K-complex detection using sparse optimization
This work describes a method using sparse optimization for the detection of K-complex in sleep EEG. K-complex is an important feature in sleep stage identification, which is helpful to sleep disorder diagnostics process. In this work, a discrete-time sleep EEG signal Y ϵ R<sup>N</sup> is modeled as: Y=X<sub>1</sub>+X<sub>2</sub>+W, (1) where x<sub>1</sub> is the baseline trend, X<sub>2</sub> is composed of K-complexes, and w presents noise. More specifically, x<sub>1</sub> is piecewise smooth comprising a lowpass signal component f, and a sparse order-K<sub>1</sub> derivative component g<sub>1</sub> i.e., x<sub>1</sub> = f + g<sub>1</sub>. Further, x<sub>2</sub> is assumed as a transient waveform with a negative wave followed by a positive wave, and modeled as a `wavelet' (e.g. Fig. 1(b)). Moreover, x<sub>2</sub> is modeled as the output of a high-pass filter, i.e., x<sub>2</sub> = H<sub>2</sub>g<sub>2</sub>. In addition, we assume the order-K<sub>1</sub> derivative of g<sub>1</sub> is sparse, and we likewise assume the order-K<sub>2</sub> derivative of g<sub>2</sub> is sparse. In another word, g<sub>1</sub> and g<sub>2</sub> are sparse-derivative signals, where u<sub>1</sub> = D<sub>1</sub>g<sub>1</sub>, and u<sub>2</sub> = D<sub>2</sub>g<sub>2</sub> are both sparse. Adopting the zero-phase filter design techniques discussed in Ref. [3], and the idea of morphological component analysis (MCA) [4], we formulate the optimization problem: {u<sub>1</sub>*, u<sub>2</sub>*} = arg min <sub>u1, u2</sub> 1/2|| H<sub>1</sub>y - A<sub>1</sub><sup>-1</sup>B<sub>1</sub>u<sub>1</sub> - A<sub>2</sub><sup>-1</sup>B<sub>2</sub>u<sub>2</sub>||<sup>2</sup><sub>2</sub> + λ<sub>1</sub>Σ<sub>n</sub> ρ<sub>1</sub>([u<sub>1</sub>]<sub>n</sub>)+ λ<sub>2</sub> Σ<sub>n</sub>ρ<sub>2</sub>([u<sub>2</sub>]n). (2) where P<sub>1</sub> and P<sub>2</sub> denote penalty functions. The high-pass filters are expressed as H<sub>1</sub> = A<sub>1</sub><sup>-1</sup>B, H<sub>2</sub>= A<sub>2</sub><sup>-1</sup>B, with B = B<sub>1</sub>D<sub>1</sub> = B<sub>2</sub>D<sub>2</sub>. Using the solution from (2) we recover x<sub>1</sub> and x<sub>2</sub> by: x<sub>1</sub> = Y - H<sub>1y</sub> + A<sub>1</sub><sup>-1</sup>B<sub>1</sub>u<sub>1</sub>*, = x<sub>2</sub>=A<sub>2</sub><sup>-1</sup>B<sub>2</sub>u<sub>2</sub>*. Problem (2) both decomposes the data y into x<sub>1</sub> and X<sub>2</sub>, and performs denoising. It can be solved iteratively by majorization-minimization (MM) [2]. Further, the proposed algorithm is computationally efficient as it makes use of banded matrices. We use an asymmetric penalty function to capture the morphology of K-complex, and implement a simple detector by thresholding the local energy of X<sub>2</sub>. We test the proposed method by the public dataset collected in [1]. It achieves a better accuracy (F-measurement) than the result reported in [1]. An example is illustrated in Fig. 2.