AReallyFriendlyGuidetoWavelets-oc.Valens1999-c.valensamindless.com deals with the general properties of the wavelets and wavelet transforms only. It defines a framework within one can design wavelets to taste and wishes 3. Wavelet properties The most important properties of wavelets are the admissibility and the regularity conditions and these are the properties which gave wavelets their name. It can be shown [ She96] that square integrable functions y(n) satisfying the admissibility condition p(o) do<+∞, can be used to first analyze and then reconstruct a signal without loss of information. In(4)p(o) stands for the Fourier transform of w(o). The admissibility condition implies that the Fourier transform of w(t) vanishes at the zero frequency. i.e IY(O)I=0 This means that wavelets must have a band-pass like spectrum. This is a very important observation, which we will e later on to build an efficient wavelet transform A zero at the zero frequency also means that the average value of the wavelet in the time domain must be zero dt=0 (6) nd therefore it must be oscillatory. In other words, w (o) must be a wave be seen from(1)the wavelet transform of a one-dimensional function is two-dimensional; the wavelet transform of a two-dimensional function is four-dimensional. The time-bandwidth product of the wavelet transform is the square of the input signal and for most practical applications this is not a desirable property. Therefore one mposes some additional conditions on the wavelet functions in order to make the wavelet transform decrease quickly with decreasing scale s. These are the regularity conditions and they state that the wavelet function should have some smoothness and concentration in both time and frequency domains. Regularity is a quite complex concept nd we will try to explain it a little using the concept of vanishing moments If we expand the wavelet transform(1)into the Taylor series at t=0 until order n (let t=0 for simplicity)we get +O(n+1
A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 6 deals with the general properties of the wavelets and wavelet transforms only. It defines a framework within one can design wavelets to taste and wishes. 3. Wavelet properties The most important properties of wavelets are the admissibility and the regularity conditions and these are the properties which gave wavelets their name. It can be shown [She96] that square integrable functions 5(t) satisfying the admissibility condition, ω < +∞ ω Ψ ω ∫ d | | | ( ) | 2 , (4) can be used to first analyze and then reconstruct a signal without loss of information. In (4) 4(7) stands for the Fourier transform of 5(t). The admissibility condition implies that the Fourier transform of 5(t) vanishes at the zero frequency, i.e. | ( ) | 0 0 2 Ψ ω = ω= . (5) This means that wavelets must have a band-pass like spectrum. This is a very important observation, which we will use later on to build an efficient wavelet transform. A zero at the zero frequency also means that the average value of the wavelet in the time domain must be zero, ∫ψ(t)dt = 0 , (6) and therefore it must be oscillatory. In other words, 5(t) must be a wave. As can be seen from (1) the wavelet transform of a one-dimensional function is two-dimensional; the wavelet transform of a two-dimensional function is four-dimensional. The time-bandwidth product of the wavelet transform is the square of the input signal and for most practical applications this is not a desirable property. Therefore one imposes some additional conditions on the wavelet functions in order to make the wavelet transform decrease quickly with decreasing scale s. These are the regularity conditions and they state that the wavelet function should have some smoothness and concentration in both time and frequency domains. Regularity is a quite complex concept and we will try to explain it a little using the concept of vanishing moments. If we expand the wavelet transform (1) into the Taylor series at t = 0 until order n (let τ = 0 for simplicity) we get [She96]: + + γ = ∑ ψ ∫ = ( 1) ! (0) 1 ( ,0) 0 ( ) dt O n s t p t f s s n p p p . (7)
AReallyFriendlyGuidetoWavelets-oc.Valens1999-c.valensamindless.com Here f)stands for the p" derivative of f and an+l)means the rest of the expansion. Now, if we define the moments of the wavelet by M then we can rewrite(7)into the finite development X60|/0tns+fM2+(2(My3 +…+C"oMny+ (9) 2 From the admissibility condition we already have that the oh moment Mo=0 so that the first term in the right-hand side of (9)is zero. If we now manage to make the other moments up to M, zero as well, then the wavelet transform coefficients Ys, t)will decay as fast as sfor a smooth signal f(t). This is known in literature as the vanishing moments or approximation order. If a wavelet has N vanishing moments, then the approximation order of the wavelet transform is also N. The moments do not have to be exactly zero, a small value is often good enough. In fact, experimental research suggests that the number of vanishing moments required depends heavily on the application Summarizing, the admissibility condition gave us the wave, regularity and vanishing moments gave us the fast decay or the let, and put together they give us the wavelet. More about regularity"can be found for instance in [Bur98]and [Dau92 4. Discrete wavelets Now that we know what the wavelet transform is, we would like to make it practical. However, the wavelet transform as described so far still has three properties that make it difficult to use directly in the form of (1). The first is the redundancy of the CWT In (1)the wavelet transform is calculated by continuously shifting a continuously scalable function over a signal and calculating the correlation between the two. It will be clear that these scaled functions will be nowhere near an orthogonal basis and the obtained wavelet coefficients will therefore be highly edundant. For most practical applications we would like to remove this redundancy Even without the redundancy of the CWT we still have an infinite number of wavelets in the wavelet transform and we would like to see this number reduced to a more manageable count. This is the second problem we have The third problem is that for most functions the wavelet transforms have no analytical solutions and they can be calculated only numerically or by an optical analog computer. Fast algorithms are needed to be able to exploit the 3 There exist functions of which all moments vanish. An example is the function e-x.sin(x+) for x 20(Kor96] ems to stem from the definition that a filter is called K-regular if its z-transform has k zeroes at ze ies to the scaling filter(which has not been mentioned yet)and it is possible only if all wavelet moments to K he scaling filter is formed by the coefficients h(h)in equation( The CWt behaves just like an orthogonal transform in the sense that the inverse wavelet transform permits us to reconstruct the signal by an integration of all the projections of the signal onto the wavelet basis. This is called quasi-orthogonality [She96]
A Really Friendly Guide to Wavelets – © C. Valens, 1999 – c.valens@mindless.com 7 Here ƒ(p) stands for the p th derivative of ƒ and "(n+1) means the rest of the expansion. Now, if we define the moments of the wavelet by Mp, ∫ M = t ψ t dt p p ( , (8) ) then we can rewrite (7) into the finite development γ = + + + + + + + ( ) ! (0) ... 2! (0) 1! (0) (0) 1 ( ,0) 1 2 ( ) 3 2 (2) 2 1 (1) 0 n n n n M s O s n f M s f M s f f M s s s . (9) From the admissibility condition we already have that the 0th moment M0 = 0 so that the first term in the right-hand side of (9) is zero. If we now manage to make the other moments up to Mn zero as well, then the wavelet transform coefficients γ(s,-) will decay as fast as s n+2 for a smooth signal ƒ(t). This is known in literature as the vanishing moments3 or approximation order. If a wavelet has N vanishing moments, then the approximation order of the wavelet transform is also N. The moments do not have to be exactly zero, a small value is often good enough. In fact, experimental research suggests that the number of vanishing moments required depends heavily on the application [Cal96]. Summarizing, the admissibility condition gave us the wave, regularity and vanishing moments gave us the fast decay or the let, and put together they give us the wavelet. More about regularity4 can be found for instance in [Bur98] and [Dau92]. 4. Discrete wavelets Now that we know what the wavelet transform is, we would like to make it practical. However, the wavelet transform as described so far still has three properties that make it difficult to use directly in the form of (1). The first is the redundancy of the CWT. In (1) the wavelet transform is calculated by continuously shifting a continuously scalable function over a signal and calculating the correlation between the two. It will be clear that these scaled functions will be nowhere near an orthogonal basis5 and the obtained wavelet coefficients will therefore be highly redundant. For most practical applications we would like to remove this redundancy. Even without the redundancy of the CWT we still have an infinite number of wavelets in the wavelet transform and we would like to see this number reduced to a more manageable count. This is the second problem we have. The third problem is that for most functions the wavelet transforms have no analytical solutions and they can be calculated only numerically or by an optical analog computer. Fast algorithms are needed to be able to exploit the 3 There exist functions of which all moments vanish. An example is the function ) sin( 4 1/ 4 e x x ⋅ − for x ≥ 0 [Kör96]. 4 The term regularity seems to stem from the definition that a filter is called K-regular if its z-transform has K zeroes at z=eiπ . In wavelet theory this applies to the scaling filter (which has not been mentioned yet) and it is possible only if all wavelet moments up to K-1 vanish [Bur98]. The scaling filter is formed by the coefficients h(k) in equation (17). 5 The CWT behaves just like an orthogonal transform in the sense that the inverse wavelet transform permits us to reconstruct the signal by an integration of all the projections of the signal onto the wavelet basis. This is called quasi-orthogonality [She96]