Underdetermined Audio Source Separation Using Laplacian Mixture Modelling

Abstract

The problem of underdetermined audio source separation has been explored in the literature for many years. The instantaneous $K$-sensors, $L$-sources mixing scenario (where $K<L$) has been tackled by many different approaches, provided the sources remain quite distinct in the virtual positioning space spanned by the sensors. In this case, the source separation problem can be solved as a directional clustering problem along the source position angles in the mixture. The use of Laplacian Mixture Models in order to cluster and thus separate sparse sources in underdetermined mixtures will be explained in detail in this chapter. The novel Generalised Directional Laplacian Density will be derived in order to address the problem of modelling multidimensional angular data. The developed scheme demonstrates robust separation performance along with low processing time.

Appendices

Appendix 1

Calculation of the Normalisation Parameter for the Generalised DLD

To estimate the normalisation coefficient $c_p(k)$ of (7.27), we need to solve the following equation:

$$\begin{aligned} \int \limits _{\mathbf {x}\in \fancyscript{S}^{p-1}} c_p(k)e^{-k\sqrt{1-(\mathbf {m}^T\mathbf {x})^2}} \mathrm{d}\mathbf {x}=1. \end{aligned}$$

(7.43)

Following Eq. (B.8) and in a similar manner to the analysis in Appendix B.2 in [18], we can rewrite the above equation as follows:

$$\begin{aligned} c_p(k)\int \limits _{0}^{\pi }\mathrm{d}\theta _{p-1}\int \limits _{0}^{\pi } e^{-k\sqrt{1-\cos ^2\theta _1}}\sin ^{p-2}\theta _1 \mathrm{d}\theta _1 \prod _{j=3}^{p-1}\int \limits _{0}^{\pi }\sin ^{p-j}\theta _{j-1} \mathrm{d}\theta _{j-1}=1. \end{aligned}$$

(7.44)

Following a similar methodology to Appendix B.2 in [18], the above yields:

$$\begin{aligned} c_p(k)\pi \int \limits _{0}^{\pi } e^{-k\sin \theta _1}\sin ^{p-2}\theta _1\mathrm{d}\theta _1 \frac{\pi ^{\frac{p-3}{2}}}{\varGamma \left( \frac{p-1}{2}\right) }=1. \end{aligned}$$

(7.45)

Using the definition of $I_p(k)$, we can write

$$\begin{aligned} c_p(k)I_{p-2}(k)\frac{\pi ^{\frac{p+1}{2}}}{\varGamma \left( \frac{p-1}{2}\right) }=1\Rightarrow c_p(k)=\frac{\varGamma \left( \frac{p-1}{2}\right) }{\pi ^{\frac{p+1}{2}}I_{p-2}(k)}. \end{aligned}$$

(7.46)

Appendix 2

Gradient Updates for $\mathbf {m}$ and $k$ for the MDDLD

The first-order derivative of the log-likelihood in (7.28) for the estimation of $\mathbf {m}$ are calculated below:

$$\begin{aligned} \frac{\partial J(\mathbf {X},\mathbf {m},k)}{\partial \mathbf {m}}&= -k\sum _{n-1}^N\frac{-2\mathbf {m}^T\mathbf {x}_n}{2\sqrt{1-(\mathbf {m}^T\mathbf {x}_n)^2}}\mathbf {x}_n\nonumber \\&= k\sum _{n=1}^N\frac{\mathbf {m}^T\mathbf {x}_n}{\sqrt{1-(\mathbf {m}^T\mathbf {x}_n)^2}}\mathbf {x}_n. \end{aligned}$$

(7.47)

Before we estimate $k$ from the log-likelihood (7.28), we derive the following property:

$$\begin{aligned} \frac{\partial }{\partial k}I_0(k)= -\frac{1}{\pi }\int \limits _{0}^{\pi } e^{-k\sin \theta }\sin \theta \mathrm{d}\theta =-I_1(k). \end{aligned}$$

(7.48)

The above property can be generalised as follows:

$$\begin{aligned} \frac{\partial ^p}{\partial k^p}I_0(k)=(-1)^{p}\frac{1}{\pi }\int \limits _{0}^{\pi }\sin ^p\theta e^{-k\sin \theta }\mathrm{d}\theta =(-1)^{p}I_p(k). \end{aligned}$$

(7.49)

The first-order derivative of the log-likelihood in (7.28) for the estimation of $k$ is then calculated below:

$$\begin{aligned} \frac{\partial J(\mathbf {X},\mathbf {m},k)}{\partial k}=N\frac{I_{p-1}(k)}{I_{p-2}(k)}-\sum _{n=1}^{N}\sqrt{1-(\mathbf {m}^T\mathbf {x}_n)^2}. \end{aligned}$$

(7.50)

Appendix 3

A Directional K-Means Algorithm

Assume that $K$ is the number of clusters, $\fancyscript{C}_i,\mathrm i=1,\dots ,K$ are the clusters, $\mathbf {m}_i$ are the cluster centres and $\mathbf {X}=\{\mathbf {x}_1,\dots ,\mathbf {x}_n,\dots ,\mathbf {x}_N\}$ is a $p$-D angular dataset lying on the half-unit $p$-D sphere. The original $K$-means [36] minimises the following non-directional error function:

$$\begin{aligned} Q=\sum _{n=1}^N\sum _{i=1}^K||\mathbf {x}_n-\mathbf {m}_i||^2 \end{aligned}$$

(7.51)

where $||\cdot ||$ represents the Euclidean distance. Instead of using the squared Euclidean distance for the $p$-D Directional $K$-Means, we introduce the following distance function:

$$\begin{aligned} D_l(\mathbf {x}_n,\mathbf {m}_i)=\sqrt{1-(\mathbf {m}_i^T\mathbf {x}_n)^2}. \end{aligned}$$

(7.52)

The novel function $D_l$ is similarly monotonic as the original distance but emphasises more on the contribution of points closer to the cluster centre. In addition, $D_l$ is periodic with period $\pi$. The $p$-D Directional $K$-Means can thus be described as follows:  

• 1. Randomly initialise $K$ cluster centres $\mathbf {m}_i$, where $||\mathbf {m}_i||=1$.

• 2. Calculate the distance of all points $\mathbf {x}_n$ to the cluster centres $\mathbf {m}_i$, using $D_l$.

• 3. The points with minimum distance to the centres $\mathbf {m}_i$ form the new clusters $\fancyscript{C}_i$.

• 4. The clusters $\fancyscript{C}_i$ vote for their new centres $\mathbf {m}_i^+$. To avoid averaging mistakes with directional data, vector averaging is employed to ensure the validity of the addition. The resulting average is normalised to the half-unit $p$-D sphere:

  $$\begin{aligned} \mathbf {m}_i^+=\frac{1}{k_i}\sum _{\mathbf {x}_n \in k_i }\mathbf {x}_n \end{aligned}$$

  (7.53)
  $$\begin{aligned} \mathbf {m}_i^+\leftarrow \mathbf {m}_i^+/||\mathbf {m}_i^+|| \end{aligned}$$

  (7.54)

• 5. Repeat steps (2)–(4) until the means $\mathbf {m}_i$ have converged.