Perceptual declipping of audio signals through compressed sensing:
algorithm design and evaluation
Tussentijdse presentatie
Naim Mansour Promotor: Prof. dr. ir. Marc MoonenAssistent: Ir. Bruno Defraene
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Overzicht• Onderwerp & doelstellingen (vermelding Steven) – 3 min.• Compressed sensing – 5 min.
– Wat?– Theoretisch– Declipping (don’t forget perfect reconstruction)
• CS & Declipping – 4 min.– Specifieke theorie– Eerder werk (INRIA, AxBe)– Kort: perceptuele component
• Toelichting gemaakte keuzes & motivatie (2 keuzes) – 3 min.– Don’t forget frame length (basically all details)
• Implementatie & resultaten (demo) – 5 min.• Planning, en plannen voor fase 2 – 2 min.• Dank & vragen
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Overview• Subject• Compressed Sensing• CS & Declipping• Perceptual components• Extra: IRL1• Implementation
• Evaluation
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Subject
• Declipping of audio signals
• Through compressed sensing
• Perceptual
• Algorithm design & evaluation
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Compressed Sensing: general• Candès, Romberg, Tao – 2006• Recover sparse signal from sub-Nyquist rate sampled measurements• Consider the signal s, sparse in a fixed basis :
• Measurement basis selects reliable values from s according to ( is known as the sensing base):
• Reconstruction through constrained L0/L1 minimization:
𝒚𝑀× 1=𝜱𝑀×𝑁𝜳 𝑁×𝑁 𝒙𝑁 ×1=𝑨𝒙
𝒔=∑𝑖=1
𝑁
Ψ 𝑖𝑥 𝑖=𝜳 𝒙
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• Solution equals translation of null(A)-plane by vector z• L0 & L1 lead to sparse solutions, L2 doesn’t
• L1 minimization is convex -> convex optimization,
• L0 minimization non-convex -> greedy opt.
𝒙 ′=𝑎𝑟𝑔𝑚𝑖𝑛‖𝒛‖𝟎𝑠 .𝑡 .𝑨𝒛=𝒚𝒙 ′=𝑎𝑟𝑔𝑚𝑖𝑛‖𝒛‖𝟐𝑠 .𝑡 .… 𝒙 ′=𝑎𝑟𝑔𝑚𝑖𝑛‖𝒛‖𝟏𝑠 .𝑡 .…
Compressed Sensing: Choice of Lp
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Compressed Sensing: AxBe model• Other possible model (Bölcskei & Studer, – 2011)
• In case of clipping, we consider to be measurement including clipped samples (). No explicit measurement matrices, and , to obtain sparse error base ().
• Recovery through projected Lp minimization:
𝒚𝑀× 1=𝜱𝑀×𝑁 𝑠𝜳𝑁 𝑠×𝑁𝑠
𝒙𝑁𝑠× 1+𝜣𝑀×𝑁 𝑒
𝜠𝑁 𝑒×𝑁𝑒𝒆𝑁𝑒×1
=𝑨𝒙+𝑩𝒆
𝒚=𝒔+𝒆 ,𝒆=𝒔𝒄−𝒔 , 𝒔𝒄=𝑐𝑙𝑖𝑝𝑝𝑒𝑑𝑣𝑒𝑟𝑠𝑖𝑜𝑛𝑜𝑓 𝒔
𝒚𝑀× 1=𝜳𝑀 ×𝑀 𝒙𝑀 ×1+𝜠𝑀 ×𝑀 𝒆𝑀 ×1=𝑨𝒙+ 𝑰𝒆
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Compressed Sensing: Recovery• Certain theoretical bounds for perfect recovery of signal• Classical model (no noise assumption):
• AxBe model:
• Coherence of a basis: measure of decorrelation in analysis domain
• Fourier base: DCT base:
,
𝑛𝑥𝑛𝑒<( 1𝜇𝐴❑ )
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,𝑛𝑒=‖𝑒‖0
𝜇𝐴❑= max
𝑘, 𝑙 ,𝑘≠𝑙|𝒂𝑘
𝐻𝒂𝑙|
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CS & Declipping: recovery• Recovery ability dependent on coherence of sensing base• Classical CS: Usage of pseudorandom measurement matrices
(e.g. iid Gaussian sampling) leads to very low coherence• Declipping: reliable, “sampled” values in signal are unclipped ones
-> clearly not pseudorandom!• Coherence of combined Fourier/DCT base with clipping sensing base
= coherence Fourier/DCT• Recovery guarantees for DCT base ( reliable samples):
• Perfect recovery of real audio signals practically always impossible, since
M 900 800 700 600 500 400 300
nx (max) 8,6699 4,6459 3,3281 2,7202 2,2953 2,1393 1,89
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CS & Declipping: • Missing samples will always lie beyond the clipping threshold • Lp minimization can be improved through introduction of additional
linear constraints
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CS & Declipping: previous work• INRIA• Bölcskei
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Perceptual components• Perceptual weighting matrix based on acoustic loudness perception
• Psychoacoustically optimized (adaptive) basis
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Extra: IRL1• Iteratively reweighted L1 minimization (Candès, Wakin, Boyd – 2007)
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Implementation: general• 2 main choices
– PCS through bounded L1 minimization, using perceptual weighting, Axy & AxBe models (further improvement through IRL1)
– PCS through bounded L0 minimization, using psychoacoustic wavelet basis, Axy & AxBe models
• Incremental design: implementation & evaluation with & without bounds, with & without perceptual components,…
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Implementation: Clipping
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Evaluation: general
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Evaluation: SNR vs. PEAQ
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• SNR no guarantee for audio quality!!
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Planning & future prospects
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Planning & future prospects
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Planning & future prospects
• Semester 2– Execute psychoacoustic experiments– Finish algorithms – Write final texts
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References
• http://people.ee.duke.edu/~willett/SSP//Tutorials/ssp07-cs-tutorial.pdf
• Recovery of Sparsely Corrupted Signals blablabla
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?
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Zalig Kerstfeest!
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