Computer Sciences, and Statistics Sam Houston State University Huntsville, Texas, USA Abstract. We give a review of applications of spline wavelets in the resolution of partial differential equations. Two typical methods for numerical solutions of partial dif- ferential equations are Galerkin method and collocation method. Corresponding to these two methods, we present the constructions of semi-orthogonal spline wavelets and semi- interpolation spline wavelets respectively. We also show how to use them in the numerical resolution of various partial differential equations. 1 Introduction In this paper, we show how wavelets, particularly the spline wavelets, can be used in the nu- merical resolution of partial differential equations (PDE’s). To solve a PDE numerically, we first need to find a finite-dimensional approximation space for the solutions, then discretize the PDE to a system of algebraic equations in this space so that the numerical solutions can be obtained. In many cases, the wavelets provide better bases of the approximation spaces than other bases in the following sense. First, the representations of the differen- tial operators are almost diagonal on the wavelet bases, that improves the conditioning of the discrete algebraic equations. Second, the wavelet representations are effective in the adaptive procedures so that the complexity of calculations can be reduced. Furthermore, when the solution has a certain singularity, its wavelet representation can automatically capture the singularity. Hence, the advantage of wavelet methods in numerical resolution of PDE’s is quite significant. However, in general, the wavelets do not have a localization as sharp as finite element method or finite difference method.
    [Show full text]
  • Wavelet Transforms Generated by Splines
    Wavelet transforms generated by splines Amir Z. Averbuch Valery A. Zheludev School of Computer Science Tel Aviv University Tel Aviv 69978, Israel Abstract In this paper we design a new family of biorthogonal wavelet transforms that are based on polynomial and discrete splines. The wavelet transforms are constructed from various types of interpolatory and quasi-interpolatory splines. The transforms use finite and infinite impulse response filters and are implemented in a fast lifting mode. We analyze properties of the generated wavelets. We describe successful applications of the designed transforms to still image compression. 1 Introduction In this paper we describe a new generic technique for the design of biorthogonal wavelet transforms. Some of the results that are presented in this paper have already appeared in [6]–[9]. However, this paper contains a unified theory that combines a full theoretical justifi- cations of the previous results with new facts about the spline-based wavelet transforms. The developed technique enables us to construct a wide family of transforms with various prop- erties. It supports flexible adaptation of the transforms to the problems under consideration. In particular, the newly designed transforms prove to be efficient for distinct computational problems such as image compression, feature extraction for signal identification, to name a few. The performance of the suggested transforms for still image compression is similar to the performance of the transform with 9/7 wavelets ([4]) on most of the benchmark images. Our 1 approach combines custom-design capabilities which are inherent in the lifting schemes [46] with the usage of the well-developed theory of interpolatory, quasi-interpolatory, continuous and discrete splines [34, 43, 55, 56].
    [Show full text]