![Page 1: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/1.jpg)
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
![Page 2: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/2.jpg)
2
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
![Page 3: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/3.jpg)
3
Overview• Subject• Compressed Sensing• CS & Declipping• Perceptual components• Extra: IRL1• Implementation
• Evaluation
![Page 4: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/4.jpg)
4
Subject
• Declipping of audio signals
• Through compressed sensing
• Perceptual
• Algorithm design & evaluation
![Page 5: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/5.jpg)
5
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
![Page 6: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/6.jpg)
6
• 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
![Page 7: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/7.jpg)
7
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
![Page 8: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/8.jpg)
8
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❑ )
2
,𝑛𝑒=‖𝑒‖0
𝜇𝐴❑= max
𝑘, 𝑙 ,𝑘≠𝑙|𝒂𝑘
𝐻𝒂𝑙|
![Page 9: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/9.jpg)
9
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
![Page 10: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/10.jpg)
10
CS & Declipping: • Missing samples will always lie beyond the clipping threshold • Lp minimization can be improved through introduction of additional
linear constraints
![Page 11: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/11.jpg)
11
CS & Declipping: previous work• INRIA• Bölcskei
![Page 12: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/12.jpg)
12
Perceptual components• Perceptual weighting matrix based on acoustic loudness perception
• Psychoacoustically optimized (adaptive) basis
![Page 13: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/13.jpg)
13
Extra: IRL1• Iteratively reweighted L1 minimization (Candès, Wakin, Boyd – 2007)
![Page 14: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/14.jpg)
14
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,…
![Page 15: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/15.jpg)
15
![Page 16: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/16.jpg)
16
Implementation: Clipping
![Page 17: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/17.jpg)
17
Evaluation: general
![Page 18: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/18.jpg)
18
Evaluation: SNR vs. PEAQ
![Page 19: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/19.jpg)
19
• SNR no guarantee for audio quality!!
![Page 20: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/20.jpg)
20
Planning & future prospects
![Page 21: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/21.jpg)
21
Planning & future prospects
![Page 22: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/22.jpg)
22
Planning & future prospects
• Semester 2– Execute psychoacoustic experiments– Finish algorithms – Write final texts
![Page 23: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/23.jpg)
23
References
• http://people.ee.duke.edu/~willett/SSP//Tutorials/ssp07-cs-tutorial.pdf
• Recovery of Sparsely Corrupted Signals blablabla
![Page 24: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/24.jpg)
24
?
![Page 25: Perceptual declipping of audio signals through compressed sensing: algorithm design and evaluation Tussentijdse presentatie Naim MansourPromotor: Prof](https://reader035.vdocuments.mx/reader035/viewer/2022062312/551a7516550346b52d8b510b/html5/thumbnails/25.jpg)
25
Zalig Kerstfeest!