Exponential b splines scale space and wavelet representations

Exponential B-Splines: Scale-Space and Wavelet Representations Lor Choon Yee B.Sc.(Hons), NTU A Thesis Submitted for the Degree of Master of Science Department of Mathematics National University of Singapore 2012 i Abstract Exponential B-splines have been well studied in the context of interpolation and approximation theory as a generalization of the classical polynomial B-splines. The focus of this thesis is on two new aspects, namely the scale-space and wavelet representations by exponential B-splines. First, the concept of elementary symmetric polynomials is introduced in order to express derivatives of exponential B-spline in terms of lower order ones. An exponential B-spline is known to satisfy a refinement equation. This refinable property is utilized to formulate the exponential B-spline scale-space representation at rational scale. A simple algorithm for scale-space computation is presented together with some numerical examples. This algorithm can be modified for computation of wavelet transforms defined by derivatives of exponential B-splines. Finally, a semi-discrete wavelet representation for the difference of exponential B-spline scale-space at two consecutive dyadic scales is analyzed. ii Acknowledgements First of all, I would like to express my gratitude to my supervisor Professor Lee Seng Luan for his kind guidance and insightful discussion throughout the course of this thesis. I would like to thank all the lecturers who have taught me during my graduate and undergraduate studies. Special thank goes to Assoc. Prof. Goh Say Song for his advice and encouragement during my study in NUS. I would also like to thank Assoc. Prof. Victor Tan for mentoring my teaching assignment as graduate tutor. I am grateful to all my friends and coursemates for their comradeship throughout the years. Special thanks go to Dr. Lim Kay Jin and Dr. Ku Cheng Yeaw for their companion during my days in NUS. May all of you be successful in your future endeavours. Last but not least, I shall reserve my greatest gratitude to my mother for her compact support. Modifying Pierre de Fermat’s famous quote, “the margin here is too small to contain all my gratitude and love for her.” Lor Choon Yee May 2012 iii Contents Abstract ii Acknowledgements iii Contents iv Summary vi List of Figures viii List of Notations ix 1 Preliminaries 1 1.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 B-Splines 7 1.3 Elementary Symmetric Polynomials . . . . . . . . . . . . . . . . . . . 10 1.4 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 2 Exponential B-splines 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Refinement Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Scale-Space Representation 37 3.1 Introducing Scale-Space . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Exponential B-spline Scale-Space . . . . . . . . . . . . . . . . . . . . 40 3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Wavelet Representation 48 4.1 Continuous Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . 48 4.2 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Semi-Discrete Wavelet Representation . . . . . . . . . . . . . . . . . . 57 5 Concluding Remarks 63 Bibliography 65 A Proof of Theorem 2.1.1 68 v Summary In this thesis, we consider a special class of splines, called the exponential B-splines, which are well-developed from theoretical point of view as generalization of classical B-splines, but have yet to be fully studied in the context of scale-space and the wavelet transforms. The new results in this thesis, to the knowledge of the author, are labelled as Propositions. This thesis is organized as follows. Chapter 1 collects some preliminary materials that will be used throughout the thesis. We state, without proof, some fundamental results in Fourier analysis and (polynomial) B-splines. In addition, we give a quick review of elementary symmetric polynomials and diffferential operators which arise naturally as we generalize the results of polynomial B-splines to exponential B-splines. In Chapter 2, we formally introduce the exponential B-splines with real parameters. We give an explicit formula for exponential B-splines of arbitrary order in Section 2.1 with a complete proof in Appendix A. This corrects the formula given in page 4 of [3]. The basic properties such as partition of unity and Fourier transform of exponential B-splines are summarized in Section 2.2. In the same section, we vi also provide an explicit formula for the first moment of exponential B-splines and investigate the mirroring effect of changing the sign of the parameters. In Section 2.3, we show that derivatives of exponential B-splines can be expressed in terms of lower order ones with the aid of elementary symmetric polynomials. In the last section, we prove that an exponential B-spline is refinable. This result bridges the theory of exponential B-splines to scale-space and wavelet transforms. In Chapter 3, we give a brief introduction to scale-space representation in Section 3.1 before we generalize the B-spline scale-space to exponential B-spline scale-space in Section 3.2. We also derive a simple algorithm for computation of scale-space at rational scale with applications to modelling and smoothing data. From various examples given in Section 3.3, we observe that exponential B-spline scale-space gives an extra degree of flexibility in choosing the parameters but its difference from polynomial B-spline scale-space is not significant. In the first section of Chapter 4, we summarize some recent results in [9], where the theory of scale-space is linked to continuous wavelet transforms. In Section 4.2, we modify the scale-space algorithm for computation of continuous wavelet transforms defined by derivatives of exponential B-splines. In the last section, we investigate the semi-discrete wavelet representation for the difference of exponential B-spline scale-space at two consecutive dyadic scales. The last chapter contains some concluding remarks on the work in this thesis and points out some potential future work in this area. Appendix A contains a proof of Theorem 2.1.1 with a reformulation using the modified elementary symmetric polynomials. vii List of Figures → 2.1 (t) with α= (α, · · · , α), α = 0.25. . . . . . . . 19 (a) E(α1 ,α2 ) (t), (b) En,→ α 2.2 E4 (t) with parameter (a) α= (−2, −1, 1, 2), (b) α= (0.5, 1, 1.5, 2). . . 21 2.3 (a) E(1,2,3) (t), (b) E(−1,0,1) (t), (c) E(−1,−2,−3) (t). . . . . . . . . . . . . . 27 2.4 (a) E4 (t), (b) E4 (t), (c) E4 (t), with parameter α= (−2, −1, 1, 2). . . . 29 3.1 (a) s = 25 , (b) s = 3.2 3D scale-space curves. . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Scale-space SE4 f with f modelled by M2 at scale (a) s = 25 , (b) s = 22 . 5 46 3.4 Scale-space SE4 f with f modelled by E3 at scale (a) s = 25 , (b) s = 22 . 5 47 4.1 (a) Original data, (b) Scale-space SM5 at scale s = 25 , (c) Wavelet → → → 12 , 5 (c) s = 42 . 5 . . . . . . . . . . . . . . . . . . . . . 44 transform WM5 at scale s = 25 . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 (a) Scale-space SM5 f ( 12 , ·), (b) Wavelet transform WM5 f ( 12 , ·). 5 5 4.3 (a) Scale-space SE5 f ( 52 , ·), (b) Wavelet transform WE5 f ( 25 , ·). . . . . . 56 viii . . . 55 List of Notations N Z Q R C C(R) Cc (R) C k (R) Lp (R) · p f (t) F{f }(u), f (u) supp(f ) (f ∗ g)(x) χE Ej Ej,i Mn (t) En,→ (t), En (t) α En,→ (t) α Sφ f (s, x) Wψ f (s, x) The set {1, 2, 3, · · · }. The set of integers. The set of rational numbers. The set of real numbers. The set of complex numbers. The space of continuous functions on R. The space of continuous functions with compact support on R. The space of functions with continuous k-th derivatives. The Lp space on R. The Lp -norm. Complex conjugation of f (t). Fourier transform of f (t). Support of f (t). Convolution of f and g. Characteristic function of a set E. Elementary symmetric polynomials. Modified elementary symmetric polynomials. Polynomial B-splines of order n. → Exponential B-splines of order n with parameter α. → Normalized exponential B-splines of order n with parameter α. Scale-space defined by kernel φ at scale s. Continuous wavelet transform defined by mother wavelet ψ at scale s. ix Chapter 1 Preliminaries Fourier is a mathematical poem. - Lord Kelvin. In this chapter, we put together preliminary materials that will be used frequently in the subsequent chapters. In the first section, we review the classical results of Fourier transform and convolution. The next section summarizes classical results on B-splines that we shall generalize to exponential B-splines in the next chapter. The third section gives a quick introduction to elementary symmetric polynomials, which is ubiquitous in many results for exponential B-splines. The last section addresses the differential operators that are commonly encountered in the study of exponential B-splines in later chapters. 1 CHAPTER 1. PRELIMINARIES 1.1 2 Fourier Transform For 1 ≤ p < ∞, let Lp (R) denotes the space of Lebesgue measurable functions f on R such that the Lebesgue integral ∞ |f (x)|p dx < ∞. −∞ Let L∞ (R) be the class of almost everywhere bounded functions, that is, functions bounded everywhere except on a set of Lebesgue measure zero. Lp (R), 1 ≤ p ≤ ∞, is a Banach space with Lp norm defined as, f p :=     ∞ −∞ p |f (x)| dx   ess supx∈R |f (x)|, 1 p , 1 ≤ p < ∞, p = ∞. In particular, L2 (R) is a Hilbert space whereby the inner product is defined by ∞ f (x)g(x) dx, f, g := f, g ∈ L2 (R). −∞ In the following, we first restrict our attention to the space L1 (R), which consists of all absolutely integrable functions on R. We begin with the Fubini’s theorem which allows the change of order of integration. Theorem 1.1.1. (Fubini’s theorem) If f ∈ L1 (R2 ), then ∞ ∞ −∞ −∞ ∞ ∞ −∞ −∞ f (x, y)dx dy = f (x, y)dy dx = f (x, y)d(x, y). R2 CHAPTER 1. PRELIMINARIES 3 Definition 1.1.1. The Fourier transform of a function f ∈ L1 (R) is defined as ∞ f (x)e−iux dx, u ∈ R. F{f (x)}(u) = f (u) := (1.1.1) −∞ The basic properties of Fourier transform are summarized in the following theorem. The proofs are omitted here but can be found in [4, 8]. Theorem 1.1.2. Let f ∈ L1 (R), then its Fourier transform f (u) satisfies: (a) f (u) is uniformly continuous on R, (b) lim f (u) = 0, u→±∞ (c) if f exists and f ∈ L1 (R), then f (u) = (iu)f (u). The following lemmas will be useful in later chapters. Lemma 1.1.1. For any a ∈ R\{0}, b ∈ R and f ∈ L1 (R), F{f (ax − b)}(u) = e−iub/a u f , u ∈ R. |a| a Proof. First, suppose a > 0. Using a change of variable y = ax − b, ∞ F{f (ax − b)}(u) = f (y) −∞ e−iu(y+b)/a e−iub/a dy = a a ∞ f (y)e−iuy/a dy. −∞ The case a < 0 is done similarly. Lemma 1.1.2. If f is continuous on [a, b], then F{xf (x)}(u) = i d f (u). du (1.1.2) CHAPTER 1. PRELIMINARIES 4 Proof. Let F (x, u) = f (x)e−iux . If f is continuous on [a, b], then both F (x, u) and ∂ F (x, u) ∂u are continuous on [a, b]×R. Hence, we can differentiate under the integral sign, d f (u) = du ∞ −∞ ∂ f (x)e−iux dx = ∂u ∞ (−ix)f (x)e−iux dt = −iF{xf (x)}(u). −∞ Note that f is not necessarily in L1 (R) even though f ∈ L1 (R). However, if both f and f are in L1 (R), then we can define inverse Fourier transform of f (u) as F −1 {f (u)}(x) := 1 2π ∞ f (u)eiux du. (1.1.3) −∞ The central question regarding inverse Fourier transform is when can f be recovered from f ? In other words, when does F −1 {f (u)}(x) = f (x) hold? The answer is provided by the following theorem, where the proof is again omitted here but can be found in [4]. Theorem 1.1.3. Let f ∈ L1 (R) such that f ∈ L1 (R). Then, F −1 {f (u)}(x) = f (x), at every point x where f is continuous. Convolution is an important concept in both mathematics and engineering, especially in signal processing. In fact, we shall see shortly that B-splines, the central objects of our study, can be defined in terms of convolution. CHAPTER 1. PRELIMINARIES 5 Definition 1.1.2. Let f, g ∈ L1 (R). The convolution of f and g is defined by ∞ f (x − y)g(y) dy. (f ∗ g)(x) := (1.1.4) −∞ Note that f ∗ g is well-defined and also belongs to L1 (R) since f ∗ g f 1 g 1 1 ≤ by Fubini’s theorem. In fact, this is a special case of the following more general inequality of Young. The proof is omitted here but is available in [8]. Theorem 1.1.4. (Young’s inequality for convolution) Suppose f ∈ Lp (R) and g ∈ Lq (R), where 1 p + 1 q = 1 r + 1 with 1 ≤ p, q, r ≤ ∞. Then, f ∗ g ∈ Lr (R) and f ∗g r ≤ f p g q. The support of a function f ∈ C(R), denoted by supp(f ), is the closure of the set {x ∈ R : f (x) = 0}. We say that f has compact support if supp(f ) is compact. We define Cc (R) := {f ∈ C(R) : supp(f ) is compact}. The elementary properties of convolution are summarized in the following theorem, where the proof is available in [8]. Theorem 1.1.5. Suppose f, g, h ∈ L1 (R). Then, (a) f ∗ g = g ∗ f . (b) f ∗ (g ∗ h) = (f ∗ g) ∗ h. (c) supp(f ∗ g) ⊆ A, where A is the closure of {x + y : x ∈ supp(f ), y ∈ supp(g)}. (d) If f, g ∈ Cc (R), then f ∗ g ∈ Cc (R). CHAPTER 1. PRELIMINARIES 6 The next theorem underlies many applications of convolution and will be useful in Chapter 4. Suppose φ ∈ L1 (R) and let ε > 0, define 1 x φε (x) := φ . ε ε Observe that φε = φ. Since φε (u) → 1 as ε → 0, φε is sometimes referred to as an approximate identity. The proof is omitted here but is available in [8]. Theorem 1.1.6. Suppose φ ∈ L1 (R) and ∞ −∞ φ(x) dx = c, where c > 0. (a) If f ∈ Lp (R), 1 ≤ p < ∞, then f ∗ φε → cf p → 0 as ε → 0. (b) If f is bounded and uniformly continuous, then f ∗ φε → cf uniformly as ε → 0. Remark 1.1.1. Note that φ can be easily normalized so that φ = 1 and the above theorem still holds mutatis mutandis. Convolution plays a vital role in Fourier transform and their close relationship is culminated in the following theorem, which says that the convolution of two functions is equivalent to the pointwise product of their Fourier transform. The proof is a direct application of Fubini’s theorem and thus omitted here. Theorem 1.1.7. (Convolution theorem) If f, g ∈ L1 (R), then (f ∗ g)(u) = f (u)g(u), u ∈ R. CHAPTER 1. PRELIMINARIES 1.2 7 B-Splines In general terms, splines are piecewise polynomials. In this section, we focus on a special class of splines, namely the B-splines which have many important properties. The term B-spline was first coined by Schoenberg and stands for basis splines. They were first studied for interpolation dated back in 1940s and later rediscovered in 1970s for geometric modeling in computer-aided design. In 1990s, B-splines were in the limelight again in connection with wavelets and applications in signal processing. Definition 1.2.1. The first order B-spline, denoted by M1 (t), is the characteristic function of the interval [0, 1), M1 (t) = χ[0,1) :=    1, 0 ≤ t < 1,   0, otherwise. The B-spline of order n, denoted by Mn (t), is defined recursively as convolution with lower order ones, 1 Mn (t) := (Mn−1 ∗ M1 )(t) = Mn−1 (t − x)dx, n ≥ 2. (1.2.1) 0 Remark 1.2.1. There is an alternative explicit definition of B-spline Mn using truncated powers, Mn (t) = 1 n−1 ∆n t+ . (n − 1)! But these two definitions can be shown to be equivalent as given in [4]. The main advantage of defining Mn as (1.2.1) is that many crucial properties of Mn can be derived from it easily, especially those that involve Fourier transform. CHAPTER 1. PRELIMINARIES 8 The basic properties of B-splines are summarized in the following theorem. The proof is omitted here but is available in [4]. Theorem 1.2.1. The n-th order B-splines Mn satisfy: (a) supp(Mn ) = [0, n]. (b) Mn (t) > 0 for t ∈ (0, n). Mn (t − j) = 1 for all t ∈ R. (c) j∈Z (d) Mn (t) = t Mn−1 (t) n−1 − n−t Mn−1 (t n−1 − 1), n ≥ 2. (e) Mn (t) = Mn−1 (t) − Mn−1 (t − 1). (f) Mn ∈ C n−2 . Remark 1.2.2. Property (c) is often referred to as partition of unity. Property (d) shows that Mn is a polynomial of degree n − 1 on each interval [k, k + 1], k = 0, · · · , n − 1. Repeated applications of Property (e) show that for 1 ≤ k ≤ n − 1, k Mn(k) (t) (−1)j = j=0 k Mn−k (t − j). j Fourier transform is an essential tool to work with B-splines. A direct computation shows that the Fourier transform of M1 is given by M1 (u) = 1 − e−iu , u ∈ R. iu Fourier transform of higher order B-splines follows from repeated applications of CHAPTER 1. PRELIMINARIES 9 Theorem 1.1.7, Mn (u) = 1 − e−iu iu n , u ∈ R. Using Fourier transform, we can easily compute the zeroth and first moments of B-splines as shown below. Theorem 1.2.2. The n-th order B-splines Mn satisfy: ∞ (a) Mn (t) dt = 1. −∞ ∞ (b) tMn (t) dt = −∞ n . 2 Proof. (a) Applying L’Hospital’s rule, ∞ Mn (t) dt = Mn (0) = lim u→0 −∞ 1 − e−iu iu n = lim e−inu = 1. u→0 (b) d Mn (0) = lim n u→0 du 1 − e−iu iu n−1 −ue−iu − i(1 − e−iu ) −u2 Applying L’Hospital’s rule twice, −i −ue−iu − i(1 − e−iu ) = . u→0 −u2 2 lim Hence, from Lemma 1.1.2, ∞ tMn (t) dt = F{tMn (t)}(0) = i −∞ d −i n Mn (0) = in = . du 2 2 CHAPTER 1. PRELIMINARIES 1.3 10 Elementary Symmetric Polynomials In this section, we introduce the elementary symmetric polynomials which is sort of a generalization of binomial coefficients. In fact, as we shall see in Theorem 1.3.1, the elementary symmetric polynomials are coefficients obtained when expanding n distinct products, whereas binomial coefficients are obtained when expanding n identical products. Hence, elementary symmetric polynomials appear naturally for exponential B-splines as the Fourier transform of an exponential B-spline of order n is given by n distinct products (see Theorem 2.2.4). Definition 1.3.1. The elementary symmetric polynomial Ej (x1 , · · · , xn ) is the sum of all products of j distinct variables chosen from the set {x1 , · · · , xn }, for 1 ≤ j ≤ n. It can be expressed compactly as Ej (x1 , · · · , xn ) := xi 1 · · · xi j . (1.3.1) 1≤i1 0 and x ∈ R, t Sφ f (t, x) − Sφ f (r, x) = r 1 Wψ f (s, x)ds, s (4.1.5) where ψ(t) = φ(t) + tφ (t), t ∈ R. Theorem 4.1.4. Let φ : R → R be a bounded integrable function with ∞ −∞ φ = 1. Suppose tφ (t) is integrable and bounded on R. Define ψ(t) := φ(t) + tφ (t). (a) If f ∈ Lp (R), 1 ≤ p < ∞, then t lim f − t→∞ r→0 r 1 Wψ f (s, ·) ds s = 0. p (b) If f ∈ Lp (R), 1 ≤ p < ∞, is bounded and continuous on R, then ∞ Wψ f (s, x) ds, x ∈ R. f (x) = 0 (c) If f ∈ Lp (R), 1 ≤ p < ∞, is bounded and uniformly continuous on R, then the convergence in (b) is uniform. The proofs of all results in this section are available in [9]. CHAPTER 4. WAVELET REPRESENTATION 4.2 52 Numerical Computation We can readily verify that derivatives of B-splines are admissible as mother wavelets. Moreover, we can express the derivatives of polynomial B-splines in terms of lower order splines by using (2.3.3). Hence, with the derivatives of B-splines as mother wavelets, we can formulate the continuous wavelet transforms in terms of lower order scale-space. Consequently, we can modify the algorithm of scale-space for computation of continuous wavelet transforms. (k) Lemma 4.2.1. Let ψk (t) = Mn (t), 1 ≤ k ≤ n − 1, then k (−1)j Wψk f (s, t) = j=0 k SMn−k f (s, t − j). j (4.2.1) Proof. Substitute (2.3.3) in (4.1.2) and (3.1.1). (k) Proposition 4.2.1. Suppose f (t) = j p(j)Mm (t − j) and ψk (t) = Mn (t) with 1 ≤ k ≤ n−1. The wavelet transform WMn(k) f (s, t) at rational scale s = a/b, a, b ∈ N is given by WMn(k) f a ,t = b h(j)Mm+n−k (at − j), t ∈ R, (4.2.2) j where k [(−1)l b a h(j)z j = aBn−k,α (z)Bm,β (z) j j l=0 k p(j − l)]z aj , z = e−iu/a . (4.2.3) l Proof. Taking Fourier transform of (4.2.1), for u ∈ R, a Wψk f ,u = b k (−1)l l=0 k −ilu a e SMn−k f ,u . l b CHAPTER 4. WAVELET REPRESENTATION 53 Using the expression (3.2.6), with z = e−iu/a , a Wψk f ,u = b k (−1)l l=0 k al u b a z P (z a )Bn−k,α (z)Bm,β (z)Mm+n−k . l a Simplifying with (4.2.3) and taking inverse Fourier transform gives (4.2.2). The equation (4.2.3) gives a simple algorithm for computation of a continuous wavelet transform at rational scale with mother wavelet defined by derivatives of Bsplines. In fact, we only need to modify the Algorithm 3.2.1 for B-spline scale-space. Algorithm 4.2.1. Inputs: a, b, m, n, p(j), j = 0, 1, · · · , T. 1. Compute r(j) := k l k l=0 (−1) l p(j − l). 2. For j = 0, 1, · · · , T , i = 0, 1, · · · , a − 1, do r1 (ja + i) := r(j). 3. For l = 1, 2, · · · , m − 1, j = l(a − 1), · · · , aT + a − 1, do rl+1 (j) := 1 a a−1 rl (j − i). i=0 4. For l = m, · · · , m+n−k−1, j = (m−1)(a−1)+(l−m+1)(b−1), · · · , aT +a−1, do 1 rl+1 (j) := b Output: h(j) := pm+n−k (j). b−1 rl (j − i). i=0 CHAPTER 4. WAVELET REPRESENTATION 54 Example 4.2.1. Let ψ2 (t) = Mn (t), then WMn f (s, t) = SMn−2 f (s, t) − 2SMn−2 f (s, t − 1) + SMn−2 f (s, t − 2). Figure 4.1 shows the scale space transform of 100 randomly generated data at scale s = 2 5 with kernel φ(t) = M5 (t) and the wavelet transform with mother wavelet defined by the second derivatives ψ(t) = M5 (t). Figure 4.2 shows the smoothing effect when the scale is increased to s = 12 . 5 Figure 4.1: (a) Original data, (b) Scale-space SM5 at scale s = transform WM5 at scale s = 52 . 2 , 5 (c) Wavelet CHAPTER 4. WAVELET REPRESENTATION 55 Figure 4.2: (a) Scale-space SM5 f ( 12 , ·), (b) Wavelet transform WM5 f ( 12 , ·). 5 5 Remark 4.2.1. We have not presented general result for exponential B-splines En (t) (k) since their derivatives En (t) cannot be entirely expressed in terms of lower order splines En−k (t) without paying special attention to the parameters. See Remark 2.3.2 for further discussion. Thus, we only focus on special cases where explicit formula of derivatives can be formulated, such as (2.3.5). Example 4.2.2. Let ψ2 (t) = E 5, → (t) with α= (0, 0, 0, α, −α). From (2.3.2), α → E5 (t) = E3 (t) − (1 + 2 cosh(α))E3 (t − 1) + (1 + 2 cosh(α))E3 (t − 2), where E3 (·) = E(0,α,−α) (·) and E1 (1, eα , e−α ) = E2 (1, eα , e−α ) = 1 + 2 cosh(α). Thus, WE f (s, t) = SE3 f (s, t)−(1+2 cosh(α))SE3 f (s, t−1)+(1+2 cosh(α))SE3 f (s, t−2). 5 CHAPTER 4. WAVELET REPRESENTATION 56 We consider a set of data p(j) generated from Brownian motion and we modelled it by f (t) = j p(j)M2 (t − j). We use exponential B-spline scale-space kernel → (t) with α= (0, 0, 0, 0.25, −0.25). Figure 4.3 shows the scale space φ(t) = E5,→ α transform and the wavelet transform with mother wavelet defined by its second derivatives ψ(t) = E5 (t) at scale s = 52 . Figure 4.3: (a) Scale-space SE5 f ( 52 , ·), (b) Wavelet transform WE5 f ( 25 , ·). CHAPTER 4. WAVELET REPRESENTATION 4.3 57 Semi-Discrete Wavelet Representation Theorem 4.1.3 provides the representation for the difference of scale-space at two arbitrary scales t > r > 0 in terms of a continuous wavelet transform. Now, we wish to derive an analogous result at two discrete scales αm and αn , where α = 1 is a fixed positive number and n > m are integers. Such a representation is called a semi-discrete wavelet representation. In similar fashion as the continuous wavelet transforms, a slightly different admisibility conditions for the semi-discrete wavelet transforms are given in [9], ∞ n Ψα (u/αk ) ≤ C a.e. k Ψα (u/α ) = 1, k=m k=−∞ In fact, if Ψα (t) := αφ(αt) − φ(t), α = 1, is constructed from the scale-space kernel φ, then it satisfies the above admissibility conditions and WΨα is defined by ∞ WΨα f (αk , x) := αk Ψα (αk (x − t))f (t) dt, (4.3.1) −∞ is a semi-discrete wavelet transform. See [9] for a proof and further discussion. The next two results are semi-discrete analogues of Theorem 4.1.3 and Theorem 4.1.4 respectively. Theorem 4.3.1. Let φ : R → R be a bounded integrable function with Let Ψα (t) = αφ(αt) − φ(t), α = 1. For f ∈ Lp , 1 ≤ p < ∞, n−1 Sφ f (αn , x) − Sφ f (αm , x) = WΨα f (αk , x), x ∈ R, k=m where WΨα is defined as in (4.3.1) ∞ −∞ φ = 1. CHAPTER 4. WAVELET REPRESENTATION 58 Theorem 4.3.2. Let φ : R → R be a bounded integrable function with ∞ −∞ φ=1 and Ψα (t) := αφ(αt) − φ(t). (a) If f ∈ Lp (R), 1 ≤ p < ∞, then n−1 lim n→∞ m→−∞ WΨα f (αk , ·) f− k=m = 0. p (b) If f ∈ Lp (R), 1 ≤ p < ∞, is bounded and continuous on R, then ∞ WΨα f (αk , x), x ∈ R. f (x) = k=−∞ (c) If f ∈ Lp (R), 1 ≤ p < ∞, is bounded and uniformly continuous on R, then the convergence in (b) is uniform. In the remaining part of this section, we shall work with the centered exponen(t), which are shifted version of the usual exponential B-splines. tial B-splines En,→ α They have support on [−n/2, n/2] with Fourier transform given by n En,→ (u) = α αk − iu sinh θk , where θk = , u ∈ R. θ 2 k k=1 (4.3.2) We refrain from introducing new notation to denote the centered exponential Bsplines as all the results in this section are self-contained. First, we establish a formula for the difference of centered exponential B-splines at two consecutive dyadic scales. CHAPTER 4. WAVELET REPRESENTATION 59 → Proposition 4.3.1. Let α= (α1 , · · · , αm ). For f ∈ Lp (R), 1 ≤ p < ∞, x ∈ R, m 2−2k Lk [E2m+2k ](x), E2m,(2 α ,2 α ) (x/2) − 2E2m,( α , α ) (x) = 2 → → →→ (4.3.3) k=1 where Lk := Ek ((D − α1 I)2 , · · · , (D − αm I)2 ), (4.3.4) and Ek are the elementary symmetric polynomials. Proof. From the Fourier transform (4.3.2), 2m → (2u) = E2m,(2→ α ,2 α ) sinh 2θk , u ∈ R. 2θ k k=1 Note that each parameter αk is repeated twice in the product and thus can be collected together. Applying indentities for hyperbolic functions, we have m → (2u) = E2m,(2→ α ,2 α ) sinh2 2θk (2θk )2 k=1 m m sinh2 θk = θk2 k=1 cosh2 θk k=1 m → (u) = E2m,(→ α,α) [1 + sinh2 θk ] k=1 Expanding the product using (1.3.4), we have m m 2 Ek (sinh2 θ1 , · · · , sinh2 θm ). [1 + sinh θk ] = 1 + k=1 k=1 CHAPTER 4. WAVELET REPRESENTATION 60 Therefore, m → (u). (4.3.5) Ek (sinh2 θ1 , · · · , sinh2 θm )E2m,(→ α,α) E2m,(2 α ,2 α ) (2u) − E2m,( α , α ) (u) = → → →→ k=1 Note that for each 1 ≤ k ≤ m, we can rewrite sinh2 θk = 41 (iu − αk )2 E2 . Applying Lemma 1.3.1 and using the property that (E2 )k E2m = E2m+2k , we can simplify the right hand side of (4.3.5). For each 1 ≤ k ≤ m, Ek (sinh2 θ1 , · · · , sinh2 θm )E2m (u) = 2−2k Ek ((iu − α1 )2 E2 , · · · , (iu − αm )2 E2 )E2m (u) = 2−2k Ek ((iu − α1 )2 , · · · , (iu − αm )2 )E2m+2k (u) = 2−2k Lk [E2m+2k ](u), where Lk is defined as in (4.3.4) and its Fourier transform follows from Lemma 1.4.1. Substituting back to (4.3.5), m 2−2k Lk [E2m+2k ](u). → (2u) − E → → (u) = E2m,(2→ α ,2 α ) 2m,( α , α ) k=1 Since E2m and Lk [E2m+2k ] are both in L1 (R), taking inverse Fourier transform on both sides gives (4.3.3). Remark 4.3.1. Note that Lk [E2m+2k ](t) is not admissible as mother wavelet since 2 Lk [E2m+2k ](0) = Ek (α12 , · · · , αm )E2m+2k (0) = 0. Therefore, we consider 2 Ψk (t) := Lk [E2m+2k ](t) − Ek (α12 , · · · , αm )E2m+2k (t), so that Ψk (0) = 0. (4.3.6) CHAPTER 4. WAVELET REPRESENTATION 61 → Proposition 4.3.2. Let α= (α1 , · · · , αm ). For f ∈ Lp (R), 1 ≤ p < ∞, and scale s > 0, m SE → 2m,2 α f (s, x) − SE → 2m, α 2 )SE 2−2k Ek (α12 , · · · , αm f (2s, x) = → 2m+2k, α f (2s, x) k=1 m 2−2k WΨk f (2s, x), + (4.3.7) k=1 where Ψk is defined as in (4.3.6). Proof. Replacing x by 2st and substituting (4.3.6) to (4.3.3) give m 2 2−2k Ek (α12 , · · · , αm )E2m+2k (2st) → (st) − 2E → → (2st) = 2 E2m,(2→ α ,2 α ) 2m,( α , α ) k=1 m 2−2k Ψk (2st). +2 k=1 Multiplying both sides with s and taking convolution with f using (3.1.1) and (4.1.2) yield (4.3.7). → Corollary 4.3.1. If α= (α, · · · , α), then m SE → 2m,2 α f (s, x) − SE → 2m, α 2−2k α2k f (2s, x) = k=1 m SE → f (2s, x) 2m+2k, α k m 2−2k WΨk f (2s, x), + k=1 where Ψk = (D − αI)2k − α2k m E2m+2k . k CHAPTER 4. WAVELET REPRESENTATION → 62 → Corollary 4.3.2. If α= 0 , then m 2−2k SM2m f (s, x) − SM2m f (2s, x) = k=1 → m WM (2k) f (2s, x). 2m+2k k → (4.3.8) Proof. If α= 0 , then Ek (0, · · · , 0) = 0 and the terms involving SE2m+2k on the right m (2k) hand side of (4.3.7) vanish. Moreover, En (t) = Mn (t) and Ψk (t) = M2m+2k (t) k follows from (4.3.6). Hence, (4.3.7) reduces to (4.3.8), which is the semi-discrete wavelet representation by derivatives of B-splines as recently reported in [9]. Chapter 5 Concluding Remarks What we know is not much. What we do not know is immense. - Pierre-Simon Laplace In this thesis, we mainly focus on univariate exponential B-splines with real parameters. We manage to generalize several results of polynomial B-splines to exponential B-splines, often with the help of elementary symmetric polynomials. This tool, to the knowledge of the author, has not been used in the study of exponential B-splines previously. In Appendix A, we introduce the modified elementary symmetric polynomials to aid the proof of the explicit formula for exponential B-splines. They can be viewed as a variant of elementary symmetric polynomials subject to a constraint on the indices. 63 CHAPTER 5. CONCLUDING REMARKS 64 We have seen in Section 2.3 that derivatives of exponential B-splines are much more involved than their counterparts for polynomial B-splines. This complication prevents the exponential B-spline scale-space algorithm to be directly modified for continuous wavelet transforms defined by derivatives of exponential B-splines, except for special cases that can be considered separately. The same problem also hinders the difference of exponential B-spline scale-space at two consecutive dyadic scales to be represented in terms of wavelet transforms alone, since there is an additional component of higher order scale-space as shown in (4.3.7). There are several directions of further generalization that can be pursued, even though the real parameters case that we consider here have substantially generalized the results for polynomial case. For instance, complex parameters are considered in [17] and multivariate version of exponential B-splines, called the exponential box splines, are introduced in [13]. However, these generalizations require much more effort to handle the delicate parameters. It is well-known that B-splines converge to Gaussian function and derivatives of B-splines converge to derivatives of Gaussian function (see [1], [2], [16]). But it is unknown, to the knowledge of the author, which type of distribution does the exponential B-splines converge to. Intuitively, the author conjectures that the exponential B-splines converge to the exponential family, which is a large class of distribution that includes the Gaussian and many other common distributions. A further probe along this direction would be interesting. Bibliography [1] R. Brinks. On the convergence of derivatives of B-splines to derivatives of the Gaussian function. Comput. Appl. Math., 27:79–92, 2008. [2] L.H.Y. Chen, T.N.T. Goodman, and S.L. 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David Hilbert In the first section of this chapter, we introduce the exponential < /b> B- splines as a generalization of the classical polynomial B- splines and provide an explicit formula for the exponential < /b> B- splines with distinct parameters The next two sections collect various properties and study the derivatives of exponential < /b> B- splines respectively In the last section, we show that an exponential < /b> B- spline... we are interested in exponential < /b> B- splines, which are generalization of polynomial B- splines but at the same time are special cases of L -splines Analogously to the definition of polynomial B- splines (1.2.1), we can introduce the first order exponential < /b> B- spline as an exponential < /b> function on the unit interval Subsequently, higher order exponential < /b> B- splines are formed by taking convolution with lower... observe that if the parameters are not symmetrical, neither does the graph See Theorem 2.2.6 for further discussion on symmetrical property of exponential < /b> B- splines → → Figure 2.2: E4 (t) with parameter (a) α= (−2, −1, 1, 2), (b) α= (0.5, 1, 1.5, 2) CHAPTER 2 EXPONENTIAL < /b> B- SPLINES 2.2 22 Basic Properties The basic properties of exponential < /b> B- splines are similar to those of polynomial B- splines We begin... (u)g(u), u ∈ R CHAPTER 1 PRELIMINARIES 1.2 7 B- Splines In general terms, splines are piecewise polynomials In this section, we focus on a special class of splines, namely the B- splines which have many important properties The term B- spline was first coined by Schoenberg and stands for basis splines They were first studied for interpolation dated back in 1940s and later rediscovered in 1970s for geometric... 2 EXPONENTIAL < /b> B- SPLINES 2.1 17 Introduction B- splines, henceforth referred to as polynomial B- splines, have been widely studied and applied in various fields from interpolation to signal processing Since the pioneering work of Schoenberg in 1946 [14], there have been many theoretical advances to non-uniform grids, non-polynomial basis and higher dimensions [15] In this thesis, we are interested in exponential.< /b> .. of binomial coefficients In fact, as we shall see in Theorem 1.3.1, the elementary symmetric polynomials are coefficients obtained when expanding n distinct products, whereas binomial coefficients are obtained when expanding n identical products Hence, elementary symmetric polynomials appear naturally for exponential < /b> B- splines as the Fourier transform of an exponential < /b> B- spline of order n is given by... B- splines A direct computation shows that the Fourier transform of M1 is given by M1 (u) = 1 − e−iu , u ∈ R iu Fourier transform of higher order B- splines follows from repeated applications of CHAPTER 1 PRELIMINARIES 9 Theorem 1.1.7, Mn (u) = 1 − e−iu iu n , u ∈ R Using Fourier transform, we can easily compute the zeroth and first moments of B- splines as shown below Theorem 1.2.2 The n-th order B- splines. .. be used frequently in the subsequent chapters In the first section, we review the classical results of Fourier transform and convolution The next section summarizes classical results on B- splines that we shall generalize to exponential < /b> B- splines in the next chapter The third section gives a quick introduction to elementary symmetric polynomials, which is ubiquitous in many results for exponential < /b> B- splines. .. be useful in later chapters Lemma 1.1.1 For any a ∈ R\{0}, b ∈ R and f ∈ L1 (R), F{f (ax − b) }(u) = e−iub/a u f , u ∈ R |a| a Proof First, suppose a > 0 Using a change of variable y = ax − b, ∞ F{f (ax − b) }(u) = f (y) −∞ e−iu(y +b) /a e−iub/a dy = a a ∞ f (y)e−iuy/a dy −∞ The case a < 0 is done similarly Lemma 1.1.2 If f is continuous on [a, b] , then F{xf (x)}(u) = i d f (u) du (1.1.2) CHAPTER 1 PRELIMINARIES... processing In fact, we shall see shortly that B- splines, the central objects of our study, can be defined in terms of convolution CHAPTER 1 PRELIMINARIES 5 Definition 1.1.2 Let f, g ∈ L1 (R) The convolution of f and g is defined by ∞ f (x − y)g(y) dy (f ∗ g)(x) := (1.1.4) −∞ Note that f ∗ g is well-defined and also belongs to L1 (R) since f ∗ g f 1 g 1 1 ≤ by Fubini’s theorem In fact, this is a special ... namely the scale-space and wavelet representations by exponential B-splines First, the concept of elementary symmetric polynomials is introduced in order to express derivatives of exponential. .. of polynomial B-splines to exponential B-splines In Chapter 2, we formally introduce the exponential B-splines with real parameters We give an explicit formula for exponential B-splines of arbitrary... theory of exponential B-splines to scale-space and wavelet transforms In Chapter 3, we give a brief introduction to scale-space representation in Section 3.1 before we generalize the B-spline scale-space