Frequency-Domain Design and Implementation of Overcomplete Rational-Dilation Wavelet Transforms Ivan Selesnick and Ilker Bayram Polytechnic Institute of New York University Brooklyn, New York 1
Rational-Dilation Wavelet Transforms 1. K. Nayebi, T. P. Barnwell III and M. J. T. Smith (1991) 2. P. Auscher (1992) 3. J. Kovacevic and M. Vetterli (1993) 4. T. Blu (1993, 1996, 1998) 5. A. Baussard, F. Nicolier and F. Truchetet (2004) 6. G. F. Choueiter and J. R. Glass (2007) Prior work on rational-dilation wavelet transforms addresses the critically-sampled case. We develop overcomplete rational-dilation wavelet transforms. 2
Rational-Dilation Wavelet Transforms 10 0 DYADIC WAVELET TRANSFORM (CRITICALLY SAMPLED) frequency 10 1 0 8 16 24 32 40 48 56 64 72 80 time 10 0 RATIONAL DILATION (1.5) WAVELET TRANSFORM (CRITICALLY SAMPLED) frequency 10 1 0 8 16 24 32 40 48 56 64 72 80 time 3
Rational-Dilation Wavelet Transforms 10 0 RATIONAL DILATION (1.5) WAVELET TRANSFORM (3X OVERCOMPLETE) frequency 10 1 0 8 16 24 32 40 48 56 64 72 80 time The overcomplete rational-dilation wavelet transform samples the time-frequency plane more densely in both time and frequency. 4
Oversampled-Sampled Rational Filter Bank The overcomplete rational-dilation wavelet transform is based on this filter bank. p H(ω) q q H (ω) p x(n) + y(n) G(ω) s s G (ω) where Y (ω) = q 1 k=0 L k (ω) = 1 p q M k (ω) = 1 s ( L k (ω) X ω + p k 2π q p 1 n=0 [ G ) + s 1 k=0 ( ω H p + k2π q + n2π p ( ω + k 2π ) s ] G (ω). ( M k (ω) X ω + k 2π s ) ( ω H p + n2π p ), ), There are no perfect reconstruction filters with rational-transfer functions unless the filter bank is orthonormal (p + 1 = q = s). 5
Oversampled-Sampled Rational Filter Bank p H(ω) q q H (ω) p x(n) + y(n) G(ω) s s G (ω) Perfect reconstruction is attained by ( pq ω 1 1 π s) p 0 ω ( 1 s) 1 π H(ω) = 0 ω [ G(ω) = π q, π] [ s ω p qπ, π] provided the transition-bands of H 0 (ω) and G 0 (ω) are chosen so as to satisfy 1 ( ω H p q p ) 2 + 1 s G (ω) 2 = 1. 6
The bands of the proposed perfect reconstruction filters are: pq H(ω) G(ω) s (s 1)π π (s 1)π 0 q p s s q π ω Let s use Daubechies-2vm frequency response for the transition-band for complementary behaviour. With q = 3, p = 2, s = 2: pπ 6 H(ω) G(ω) 2 0 0!/4!/3!/2 2!/3! ω 7
Equivalent Filters q p G(ω) s H(ω) (s 1)π 0 s q π ω pπ 3/2 H(ω) G(ω) 2 0 0!/2 2!/3! ω 8
Rational-Dilation WT Frequency Response: Low-Q 4 3 dilation q/p = 3/2, s = 2 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω / π 1.5 dilation q/p = 3/2, s = 1 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω / π 9
Rational-Dilation WT Frequency Response: Higher-Q 5 4 dilation q/p = 6/5, s = 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω / π 3 dilation q/p = 6/5, s = 2 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω / π 10
Rational-Dilation Wavelets dilation q/p = 3/2, s = 2 dilation q/p = 3/2, s = 1 dilation q/p = 6/5, s = 4 dilation q/p = 6/5, s = 2 The parameters p, q, and s affect the Q-factor and the degree of ringing. 11
Low Q-Factor Wavelet dilation q/p = 3/2, s = 1 For dilation factors close to 1 and s = 1, the wavelet more closely resembles the Mexican hat function. 12
High Q-Factor Wavelet dilation q/p = 6/5, s = 2 13
Redundancy Factor The redundancy of the j-level iterated filter bank is Red j (p, q, s) = 1 1 (p/q) j+1 s 1 p/q The redundancy of the wavelet transform is ( p + q ) j+1. Red(p, q, s) = lim j Red j (p, q, s) Red(p, q, s) = 1 s 1 1 p/q 14
Decomposition of a Signal The following pages illustrate the decomposition of a signal using the rational-dilation wavelet transform with s = 1 and different dilation factors. With s = 1 the wavelet is similar to the Mexican hat function. The first example has dilation factor of 2 (a dyadic WT). The subsequent examples have dilation factor closer to 1. The signal is a single row from a photographic image. 15
DILATION FACTOR = 2/1 16
DILATION FACTOR = 3/2 17
DILATION FACTOR = 4/3 18
DILATION FACTOR = 5/4 19
DILATION FACTOR = 6/5 20
Application to EEG Signal Decomposition The dyadic wavelet transforms is effective for the representation of piecewise smooth signals. However, its poor frequency resolution limits its effectiveness for representing oscillatory signals. High Q-factor WTs provide a more efficient representation of oscillatory signals. Many measured signals have both an oscillatory and a non-oscillatory component, for example EEG. 40 20 0 20 EEG SIGNAL 40 0 1 2 3 4 5 6 7 8 9 10 TIME (SECONDS) We model the transient component of an EEG as a piecewise smooth signal which can be sparsely represented using a low Q-factor wavelet transform. We model the rhythmic component as a signal that can be sparsely represented using a high Q-factor wavelet transform. 21
Morphological Component Analysis Morphological component analysis (MCA) provides an optimality principal for the decomposition of a signal s(t) into distinct components, s(t) = s 1 (t) + s 2 (t), provided each component is sparse in the domain of a specified transform. MCA assumes s 1 (t) can be sparsely represented in the frame {f i } i Z and s 2 (t) can be sparsely represented in the frame {g i } i Z. MCA proposes that s 1 (t) and s 2 (t) can be (approximately) recovered by: min a 1 0 + a 2 0, a 1,a 2 s.t. s(t) = i Λ 1 a 1 (i) f i (t) + i Λ 2 a 2 (i)g i (t) where a 0 denotes the number of non-zero values of the vector a. Image Decomposition via the Combination of Sparse Representations and a Variational Approach, J.-L. Starck, M. Elad and D. L. Donoho, IEEE Trans. on Image Proc., Oct 2005. 22
Decomposing EEG Signals into Rhythmic and Transient Components The transient and rhythmic components of an EEG signal can be separated using MCA with low Q-factor and high Q-factor wavelet transforms. 40 20 0 20 EEG SIGNAL 40 0 1 2 3 4 5 6 7 8 9 10 40 20 0 20 RHYTHMIC COMPONENT 40 0 1 2 3 4 5 6 7 8 9 10 40 20 0 20 TRANSIENT COMPONENT 40 0 1 2 3 4 5 6 7 8 9 10 23
Multi-resonance Signal Representation The use of both low Q-factor and high Q-factor wavelet transforms for sparse signal representation is a practical method to extract the oscillatory component from measured signals. Note the oscillatory component is not necessarily a high-pass signal; it may contain both low and high frequencies. Similarly, the non-oscillatory component is not necessarily a low-pass signal; it may contains sharp bumps and jumps. Therefore, the decomposition is not a frequency-based decomposition. Instead it is multi-resonance signal decomposition. A multi-resonance signal representation may be a useful preprocessing step for the frequency analysis of vibration phenomenon which can only be measured in the presence of a non-stationary disturbance. 24
Multi-resonance Signal Representation The continuous wavelet transform (CWT) can use many different wavelet functions including high Q-factor wavelets. However, the CWT is highly redundant and an implementation for discretetime data is not always easily inverted. Because an efficient stable inverse is important for MCA, the rational-dilation WT is preferable. The short-time Fourier transform, cosine modulated filter banks, and other transforms with uniform frequency decomposition provide finer resolution than the dyadic wavelet transform. However, the low-frequency analysis/synthesis functions of these transforms may be similar to the low-frequency wavelets of the rational-dilation WT. Because the success of MCA depends on the low coherence of the two frames utilized, the rational-dilation WT is preferable 25
2-D Radial Rational-Dilation Wavelet Transform Dilation = 3/2 Dilation = 4/3 Redundancy = 1/(1 (2/3) 2 ) = 1.8 Redundancy = 1/(1 (3/4) 2 ) = 2.28 26
2-D Radial Rational-Dilation Wavelet Transform The gradual change in scale between levels facilitates the tracking of wavelet-maxima across scales for edge detection and edge-based algorithms. Therefore it facilitates scale-space processing with analysis/synthesis algorithms. Isotropic transform with dilation factor q/p and redundancy 1 1 (p/q) 2. Perfect reconstruction requires s = 1. Example image: American eagle carousel ride. 27
dilation factor = 2, redundancy = 1.33 28
dilation factor = 3/2, redundancy = 1.8 29
dilation factor = 4/3, redundancy = 2.28 30
2-D Rational-Dilation Steerable Pyramid Substituting the rational-dilation isotropic transform for the dyadic one in the steerable pyramid gives a rationaldilation steerable pyramid. Dilation q/p = 3/2 and four directions (two directions shown): 31
Rational-Dilation Wavelet Transforms Self-inverting overcomplete wavelet transforms based on rational dilations can be developed using the frequency-domain design of over-sampled filter banks. The overcomplete rational-dilation wavelet transform: 1. Samples the time-frequency plane more densely in both time and frequency. = Exactly invertible, fully-discrete approximation of the continuous WT. 2. Can attain higher Q-factors (or the same low Q-factor) of the dyadic WT. = Can achieve the higher-frequency resolution needed for oscillatory signals. 3. Can attain a range of redundancies. 4. The wavelet can be well localized in the time-frequency plane. 5. Provides a more gradual scaling than the dyadic transform. 32