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Mohammad Hadi Kefayati Xlets Mohammad Hadi Kefayati Xlets Report SSP-180112 V1.1 Sparse Signal Processing Group, Digital Media Lab (DML), SUT, Tehran, Iran [email protected] http://ssp.dml.ir Table of Contents Xlets :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 Mohammad Hadi Kefayati I A Brief Introduction to the Wavelets 2 Introduction ::::::::::::::::::::::::::::::::::::::::::::::: 6 1 Fourier Analysis . 10 1.1 Variation and oscillation . 10 1.2 Transfer Functions. 10 2 From Fourier Transform to Wavelets . 10 2.1 Heisenberg Uncertainty . 10 3 Wavelets...................................................... 13 3.1 Local Cosine Bases . 13 4 General Theorem of Sampling and linear analogue conversions . 15 II The leap to two dimensional transforms and Xlets 5 Regularity analysis . 19 6 Ridglet . 21 7 Curvelets . 22 7.1 The First Generation Curvelet Transform . 24 7.2 Second Generation Curvelet . 24 8 Shearlet . 26 9 Contourlet . 27 9.1 Laplacian pyramid. 27 9.2 Directional Decomposition . 28 9.3 Contourlet. 29 10 Bandlets . 29 11 Grouplets . 31 12 Wrapping up the Xlets trend line . 32 III Appendix A Brief introduction to Frame Theory . 37 A.1 Signal Expansion . 37 B Hough Transform . 38 C Radon Transform . 39 D History of Xlets in one shot . 41 Part I A Brief Introduction to the Wavelets 5 This introduction does not intended to give a deep understanding of the the- ory of wavelets and the filter banks, however we will point the interested reader to good references where he/she could gain deeper insight into the presented material. To avoid any further requirements, we strongly recommend the reader to gain a prior knowledge of discrete signal processing ([14]). 2 Introduction Definition 1. [5] A vector space over the set of complex or real numbers, C or R, is a set of vectors, E, together with addition and scalar multiplication, which, for general x; y in E, and α; β in C or R, satisfy the following: 1. Commutativity: x + y = y + x 2. Associativity: (x + y) + z = x + (y + z); (αβ)x = α(βx) 3. Distributivity: α(x + y) = αx + αy; (α + β)x = αx + βx 4. Additive identity: there exists 0 in E, such that x + 0 = x, for all x in E 5. Additive inverse: for all x in E, there exists a (=x) in E, such that x+(=x) = 0 6. Multiplicative identity: 1 · x = x for all x in E Definition 2. [2] A subset x1; :::; xn of a vector space E is called a basis for E, when E = span(x1; :::; xn) and x1; :::; xn are linearly independent. Definition 3. [5] A sequence of vectors fxng is called a Cauchy sequence, if kxn − xmk ! 0, when n; m ! 1. If every Cauchy sequence in E, converges to a vector in E, then E is called complete. Definition 4. [5] Properties that yeild V {h:; :i : V × V ! C} to be an Inner product space: (x; y; z 2 V ) 1. Conjugate-symmetry: hx; yi = hy; xi 2. Linearity in the first term: hx + y; zi = hx; zi + hy; zi 3. Positive-definiteness: hx; xi ≥ 0, with equality iff x = 0. Definition 5. [5] The following three equations hold for inner products over a vector space. – Cauchy-Schwarz inequality jhx; yij ≤ kxk kyk with equality if and only if x = αy . – Triangle inequality kx + yk ≤ kxk + kyk 7 with equality if and only if x = αy, where α is a positive real constant. – Parallelogram law ( ) kx + yk2 + kx − yk2 = 2 kxk2 + kyk2 (1) Definition 6. [5] Banach spaces is a complete normed vector spaces. This means that a Banach space is a vector space V with a norm |||| such that every Cauchy sequence in V has a limit in V (with respect to the topology induced by that metric). Definition 7. [5] A Hilbert space is : p An inner product space is that is complete under the induced norm k:k = h:; :i. A very good example is the obvious inner product hx; yi = xy. Every Hilbert space is a Banach space. Because a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if kvk2 = (v; v) for all v. In inverse a necessary and sufficient condition for a Banach space V to be associated to an inner product (which will then necessarily make V into a Hilbert space) is the parallelogram identity (eq. 1). Definition 8. [5] A Hilbert space is separable if and only if it contains a countable orthonormal basis. Definition 9. [5] The space of L2-integrable of functions on the interval 2 [a; b] is a Hilbert space´ (denoted L (a; b)) under the standard inner product, h i b defined as f; g = a f(x)g(x)dx. f g H Definition 10. [5] An orthonormal basis en n2Z for the Hilbert space is a sequence of mutually orthogonal unit vectors whose closed span is the whole space. That is, h i f g H ej; ek = δj;k and span = en n2Z = where δj;k is the Dirac delta function. f g Theorem 11. (Perfect Reconstruction) [5] If en n2Z is an orthonormal basis for a Hilbert space H, then X x = hx; eni en for all x 2 H n2Z 8 1 f g Theorem 12. (Parseval’s Identity [5] ) If en n2Z is an orthonormal basis for a Hilbert space H, then X 2 2 jhx; enij = kxk for all x 2 H n2Z f g H Definition 13. [5] A family fn n2Z of elements of a Hilbert space is a Parseval frame if it satisfies Parseval’s identity, that is, if X 2 2 jhx; fnij = kxk for all x 2 H n2Z NOTE 1 : fn stands for frame basis and en for orthonormal basis. NOTE 2 : We can have perfect reconstruction while the basis used were not orthonormal, if the frame is a parseval’s frame. f g H Definition 14. [5] A family fn n2Z of elements of a Hilbert space is a frame for H if there exist positive constant A and B such that X 2 2 2 A kfk ≤ jhf; fnij ≤ B kfk n2Z for all f 2 H. NOTE 3 : The lower frame condition (bound) ensures that a frame is com- plete (its closed span is the whole space). NOTE 4 : The upper frame condition ensures that the following map is well defined. Definition 15. [5] Bessels Inequality : X 2 2 8y 2 E ) kyk ≥ jhy; ekij k f g Where ek k2Z are a set of orthonormal basis. f g H 2 Definition 16. [5] Let fn n2Z be a frame for . The Bessel map is a function T ∗ : H! `2 (N) defined by ∗ 7−! fh ig1 T : f f; fn n=1 1 Also known as Pythagorean Theorem 2 analysis operator 9 Definition 17. [5] A tight frame is a frame whose bounds are equal (i.e. A = B). A 1-tight frame has bounds A = B = 1. Note that a 1-tight frame is the same as a Parseval frame. f g H 3 Definition 18. [5] Let fn n2N be a frame for . The pre-frame operator is a map T : `2 (N) !H defined by4 X1 T : fcng ! cnfn: n=1 f g H Definition 19. [5] Let fn n2Z be a frame for . Then the frame operator S : H!H is defined as TT ∗, that is, X S : f ! hf; fni fn n2Z Fig. 1. The frame operator[5] Theorem 20. [5] The frame operator S, as defined above,is well-defined. Furthermore, S is bounded, invertible, self-ad joint and positive. f g H Theorem 21. [5] If fn{ n2Z is} a frame for with corresponding frame −1 H operator S, then the family S fn n2Z is also a frame for , and has frame 1 1 1 1 bounds B and A . In fact, if A and B are optimal, then so are B and A . f g NOTE 5 : In theoryn weo can always transform the frame fn n2Z into the − 1 Parseval frame S 2 fn . n2Z 3 synthesis operator 4 More naturals to note that the pre-frame operator is the ad joint of the Bessel map. This makes the definition of the frame operator almost second-nature. 10 1 Fourier Analysis 1.1 Variation and oscillation The Fourier{ } Transform of f denoted as f^(!) is the projection of f on Fourier basis e−i!t and is determined by equation 2. Equation 2 is also interpreted as a measurement of oscillations existing is in f at the frequency !. ˆ +1 f^(!) = f(t)e−i!tdt (2) −∞ As eq.2 shows, the Fourier transform is insensitive to the temporal changes in the frequency of oscillations in f . 1.2 Transfer Functions A simplifying thing about LTI systems is their transfer function. Transfer func- tions ease finding responses for wide types of inputs. This is done by determining the convolution between the input signal and the transfer function. It may be seen below that complex exponentials, ei!t, and h^(!) are respectively the eigen- vectors is the eigenvalues of the convolution operator. ˆ ˆ +1 +1 Lei!t = h(u)ei!(t−u)du = ei!t h(u)e−i!udu = ei!t h^(!) −∞ −∞ 2 From Fourier Transform to Wavelets 2.1 Heisenberg Uncertainty To deal with non-stationary waveforms, we need to build a dictionary of local- f g ized time and frequency atoms (D = φγ γ2Γ ). The dictionary is normal if all the atoms in the dictionary have length one (kφγ k = 1). To determine the localization parameter of the frame, φγ , time localization u and frequency local- ization ξ are calculated in the following manner.
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