Complex and Quaternionic Principal Component Pursuit and Its Application to Audio Separation

Abstract

Abstract-Recently, the principal component pursuit has received increasing attention in signal processing research ranging from source separation to video surveillance. So far, all existing formulations are real-valued and lack the concept of phase, which is inherent in inputs such as complex spectrograms or color images. Thus, in this letter, we extend principal component pursuit to the complex and quaternionic cases to account for the missing phase information. Specifically, we present both complex and quaternionic proximity operators for the 1-and trace-norm regularizers. These operators can be used in conjunction with proximal minimization methods such as the inexact augmented Lagrange multiplier algorithm. The new algorithms are then applied to the singing voice separation problem, which aims to separate the singing voice from the instrumental accompaniment. Results on the iKala and MSD100 datasets confirmed the usefulness of phase information in principal component pursuit.

Frequency spectrum regularization for pattern noise removal based on image decomposition

“…As for the complex vector, basically, we can regard the real number and the imaginary number as a group variables [3].…”

“…The RPCA [1], [3] regards luminance change (specular reflection) caused by reflected light as an artifact component, and characterize it so as to minimize the sum of absolute luminance values over the whole image (i.e., ℓ 1 norm). The reflection component removal [2] deals with a faint and blurred reflection image component, and characterize it so In the spectral image, the 1st quadrant is shown and the intensity range is normalized for displaying the detail.…”

Polar $n$-Complex and $n$-Bicomplex Singular Value Decomposition and Principal Component Pursuit

“…The extra √ n scaling for the proximity operators is due to the fact that MATLAB's fft is unnormalized. 7…”

“…+ (Re X n )e 2n−2 + (Im X n )e 2n−1 , where X i contains the complex spectrogram for the i-th channel. 3) Quaternionic PCP (if applicable) [7]: the two-channel audio is represented using X 1 +X 2 , where X i contains the complex spectrogram for the i-th channel. 4) Tensor RPCA [15]: the same spectrograms are represented by complex matrices of tubes.…”

“…For example in parametric spatial audio, spectrograms have not only spectral phases but inter-channel phases as well. For that reason alone, we have recently extended the PCP to the complex and the quaternionic cases [7]. However, there exists inputs with dimensionality greater than four, such as microphone array data, surveillance video from multiple cameras, or electroencephalogram (EEG) signals, which exceed the capability of quaternions.…”