TIME-DEPENDENT PARAMETRIC AND HARMONIC TEMPLATES IN NON-NEGATIVE MATRIX FACTORIZATION

Transcription

1 TIME-DEPENDENT PARAMETRIC AND HARMONIC TEMPLATES IN NON-NEGATIVE MATRIX FACTORIZATION 13 th International Conference on Digital Audio Effects Romain Hennequin, Roland Badeau and Bertrand David Telecom ParisTech September 8, 2010 Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 1/26

2 Introduction Musical spectrograms decomposition (on a basis of notes) Decomposition based on Non-negative Matrix Factorization (NMF) s are introduced into decomposition methods: parametric harmonic atoms makes it possible to model slight pitch variations Potential applications: Multipitch estimation/transcription Source separation Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 2/26

3 Sommaire Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 3/26

4 Contents Introduction Principle Issues Proposed solution 1 Principle Issues Proposed solution 2 3 Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 4/26

5 Principle of NMF Introduction Principle Issues Proposed solution Low-rank approximation: R V ˆV = WH ˆV ft = W fr H rt r=1 Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 5/26

6 Issues with NMF Introduction Principle Issues Proposed solution Pitch variations Low-rank approximation does not permit to model variations over time, such as slight pitch variations (vibrato...). Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 6/26

7 Issues with NMF Introduction Principle Issues Proposed solution Original spectrogram NMF spectrogram R = frequency (khz) 3 2 frequency (khz) time (frames) time (frames) Note with vibrato: Decomposition with a single atom. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 7/26

8 Issues with NMF Introduction Principle Issues Proposed solution Original spectrogram NMF spectrogram R = frequency (khz) 3 2 frequency (khz) time (frames) time (frames) Note with vibrato: Decomposition with 3 atoms. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 8/26

9 Proposed solution Introduction Principle Issues Proposed solution What does an atom look like in a musical spectrogram? In a musical spectrogram most of the (non-percussive) elements are instruments notes which are generally harmonic tones. Parameters of interest are generally the fundamental frequency of these tones, and the shape of the amplitudes of the harmonics. Proposed method: parametric model of spectrogram with harmonic atoms. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 9/26

10 Contents Introduction Parametric spectrogram Parametric atoms Algorithm 1 2 Parametric spectrogram Parametric atoms Algorithm 3 Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 10/26

11 Parametric spectrogram Parametric spectrogram Parametric atoms Algorithm Time-varying atoms in NMF: ˆV ft = R W fr H rt ˆV ft = r=1 R r=1 W θrt fr H rt θ rt is a time-varying parameter associated to each atom. In this paper, θ rt is the fundamental frequency f rt 0 of each atom. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 11/26

12 Parametric atoms Introduction Parametric spectrogram Parametric atoms Algorithm Parametric harmonic atom construction n h (f rt W f 0 rt fr = 0 ) k=1 a k g(f kf rt 0 ) Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 12/26

13 Parametric spectrogram Parametric spectrogram Parametric atoms Algorithm Hypotheses of the model The harmonic part of notes is supposed to be stationary within an analysis frame. Interferences between harmonics are supposed to be negligible. Classical hypothesis of NMF about positive summation of parts. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 13/26

14 Algorithm Introduction Parametric spectrogram Learnt parameters ˆV ft = Parametric spectrogram Parametric atoms Algorithm R n h a k g(f kf rt r=1 k=1 0 ) h rt }{{} W f 0 rt fr A divergence between V and ˆV is to be minimized w.r.t.: f0 rt : the fundamental frequency of each atom at each frame a k : the amplitudes of harmonics (Atoms share the same set of amplitudes) h rt : the activation of each atom at each frame Cost function: C(f rt 0, a k, h rt ) = D(V ft ˆV ft ) Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 14/26

15 Algorithm Introduction Parametric spectrogram Parametric atoms Algorithm Minimization Global optimization w.r.t. f rt 0 is impossible (numerous local minima in C). one atom is introduced for each MIDI note. Optimization thus becomes local (fine estimate of f rt 0 ). Minimization achieved with multiplicative update rules. Remark The proposed method is no longer a rank-reduction method but still reduces the data dimension. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 15/26

16 Contents Introduction Decomposition Improvement Estimated frequency Real signals Decomposition Improvement Estimated frequency Real signals Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 16/26

17 Decomposition Improvement Estimated frequency Real signals Decomposition of a synthetic spectrogram Original power spectrogram Frequency (khz) Time (frame) Spectrogram of the first bars of JS Bach s first prelude played by a synthesizer. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 17/26

18 Obtained decomposition Decomposition Improvement Estimated frequency Real signals Semitones Frames 35 Activations for each MIDI note. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 18/26

19 Obtained decomposition Decomposition Improvement Estimated frequency Real signals Decomposition Notes appear at the right place with decreasing amplitudes Numerous atoms activated at onset time Notes activated at octave, twelfth and double octave of the right note (note with many common partials). Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 19/26

20 Improvement Introduction Decomposition Improvement Estimated frequency Real signals Onset A few standard NMF atoms can be used to model onsets: ˆV ft = R r=1 W θrt fr H rt + K A fk B kt k=1 Octaves, twelfths... Add constraints to the cost function: Sparsity constraints on activations Decorrelation constraints (between activations of octaves...) Smoothness constraints on amplitudes Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 20/26

21 Obtained decomposition Decomposition Improvement Estimated frequency Real signals Semitones Frames Activations for each MIDI note. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 21/26

22 Time/frequency representation Decomposition Improvement Estimated frequency Real signals Semitones Frames Activations centered on estimated frequency for each MIDI note: vibrato appears. Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 22/26

23 Issues with real signals Decomposition Improvement Estimated frequency Real signals Semitones Frames Activations for each MIDI note. (Piano sound) Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 23/26

24 Issues with real signals Decomposition Improvement Estimated frequency Real signals Issues The model of amplitudes of harmonics is quite rough Issues with onsets and octaves are more important Noisy components (breath... ) Some instruments are not perfectly harmonic (piano... ) Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 24/26

25 Summary New way of decomposing musical spectrograms with slight pitch variations in constituting elements. Parametric thus flexible model. Perspectives Improve decomposition to make it more adapted to real data: Better modeling of harmonic amplitudes Supervised learning of amplitudes Better onset and noise modeling Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 25/26

26 Any questions? Romain Hennequin, Roland Badeau and Bertrand David Time-dependent parametric templates in NMF - slide 26/26