Recent Advances in Multichannel Source Separation and Denoising Based on Source Sparseness

Abstract

This chapter deals with multichannel source separation and denoising based on sparseness of source signals in the time-frequency domain. In this approach, time-frequency masks are typically estimated based on clustering of source location features, such as time and level differences between microphones. In this chapter, we describe the approach and its recent advances. Especially, we introduce a recently proposed clustering method, observation vector clustering, which has attracted attention for its effectiveness. We introduce algorithms for observation vector clustering based on a complex Watson mixture model (cWMM), a complex Bingham mixture model (cBMM), and a complex Gaussian mixture model (cGMM). We show through experiments the effectiveness of observation vector clustering in source separation and denoising.

Appendices

Appendix 1 Derivation of cWMM-Based Mask Estimation Algorithm

Here we derive the cWMM-based mask estimation algorithm in Sect. 11.3.1. The derivation of the E-step is straightforward and omitted. The update rules for the M-step is obtained by maximizing the following Q-function with respect to $\varTheta _{\text {W},f}$:

$$\begin{aligned} Q(\varTheta _{\text {W},f})&\triangleq \sum _{t=1}^T\sum _k\tilde{\gamma }^{(k)}_{tf}\ln \Biggl [\alpha ^{(k)}_fp_\text {W}\Bigl (\mathbf {z}_{tf};\mathbf {a}^{(k)}_f,\kappa ^{(k)}_f\Bigr )\Biggr ] \end{aligned}$$

(11.52)

$$\begin{aligned}&=\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln \alpha ^{(k)}_f -\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln \mathscr {K}\Bigl (1,M;\kappa ^{(k)}_{f}\Bigr ) \end{aligned}$$

(11.53)

$$\begin{aligned}&{=}+\sum _k\kappa ^{(k)}_{f}\mathbf {a}^{(k)\textsf {H}}_f\biggl ( \sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\mathbf {z}_{tf}\mathbf {z}^{\textsf {H}}_{tf} \biggr )\mathbf {a}^{(k)}_f+C \nonumber \\&=\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\Biggl [\ln \alpha ^{(k)}_f -\ln \mathscr {K}\Bigl (1,M;\kappa ^{(k)}_{f}\Bigr )+\kappa ^{(k)}_{f}\mathbf {a}^{(k)\textsf {H}}_f\mathbf {R}^{(k)}_f\mathbf {a}^{(k)}_f\Biggr ]+C. \end{aligned}$$

(11.54)

Here, $\mathbf {R}^{(k)}_f$ is defined by

$$\begin{aligned} \mathbf {R}^{(k)}_f\triangleq \frac{ \sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\mathbf {z}_{tf}\mathbf {z}^{\textsf {H}}_{tf}}{\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}}, \end{aligned}$$

(11.55)

and C denotes a constant independent of $\varTheta _{\text {W},f}$.

The update rule for $\alpha ^{(k)}_f$ is obvious: note the constraint (11.18) and apply the Lagrangian multiplier method.

The update rule for $\mathbf {a}^{(k)}_f$ is obtained by maximizing $Q(\varTheta _{\text {W},f})$ subject to (11.19). Noting (11.26), we see that this is equivalent to maximizing $\mathbf {a}^{(k)\textsf {H}}_f\mathbf {R}^{(k)}_f\mathbf {a}^{(k)}_f$ subject to (11.19). From the linear algebra, $\mathbf {a}^{(k)}_f$ is therefore a unit-norm principal eigenvector of $\mathbf {R}^{(k)}_f$.

The update rule for $\kappa ^{(k)}_f$ is obtained by maximizing

$$\begin{aligned} -\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln \mathscr {K}\Bigl (1,M;\kappa ^{(k)}_{f}\Bigr )+ \biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\kappa ^{(k)}_{f}\mathbf {a}^{(k)\textsf {H}}_f\mathbf {R}^{(k)}_f\mathbf {a}^{(k)}_f. \end{aligned}$$

(11.56)

Since $\mathbf {a}^{(k)}_f$ is a unit-norm principal eigenvector of $\mathbf {R}^{(k)}_f$, we have

$$\begin{aligned} \mathbf {a}^{(k)\textsf {H}}_f\mathbf {R}^{(k)}_f\mathbf {a}^{(k)}_f=\lambda ^{(k)}_f, \end{aligned}$$

(11.57)

where $\lambda ^{(k)}_f$ is the principal eigenvalue of $\mathbf {R}^{(k)}_f$. Therefore, we have the following nonlinear equation for $\kappa ^{(k)}_{f}$:

$$\begin{aligned} \frac{\partial }{\partial \kappa ^{(k)}_{f}}\mathscr {K}\Bigl (1,M;\kappa ^{(k)}_{f}\Bigr )=\lambda ^{(k)}_f\mathscr {K}\Bigl (1,M;\kappa ^{(k)}_{f}\Bigr ). \end{aligned}$$

(11.58)

Using (3.8) in [37], (11.58) is approximately solved as follows:

$$\begin{aligned} \kappa ^{(k)}_{f}=\frac{M\lambda ^{(k)}_f-1}{2\lambda ^{(k)}_f\Bigl (1-\lambda ^{(k)}_f\Bigr )}\Biggl [1+\sqrt{\displaystyle 1+\frac{4(M+1)\lambda ^{(k)}_f\Bigl (1-\lambda ^{(k)}_f\Bigr )}{M-1}}\Biggr ]. \end{aligned}$$

(11.59)

Appendix 2 Derivation of cBMM-Based Mask Estimation Algorithm

Here we derive the cBMM-based mask estimation algorithm in Sect. 11.3.2. The update rule for the E-step is obvious. The update rules for the M-step is obtained by maximizing the following Q-function with respect to $\varTheta _{\text {B},f}$:

$$\begin{aligned} Q(\varTheta _{\text {B},f})&\triangleq \sum _{t=1}^T\sum _k\tilde{\gamma }^{(k)}_{tf}\ln \Biggl [\alpha ^{(k)}_fp_\text {B}\Bigl (\mathbf {z}_{tf};\mathbf {B}^{(k)}_f\Bigr )\Biggr ] \end{aligned}$$

(11.60)

$$\begin{aligned}&=\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln \alpha ^{(k)}_f -\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln c\Bigl (\mathbf {B}^{(k)}_{f}\Bigr ) \end{aligned}$$

(11.61)

$$\begin{aligned}&{=}+\sum _k\text {tr}\Biggl [\mathbf {B}^{(k)}_{f}\biggl ( \sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\mathbf {z}_{tf}\mathbf {z}^{\textsf {H}}_{tf} \biggr )\Biggr ] \nonumber \\&=\sum _k\biggl (\sum _{t=1}^T{\gamma }^{(k)}_{tf}\biggr )\Biggl [\ln \alpha ^{(k)}_f-\ln c\Bigl (\mathbf {B}^{(k)}_{f}\Bigr )+\text {tr}\Bigl (\mathbf {B}^{(k)}_{f}\mathbf {R}^{(k)}_{f}\Bigr )\Biggr ]. \end{aligned}$$

(11.62)

Here, $c(\mathbf {B})$ is defined by (11.36), and $\mathbf {R}^{(k)}_f$ by (11.55).

The update rule for $\alpha ^{(k)}_f$ is obvious.

To derive the update rule for $\mathbf {B}^{(k)}_{f}$, let us denote the mth largest eigenvalue of $\mathbf {R}^{(k)}_{f}$ by $\lambda ^{(k)}_{fm}$ and a corresponding unit-norm eigenvector by $\mathbf {v}^{(k)}_{fm}$. We assume that $\lambda ^{(k)}_{fm},$ $m=1,\dots ,M,$ are all distinct and positive, which is always true in practice. $\mathbf {R}^{(k)}_{f}$ is represented as

$$\begin{aligned} \mathbf {R}^{(k)}_{f}=\sum _{m=1}^M\lambda ^{(k)}_{fm}\mathbf {v}^{(k)}_{fm}\mathbf {v}^{(k)\textsf {H}}_{fm}. \end{aligned}$$

(11.63)

From a result in [38], $\mathbf {v}^{(k)}_{fm},$ $m=1,\dots ,M$, are also the eigenvectors of $\mathbf {B}^{(k)}_{f}$. Hence, $\mathbf {B}^{(k)}_{f}$ is represented in the form

$$\begin{aligned} \mathbf {B}^{(k)}_{f}=\sum _{m=1}^M\beta ^{(k)}_{fm}\mathbf {v}^{(k)}_{fm}\mathbf {v}^{(k)\textsf {H}}_{fm}. \end{aligned}$$

(11.64)

Substituting (11.63) and (11.64) into (11.62) and disregarding terms independent of $\beta ^{(k)}_{fm}, m=1,\dots ,M,$ we have

$$\begin{aligned} \biggl (\sum _{t=1}^T{\gamma }^{(k)}_{tf}\biggr )\Biggl [-\ln c\Bigl (\mathbf {B}^{(k)}_{f}\Bigr )+\sum _{m=1}^M \lambda ^{(k)}_{fm}\beta ^{(k)}_{fm} \Biggr ]. \end{aligned}$$

(11.65)

Therefore, we have

$$\begin{aligned} \frac{\partial \ln c\Bigl (\mathbf {B}^{(k)}_{f}\Bigr )}{\partial \beta ^{(k)}_{fm}}=\lambda ^{(k)}_{fm}. \end{aligned}$$

(11.66)

Using an approximation in [38], this nonlinear equation can be approximately solved as follows:

$$\begin{aligned} \beta ^{(k)}_{fm}\sim -\frac{1}{\lambda ^{(k)}_{fm}}. \end{aligned}$$

(11.67)

Substituting (11.67) into (11.64) and adding a matrix of the form $\xi \mathbf {I}$ so that the largest eigenvalue of $\mathbf {B}^{(k)}_{f}$ is zero, we obtain the following update rule for $\mathbf {B}^{(k)}_{f}$:

$$\begin{aligned} \mathbf {B}^{(k)}_{f}\leftarrow \sum _{m=1}^M\Biggl (-\frac{1}{\lambda ^{(k)}_{fm}}+\frac{1}{\lambda ^{(k)}_{f1}}\Biggr )\mathbf {v}^{(k)}_{fm}\mathbf {v}^{(k)\textsf {H}}_{fm}. \end{aligned}$$

(11.68)

Appendix 3 Derivation of cGMM-Based Mask Estimation Algorithm

Here we derive the cGMM-based mask estimation algorithm in Sect. 11.3.3. The derivation of the E-step is straightforward and omitted. The update rules for the M-step is obtained by maximizing the following Q-function with respect to $\varTheta _{\text {G},f}$:

$$\begin{aligned} Q(\varTheta _{\text {G},f})&\triangleq \sum _{t=1}^T\sum _k\tilde{\gamma }^{(k)}_{tf}\ln \biggl [\alpha ^{(k)}_fp_\text {G}\Bigl (\mathbf {y}_{tf};0,\phi ^{(k)}_{tf}\mathbf {B}^{(k)}_f\Bigr )\biggr ] \end{aligned}$$

(11.69)

$$\begin{aligned}&=\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln \alpha ^{(k)}_f -M\sum _{t=1}^T\sum _k\tilde{\gamma }^{(k)}_{tf}\ln \phi ^{(k)}_{tf} \end{aligned}$$

(11.70)

$$\begin{aligned}&{=}-\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln \det \mathbf {B}^{(k)}_{f}-\nonumber \sum _{t=1}^T\sum _k \frac{\tilde{\gamma }^{(k)}_{tf}}{\phi ^{(k)}_{tf}}\mathbf {y}_{tf}^\textsf {H}\Bigl (\mathbf {B}^{(k)}_f\Bigr )^{-1}\mathbf {y}_{tf}+C\nonumber \\&=\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln \alpha ^{(k)}_f -M\sum _{t=1}^T\sum _k\tilde{\gamma }^{(k)}_{tf}\ln \phi ^{(k)}_{tf} \\&{=}-\sum _k\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\ln \det \mathbf {B}^{(k)}_{f}-\nonumber \sum _k\text {tr}\Biggl [\Bigl (\mathbf {B}^{(k)}_f\Bigr )^{-1}\Biggl (\sum _{t=1}^T\frac{\tilde{\gamma }^{(k)}_{tf}}{\phi ^{(k)}_{tf}}\mathbf {y}_{tf}\mathbf {y}_{tf}^\textsf {H}\Biggr )\Biggr ]+C.\nonumber \end{aligned}$$

(11.71)

Here, C denotes a constant independent of $\varTheta _{\text {G},f}$.

The update rule for $\alpha ^{(k)}_f$ is obvious.

From (11.70), the update rule for $\phi ^{(k)}_{tf}$ is given by

$$\begin{aligned} \phi ^{(k)}_{tf}=\frac{1}{M}\mathbf {y}_{tf}^\textsf {H}\Bigl (\mathbf {B}^{(k)}_f\Bigr )^{-1}\mathbf {y}_{tf}. \end{aligned}$$

(11.72)

As for $\mathbf {B}^{(k)}_{f}$, it should satisfy

$$\begin{aligned} -\biggl (\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\biggr )\Bigl (\mathbf {B}^{(k)}_{f}\Bigr )^{-1}+\Bigl (\mathbf {B}^{(k)}_f\Bigr )^{-1}\Biggl (\sum _{t=1}^T\frac{\tilde{\gamma }^{(k)}_{tf}}{\phi ^{(k)}_{tf}}\mathbf {y}_{tf}\mathbf {y}_{tf}^\textsf {H}\Biggr )\Bigl (\mathbf {B}^{(k)}_f\Bigr )^{-1}=0. \end{aligned}$$

(11.73)

Therefore, the update rule for $\mathbf {B}^{(k)}_{f}$ is

$$\begin{aligned} \mathbf {B}^{(k)}_f=\frac{\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}\mathbf {y}_{tf}\mathbf {y}^\textsf {H}_{tf}/\phi ^{(k)}_{tf}}{\sum _{t=1}^T\tilde{\gamma }^{(k)}_{tf}}. \end{aligned}$$

(11.74)