An Adaptive Approach for Contextual Audio ... - Semantic Scholar
An Adaptive Approach for Contextual Audio Denoising using Local Fisher Information Alexandre L. M. Levada1 http://www.dc.ufscar.br/˜alexandre D´ebora C. Corrˆea2 http://cyvision.ifsc.usp.br/˜deboracorrea 1 Computer
Science Department, Federal University of S˜ ao Carlos (UFSCar)
2 Physics
Institute of S˜ ao Carlos, University of S˜ ao Paulo (USP)
2011
Overview
1
Introduction
2
Adaptive Gaussian Filtering
3
Fisher Information on MRF Models
4
The Proposed Wiener-Fisher Filter
5
Experiments and Results
6
Conclusions
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Motivation
Lately, with the advent of VOIP applications audio denoising has regained popularity. In computerized tomography, radiologists are becoming aware of cancer risks and have been working with low radiation doses, which causes noise in the 1-D projections. Projections show a significant increase in the number of cancers due to excessive X-ray exposure
Nowadays, signal denoising is a crucial application in several areas of science and engineering.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Objectives
Propose a generalized improved version of the pointwise adaptive Wiener filter, by exploring an information-theoretic measure Basically, our approach is based on the minimization of the observed local Fisher information in a Gaussian Markov Random Field model with respect to the dependency parameter (β).
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Adaptive Gaussian Filtering We consider signals corrupted by additive Gaussian noise. zi = xi + ηi
(1)
where ηi satisfies E [ηi ] = 0 and E [ηi ηj ] = σn2 δi,j According to (1) we have
E [xi ] = E [zi ] h i h i E (xi − x¯i )2 = E (zi − z¯i )2 − σn2
(2) (3)
where the statistical moments are estimated by the sample mean (¯ zi ) and the sample variance (¯ σz2 ) using an adaptive windowing scheme. ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
The Pointwise Adaptive Wiener Filter
Under these hypothesis, a local linear minimum mean square error filter (LLMMSE) can be derived (Lee, 1980) xˆi = x¯i +
σ ¯x2 (zi − x¯i ) σ ¯x2 + σn2
(4)
where xˆi is always a value between the sample mean x¯i and the observation zi . In the Bayesian approach, this is the optimum filter in case of Gaussian priors.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information in MRF Models Fisher Information A measure of information that an observation conveys about an unknown parameter For i.i.d random variables this information is the same for every observation In MRF’s, different configuration patterns provide distinct contribution to the global Fisher Information
" I (θ) = Eθ
2 # 2 ∂ ∂ ` (θ) = −Eθ ` (θ) ∂θ ∂θ2
(5)
In our approach, ` (θ) represents the logarithm of the pseudo-likelihood function (local model)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Observed Fisher Information
The expected Fisher Information is a global measure An alternative is to use the observed Fisher Information
∂ Iobs (θ) = log PL(θ) ∂θ
2 =−
∂2 log PL(θ) ∂θ2
(6)
An unbiased estimator for the observed Fisher Information is 2 N X ∂ 1 ˆI 1 (θ) = log p(xi |ηi , θ) obs N ∂θ i=1
ISCAS 2011 - International Symposium on Circuits and Systems
(7)
θ=θˆ
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information as a Likelihood Measure Defining the local observed Fisher information by ϕβ (xi ) =
∂ log p xi |ηi , β, θ~ ∂β
2
(8) ~ θ~ˆ ˆ θ= β=β,
we have that ϕβ (xi ) is nothing more than the square of the slope ˆ of the tangent line to the local likelihood in the point β.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information as a Likelihood Measure Defining the local observed Fisher information by ϕβ (xi ) =
∂ log p xi |ηi , β, θ~ ∂β
2
(8) ~ θ~ˆ ˆ θ= β=β,
we have that ϕβ (xi ) is nothing more than the square of the slope ˆ of the tangent line to the local likelihood in the point β.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
The Gaussian MRF Model Especially suitable for continuous data Completely characterized by a set of local conditional density functions
2 X 1 1 p xi |ηi , µ, σ 2 , β = √ exp − 2 xi − µ − β (xj − µ) 2πσ 2 2σ xj ∈ηi
β is the parameter that controls the spatial dependency between neighboring elements and ηi denotes the neighborhood system
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information on GMRF Models
Replacing the LCDF in equation (7) leads to observed Fisher information 2 N X X X 1 1 1 ˆIobs xi − µ − (β) = β (x − µ) (x − µ) j j σ2 N x ∈η x ∈η i=1
j
i
j
i
and a trivial condition for its minimization is xi = µ +
X
β (xj − µ)
(9)
xj ∈ηi
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information Distribution What is the typical behavior of the observed Fisher information in audio signals ?
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Generalizing the pointwise Wiener Filter
From equation (9), considering a non-stationary MRF with βi σi2 , σn2 =
σi2 σi2 + σn2
we have: xi = µi +
σi2 σi2 + σn2
X
(xj − µi )
This, comparing to the pointwise adaptive Wiener filter σi2 xˆi = µi + (zi − µi ) σi2 + σn2
ISCAS 2011 - International Symposium on Circuits and Systems
(10)
xj ∈ηi
(11)
Adaptive Approach for Audio Denoising using Fisher Information
The Wiener-Fisher Filter Applying a linear convex combination on equations (11) and (10) xˆi = x¯i +
σ ¯x2 2 σ ¯x + σn2
α (zi − x¯i ) + (1 − α)
X
(zj − x¯i )
(12)
zj ∈ηi
α controls the tradeoff between minimization of the local MSE and minimization of the local Fisher information In homogeneous regions, the filter is quite similar to the pointwise version (ˆ xi → µ i ) In heteregenous regions (points with high Fisher information), the performance of the proposed filter is significantly different
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Experiments and Results Audio signal corrupted by additive white noise (32.748 dB) Four metrics to evaluate the results: PSNR ISNR NRR (Noise-Reduction Ratio) SDR (Signal-to-Distiortion Ration)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Results
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Noise Reduction x Distortion Distortion is bigger in the proposed method SDR x NRR for different values of α Best tradeoff between noise reduction and distortion (PSNR)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
ISNR as a funtion of the α parameter Proposed filter shows superior performance for a wide range of α values. Incorporates dependency between neighboring elements (signal samples are not completely independent)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Graphical Comparison
Signal filtered by Wiener-Fisher is smoother
(a) Wiener filter
ISCAS 2011 - International Symposium on Circuits and Systems
(b) Proposed Contextual filter
Adaptive Approach for Audio Denoising using Fisher Information
Conclusions
Tradeoff between the minimization of the local MSE and the minimization of the local observed Fisher information can significantly improve the signal denoising performance Despite its simplicity, the proposed Wiener-Fisher filter is effective, since it can improve denoising performance without any considerable computational cost Future works may include a study on how the local Fisher information can be incorporated into different noise models and filters as well as the use of speech signals.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Questions ?
Thank you very much!!!
Alexandre L. M. Levada Federal University of S˜ ao Carlos (UFSCar) http://www.dc.ufscar.br/˜alexandre
D´ebora C. Corrˆea University of S˜ ao Paulo (USP) http://cyvision.ifsc.usp.br/˜deboracorrea
”The mind that opens to a new idea never returns to its original size.” (Albert Einstein)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Science Department, Federal University of S˜ ao Carlos (UFSCar)
2 Physics
Institute of S˜ ao Carlos, University of S˜ ao Paulo (USP)
2011
Overview
1
Introduction
2
Adaptive Gaussian Filtering
3
Fisher Information on MRF Models
4
The Proposed Wiener-Fisher Filter
5
Experiments and Results
6
Conclusions
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Motivation
Lately, with the advent of VOIP applications audio denoising has regained popularity. In computerized tomography, radiologists are becoming aware of cancer risks and have been working with low radiation doses, which causes noise in the 1-D projections. Projections show a significant increase in the number of cancers due to excessive X-ray exposure
Nowadays, signal denoising is a crucial application in several areas of science and engineering.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Objectives
Propose a generalized improved version of the pointwise adaptive Wiener filter, by exploring an information-theoretic measure Basically, our approach is based on the minimization of the observed local Fisher information in a Gaussian Markov Random Field model with respect to the dependency parameter (β).
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Adaptive Gaussian Filtering We consider signals corrupted by additive Gaussian noise. zi = xi + ηi
(1)
where ηi satisfies E [ηi ] = 0 and E [ηi ηj ] = σn2 δi,j According to (1) we have
E [xi ] = E [zi ] h i h i E (xi − x¯i )2 = E (zi − z¯i )2 − σn2
(2) (3)
where the statistical moments are estimated by the sample mean (¯ zi ) and the sample variance (¯ σz2 ) using an adaptive windowing scheme. ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
The Pointwise Adaptive Wiener Filter
Under these hypothesis, a local linear minimum mean square error filter (LLMMSE) can be derived (Lee, 1980) xˆi = x¯i +
σ ¯x2 (zi − x¯i ) σ ¯x2 + σn2
(4)
where xˆi is always a value between the sample mean x¯i and the observation zi . In the Bayesian approach, this is the optimum filter in case of Gaussian priors.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information in MRF Models Fisher Information A measure of information that an observation conveys about an unknown parameter For i.i.d random variables this information is the same for every observation In MRF’s, different configuration patterns provide distinct contribution to the global Fisher Information
" I (θ) = Eθ
2 # 2 ∂ ∂ ` (θ) = −Eθ ` (θ) ∂θ ∂θ2
(5)
In our approach, ` (θ) represents the logarithm of the pseudo-likelihood function (local model)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Observed Fisher Information
The expected Fisher Information is a global measure An alternative is to use the observed Fisher Information
∂ Iobs (θ) = log PL(θ) ∂θ
2 =−
∂2 log PL(θ) ∂θ2
(6)
An unbiased estimator for the observed Fisher Information is 2 N X ∂ 1 ˆI 1 (θ) = log p(xi |ηi , θ) obs N ∂θ i=1
ISCAS 2011 - International Symposium on Circuits and Systems
(7)
θ=θˆ
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information as a Likelihood Measure Defining the local observed Fisher information by ϕβ (xi ) =
∂ log p xi |ηi , β, θ~ ∂β
2
(8) ~ θ~ˆ ˆ θ= β=β,
we have that ϕβ (xi ) is nothing more than the square of the slope ˆ of the tangent line to the local likelihood in the point β.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information as a Likelihood Measure Defining the local observed Fisher information by ϕβ (xi ) =
∂ log p xi |ηi , β, θ~ ∂β
2
(8) ~ θ~ˆ ˆ θ= β=β,
we have that ϕβ (xi ) is nothing more than the square of the slope ˆ of the tangent line to the local likelihood in the point β.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
The Gaussian MRF Model Especially suitable for continuous data Completely characterized by a set of local conditional density functions
2 X 1 1 p xi |ηi , µ, σ 2 , β = √ exp − 2 xi − µ − β (xj − µ) 2πσ 2 2σ xj ∈ηi
β is the parameter that controls the spatial dependency between neighboring elements and ηi denotes the neighborhood system
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information on GMRF Models
Replacing the LCDF in equation (7) leads to observed Fisher information 2 N X X X 1 1 1 ˆIobs xi − µ − (β) = β (x − µ) (x − µ) j j σ2 N x ∈η x ∈η i=1
j
i
j
i
and a trivial condition for its minimization is xi = µ +
X
β (xj − µ)
(9)
xj ∈ηi
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Fisher Information Distribution What is the typical behavior of the observed Fisher information in audio signals ?
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Generalizing the pointwise Wiener Filter
From equation (9), considering a non-stationary MRF with βi σi2 , σn2 =
σi2 σi2 + σn2
we have: xi = µi +
σi2 σi2 + σn2
X
(xj − µi )
This, comparing to the pointwise adaptive Wiener filter σi2 xˆi = µi + (zi − µi ) σi2 + σn2
ISCAS 2011 - International Symposium on Circuits and Systems
(10)
xj ∈ηi
(11)
Adaptive Approach for Audio Denoising using Fisher Information
The Wiener-Fisher Filter Applying a linear convex combination on equations (11) and (10) xˆi = x¯i +
σ ¯x2 2 σ ¯x + σn2
α (zi − x¯i ) + (1 − α)
X
(zj − x¯i )
(12)
zj ∈ηi
α controls the tradeoff between minimization of the local MSE and minimization of the local Fisher information In homogeneous regions, the filter is quite similar to the pointwise version (ˆ xi → µ i ) In heteregenous regions (points with high Fisher information), the performance of the proposed filter is significantly different
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Experiments and Results Audio signal corrupted by additive white noise (32.748 dB) Four metrics to evaluate the results: PSNR ISNR NRR (Noise-Reduction Ratio) SDR (Signal-to-Distiortion Ration)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Results
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Noise Reduction x Distortion Distortion is bigger in the proposed method SDR x NRR for different values of α Best tradeoff between noise reduction and distortion (PSNR)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
ISNR as a funtion of the α parameter Proposed filter shows superior performance for a wide range of α values. Incorporates dependency between neighboring elements (signal samples are not completely independent)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Graphical Comparison
Signal filtered by Wiener-Fisher is smoother
(a) Wiener filter
ISCAS 2011 - International Symposium on Circuits and Systems
(b) Proposed Contextual filter
Adaptive Approach for Audio Denoising using Fisher Information
Conclusions
Tradeoff between the minimization of the local MSE and the minimization of the local observed Fisher information can significantly improve the signal denoising performance Despite its simplicity, the proposed Wiener-Fisher filter is effective, since it can improve denoising performance without any considerable computational cost Future works may include a study on how the local Fisher information can be incorporated into different noise models and filters as well as the use of speech signals.
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
Questions ?
Thank you very much!!!
Alexandre L. M. Levada Federal University of S˜ ao Carlos (UFSCar) http://www.dc.ufscar.br/˜alexandre
D´ebora C. Corrˆea University of S˜ ao Paulo (USP) http://cyvision.ifsc.usp.br/˜deboracorrea
”The mind that opens to a new idea never returns to its original size.” (Albert Einstein)
ISCAS 2011 - International Symposium on Circuits and Systems
Adaptive Approach for Audio Denoising using Fisher Information
