K-complex detection using sparse optimization
Yin Ding, Ivan Selesnick · 2014
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 RNis modeled as: Y=X1+X2+W, (1) where x1is the baseline trend, X2is composed of K-complexes, and w presents noise. More specifically, x1is piecewise smooth comprising a lowpass signal component f, and a sparse order-K1derivative component g1i.e., x1= f + g1. Further, x2is 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, x2is modeled as the output of a high-pass filter, i.e., x2= H2g2. In addition, we assume the order-K1derivative of g1is sparse, and we likewise assume the order-K2derivative of g2is sparse. In another word, g1and g2are sparse-derivative signals, where u1= D1g1, and u2= D2g2are 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: {u1*, u2*} = arg minu1, u21/2|| H1y - A1-1B1u1- A2-1B2u2||22+ λ1Σnρ1([u1]n)+ λ2Σnρ2([u2]n). (2) where P1and P2denote penalty functions. The high-pass filters are expressed as H1= A1-1B, H2= A2-1B, with B = B1D1= B2D2. Using the solution from (2) we recover x1and x2by: x1= Y - H1y+ A1-1B1u1*, = x2=A2-1B2u2*. Problem (2) both decomposes the data y into x1and X2, 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 X2. 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.