## The phaselet transform - an integral redundancy nearly shift-invariant wavelet transform (2003)

Venue: | IEEE Trans. on Signal Proc |

Citations: | 12 - 1 self |

### BibTeX

@ARTICLE{Gopinath03thephaselet,

author = {Ramesh A. Gopinath},

title = {The phaselet transform - an integral redundancy nearly shift-invariant wavelet transform},

journal = {IEEE Trans. on Signal Proc},

year = {2003},

volume = {51},

pages = {1792--1805}

}

### OpenURL

### Abstract

This paper introduces an approximately shift invariant redundant dyadic wavelet transform- the phaselet transform- that includes the popular dual-tree complex wavelet transform of Kingsbury [1] as a special case. The main idea is to use a finite set of wavelets that are related to each other in a special way- and hence called phaselets-to achieve approximate shift-redundancy; bigger the set better the approximation. A sufficient condition on the associated scaling filters to achieve this is that they are frac-tional shifts of each other. Algorithms for the design of phaselets with a fixed number vanishing moments is presented- building upon the work of Selesnick [2] for the design of wavelet pairs for Kingsbury’s dual-tree complex wavelet transform. Construction of 2-dimensional directional bases from tensor products of 1-d phaselets is also described. Phaselets as a new approach to redundant wavelet transforms and their construction are both novel and should be interesting to the reader independently of the approximate shift invariance property that this paper argues they possess. 1

### Citations

1587 | Biorthogonal bases of compactly supported wavelets
- Cohen, Daubechies, et al.
- 1992
(Show Context)
Citation Context ...atisfying (6). In particular, this section constructs a special family of phaselet frames which are close in spirit to Daubechies’ construction of biorthogonal wavelet bases with linear phase filters =-=[21, 10]-=-. Such a wavelet basis is associated with pairs of scaling and wavelet filters that are duals of each other. As described by Selesnick in [2] the reason to consider this case is that one can obtain wa... |

853 |
Multirate Systems and Filter Banks
- Vaidyanathan
- 1992
(Show Context)
Citation Context ...ssociated real-valued filter coefficients. A dyadic wavelet tight frame can be constructed from a two-channel unitary filter bank when the (synthesis) filters (h0, h1) satisfy the following relations =-=[10, 11, 12]-=-: � h0(n)h0(2k + n) = δ(k), n � h0(n) = n √ 2, h1(n) = (−1) n h0(−n + 1). (2) In terms of Z-transforms H0(z)H0(z −1 ) + H0(−z)H0(−z −1 ) = 2, H0(1) = √ 2, H1(z) = −z −1 H0(−z −1 ). (3) In particular H... |

543 |
Orthogonal polynomials
- Szegö
- 1991
(Show Context)
Citation Context ...lated to the L th Jacobi polynomial, specifically P τ,−τ L a bilinear transformation. Jacobi polynomials are orthogonal polynomials on [−1, 1] with respect to the weight function (1 − x) α (1 + x) β (=-=[23]-=- page 68): P α,β ⎛ ⎞ ⎛ ⎞ L� L + β L + α L (x) = 2−L ⎝ ⎠ ⎝ ⎠ (x − 1) i=0 i L − i i (x + 1) L−i , (39) where ⎛ ⎝ a ⎞ ⎠ = i Also note that P α,β P τ,−τ L a(a − 1) . . . (a − i + 1) i! L (−x) = (−1)LP β,α... |

430 | Shiftable multiscale transforms
- Simoncelli, Freeman, et al.
- 1992
(Show Context)
Citation Context ...e of the total energy in the transform coefficients at a particular scale. Many approaches have been suggested to address this problem - the common theme being the use of redundant wavelet transforms =-=[3]-=-. The 1ssimplest redundant wavelet transform is the undecimated wavelet transform that has a redundancy of J where J is the number of wavelet scales used in the transform (typically J ≈ log N, where N... |

333 |
Introduction to Wavelets and Wavelet Transforms: A
- Burrus, Gopinath, et al.
- 1997
(Show Context)
Citation Context ...ssociated real-valued filter coefficients. A dyadic wavelet tight frame can be constructed from a two-channel unitary filter bank when the (synthesis) filters (h0, h1) satisfy the following relations =-=[10, 11, 12]-=-: � h0(n)h0(2k + n) = δ(k), n � h0(n) = n √ 2, h1(n) = (−1) n h0(−n + 1). (2) In terms of Z-transforms H0(z)H0(z −1 ) + H0(−z)H0(−z −1 ) = 2, H0(1) = √ 2, H1(z) = −z −1 H0(−z −1 ). (3) In particular H... |

262 |
Complex wavelets for shift invariant analysis and filtering of signals
- Kingsbury
- 2001
(Show Context)
Citation Context ...avelets (i.e., wavelets whose Fourier transform is supported on IR + ). The importance of Hardy wavelets in signal processing applications is clearly well established in the seminal work of Kingsbury =-=[17, 15, 1]-=-. Let (ψ 0 1 , ψ1 1 ) be a phaselet pair. Without loss of generality let τ 0 = 0 and τ 1 = τ. From (16) ψ1 1 = Uτ ψ0 1 1 . If τ = 2 , then, the phaselets form a Hilbert transform pair and therefore ψ0... |

123 | 2000,”The dual-tree complex wavelet transform with improved orthogonality and symmetry properties
- Kingsbury
(Show Context)
Citation Context ...k∈ Z value of l, 0 ≤ l ≤ n − 1, ψl � � �� 1jk , ˜ψ l 1jk also constitutes a pair of dual wavelet j,k∈ Z frames. We now show that phaselets generalize the notion of Hilbert transform pairs of wavelets =-=[1, 15, 16, 2]-=-. Consider a Hilbert transform pair of wavelets - two wavelets ψ 0 1 and ψ1 1 such that ˆ ψ 1 1 (ω) = −ısign(ω) ˆ ψ 0 1 (ω). Clearly the functions ψ0 and ψ1 form a redundancy 2 phaselet family - in (1... |

82 |
Compactly supported tight and sibling frames with maximum vanishing moments
- Chui, He, et al.
(Show Context)
Citation Context ...ion based dyadic wavelet bases have been generalized to the case where dilates and translates of not just one but a finite set of wavelets, referred to as framelets, generate a frame (or tight frame) =-=[4, 5, 6, 7]-=-. Framelets and the associated framelet transforms are related to perfect reconstruction over-sampled filter banks with a decimation factor of 2. The extension of phaselets to framelets is described i... |

74 |
Splitting the unit delay
- Laasko, Valimaki, et al.
- 1996
(Show Context)
Citation Context ...an filter is given by ⎛ L� F0,τ,L(z) = 1 + ⎝ L ⎞ (L − τ)(L − 1 − τ) . . . (L − i + 1 − τ) ⎠ z i (τ + 1)(τ + 2) . . . (τ + i) −i . (21) i=1 The associated L th order Thiran all-pass filter is given by =-=[18, 19]-=- Aτ,L(z) = z−L F0,τ,L(z −1 ) F0,τ,L(z) ∆ = F1,τ,L(z) ≈ z−τ F0,τ,L(z) Aτ,L(z) is allpass and approximates the delay by τ samples. near z = 1. (22) We now give a simple recipe for generating products of... |

60 |
Wavelets and recursive filter banks
- Herley, Vetterli
- 1993
(Show Context)
Citation Context ...nstruction of phaselet tight frames and frames associated with filter banks with infinite impulse response (IIR) filters. The methodology is similar to the construction of IIR orthonormal wavelets in =-=[22, 2]-=-. 7.1 IIR Phaselet Tight Frames Let the filters h l 0 be of the form H l 0(z) = 1 Q(z2 ) (1 + z−1 ) K k−1 � Fbl i ,2 i=0 i /n,Li (z) ∆ = 1 Q(z2 ) El (z). (33) They clearly satisfy the requirements of ... |

53 | Hilbert transform pairs of wavelet bases
- Selesnick
- 2001
(Show Context)
Citation Context ...k∈ Z value of l, 0 ≤ l ≤ n − 1, ψl � � �� 1jk , ˜ψ l 1jk also constitutes a pair of dual wavelet j,k∈ Z frames. We now show that phaselets generalize the notion of Hilbert transform pairs of wavelets =-=[1, 15, 16, 2]-=-. Consider a Hilbert transform pair of wavelets - two wavelets ψ 0 1 and ψ1 1 such that ˆ ψ 1 1 (ω) = −ısign(ω) ˆ ψ 0 1 (ω). Clearly the functions ψ0 and ψ1 form a redundancy 2 phaselet family - in (1... |

38 |
Recursive digital filters with maximally flat group delay
- Thiran
- 1971
(Show Context)
Citation Context ...an filter is given by ⎛ L� F0,τ,L(z) = 1 + ⎝ L ⎞ (L − τ)(L − 1 − τ) . . . (L − i + 1 − τ) ⎠ z i (τ + 1)(τ + 2) . . . (τ + i) −i . (21) i=1 The associated L th order Thiran all-pass filter is given by =-=[18, 19]-=- Aτ,L(z) = z−L F0,τ,L(z −1 ) F0,τ,L(z) ∆ = F1,τ,L(z) ≈ z−τ F0,τ,L(z) Aτ,L(z) is allpass and approximates the delay by τ samples. near z = 1. (22) We now give a simple recipe for generating products of... |

36 | The design of approximate Hilbert transform pairs of wavelet bases - Selesnick |

26 |
Image processing with complex wavelets,” Philos
- Kingsbury
- 1999
(Show Context)
Citation Context ...bstract This paper introduces an approximately shift invariant redundant dyadic wavelet transform - the phaselet transform - that includes the popular dual-tree complex wavelet transform of Kingsbury =-=[1]-=- as a special case. The main idea is to use a finite set of wavelets that are related to each other in a special way - and hence called phaselets - to achieve approximate shift-redundancy; bigger the ... |

8 |
Wavelet and Filter Banks - New Results and Applications
- Gopinath
- 1992
(Show Context)
Citation Context ...iewpoint. The Fourier transform at a given frequency can be identified with a point in IR 2 � . Now, every tight frame ϕl� n−1 of unit-norm vectors, i.e., � l=0 � �ϕl�� � = 1, in IR 2 is of the form (=-=[13]-=- page 122) ϕ l ⎡ cos πτ l = ⎣ sin πτ l ⎤ ⎦ , l ∈ {0, 1, . . . , n − 1} , such that n−1 � l=0 ı2πτ l e = 0, n ≥ 2. (13) In particular for uniformly spaced τ l in [0, 1) - corresponding to nth roots of ... |

7 | Phaselets of framelets
- Gopinath
(Show Context)
Citation Context ...associated framelet transforms are related to perfect reconstruction over-sampled filter banks with a decimation factor of 2. The extension of phaselets to framelets is described in a companion paper =-=[8]-=-. An extension of Hilbert transform pairs of wavelets to a framelets is also given in [9]. The rest of the paper is organized as follows. Section 2 gives the notation and background. Section 3 looks a... |

6 |
An equivalence relation between multiresolution analyses
- Papadakis, Stavropoulos, et al.
- 1995
(Show Context)
Citation Context ...kψ1 = D 2 jTkUτ ψ1 = (Uτ ψ1)jk. Equivalently it suffices to show that Uτ commutes with the translation operator Tk and dilation operator D 2 j for j, k ∈ ZZ. For Uτ defined in (16) this follows from (=-=[14]-=-, Proposition 2.3). Indeed directly from (16) for any function f, translation Tδ and dilation Da, f ∈ L 2 (IR), δ ∈ IR and a ∈ IR + , (Uτ �Tδf)(ω) = e −ıπτsign(ω) Tδf(ω) � = e −ıωδ Uτ �f(ω) = � (TδUτ ... |

4 |
Affine systems in L 2 (IR d ) II: dual system
- Ron, Shen
- 1997
(Show Context)
Citation Context ...ion based dyadic wavelet bases have been generalized to the case where dilates and translates of not just one but a finite set of wavelets, referred to as framelets, generate a frame (or tight frame) =-=[4, 5, 6, 7]-=-. Framelets and the associated framelet transforms are related to perfect reconstruction over-sampled filter banks with a decimation factor of 2. The extension of phaselets to framelets is described i... |

3 |
Framelets: MRA based Construction of Wavelet Frames
- Daubechies, Han, et al.
(Show Context)
Citation Context ...ion based dyadic wavelet bases have been generalized to the case where dilates and translates of not just one but a finite set of wavelets, referred to as framelets, generate a frame (or tight frame) =-=[4, 5, 6, 7]-=-. Framelets and the associated framelet transforms are related to perfect reconstruction over-sampled filter banks with a decimation factor of 2. The extension of phaselets to framelets is described i... |

3 |
The Double Density Dual Tree Wavelet transform
- Selesnick
(Show Context)
Citation Context ...banks with a decimation factor of 2. The extension of phaselets to framelets is described in a companion paper [8]. An extension of Hilbert transform pairs of wavelets to a framelets is also given in =-=[9]-=-. The rest of the paper is organized as follows. Section 2 gives the notation and background. Section 3 looks at shift-invariance of the wavelet transform, defines phaselets and describes the construc... |

2 |
Affine Frames in L 2 (IR d ): Analysis of the Analysis Operator
- Ron, Shen
- 1997
(Show Context)
Citation Context |