Sparse signal representation and the tunable Q-factor wavelet - PowerPoint PPT Presentation

Sparse signal representation and the tunable Q-factor wavelet transform Ivan Selesnick Polytechnic Institute of New York University Brooklyn, New York 1

Introduction Problem: Decomposition of a signal into the sum of two components: 1. Oscillatory (rhythmic, tonal) component 2. Transient (non-oscillatory) component Outline 1. Signal resonance and Q-factors 2. Morphological component analysis (MCA) 3. Tunable Q-factor wavelet transform (TQWT) 4. Split augmented Lagrangian shrinkage algorithm (SALSA) 5. Examples References (MDCT, etc) S. N. Levine and J. O. Smith III. A sines+transients+noise audio representation for data compression and time/pitch scale modications . (1998) L. Daudet and B. Torr´ esani. Hybrid representations for audiophonic signal encoding . (2002) S. Molla and B. Torr´ esani. An hybrid audio coding scheme using hidden Markov models of waveforms . (2005) M. E. Davies and L. Daudet. Sparse audio representations using the MCLT. (2006) 2

Oscillatory (rhythmic) and Transient Components in EEG Many measured signals have both an oscillatory and a non-oscillatory component. EEG SIGNAL 40 20 0 − 20 − 40 0 1 2 3 4 5 6 7 8 9 10 TIME (SECONDS) Rhythms of the EEG: Delta 0 - 3 Hz Theta 4 - 7 Hz Alpha 8 - 12 Hz Beta 12 - 30 Hz Gamma 26 - 100 Hz Transients in EEG due to: 1) unwanted measurement artifacts 2) non-rhythmic brain activity (spikes, spindles, and vertex waves) 3

Signal resonance and Q-factor PULSE 1 SPECTRUM 1 10 Q − factor = 1.15 0 5 − 1 0 100 200 300 400 500 0 0 0.02 0.04 0.06 0.08 0.1 0.12 PULSE 2 1 40 Q − factor = 4.6 0 20 − 1 0 0 100 200 300 400 500 0 0.02 0.04 0.06 0.08 0.1 0.12 PULSE 3 1 20 Q − factor = 1.15 0 10 − 1 0 0 100 200 300 400 500 0 0.02 0.04 0.06 0.08 0.1 0.12 PULSE 4 100 1 Q − factor = 4.6 0 50 − 1 0 0 100 200 300 400 500 0 0.02 0.04 0.06 0.08 0.1 0.12 TIME (SAMPLES) FREQUENCY (CYCLES/SAMPLE) (a) Signals. (b) Spectra. Figure 1: The resonance of an isolated pulse can be quantified by its Q-factor, defined as the ratio of its center frequency to its bandwidth. Pulses 1 and 3, essentially a single cycle in duration, are low-resonance pulses. Pulses 2 and 4, whose oscillations are more sustained, are high-resonance pulses. 4

Resonance-based signal decomposition 2 2 TEST SIGNAL TEST SIGNAL 1 1 0 0 − 1 − 1 − 2 − 2 0 100 200 300 400 500 0 100 200 300 400 500 2 2 HIGH − RESONANCE COMPONENT LOW − PASS FILTERED 1 1 0 0 − 1 − 1 − 2 − 2 0 100 200 300 400 500 0 100 200 300 400 500 2 2 LOW − RESONANCE COMPONENT BAND − PASS FILTERED 1 1 0 0 − 1 − 1 − 2 − 2 0 100 200 300 400 500 0 100 200 300 400 500 2 2 RESIDUAL HIGH − PASS FILTERED 1 1 0 0 − 1 − 1 − 2 − 2 0 100 200 300 400 500 0 100 200 300 400 500 TIME (SAMPLES) TIME (SAMPLES) (a) Resonance-based decomposition. (b) Frequency-based filtering. Figure 2: Resonance- and frequency-based filtering. (a) Decomposition of a test signal into high- and low-resonance components. The high-resonance signal component is sparsely represented using a high Q-factor WT. Similarly, the low-resonance signal component is sparsely represented using a low Q-factor WT. (b) Decomposition of a test signal into low, mid, and high frequency components using LTI discrete-time filters. 5

Resonance-based signal decomposition must be nonlinear SIGNAL LOW − RESONANCE COMPONENT HIGH − RESONANCE COMPONENT = + = + = + = + = + = + = + Figure 3: Resonance-based signal decomposition must be nonlinear: The signal in the bottom left panel is the sum of the signals above it; however, the low-resonance component of a sum is not the sum of the low-resonance components. The same is true for the high-resonance component. Neither the low- nor high-resonance components satisfy the superposition property. F ( s 1 + · · · + s 6 ) 6 = F ( s 1 ) + · · · + F ( s 6 ) 6

Rational-dilation wavelet transform (RADWT) Prior work on rational-dilation wavelet transforms addresses the critically-sampled case. 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) RADWT (2009) gives a solution for the overcomplete case. Reference: Bayram, Selesnick. Frequency-domain design of overcomplete rational-dilation wavelet transforms. IEEE Trans. on Signal Processing, 57, August 2009. New: Tunable Q-factor wavelet transform — dilation need not be rational. 7

Low-pass Scaling x ( n ) y ( n ) LPS ↵ X( ω ) X( ω ) 1 1 0.5 0.5 0 0 0 0 −π −απ απ π −π π FREQUENCY ( ω ) FREQUENCY ( ω ) Y( ω ) Y( ω ) 1 1 0.5 0.5 0 0 0 −π / α 0 π / α −π π −π π FREQUENCY ( ω ) FREQUENCY ( ω ) Low-pass scaling with ↵ < 1 Low-pass scaling with ↵ > 1 8

High-pass Scaling x ( n ) HPS � y ( n ) X( ω ) X( ω ) 1 1 0.5 0.5 0 0 −π − (1 −β ) π 0 (1 −β ) π π −π 0 π FREQUENCY ( ω ) FREQUENCY ( ω ) Y( ω ) Y( ω ) 1 1 0.5 0.5 0 0 −π π −π − (1 − 1/ β ) π (1 − 1/ β ) π π 0 0 FREQUENCY ( ω ) FREQUENCY ( ω ) High-pass scaling with � < 1 High-pass scaling with � > 1 9

Tunable Q-factor wavelet transform (TQWT) v 0 ( n ) y 0 ( n ) H ⇤ H 0 ( ! ) LPS 1 / ↵ 0 ( ! ) LPS ↵ x ( n ) y ( n ) + H ⇤ H 1 ( ! ) HPS � HPS 1 / � 1 ( ! ) v 1 ( n ) y 1 ( n ) 0 < �  1 , 0 < ↵ < 1 , ↵ + � > 1 8 | H 0 ( ! ) | 2 X ( ! ) | ! |  ↵ ⇡ > < Y 0 ( ! ) = > 0 ↵ ⇡ < | ! |  ⇡ : 8 0 | ! | < (1 � � ) ⇡ > < Y 1 ( ! ) = | H 1 ( ! ) | 2 X ( ! ) > (1 � � ) ⇡  | ! |  ⇡ : 8 | H 0 ( ! ) | 2 X ( ! ) | ! | < (1 � � ) ⇡ > > > > > > < ( | H 0 ( ! ) | 2 + | H 1 ( ! ) | 2 ) X ( ! ) Y ( ! ) = (1 � � ) ⇡  | ! | < ↵ ⇡ > > > | H 1 ( ! ) | 2 X ( ! ) > > ↵ ⇡  | ! |  ⇡ > : 10

For perfect reconstruction, the filters should satisfy | H 0 ( ! ) | = 1 , H 1 ( ! ) = 0 , | ! |  (1 � � ) ⇡ H 0 ( ! ) = 0 , | H 1 ( ! ) | = 1 , ↵ ⇡  | ! |  ⇡ The transition bands of H 0 ( ! ) and of H 1 ( ! ) must be chosen so that | H 0 ( ! ) | 2 + | H 1 ( ! ) | 2 = 1 (1 � � ) ⇡ < | ! | < ↵ ⇡ . (1) H0( ω ) 1 0.5 0 −π −απ − (1 −β ) π (1 −β ) π απ π 0 FREQUENCY ( ω ) H1( ω ) 1 0.5 0 −π −απ − (1 −β ) π 0 (1 −β ) π απ π FREQUENCY ( ω ) 11

The transition band of H 0 ( ! ) and H 1 ( ! ) can be constructed using any power-complementary function, ✓ ( ! ) , ✓ 2 ( ! ) + ✓ 2 ( ⇡ � ! ) = 1 , (2) We use the Daubechies filter frequency response with two vanishing moments, ✓ ( ! ) = 1 p 2 (1 + cos( ! )) 2 � cos( ! ) , | ! |  ⇡ . (3) Scale and dilate ✓ ( ! ) to obtain transition bands for H 0 ( ! ) and H 1 ( ! ) . 12

X( ω ) 1 0.5 0 −π −απ − (1 −β ) π 0 (1 −β ) π απ π ω (a) Fourier transform of input signal, X ( ω ) . H0( ω ) H1( ω ) 1 1 0.5 0.5 0 0 −π −απ − (1 −β ) π 0 (1 −β ) π απ π −π −απ − (1 −β ) π 0 (1 −β ) π απ π FREQUENCY ( ω ) FREQUENCY ( ω ) (b) Frequency responses H 0 ( ω ) and H 1 ( ω ) . H0( ω ) X( ω ) H1( ω ) X( ω ) 1 1 0.5 0.5 0 0 −π −απ − (1 −β ) π (1 −β ) π απ π −π −απ − (1 −β ) π (1 −β ) π απ π 0 0 ω ω (c) Fourier transforms of input signal after filtering. V0( ω ) V1( ω ) 1 1 0.5 0.5 0 0 − 0.5 π 0 0.5 π − 0.5 π 0 0.5 π −π π −π π ω ω (d) Fourier transforms after scaling. 13

Iterated Filters c ( n ) stage 3 stage 2 d 3 ( n ) x ( n ) stage 1 d 2 ( n ) d 1 ( n ) Redundancy (total oversampling rate) for many stages: � r = 1 � ↵ When ↵  1 , �  1 , we have the system equivalence: x ( n ) H 0 ( ! ) · · · H 0 ( ! ) H 1 ( ! ) HPS � LPS ↵ LPS ↵ j � 1 stages H ( j ) ⌘ LPS ↵ j � 1 HPS � d j ( n ) 1 ( ! ) 14

The equivalent filter is 8 | ! | < (1 � � ) ↵ j � 1 ⇡ 0 > > > > j � 2 > > < (1 � � ) ↵ j � 1 ⇡  | ! |  ↵ j � 1 ⇡ Y H ( j ) H 1 ( ! / ↵ j � 1 ) H 0 ( ! / ↵ m ) 1 ( ! ) := (4) m =0 > > > > ↵ j � 1 ⇡ < | ! |  ⇡ . > > 0 : H j 1 ( ! ) ! ↵ j � 1 ⇡ (1 � � ) ↵ j � 1 ⇡ ⇡ 0 Q-factor: BW = 2 � � Q = ! c � Scaling factors ↵ , � : 2 ↵ = 1 � � � = Q + 1 , r 15

LOW Q-FACTOR WT HIGH Q-FACTOR WT FREQUENCY RESPONSES − 4 LEVELS FREQUENCY RESPONSES − 13 LEVELS alpha = 0.67, beta = 1.00, Q = 1.00, R = 3.00 alpha = 0.87, beta = 0.40, Q = 4.00, R = 3.00 1 1 0.5 0.5 0 0 0 π /4 π /2 3 π /4 0 π /4 π /2 3 π /4 π π FREQUENCY ( ω ) FREQUENCY ( ω ) WAVELET WAVELET 0.1 0.1 0.05 0.05 0 0 − 0.05 − 0.05 − 0.1 − 0.1 0 50 100 150 200 250 300 0 50 100 150 200 250 300 TIME (SAMPLES) TIME (SAMPLES) Q-factor = 1 Q-factor = 4 Redundancy = 3 Redundancy = 3 16

Finite-length Signals... The TWQT can be implemented for finite-length signals using the DFT (FFT).... v 0 ( n ) y 0 ( n ) H ⇤ H 0 ( k ) 0 ( k ) LPS N : N 0 LPS N 0 : N x ( n ) y ( n ) + H ⇤ H 1 ( k ) 1 ( k ) HPS N : N 1 HPS N 1 : N v 1 ( n ) y 1 ( n ) 17