Abstract
Aims. A modulation equation relates the observed data to the object where the observation is approximated by a linear system. Reconstructing the object from the observed data is therefore equivalent to solving the modulation equation. In this work we present the synthetic direct demodulation (synDD) method to reduce the dimensionality of a general modulation equation and solve the equation in its sparse representation. Methods. A principal component analysis is used to reduce the dimensionality of the kernel matrix and k-means clustering is applied to its sparse representation in order to decompose the kernel matrix into a weighted sum of a series of circulant matrices. The matrix-vector and matrix-matrix multiplication complexities are therefore reduced from polynomial time to linear-logarithmic time. A general statistical solution of the modulation equation in sparse representation is derived. Several data-analysis pipelines are designed for the Hard X-ray modulation Telescope (Insight-HXMT) based on the synDD method. Results. In this approach, a large set of data originating from the same object but sampled irregularly and/or observed with different instruments in multiple epochs can be reduced simultaneously in a synthetic observation model. We suggest using the proposed synDD method in Insight-HXMT data analysis especially for the detection of X-ray transients and monitoring time-varying objects with scanning observations.
Highlights
Introduction with aFredholm integral equation of the first kindThe newly launched Hard X-ray Modulation Telescope (InsightHXMT) is China’s first X-ray astronomical satellite
Insight-HXMT is based on the direct demodulation (DD) method because all telescopes onboard are position-insensitive collimated detectors and images can only be reconstructed from the observed data by offline data analysis (Li & Wu 1993, 1994; Li 2007; Zhang 2009)
In this article we provide the synthetic direct demodulation method, which features 1. a modulation-kernel-constructing process to combine kernels characterizing individual instruments, observations and/or data screening/selection into one synthetic kernel, and 2. an accelerated implementation for general cases by decomposing an arbitrary kernel matrix into a weighted sum of a series of circulant matrices so that the fast Fourier transform (FFT) can help
Summary
The newly launched Hard X-ray Modulation Telescope (InsightHXMT) is China’s first X-ray astronomical satellite. In the analysis of astronomical data, observed data can be modelled as object functions modulated by kernel functions, which characterise the observation process, mainly the instrument response. The kernel h (x, ω) characterises the observation process including the instrument response as well as non-sky backgrounds; for example, dark currents, or cosmic rays for high-energy detectors. The DD method overcomes this problem by continuously cross-correlating both sides of Eq (1) and transforming the original modulation equation into its L-fold equivalents to improve its positive definitiveness, as well as non-linear physical constraints used to regularize the iteration, thereby shrinking the pool of solutions so that the object reconstruction problem is reduced. The DD method treats the ill-posed object reconstruction problem, the lack of a modulation-kernel-constructing process or accelerated implementation for general cases prevents this method from being applied to data analysis tasks with high dimensionality or large datasets. Where N × 1 column vector d and M × 1 vector f are discrete samplings of the models of the observed data and the object, while N × M matrix H is the modulation kernel matrix, which is a discrete sampling of the modulation kernel
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