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^N is modeled as: Y=X₁+X₂+W, (1) where x₁ is the baseline trend, X₂ is composed of K-complexes, and w presents noise. More specifically, x₁ is piecewise smooth comprising a lowpass signal component f, and a sparse order-K₁ derivative component g₁ i.e., x₁ = f + g₁. Further, x₂ 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, x₂ is modeled as the output of a high-pass filter, i.e., x₂ = H₂g₂. In addition, we assume the order-K₁ derivative of g₁ is sparse, and we likewise assume the order-K₂ derivative of g₂ is sparse. In another word, g₁ and g₂ are sparse-derivative signals, where u₁ = D₁g₁, and u₂ = D₂g₂ 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: {u₁*, u₂*} = arg min _{u1, u2} 1/2|| H₁y - A₁^-1B₁u₁ - A₂^-1B₂u₂||²₂ + λ₁Σ_n ρ₁([u₁]_n)+ λ₂ Σ_nρ₂([u₂]n). (2) where P₁ and P₂ denote penalty functions. The high-pass filters are expressed as H₁ = A_{1<- sub>^-1B, H2= A2^-1B, with B = B1D1 = B2D2. Using the solution from (2) we recover x1 and x2 by: x1 = Y - H1y + A1^-1B1u1*, = x2=A2^-1B2u2*. Problem (2) both decomposes the data y into x1 and 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.}