Frames for Psychoacoustics tics Peter Balazs Erblet transform and - - PowerPoint PPT Presentation

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Frames for Psychoacoustics tics Peter Balazs Erblet transform and - - PowerPoint PPT Presentation

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Frames for Psychoacous- Frames for Psychoacoustics tics Peter Balazs Erblet transform and perceptual sparsity ARI Frame Theory Multipliers Peter Balazs joint work with T. Necciari, B. Laback, N. Holighaus, D. Stoeva, ... Perceptual

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers Perceptual Sparsity by Irrelevance Conclusions

Frames for Psychoacoustics Erblet transform and perceptual sparsity

Peter Balazs

joint work with T. Necciari, B. Laback, N. Holighaus, D. Stoeva, ...

Acoustics Research Institute (ARI) Austrian Academy of Sciences, Vienna

February Fourier Talks 2014

Peter Balazs Frames for Psychoacoustics page 1

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers Perceptual Sparsity by Irrelevance Conclusions

Acoustics Research Institute (ARI)

Peter Balazs Frames for Psychoacoustics page 2

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers Perceptual Sparsity by Irrelevance Conclusions

ARI

Interdisciplinary research in acoustics, integrating acoustic phonetics, psychoacoustics and computational physics, based

  • n a solid mathematical background.

Excellence through Synergy

Peter Balazs Frames for Psychoacoustics page 3

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers Perceptual Sparsity by Irrelevance Conclusions

ARI

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers Perceptual Sparsity by Irrelevance Conclusions

Advantage of a strong mathematical background

Application-

  • riented

mathematics True inter- disciplinarity Synergy Novel methods

Peter Balazs Frames for Psychoacoustics page 5

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers Perceptual Sparsity by Irrelevance Conclusions

Overview:

1

Acoustics Research Institute (ARI)

2

Frame Theory Time-Frequency Representation Non-stationary Gabor Transform ERBlets

3

Frame Multipliers Mathematical Background

4

Perceptual Sparsity by Irrelevance

5

Conclusions

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Signal Representations: Time-Frequency Analysis and Frames

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Spectrogram

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Short Time Fourier Transformation (STFT)

Definition (see e.g. [Gr¨

  • chenig, 2001])

Let f,g = 0 in L2 Rd , then we call Vgf(τ, ω) =

  • Rd

f(x)g(x − τ)e−2πiωxdx the Short Time Fourier Transformation (STFT) of the signal f with the window g. Sampled Version is the Gabor transform: f → Vg(f)(a · k, b · l) = f, gk,l , where gk,l(t) = g(t − ka)ei2πlbt. When is perfect reconstruction possible?

Peter Balazs Frames for Psychoacoustics page 9

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Frames

Definition

The (countable) sequence Ψ = (ψk|k ∈ K) is called a frame for the Hilbert space H if constants A > 0 and B < ∞ exist such that A · f2

H ≤

  • k

|f, ψk|2 ≤ B · f2

H , ∀ f ∈ H.

[Duffin and Schaeffer, 1952, Daubechies et al., 1986] Beautiful abstract mathematical setting: Frames = generalization of bases; can be overcomplete, allowing redundant representations.

Redundancy

Active field of research in mathematics!

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Frame Theory

Interesting for applications: Much more freedom. Finding and constructing frames can be easier and faster. Some advantageous side constraints can only be fulfilled for frames. Perfect reconstruction is guaranteed with the ‘canonical dual frame’ ˜ ψk = S−1ψk f =

  • k

< f, ψk > ˜ ψk =

  • k

< f, ˜ ψk > ψk, where S is the frame operator Sf =

k

< f, ψk > ψk.

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Frame Theory: Non-stationary Gabor transform

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Non-stationary Gabor Transform

Limitations of Standard Gabor analysis: Quality of representation highly depends on window choice, but optimal window choice is different for different signal components

Frequency (Hz) Time (s)

Standard Gabor - Wide window

1000 2000 3000 4000 5000 6000 7000 8000 0.2 0.4 0.6 0.8 1 1.2 Frequency (Hz) Time (s)

Standard Gabor - Narrow window

1000 2000 3000 4000 5000 6000 7000 8000 0.2 0.4 0.6 0.8 1 1.2

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Non-stationary Gabor Transform

Our proposition [Balazs et al., 2011]: simple extension to reduce this limitation by using windows evolving over time.

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Non-stationary Gabor Transform

Given a sequence of windows (gn)n∈Z of L2 (R) and sequences

  • f real numbers (an)n∈Z and (bn)n∈Z, the non-stationary Gabor

transform (NSGT) elements are defined, for (m, n) ∈ Z2, by: gm,n(t) = gn(t − nan)ei2πmbnt. Regular structure in frequency allows FFT implementation. An analogue construction in the frequency domain allows easy implementation of, e.g. wavelet frames; an invertible CQT [Velasco et al., 2011].

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Non-stationary Gabor Transform

Sampling grid example:

Frequency Time Windows

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Non-stationary Gabor Transform

Frame theory allows perfect reconstruction. Particularly efficient in the ’painless’ case: Theorem For every n ∈ Z, let the function gn ∈ L2(R) be compactly supported with supp(gn) ⊆ [cn, dn] such that dn − cn ≤

1 bn .

The system of functions gm,n forms a frame for L2 (R) if and

  • nly if there exists A > 0 and B < ∞, such that

A ≤

n 1 bn |gn(t − nan)|2 ≤ B. In this case, the canonical dual

frame has the same structure and is given by: ˜ gm,n(t) = gn(t)

  • k

1 bk |gk(t − kak)|2 e2πimbnt.

(1)

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

Non-stationary Gabor Transform

Bird vocalization example:

Frequency (Hz) Time (s)

Nonstationary Gabor

1000 1500 2000 2500 3000 3500 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Frequency (Hz) Time (s)

Standard Gabor - Narrow window

1000 1500 2000 2500 3000 3500 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Frequency (Hz) Time (s)

Standard Gabor - Wide window

1000 1500 2000 2500 3000 3500 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

For an overview of adapted and adaptive time-frequency representations, see [Balazs et al., 2013].

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

ERBlets

Non-stationary Gabor transform adapted to human auditory perception [Necciari et al., 2013]: ERB-scale

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

ERBlets

Non-stationary Gabor transform adapted to human auditory perception [Necciari et al., 2013]: Relative reconstruction error: < 10−15. Implementation in LTFAT [Soendergaard et al., 2012].

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

ERBlets

Filterbank:

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory

Time-Frequency Representation NSGT ERBlets

Multipliers Perceptual Sparsity by Irrelevance Conclusions

ERBlets

Dual Filterbank:

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

What is a Frame Multiplier: Analysis ⇓ Multiplication ⇓ Synthesis

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

What is a Frame Multiplier: Analysis ⇓ Multiplication ⇓ Synthesis

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

What is a Frame Multiplier: Analysis ⇓ Multiplication ⇓ Synthesis

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

Multipliers

Those are operators, that are of utmost importance in Mathematics, where they are used for the diagonalization

  • f operators [Schatten, 1960].

Physics, where they are a link between classical and quantum mechanics, so called quantization operators [Ali et al., 2000]. Signal Processing, where they are a particular way to implement time-variant filters [Matz and Hlawatsch, 2002]. Acoustics, where those time-frequency filters are used in several fields, for example in Computational Auditory Scene Analysis [Wang and Brown, 2006].

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

Example for a Multiplier

Original audio file:

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

Example for a Multiplier

Symbol:

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

Example for a Multiplier

Result of Gabor Multiplier.

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

Frame Multipliers: Definition

Definition ([Balazs, 2007]) Let (ψk)k∈K, (φk)k∈K be frames for the Hilbert spaces H1 and

  • H2. Define the frame multiplier M(mk),(φk),(ψk) : H1 → H2 as

the operator M(mk),(φk),(ψk)f =

  • k

mk f, ψk φk, where m = (mk) is called the symbol.

Generalization of Gabor multipliers [Feichtinger and Nowak, 2003].

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers

Mathematical Background

Perceptual Sparsity by Irrelevance Conclusions

Fundamental Research in the Theory of Multipliers

We have invested quite some afford into the abstract setting, in particular investigating invertible multipliers, see e.g. [Stoeva and Balazs, 2012]. Theorem (B., Stoeva; submitted) Let Φ and Ψ be frames for H, and let m be semi-normalized. Let Mm,Φ,Ψ be invertible. Then there exist a dual frame Φ† of Φ and a dual frame Ψ† of Ψ, so that for any dual frame Φd of Φ and any dual frame Ψd of Ψ we have M−1

m,Φ,Ψ = M1/m,Ψ†,Φd = M1/m,Ψd,Φ†.

(2) The frames Ψ† are uniquely determined.

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Applications in Acoustics: Perceptual Sparsity by Irrelevance

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MP3-Player

MP3: encoding / decoding scheme MPEG1/MPEG2 (Layer 3) signal processing psychoacoustical masking model

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Psychoacoustic Masking: Introduction

Masking: presence of one stimulus, the masker, decreases the detectability of another stimulus, the target. Irrelevance Filter: searches (and deletes) perceptual irrelevant (masked, inaudible) data (in complex signals) using a masking model.

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Psychoacoustic Masking: Introduction

Masking: presence of one stimulus, the masker, decreases the detectability of another stimulus, the target. Irrelevance Filter: searches (and deletes) perceptual irrelevant (masked, inaudible) data (in complex signals) using a masking model.

Peter Balazs Frames for Psychoacoustics page 32

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Perceptual Sparsity by Irrelevance

Algorithm in : Original audio file

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Psychoacoustic Masking

Algorithm in : Filtered signal Residual

Adaptive Peter Balazs Frames for Psychoacoustics page 34

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Irrelevance Filter

Existing algorithm in : Original audio file (Spectrum)

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Irrelevance Filter

Existing algorithm in : Masked signal (Spectrum)

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Psychoacoustic Masking : simultaneous masking I

Existing Model, using bark scale

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Psychoacoustic Masking : simultaneous masking I

Existing Model, using bark scale

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Psychoacoustic Masking : simultaneous masking I

Existing Model, using bark scale

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers Perceptual Sparsity by Irrelevance Conclusions

Perceptual Sparsity by Irrelevance

The irrelevance method calculates an adaptive threshold function for each spectra of a Gabor transform. This corresponds to an adaptive Gabor frame multiplier with coefficients in {0, 1}.

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Frames for Psychoacous- tics Peter Balazs ARI Frame Theory Multipliers Perceptual Sparsity by Irrelevance Conclusions

Perceptual Sparsity: Current and Future Work

Extend to True Time-Frequency Model using Multipliers: Base it on ERBlets. Use new psychoacoustical data on time-frequency masking [Necciari et al., 2012]. Use this for improved audio codec!

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Perceptual Sparsity: Current and Future Work

Perceptual OMP: OMP reduces to 400 atoms, masking removes another 73.

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References

Conclusions

Frame theory is not only a very beautiful abstract setting, but also important for applications, in particular by linking it to human hearing!

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References

Thank you for your attention!

http://www.kfs.oeaw.ac.at http://magazine.orf.at/alpha/programm/2013/ 130513.htm

Peter Balazs Frames for Psychoacoustics page 42

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References

References: I

Ali, S. T., Antoine, J.-P., and Gazeau, J.-P. (2000). Coherent States, Wavelets and Their Generalization. Graduate Texts in Contemporary Physics. Springer New York. Balazs, P. (2007). Basic definition and properties of Bessel multipliers. Journal of Mathematical Analysis and Applications, 325(1):571–585. Balazs, P., D¨

  • rfler, M., Holighaus, N., Jaillet, F., and Velasco, G. (2011).

Theory, implementation and applications of nonstationary Gabor frames. Journal of Computational and Applied Mathematics, 236(6):1481–1496. Balazs, P., D¨

  • rfler, M., Kowalski, M., and Torr´

esani, B. (2013). Adapted and adaptive linear time-frequency representations: a synthesis point of view. IEEE Signal Processing Magazine (special issue: Time-Frequency Analysis and Applications), to appear:–. Daubechies, I., Grossmann, A., and Meyer, Y. (1986). Painless non-orthogonal expansions.

  • J. Math. Phys., 27:1271–1283.

Duffin, R. J. and Schaeffer, A. C. (1952). A class of nonharmonic Fourier series.

  • Trans. Amer. Math. Soc., 72:341–366.

Peter Balazs Frames for Psychoacoustics page 43

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References

References: II

Feichtinger, H. G. and Nowak, K. (2003). A first survey of Gabor multipliers, chapter 5, pages 99–128. Birkh¨ auser Boston. Gr¨

  • chenig, K. (2001).

Foundations of Time-Frequency Analysis. Birkh¨ auser Boston. Matz, G. and Hlawatsch, F. (2002). Linear Time-Frequency Filters: On-line Algorithms and Applications, chapter 6 in ’Application in Time-Frequency Signal Processing’, pages 205–271.

  • eds. A. Papandreou-Suppappola, Boca Raton (FL): CRC Press.

Necciari, T., Balazs, P., Holighaus, N., and Søndergaard, P. (2013). The ERBlet transform: An auditory-based time-frequency representation with perfect reconstruction. In Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), pages 498–502, Vancouver, Canada. IEEE. Necciari, T., Balazs, P., Kronland-Martinet, R., Ystad, S., Laback, B., Savel, S., and Meunier, S. (2012). Auditory time-frequency masking: Psychoacoustical data and application to audio representations. volume 7172 LNCS, pages 146–171. Schatten, R. (1960). Norm Ideals of Completely Continuous Operators. Springer Berlin. Peter Balazs Frames for Psychoacoustics page 44

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References

References: III

Soendergaard, P., Torr´ esani, B., and Balazs, P. (2012). The linear time frequency analysis toolbox. International Journal of Wavelets, Multiresolution and Information Processing, 10(4):1250032. Stoeva, D. T. and Balazs, P. (2012). Invertibility of multipliers. Applied and Computational Harmonic Analysis, 33(2):292–299. Velasco, G. A., Holighaus, N., D¨

  • rfler, M., and Grill, T. (2011).

Constructing an invertible constant-Q transform with non-stationary Gabor frames. volume Paris. AudioMiner;Locatif. Wang, D. and Brown, G. J. (2006). Computational Auditory Scene Analysis: Principles, Algorithms, and Applications. Wiley-IEEE Press. Peter Balazs Frames for Psychoacoustics page 45

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References

Orthonormal Basis (ONB)

Standard aproach: orthonormal basis. Problems: Perturbation Construction Error Robustness

Peter Balazs Frames for Psychoacoustics page 46

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References

Riesz Bases

Riesz bases Problems: Perturbation Construction Error Robustness

Peter Balazs Frames for Psychoacoustics page 47

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References

Frames

Alternate approach: introduce redundancy. Problems: Perturbation Construction Error Robustness

Back Peter Balazs Frames for Psychoacoustics page 48