Sparse Time-Frequency Transforms and Applications. Bruno Torr´ esani http://www.cmi.univ-mrs.fr/~torresan LATP, Universit´ e de Provence, Marseille DAFx, Montreal, September 2006 B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 1 / 41
1 Introduction 2 Signal waveform representations Bases Frames Multiple frames More realistic time-frequency atoms ? 3 Coefficient domain models Hybrid random waveform models Estimation algorithms based on observed coefficients Estimation algorithms based on synthesis coefficients 4 Conclusion 5 References B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 2 / 41
Introduction Introduction During the last twenty years (and much more than that in fact): harmonic analysis has provided many new techniques for expanding signals into “elementary” waveforms. Redundant Gabor wavelet systems (frames) Wavelet bases MDCT and wilson bases Matching pursuit and cognates ... Most often, sparsity of the representation was a key issue. B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 3 / 41
Introduction Introduction During the last twenty years (and much more than that in fact): harmonic analysis has provided many new techniques for expanding signals into “elementary” waveforms. Redundant Gabor wavelet systems (frames) Wavelet bases MDCT and wilson bases Matching pursuit and cognates ... Most often, sparsity of the representation was a key issue. In this talk: we review a number of such approaches, in view of a few selected applications. B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 3 / 41
Introduction Introduction: What is sparsity ? A signal representation is sparse when most information is concentrated in a small amount of data (coefficients). For example, a sine wave is sparsely represented in the Fourier domain, not in the time domain. Sparsity is an “vague” concept. Ideally, the volume of data (number of coefficients for example) would be a good sparsity measure. B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 4 / 41
Introduction Introduction: What is sparsity ? A signal representation is sparse when most information is concentrated in a small amount of data (coefficients). For example, a sine wave is sparsely represented in the Fourier domain, not in the time domain. Sparsity is an “vague” concept. Ideally, the volume of data (number of coefficients for example) would be a good sparsity measure. In noisy situations, this measure is generally polluted by a large number of small coefficients, originating from noise. Other measures may be used (entropies)... but they often do not yield the same results [Jaillet & BT 2003]. B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 4 / 41
Introduction Introduction: sparsity: what for ? A sparse time-frequency representation concentrates the relevant information in a small amount of coefficients: the pdf of the coefficients is peaked at 0, and heavy tailed. Most popular applications Signal coding... if the cost of encoding the representation itself is not too high Signal modeling: expand signals into components that make sense . Denoising: most often, noise is not sparse. Source separation (exploiting dimension reduction). ... B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 5 / 41
Introduction 1 Introduction 2 Signal waveform representations Bases Frames Multiple frames More realistic time-frequency atoms ? 3 Coefficient domain models Hybrid random waveform models Estimation algorithms based on observed coefficients Estimation algorithms based on synthesis coefficients 4 Conclusion 5 References B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 6 / 41
Signal waveform representations Signal representations Signal waveform expansion: decompose a signal as a linear combination of “elementary waveforms” ψ λ , often generated using simple rules. � x ( t ) = α λ ψ λ ( t ) λ with α λ the coefficients, and ψ λ the waveforms. Examples: Time-frequency atoms (MDCT or Wilson bases, Gabor atoms,...) Time-scale atoms (wavelets, multiwavelets,...) Chirplets,... Higher dimensional versions See [Mallat 1998], [Carmona et al. 1998] or [Wickerhauser 1994]. B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 7 / 41
Signal waveform representations Bases Signal representations: bases The mathematically simplest situation: orthonormal bases. The waveform system W = { ψ λ , λ ∈ Λ } is an orthonormal basis of the signal space (inner product space, or Hilbert space) H is The atoms are mutually orthogonal and normalized: � ψ λ , ψ µ � = δ µν They form a complete set in H : if the signal x ∈ H is such that � x , ψ λ � = 0 for all λ ∈ Λ, then x = 0. Then, any signal may be written in an unique way as � x ( t ) = α λ ψ λ ( t ) , α λ = � x , ψ λ � with λ ∈ Λ Thus, analysis and synthesis involve the same atoms. In addition, the “coefficient mapping” x → { α λ , λ ∈ Λ } preserves energy (Parseval’s formula) | α λ | 2 = � x � 2 . � λ ∈ Λ B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 8 / 41
Signal waveform representations Bases Signal representations: bases MDCT basis: smooth windows modulated by a sinusoidal function. In the continuous-time setting, the following (infinite) family of functions forms an orthonormal basis of L 2 ( R ). � π � 2 � n + 1 � � u kn ( t ) = w k ( t ) cos ( t − a k ) k ∈ Z , n = 0 , 1 , 2 , . . . , ℓ k ℓ k 2 In bounded intervals, as well as finite dimensional settings, similar bases may be constructed (Malvar, Suter, ...) B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 9 / 41
Signal waveform representations Bases Signal representations: bases More precisely, the only assumption is that the window functions w k must satisfy some symmetry conditions at boundaries. In general, windows are taken as regular translates of a single one. More freedom may be introduced, as long as the symmetry conditions are fullfilled. For example, some audio coders use systems with wide and narrow windows: B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 10 / 41
Signal waveform representations Bases Signal representations: bases More precisely, the only assumption is that the window functions w k must satisfy some symmetry conditions at boundaries. In general, windows are taken as regular translates of a single one. More freedom may be introduced, as long as the symmetry conditions are fullfilled. For example, some audio coders use systems with wide and narrow windows: Simple implementations are available on the Wavelab Stanford package: http://www-stat.stanford.edu/~wavelab B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 10 / 41
Signal waveform representations Bases Signal representations: bases MDCT basis is well adapted for audio signals: the expansion of most signals is sparse. See below: pdf (log scale) of MDCT coefficients of some organ recording. Besides signal coding/compression, sparsity also helps for several applications. Application: denoising : as noise is generally not sparse in the MDCT basis, simply threshold the MDCT coefficients of the noisy signal before reconstruction. Organ signal; Noisy organ signal; Denoised organ signal. B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 11 / 41
Signal waveform representations Bases Signal representations: bases Application: source separation : Consider two mixtures (linear combinations): Mix 1; Mix 2. Below: scatter plots of the samples of mix 1 against mix 2 (left), and the mdct coefficients of mix 1 against mix 2 (right). B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 12 / 41
Signal waveform representations Bases Signal representations: bases Application: source separation : Consider two mixtures (linear combinations): Mix 1; Mix 2. Below: scatter plots of the samples of mix 1 against mix 2 (left), and the mdct coefficients of mix 1 against mix 2 (right). Method: identify the two directions, and project. Reconstructed organ; B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 12 / 41
Signal waveform representations Bases Signal representations: bases Advantages: “Optimal” in terms of redundancy. There exist bases for which fast algorithms have been developed (MDCT, Wilson, wavelets,...) Drawbacks: Being an orthonormal basis has a price: rigidity. Not any window function will generate a basis. Mathematically speaking, windows are not as smooth as one would like. Being a basis also imposes constraints on the sampling in time and frequency. No “free access” to the time-frequency domain. Question: can we make it sparser by introducing redundancy ? B. Torr´ esani (LATP Marseille) Sparse Time-Frequency Transforms September 2006 13 / 41
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