Fast Wavelet/Framelet Transform for Signal/Image Processing.

Transcription

1 Fast Wavelet/Framelet Transform for Signal/Image Processing. The following is based on book manuscript: B. Han, Framelets Wavelets: Algorithms, Analysis Applications. To introduce a discrete framelet transform, we need some definitions notation. By l(z) we denote the linear space of all sequences v=v(k)} :Z Cof complex numbers on Z. One -dimensional discrete input data or signal is often treated as an element in l(z). Similarly, by l 0 (Z) we denote the linear space of all sequences u= u(k)} : Z C on Z such that k Z : u(k) 0} is a finite set. An element in l 0 (Z) is often regarded as a finite-impulse-response (FIR) filter (also called a finitely supported mask in the literature of wavelet analysis). In this book we often use u for a general filter v for a general signal or data. It is often convenient to use the formal Fourier series (or symbol) v of a sequence v=v(k)}, which is defined as follows: v(ξ) := v(k)e ikξ, ξ R, (1) where i in this book always denotes the imaginary unit. For v l 0 (Z), v is a π-periodic trigonometric polynomial. A discrete framelet transform can be described using two linear operators the subdivision operator the transition operator. For a filter u l 0 (Z) v l(z), the subdivision operator S u : l(z) l(z) is defined to be [S u v](n) := v(k)u(n k), n Z () the transition operator T u : l(z) l(z) is defined to be [T u v](n) := v(k)u(k n), n Z. (3) The transition operator plays the role of coarsening frequency-separating the data to lower resolution levels; while the subdivision operator plays the role of refining predicting the data to higher resolution levels. In terms of Fourier series, the subdivision operator S u in () the transition operator T u in (3) can be equivalently rewritten as Ŝ u v(ξ)= v(ξ)û(ξ), ξ R (4) T u v(ξ)= v(ξ/)û(ξ/)+ v(ξ/+π)û(ξ/+π), ξ R (5) for u,v l 0 (Z), where c denotes the complex conjugate of a complex number c C. Let ã, b 1,..., b s be filters for decomposition. For a positive integer J, a J-level discrete framelet decomposition is given by v j := T ãv j 1, w l, j := T b l v j 1, l=1,...,s, j = 1,...,J, (6) where v 0 :Z Cis an input signal. The filter ã is often called a dual low-pass filter the filters b 1,..., b s are called dual high-pass filters. After a J-level discrete framelet 1

2 decomposition, the original input signal v 0 is decomposed into one sequence v J of lowpass framelet coefficients sj sequences w l, j of high-pass framelet coefficients for l=1,...,s j = 1,...,J. Such framelet coefficients are often processed for various purposes. One of the most commonly employed operations is thresholding so that the low-pass framelet coefficients v J high-pass framelet coefficients w l, j become v J ẘ l, j, respectively. More precisely, ẘ l, j (k) = η(w l, j (k)),k Z, where η :C C is a thresholding function. For example, for a given threshold value > 0, the hard thresholding function η hard η hard (z)= z, if z ; 0, otherwise soft-thresholding function η η (z)= are defined to be z ε z z, if z ; 0, otherwise. Another commonly employed operation is quantization, which can be applied after or without thresholding. For example, for a given quantization level q>0, the quantization function Q :R qz is defined to be Q(x) := q q x + 1, x R, where is the floor function such that x =n if n x<n+1 for an integer n. (7) q q Figure 1: The hard thresholding function η hard, the soft thresholding function η, the quantization function, respectively. Both thresholding quantization operations are often used to process framelet coefficients in a discrete framelet transform. Let a,b 1,...,b s be filters for reconstruction. Now a J-level discrete framelet reconstruction is s v j 1 := S a v j + S bl ẘ l, j, j = J,...,1. (8) l=1 The filter a is often called a primal low-pass filter the filters b 1,...,b s are called primal high-pass filters. We say that (ã; b 1,..., b s },a;b 1,...,b s }) is a dual framelet filter bank if it satisfies the perfect reconstruction condition: [ ][ ] ã(ξ) b 1 (ξ) b s (ξ) â(ξ) b1 (ξ) bs (ξ) = I, ã(ξ + π) b 1 (ξ + π) b s (ξ + π) â(ξ + π) b1 (ξ + π) bs (ξ + π) (9) a;b 1,...,b s }) is called a tight framelet filter bank if(a;b 1,...,b s },a;b 1,...,b s }) is a dual framelet filter bank.

3 If s=1, a dual framelet filter bank(a;b},a;b}) is called a biorthogonal wavelet filter bank. If s=1, a tight framelet filter bank a;b} is called an orthogonal wavelet filter bank. In the following, let us provide a few examples to illustrate various types of filter banks. For a filter u=u(k)} such that u(k)=0 for all \[m,n] u(m)u(n) 0, we denote by fsupp(u) := [m,n] as its filter support. To list the filter u, we shall adopt the following notation throughout the book: u=u(m),u(m+1),...,u( 1),u(0),u(1),...,u(n 1),u(n)} [m,n], (10) where we underlined boldfaced the number u(0) to indicate its position at the origin. Example 1 a; b} is an orthogonal wavelet filter bank (called the Haar orthogonal wavelet filter bank), where a= 1, 1 } [0,1], b= 1, 1 } [0,1]. (11) Example (ã; b},a; b}) is a biorthogonal wavelet filter bank, where ã= 1 8, 1 4, 3 4, 1 4, 1 8 } [,], b= 1 4, 1, 1 4 } [0,], a= 1 4, 1, 1 4 } [ 1,1], b= 1 8, 1 4, 3 4, 1 4, 1 8 } [ 1,3]. Example 3 a;b 1,b } is a tight framelet filter bank, where a= 1 4, 1, 1 4 } [ 1,1], b 1 = 4,0, 4 } [ 1,1], b = 1 4, 1, 1 4 } [ 1,1]. Example 4 (ã; b 1, b },a;b 1,b }) is a dual framelet filter bank, where ã= 1, 1 } [0,1], b 1 = 1, 1 } [ 1,0], b = 1, 1 } [0,1] a= 1 8, 3 8, 3 8, 1 8 } [ 1,], b 1 = 1 4, 1 4 } [ 1,0], b = 1 8, 3 8, 3 8, 1 8 } [ 1,]. At the end of this section, we illustrate a one-level discrete framelet transform using the Haar orthogonal wavelet filter bank in (11). Let be a test input signal. Note that v=1,0, 1, 1, 4,60,58,56} [0,7] (1) [T a v](n)=v(n+1)+v(n), [T b v](n)=v(n+1) v(n), n Z. Therefore, we have the wavelet coefficients: w 0 = 1,,56,114} [0,3], w 1 = 1,0,64, } [0,3]. 3

4 On the other h, we have Hence, we have [S a w 0 ](n)= w 0 (n), [S b w 1 ](n)= w 1 (n), [S a w 0 ](n+1)= w 0 (n), [S b w 1 ](n+1)= w 1 (n), n Z n Z. S aw 0 = 1 1,1,,,56,56,114,114} [0,7], S bw 1 = 1 1, 1,0,0, 64,64,, } [0,7]. Clearly, we have the perfect reconstruction of the original input signal v: S aw 0 + S bw 1 =1,0, 1, 1, 4,60,58,56} [0,7] = v the following energy-preserving identity w 0 l (Z) + w 1 l (Z) = = 10119= v l (Z). Next, let us describe how to efficiently implement discrete framelet/wavelet transform. The subdivision operator the transition operator in applications are often implemented through the widely used convolution operation in mathematics engineering. For u l 0 (Z) v l(z), the convolution u v is defined to be [u v](n) := u(k)v(n k), n Z. (13) Note that û v(ξ) = û(ξ) v(ξ). To implement the subdivision transition operators using the convolution operation, we also need the upsampling downsampling operators on sequences in l(z). The downsampling (or decimation) operator d : l(z) l(z) the upsampling operator d : l(z) l(z) with a sampling factor d Z\0} are given by v(n/d), if n/d is an integer; [v d](n) := v(dn) [v d](n) := (14) 0, otherwise, for n Z. For a sequence v=v(k)}, we denote its complex conjugate sequence reflected about the origin by v, which is defined to be v (k) := v( k),. Note that v (ξ) = v(ξ). Now the subdivision operator S u in () the transition operator T u in (3) can be equivalently expressed as follows: S u v=(v ) u T u v=(v u ). (15) 4

5 For u=u(k)} γ Z, we define the associated coset sequence u [γ] of u at the coset γ+ Z by u [γ] (ξ) := u(γ+ k)e ikξ, i.e., u [γ] = u(γ+ ) =u(γ+ k)}. (16) Using the coset sequences of u, we can rewrite (15) as [S u v] [0] = v u [0], [S u v] [1] = v u [1], T u v= ( v [0] (u [0] ) + v [1] (u [1] ) ). (17) ã a ã b 1 b1 a b s bs b 1 b1 input output b s bs Figure : Diagram of a two-level discrete framelet transform employing filter banks ã; b 1,..., b s } a;b 1,...,b s }. Note that T b l v = (v b l ) b l v = (v ) b l forl=1,...,s. 5