Issue 
A&A
Volume 620, December 2018



Article Number  A151  
Number of page(s)  10  
Section  Numerical methods and codes  
DOI  https://doi.org/10.1051/00046361/201732388  
Published online  11 December 2018 
Synthetic direct demodulation method and its applications in InsightHXMT data analysis^{⋆}
^{1} Qian Xuesen Laboratory of Space Technology, China Academy of Space Technology, Beijing, 100094, PR China
email: huozhuoxi@qxslab.cn
^{2} Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany
email: yzhang@cpfs.mpg.de
^{3} Leibniz Institute for Solid State and Materials Research, IFW Dresden, 01069 Dresden, Germany
^{4} Tsinghua Center for Astrophysics, Department of Physics, Tsinghua University, Beijing, 100084, PR China
Received:
30
November
2017
Accepted:
21
August
2018
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 kmeans 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 matrixvector and matrixmatrix multiplication complexities are therefore reduced from polynomial time to linearlogarithmic time. A general statistical solution of the modulation equation in sparse representation is derived. Several dataanalysis pipelines are designed for the Hard Xray modulation Telescope (InsightHXMT) 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 InsightHXMT data analysis especially for the detection of Xray transients and monitoring timevarying objects with scanning observations.
Key words: methods: data analysis / methods: numerical / techniques: image processing / Xrays: general
© ESO 2018
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The newly launched Hard Xray Modulation Telescope (InsightHXMT) is China’s first Xray astronomical satellite. InsightHXMT carries three main payloads onboard: the High Energy telescope (HE), the Medium Energy telescope (ME) and the Low Energy telescope (LE) (Zhang et al. 2014). One of its primary scientific objectives is scanning the Galactic plane to monitor timevarying objects and to discover Xray transients (Zhang et al. 2014). InsightHXMT is based on the direct demodulation (DD) method because all telescopes onboard are positioninsensitive 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 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. For example, an observed image is a spatial distribution of objects modulated by an imaging system, where a point spread function (PSF) serves as the modulation kernel function. Observed spectra are object spectra modulated by energy response matrices. The relations among the observed data, the object, and the observation process can be formulated with a Fredholm integral equation of the first kind:
where d(ω), h(x,ω) and f(x) represent models of the observed data, the kernel, and the object, respectively, and ω and x are coordinate variables for the observed data domain (e.g. pixel indices, energy channels, possible state parameters of instruments and so on) and the object domain (e.g. WCS celestial sphere coordinates), respectively. The object model f(x) represents both the sources and the sky background. The kernel h(x,ω) characterises the observation process including the instrument response as well as nonsky backgrounds; for example, dark currents, or cosmic rays for highenergy detectors. In order to solve the equation, both periodic calibration of instrument response and nonsky background estimation are necessary to estimate the kernel. Differences between the true kernel and its estimation lead to systematic errors in the solution, that is, the estimated object. Discussions of instrument response calibration, nonsky background modelling and estimation, and systematic error due to above issues are beyond the scope of this work.
The DD method is introduced by Li & Wu (1993) to solve the modulation equation and then to reconstruct the unknown object from the observed data. In Eq. (1) a model of the observed data instead of the data itself is given. Since the data itself is a random outcome of the observation process, statistical solution of the modulation equation is achieved via the DD method according to the probability distribution of the observed data implied by its model. For observed data dominated by Poisson fluctuation, the simplest implementation of the DD method degenerates into Richardson–Lucy (RL) iteration (Richardson 1972; Lucy 1974). As a nonparametric approach, reconstruction through RL iteration allows a larger pool of solutions that fit the observed data, compared with parametric fitting approaches that require explicit and known models of both signals and backgrounds (Puetter et al. 2005). However, a larger pool of solutions often results in an illposed problem, which is unstable or divergent. The DD method overcomes this problem by continuously crosscorrelating both sides of Eq. (1) and transforming the original modulation equation into its Lfold equivalents to improve its positive definitiveness, as well as nonlinear physical constraints used to regularize the iteration, thereby shrinking the pool of solutions so that the object reconstruction problem is reduced.
In realworld data analysis, it is often necessary to avoid data with a poor signaltonoise ratio (S/N) through screening and selection; for example, the Good Time Interval (GTI) auxiliary record (Blackburn 1995) is used to select time series data in Rossi Xray Timing Explorer (RXTE) as well as InsightHXMT data analysis, or, to include relevant data from other instruments or epochs in order to provide better statistics. Meanwhile, the dimensionality and irregularity of the corresponding modulation equation increases, which imposes growing complexity on the reconstruction, especially for InsightHXMT data analysis, where the modulation kernels lack circular symmetry. The computational complexity introduced by rotation modulation is reduced by an angular clustering method specifically designed for InsightHXMT allsky survey (Huo & Zhou 2013), since the rotation modulation can be represented much more sparsely with explicit angular coordinates.
Although the DD method treats the illposed object reconstruction problem, the lack of a modulationkernelconstructing process or accelerated implementation for general cases prevents this method from being applied to data analysis tasks with high dimensionality or large datasets. In this article we provide the synthetic direct demodulation method (synDD), which features

a modulationkernelconstructing process to combine kernels characterizing individual instruments, observations and/or data screening/selection into one synthetic kernel (modulation equation synthesis process), and

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 (modulation kernel matrix analysis process).
Consequently, we can not only deal with the previously mentioned rotation modulation without losing important information sampled from different position angles, as well as the projection distortion that occurs on the tangential plane of the celestial sphere, which we put aside in previous work (Huo & Zhou 2013), but can also squeeze a more complicated kernel (e.g. with screening or weighing matrices) and additional observed data into our representation of the modulation equation.
2. Method
2.1. Modulation equation
In this article, vectors and matrices are highlighted in bold, while regular type face is used for scalars, plain sequences (or, 1D array), and multiple dimensional arrays. Operations on plain arrays are elementwise if not specifically indicated. For instance, x indicates a vector of x_{1}, x_{2}, …, x_{n}, while x simply suggests a plain sequence of these elements. Modular arithmetic is used for subscripts in this article.
Consider the algebraic form of Eq. (1)
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.
2.2. Analysis and synthesis of modulation equation
Both N and M in Eq. (2) are used to indicate the size of the object reconstruction problem. Time costs in solving the modulation equation increase rapidly as the size of the problem increases unless the naive matrixmatrix and matrixvector multiplications are carefully treated. For N × N matrices and N × 1 vectors the time costs of matrixmatrix and matrixvector multiplications are proportional to N ^{3} and N ^{2}, respectively. In computer science, such costs are measured with computational complexities, denoted by O(N ^{3}) (cubic complexity) and O(N ^{2}) (quadratic complexity). If the complexity could be reduced from polynomials (cubic or quadratic) to quasilinear, for example, O(NlogN) by replacing all the naive matrix mulplications with FFTs, time costs for a typical 512 × 512 InsightHXMT image reconstruct (N = 512 × 512) can be reduced from 10^{5} s to a few seconds (Huo & Zhou 2013). In order to achieve such a treatment, we decompose the kernel matrix H, for this purpose taken to be an arbitrary matrix, into the sum of a finite sequence of circulant kernel matrices multiplied by diagonal coefficient matrices, as
where A_{k} is an N × N diagonal matrix serving as the coefficient of the kth N × M kernel matrix H_{k}, which is rowcirculant. Diagonal entries of A_{k} have values of either 0 or 1 only. A matrix H_{k} is rowcirculant if each of its row vectors is circularshifted by one element to the right relative to its preceding row; that is, H_{k, i, j} = H_{k, i + 1, j + 1} = h_{k, j − i + 1} for all i and j, where h_{k} is the first row vector of H_{k}. A rowcirculant matrix is not necessarily square. If the number of its rows is greater than the number of its columns (N > M), its excess rows are considered as circularly appended to the first M rows of the matrix, while if N < M the matrix is considered as the first N rows that truncated from a M × M square circulant matrix. In addition, it is required that
and
where I_{N} is the N × N identity matrix.
Allowing a_{k} to be a column vector of diagonal elements of A_{k}, Eq. (2) becomes
The operator ⋅ denotes the elementwise multiplication, the operator * denotes the circular convolution, and is the reverse of h_{k}, that is, . We note that a_{k} has N elements while has M elements so before calculating their elementwise multiplication the right operand is truncated (if N > M) or padded (if N < M) to the same size as the left operand a_{k}. Such truncation or padding is applied when necessary in this article.
Decomposition of d is naturally derived as
where the kth element of the observed data .
The term analysis refers to separating the original modulation equation into a finite sequence of equations by decomposing the kernel matrix as well as the observed data, while the term synthesis refers to combining a finite sequence of modulation equations to form an equivalent one. Analysis of a modulation equation is useful for reducing the computational complexity of the corresponding inverse problem. Data obtained from different observations of the same object can be synthesised as in Eq. (7) and demodulated as in Eq. (2) or Eq. (6).
2.3. Lfold correlation
The onefold correlation of a modulation equation such as Eq. (2) is achieved by leftmultiplying both sides of the equation by the transpose of the kernel matrix, as
In the correlated equation, the unknown object image remains f while the observed data as well as the kernel matrix are both transformed into onefold correlated data c_{1} = H^{T}d and kernel P_{1} = H^{T}H. The Lfold correlation is achieved recursively as
provided that c_{0} = d and P_{0} = H (Li & Wu 1994), so the Lfold correlated equation is
Although Lfold correlation is preferred when using the DD method, it is difficult to compute with naive matrix multiplication when the kernel matrix is very large; for example, a 2^{20} × 2^{20} kernel matrix, since the computational complexity of multiplication between two N × N matrices is O(N^{3}) if performed naively. With a more complicated algorithm the computational complexity can be reduced to O(N^{2.376}) (Coppersmith & Winograd 1990), however it would still take hundreds of hours to compute the multiplication between two 2^{20} × 2^{20} matrices.
We have derived a simple but efficient approach to compute the matrix multiplication from Eq. (3). First we compute the first column vector of H^{T}H, as
We can compute the remaining column vectors in the same way. The time cost of computing each column vector is proportional to KNlogN, where K is the number of circulant matrices, if the circulant convolution is calculated with the FFT algorithm. For rotation modulation, K is the number of position angle clusters, for widefield image reconstruction, K is the number of fieldofview (FoV) clusters, and for synthetic kernel, K is the number of observations. Because both the position angle and the FoV distortion vary gradually in practical observations, their clusters are always far less than pixels or bins of the observed data samples. We found that 20–100 clusters are sufficient to approximate the kernel variation where the difference between the simulated true kernel and the kernel approximated with the weighted sum of circulant matrices are negligible compared to the required kernel calibration accuracy. Therefore K is always less than N by several orders of magnitude and is independent of N. As a result, the complexity is reduced to O(N^{2}logN).
2.4. Screening and weighing
We can avoid data with poor S/N (screening) or adjust their weights accordingly by introducing a weight matrix M into Eq. (2),
where the weight matrix M is an N × N diagonal matrix. The ith element on its main diagonal m_{i} serves as the weight of the ith observed datum d_{i}. We can avoid certain observed data by assigning 0valued elements as their weights.
Modulated estimates of the object Hf′ (f′ is an estimate of the unknown object f) appear as denominators in reconstructions with the DD method as well as RL iterations. The original kernel in Eq. (1) contains both instrument response and nonsky background, while the object contains sources and the sky background; hence divisionbyzero will not happen. But possible zeroweights in M of Eq. (12) are like holes and may cause such a problem. To prevent this problem in later reconstructions, Lfold correlation is applied so that zeroholes in MHf′ are smoothed to nonzeros by the kernel and their nonzero neighbours.
2.5. Additional observation
Observations at multiple epochs or with different instruments can be joined into a single modulation equation by using a partitioned kernel matrix, as
where d_{λ} and H_{λ} (λ = 1, 2, …, Λ) are the observed data and the kernel matrix of the λth observation, respectively.
2.6. Iterative solution of synthetic modulation equation
A model of the observed data instead of the observed data themselves has been placed on the left side of Eqs. (1) and (6), where the model characterizes the statistically expected value of the observed data while the data itself is a random outcome from a specific observation. For Poisson fluctuation(photon noise)dominant data, the likelihood function of the object image f as an unknown parameter given the observed data d is
Once we find the f that maximizes the likelihood function, the maximum likelihood (ML) solution of the modulation equation is achieved. The object reconstruction problem is therefore transformed into solving the following equation.
Therefore,
and
Fixedpoint iteration is a method of finding a fixed point of a given function in a numerical analysis. x is a fixed point of the function g(x) if x = g(x). The iteration x^{(n + 1)} = g(x^{(n)}) is expected to converge to x if g is continuous. The ML estimate of the true image f is a fixed point of function
We therefore expect to find the ML estimate iteratively. Iteration at the lth step is
where the normalization factor w = ∑_{k}a_{k} * h_{k}. RL iteration can also be considered as a fixedpoint iteration that achieves an ML solution that restores an image blurred by a convolution kernel. For a perfectly known modulation kernel and statistical model of the observed data, the proper criterion to terminate the fixedpoint iteration can be found by monitoring the residuals. However when the modulation kernel is not perfectly known or the statistical model of the observed data does not perfectly describe the random nature of the data, one should terminate the iteration as soon as the required source is resolved from the data. In addition, crossvalidation and sensitivity assessment (Huo et al. 2015) are necessary to prevent overfitting or noise amplification. For InsightHXMT survey data analysis, which is mainly focused on pointlikesource detection and monitoring, the stopping criterion is not critical since the demodulated image is not the final result of object reconstruction but only serves as a hint for the following parametric fitting procedures.
Provided both the object image f and the observed data d are N × 1 and the kernel matrix is then N × N, it takes ∼2N^{2} scalar multiplications for each RL iteration of the original DD. In contrast, it takes ∼2KNlogN scalar multiplications for each iteration of demodulation synthesised from a K circulant kernel matrix with the aid of the FFT algorithm. The computational complexity is therefore reduced, provided that K ≪ N.
Let us take InsightHXMT as an example, where the level1 scientific data products consist of lists of Xray photon arrival events detected by each scientific instrument. Each event is described by properties such as time on arrival, detector identifier, energy channel identifier, anticoincidence detector counts, and so on. Satellite orbit coordinates and telescope attitude (pointing angles and position angle) on arrival of each event are interpolated from the housekeeping data provided by the satellite platform. Twodimensional discrete samples of the observed image, 1D binned samples of the observed light curve, and/or a 1D binned energy spectrum are counted from the above events as well as auxiliary housekeeping data accordingly. For object image reconstruction, each image contains too many pixels for the naive matrix multiplication in Eq. (2) and the modulation cannot be computed with spherical harmonics or fourier transforms due to the absence of circular symmetry in the kernel, i.e. the point spread function (PSF) (Huo & Zhou 2013). With the approach described here the time cost of InsightHXMT image reconstruction is reduced by orders of magnitude.
2.7. Cluster analysis of the modulation kernel
Cluster analysis of the modulation kernel is the key building block of the method we present in this article. A kernel matrix is an ordered set of row vectors, which are classified into K groups through cluster analysis. In cluster analysis we refer to each group as a cluster. Row vectors in the same cluster are similar to each other, as if they were taken from a circulant matrix. A generalized Euclidean distance ∥x_{i} − x_{j}∥ is introduced here to measure the similarity between a pair of row vectors x_{i} and x_{j}. The two row vectors are considered sufficiently similar if the distance is less than the kernel calibration accuracy. So cluster analysis is the underlying procedure through which a kernel matrix H is decomposed into a sequence of circulant matrices H_{k} and their coefficients A_{k}. The number of clusters, K, determines the extent to which the computational complexity can be reduced. The smaller the number K, the more complexity is reduced when we approximate H with . The accuracy of the approximation is determined by similarities between any two row vectors in the same cluster. Therefore, to perform a cluster analysis of a modulation kernel is to classify all row vectors of the kernel matrix into as few clusters as possible, while the similarities between any two vectors in the same cluster are acceptable.
Principal component analysis (PCA) is also used to decompose a kernel in order to reduce its dimensionality (Jolliffe 1986). We categorize the analysis of a modulation kernel in this article as a cluster analysis. Because PCA is only used to reduce the dimensionality of calculating the similarities between row vectors here, this means that instead of calculating the generalized Euclidean distance between two M × 1 row vectors, we actually calculate the distance between two representative vectors that each contain many fewer components, i.e., the principal components. Therefore, PCA serves as a preprocessing for cluster analysis. Since each row vector of the kernel matrix has the same size as the object image, expense of computing similarities between row vectors increases sharply with the image resolution, especially if the image has more than one dimension. Fortunately a row vector of a kernel could be more sparse with a certain representation than it appears with a naive pixelwise representation. PCA is used to find the basis of a sparse representation of given row vectors of a specific kernel. In PCA the basis vectors are termed principal components. By expressing a row vector as a finite linear combination of the basis vectors, its dimensionality is effectively reduced, since the number of nonzero coefficients of a sparse representation is usually less than the number of pixels of the image.
Each row of the given kernel matrix H is circularly shifted leftwards according to its row number in the matrix, that is, the ith row is shifted by i elements leftwards. In this way all row vectors of a circulant matrix would be aligned so that the shifted matrix appears as N vertically stacked copies of identical row vectors, which is ready for the following PCA processing.
Allowing x_{i} be the shifted ith row vector with pixelwise representation, we use the NIPALSPCA algorithm (Geladi & Kowalski 1986) to find the principal components of the set of shifted row vectors {x_{i}} iteratively, and transform each row vector x_{i} to a new vector t_{i} in a space with reduced dimensionality defined by the principal components.
The similarity between two row vectors x_{i} and x_{j} is measured by the Euclidean distance between their sparse representations t_{i} and t_{j}, namely, ∥t_{i} − t_{j}∥. As a result, kmeans clustering (Lloyd 1982) is used to classify {t_{i}} into clusters. A drawback of kmeans clustering is that the number of clusters is taken as an input parameter. The minimum number of clusters should therefore be determined through extra running of clustering beforehand to make sure the similarities between vectors in the same cluster are acceptable.
Once the vectors {t_{i}} are classified into K clusters, a central vector of each cluster is calculated according to the corresponding row vectors {x_{i}}, as
where N_{k} is the number of vectors in the kth cluster, while S_{k} is the set of indices of vectors in this cluster. A circulant matrix H_{k} is then constructed from h_{k} as its first row vector. Its coefficient matrix A_{k} is constructed from its diagonal elements a_{k}, as
3. Test and results
3.1. Object, modulation kernels, and observed data
We produced a model with a series of point sources located on a spiral as the object (refer to Fig. 1). Distance between adjacent sources increases from the interior to the exterior of the spiral, while the intensity decreases. We can therefore use this model object to assess both spatial resolution and sensitivity of a given observation and the corresponding reconstruction.
Fig. 1.
Model object. 

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Aperture PSF model of the high energy (HE) telescope of InsightHXMT (as shown in Fig. 2), which reflects the geometrical effects of all the collimators of HE/InsightHXMT detectors, that is, the detection efficiency of the telescope to a test point source in its FoV is brought here to build modulation kernels. The spatial coordinates of the test point source are variables of the PSF.
Fig. 2.
HE/InsightHXMT aperture PSF model, all collimated detectors combined. 

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We simulated observations in which the sky region is scanned in different directions as shown in Fig. 3. Scanning speed along each row is 0.10° s^{−1}, while the interval between adjacent scanning rows is 3.0°.
Fig. 3.
Simulated scanning paths to the same object sky region along different directions. 

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With the simulated scanning paths, PSF, object, and a uniform background of 0.3 counts s^{−1} deg^{−2}, which is consistent with the scanning paths and rates and the HE background (Li et al. 2009), we calculated the modulation kernel matrices as well as the corresponding modulated light curves, respectively. As for the detectors, we simulated Poisson fluctuation by generating pseudo Poisson random numbers with the modulated light curves as the expected values. The simulated observed light curves are shown in Fig. 4.
Fig. 4.
Simulated observed light curves in different scanning observations. 

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3.2. Reconstruction and comparison
First we reconstructed the object from a singleepoch observed light curve (as shown in the top left panel of Fig. 4) on a lowresolution pixel grid (64 by 64 pixels). The reconstructed image is shown in Fig. 5. We estimated from the marginal null region of the reconstructed image that the 3σ pixelwise sensitivity is 2.4 counts s^{−1} deg^{−2}. So pixels with image values below the estimated sensitivity were discarded.
Fig. 5.
Lowresolution image reconstructed from simulated singleepoch light curve. 

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Similarly we reconstructed the object again from simulated light curves from the four different scanning paths shown in Fig. 3 by synthesizing all modulation kernel matrices and all light curves together, according to Sect. 2.5. The reconstructed image is shown in Fig. 6. The estimated 3σ sensitivity is 1.7 counts s^{−1} deg^{−2}.
Fig. 6.
Lowresolution image reconstructed from simulated multipleepoch light curves. 

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In the above reconstructions we employed the original DD iterations with vectorextrapolation between iterations to speed up the convergence (Biggs & Andrews 1997). The vectorextrapolation is a technique in numerical analysis which predicts the subsequent step with previous steps instead of computing the iteration, meaning that less iterations are required. For the lowresolution reconstruction, we find that after 30 iterations the solutions no longer improve due to significant artefacts. For each iteration the modulation was calculated through matrix multiplication routines provided by OpenBLAS, which is a widely used open source implementation of the Basic Linear Algebra Subprograms (BLAS; Xianyi et al. 2014). This is referred to as the original DD while the solution specified in Eq. (19) is referred to as the synthetic DD. Time cost per original DD iteration is 5.3 s on a dualcore PC. We reconstructed the object from simulated light curves from the four different scanning paths shown in Fig. 3 on the same pixel grid with synthetic DD for comparison, as shown in Fig. 7.
Fig. 7.
Lowresolution image reconstructed from simulated multipleepoch light curves with synthetic DD. 

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Time cost per synthetic DD iteration is 15 milliseconds on one personal computer (PC). As we can see the reconstructed image with synthetic DD is similar to that with original DD but the time cost is reduced by orders of magnitude.
Subsequently, we tested higherresolution (512 by 512 pixels) reconstruction with synthetic DD. Original DD was not tested because of its expected time cost (at least 10 h). Time cost per original DD iteration increases polynomially and more iterations are requested with higher resolution. The image reconstructed from the simulated light curves for the four different scanning paths illustrated in Fig. 3 is shown in Fig. 8. Time cost per iteration on this pixel grid is 0.6 s.
Fig. 8.
Highresolution image reconstructed from simulated multipleepoch light curves, with synthetic DD. 

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We also observe the convergence of clustering analysis as a prerequisite of synthetic DD. The residual of the sparse representation of the kernel matrix is reduced rapidly with the first components; subsequently the marginal improvement involving more components appears to diminish, as illustrated in Fig. 9.
Fig. 9.
PC representation residuals. 

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Finally time costs of reconstructions with original DD iteration and synthetic DD iteration are shown in Fig. 10.
Fig. 10.
Time costs of reconstructions. 

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4. Conclusion
The synDD method including modulation kernel matrix analysis and modulation equation synthesis reduces the polynomial computational complexity of a timeconsuming process in solving the modulation equation to linearlogarithmic complexity, which in turn saves time costs of object reconstruction in InsightHXMT data analysis by orders of magnitude. This encourages us to include more observed data, for example, in multiple epochs or multiple energy bands, and to reconstruct the object with higher resolution. Therefore it is possible to achieve both improved sensitivity and higher resolution but with even less resources. We suggest using the proposed method in InsightHXMT data analysis especially with scanning observations from multiple payloads, different energy bands, and/or multiple epochs for transient detection, timevarying objects monitoring and so on.
Acknowledgments
Pseudocodes in this article are typeset with the package algorithms bundle. In this work we made use of SciPy (Jones et al. 2001) and PyTables (Alted et al. 2002) in numerical computing and dealing with large datasets. We thank the anonymous referee whose comments and suggestions helped improve and clarify this manuscript.
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Appendix A: Algorithm and codes
The methods we explained above are specified with several separate subroutines in pseudocodes. In Table A.1 we have summarised all the symbols and their definitions used in the following subroutines.
Vectors are implemented by 1D arrays. Matrices are implemented by 2D arrays, where rows and columns are numbered by the first and second indices respectively, that is, A[i][j] represents the matrix element A_{i, j} in the following algorithm descriptions, A[i] denotes the ith row, and A[⋯][j] denotes the jth column. Here ellipsis ⋯ between a pair of square brackets [] denotes the sequence of all indices. Similar notations are used for 3D arrays. For example, provided that A is an L × M × N array of scalar elements, A[i][j][k] represents the element A_{i, j, k}. A[⋯][j][k], A[i][⋯][k] and A[i][j] represent vectors. A[⋯][k], A[i][⋯] and A[⋯][j][⋯] represent matrices.
The circularlyshifting operation mentioned in Sect. 2.7 is specified with Subroutine ^{1}, which is usually implemented by builtin functions in numerical computing languages such as MATLAB, NumPy, and so on.
Subroutine 1Circularlyshift row vectors.
Require:M, N, H
for i = 1 to N do
for j = 1 to M do
X[i][j]←H[i][(j + i − 2)%M + 1]
end for
end for
return X
A resumable implementation of the NIPALS method for PCA is specified in Subroutine ^{2} If the principal component scores matrix T is not initialized for writing, for example, when no memory or file system resources have been allocated to it, the current call to this subroutine will be considered as the first run. In this case the necessary resources will be allocated to the score matrix T, the loading matrix P as well as the residual R inside the subroutine during the current call. Otherwise it will be considered as resumed from a previous run, where the resources that T, P and R refer to must be preserved. The expected number of principal components J is provided as an input parameter. Elements of each principal component score are computed iteratively. An iteration is stopped when the increment of the estimate of the eigenvalue becomes negligible (Geladi & Kowalski 1986).
The kmeans clustering is specified with Subroutine ^{3} We randomly choose K vectors as the initial central vectors of the K clusters. The loop is stopped when none of the central vectors changes any more.
The iterative solution of synthetic modulation equation formulated in Eq. (19) is specified in Subroutine ^{4} If M = N the observed data d itself can serve as the initial estimate, otherwise its onefold correlation H^{T}d is provided as initial estimate of the object. Convolutions in Eq. (19) are implemented with FFTs. The iteration is stopped when the difference between the current estimate of the image and the previous one is negligible. The convergence of the iteration can be accelerated via vector extrapolation (Biggs & Andrews 1997).
The workflow of algorithms used to decompose a kernel matrix H is summarised in Fig. A.1.
Subroutine 2NIPALSPCA
Require:M, N, J, X, R, P, T, ϵ
j ← 1
if resumed from previous run then
while j ≤ J and ∥T[⋯][j]∥ ≤ ϵ do
j ← j + 1
end while
else // this is the first run
for i = 1 to N do
X_{0}[i]←0
end for
for i = 1 to M do
X_{0} ← X_{0} + X[⋯][i]
end for
X_{0} ← X_{0}/M
for i = 1 to M do
R[⋯][i]←X[⋯][i]−X_{0}
end for
end if
if j > J then // all expected PC scores have been calculated
return R, P, T
end if
for j = j to J do
λ ← 0
T[⋯][j]←R[⋯][j]
repeat
for i = 1 to M do
P[i][j]← sum of R[⋯][i]⋅T[⋯][j]
end for
for i = 1 to N do
T[i][j]← sum of R[i]⋅P[⋯][j]
end for
λ_{p} ← λ
λ ← ∥T[⋯][j]∥
until
for i = 1 to M do
R[⋯][i]←R[⋯][i]−T[⋯][j]⋅P[i][j]
end for
end for
return R, P, T
Symbols in algorithm descriptions.
kmeans clustering on sparse representation.
Require:K, N, T, X, ϵ
for k = 1 to K do
i ← a random integer from1 to N
C[k]←T[i]
Y[k]←0
end for
δ ← 0
repeat
for i = 1 to N do
for k = 1 to K do
D[k][i]←∥T[i]−C[k]∥
end for
k ← index of the least element of D[⋯][i]
append i to S[k]
end for
for k = 1 to K do
L[k]← number of elements in S[k]
C_{p}[k]←C[k]
C[k]←0
for i = 1 to L[k] do
C[k]←C[k]+T[S[i]]
end for
δ ← max[δ, ∥C_{p}[k]−C[k]∥]
end for
until δ ≤ ϵ
for k = 1 to K do
for i = 1 to L[k] do
Y[k]←Y[k]+X[S[i]]
end for
end for
return A, C, Y
Fig. A.1.
Algorithms used to decompose a kernel matrix. 

Open with DEXTER 
RequireK, M, N, A, Y
f ← initial estimate
for k = 1 to K do
for i = 1 to N do
a[k]←A[k][i][i]
end for
end for
for i = 1 to M do
w[i]←0
end for
for k = 1 to K do
w ← w + IFFT(FFT(a[k])⋅FFT(Y[k]))
end for
repeat
f_{p} ← f
for i = 1 to M do
q[i]←0
end for
for i = 1 to N do
r[i]←0
end for
for k = 1 to K do
end for
for k = 1 to K do
q ← q + a[k]⋅IFFT(FFT(r)⋅FFT(Y[k]))
end for
until ∥f − fp∥≤ϵ
return f
All Tables
All Figures
Fig. 1.
Model object. 

Open with DEXTER  
In the text 
Fig. 2.
HE/InsightHXMT aperture PSF model, all collimated detectors combined. 

Open with DEXTER  
In the text 
Fig. 3.
Simulated scanning paths to the same object sky region along different directions. 

Open with DEXTER  
In the text 
Fig. 4.
Simulated observed light curves in different scanning observations. 

Open with DEXTER  
In the text 
Fig. 5.
Lowresolution image reconstructed from simulated singleepoch light curve. 

Open with DEXTER  
In the text 
Fig. 6.
Lowresolution image reconstructed from simulated multipleepoch light curves. 

Open with DEXTER  
In the text 
Fig. 7.
Lowresolution image reconstructed from simulated multipleepoch light curves with synthetic DD. 

Open with DEXTER  
In the text 
Fig. 8.
Highresolution image reconstructed from simulated multipleepoch light curves, with synthetic DD. 

Open with DEXTER  
In the text 
Fig. 9.
PC representation residuals. 

Open with DEXTER  
In the text 
Fig. 10.
Time costs of reconstructions. 

Open with DEXTER  
In the text 
Fig. A.1.
Algorithms used to decompose a kernel matrix. 

Open with DEXTER  
In the text 
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