Deterministic Constructions of Binary Measurement Matrices from Finite Geometry

Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as {provably} good measurement matrices for compressed sensing under $\ell_1$-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of $\ell_0$-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.Comment: 12 pages, 11 figure

The X-ray Nature of Nucleus in Seyfert 2 Galaxy NGC 7590

We present the result of the Chandra high-resolution observation of the Seyfert~2 galaxy NGC 7590. This object was reported to show no X-ray absorption in the low-spatial resolution ASCA data. The XMM observations show that the X-ray emission of NGC 7590 is dominated by an off-nuclear ultra-luminous X-ray source (ULX) and an extended emission from the host galaxy, and the nucleus is rather weak, likely hosting a Compton-thick AGN. Our recent Chandra observation of NGC 7590 enables to remove the X-ray contamination from the ULX and the extended component effectively. The nuclear source remains undetected at ~4x10^{-15} erg/s/cm^-2 flux level. Although not detected, Chandra data gives a 2--10 keV flux upper limit of ~6.1x10^{-15} erg/s/cm^-2 (at 3 sigma level), a factor of 3 less than the XMM value, strongly supporting the Compton-thick nature of the nucleus. In addition, we detected five off-nuclear X-ray point sources within the galaxy D25 ellipse, all with 2 -- 10 keV luminosity above 2x10^{38} erg/s (assuming the distance of NGC 7590). Particularly, the ULX previously identified by ROSAT data was resolved by Chandra into two distinct X-ray sources. Our analysis highlights the importance of high spatial resolution images in discovering and studying ULXs.Comment: 8 pages, 5 figures, RAA accepte

Proposal of a phase-shift fiber Bragg grating as an optical differentiator and an optical integrator simultaneously

We show analytically and numerically that a practically realizable phase-shift fiber Bragg grating (PS-FBG) can function as a temporal first-order optical differentiator and a temporal first-order optical integrator at the same time. The PS-FBG working in reflection implements the differentiation and working in transmission implements the integration. We provide both the generalized conditions for a PS-FBG functioning as a first-order optical differentiator and a first-order optical integrator. The proposed PS-FBG can perform the time differential and integral of the complex envelope of an arbitrary input optical signal with high accuracy, respectively