Dynamic mode decomposition for blindly separating mixed signals and decrypting encrypted images

Abstract

In this paper, we introduce the dynamic mode decomposition considering a lag $ \tau $ ($ \tau $-DMD) for solving the blind source separation (BSS) problem of chaotic signals and images. $ \tau $-DMD can be used in BSS-based image decryption with good separation performance. The unmixing problem was formulated as a modal decomposition problem. $ \tau $-DMD was applied on separating linear mixed chaotic signals showing a better separation performance than the existing blind source separation algorithms (Amuse, SOBI, FastICA and JADE). In addition, the case of adding noise in the mixing process was considered, and wavelet de-nosing before $ \tau $-DMD improved the separation performance. For the application, $ \tau $-DMD can be used to remove the noise in electrocardiograms (ECG) and the ocular artifacts in electroencephalograms (EEG). $ \tau $-DMD can also be applied in image processing, showing good separation performance of $ \tau $-DMD for both synthetic mixtures and real-life mixed texts. Structural similarity index measurement (SSIM) and peak signal-to-noise ratio (PSNR) were selected as the evaluation criteria. We tested the separation performance of $ \tau $-DMD on natural images, fingerprint images, and real-life text images and compared the results with other methods. Furthermore, $ \tau $-DMD was applied to decrypt the BSS-based encrypted images. In the process of encryption, we set up the underdetermined problem of BSS by mixing the original and key images, and then $ \tau $-DMD was used to extract the original images in the process of decryption with the secret seeds provided. Two simulations were performed to illustrate the performance of $ \tau $-DMD for image decryption, showing a better decryption results than FastICA.