Issue 
A&A
Volume 618, October 2018



Article Number  A117  
Number of page(s)  6  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201833090  
Published online  17 October 2018 
Fused CLEAN deconvolution for compact and diffuse emission
^{1}
Guizhou University, Guiyang, 550025 PR China
email: lizhang.science@gmail.com
^{2}
CAS Key Laboratory of Solar Activity, National Astronomical Observatories, Beijing, 100012 PR China
Received:
24
March
2018
Accepted:
20
July
2018
Context. CLEAN algorithms are excellent deconvolution solvers that remove the sidelobes of the dirty beam to clean the dirty image. From the point of view of the scale, there are two types: scaleinsensitive CLEAN algorithms, and scalesensitive CLEAN algorithms. Scaleinsensitive CLEAN algorithms perform excellently well for compact emission and perform poorly for diffuse emission, while scalesensitive CLEAN algorithms are good for both pointlike emission and diffuse emission but are often computationally expensive. However, observed images often contain both compact and diffuse emission. An algorithm that can simultaneously process compact and diffuse emission well is therefore required.
Aims. We propose a new deconvolution algorithm by combining a scaleinsensitive CLEAN algorithm and a scalesensitive CLEAN algorithm. The new algorithm combines the advantages of scaleinsensitive algorithms for compact emission and scalesensitive algorithms for diffuse emission. At the same time, it avoids the poor performance of scaleinsensitive algorithms for diffuse emission and the great computational load of scalesensitive algorithms for compact emission in residuals.
Methods. We propose a fuse mechanism to combine two algorithms: the AspClean2016 algorithm, which solves the computationally expensive problem of convolution operation in the fitting procedure, and the classical Högbom CLEAN (HgClean) algorithm, which is faster and works equally well for compact emission. It is called fused CLEAN (fusedClean) in this paper.
Results. We apply the fusedClean algorithm to simulated EVLA data and compare it to widely used algorithms: the HgClean algorithm, the multiscale CLEAN (MsClean), and the AspClean2016 algorithm. The results show that it performs better and is computationally effective.
Key words: methods: data analysis / techniques: image processing
© ESO 2018
1. Introduction
For many reasons, such as technical difficulties, the size of a single dish is severely limited, as is its resolution. Interferometric measurements with some imaging techniques such as deconvolution can go beyond the resolution limitation of a single telescope. Radio telescope arrays currently often use tens of telescopes to measure visibilities. However, the sampling is incomplete. These missing spatial frequencies lead to a dirty image, which is the corrupted version of the true sky image. Deconvolution algorithms are used to remove the effects of the corruption to obtain the restored image.
The most widely used deconvolution algorithms in the radio synthesis imaging field are CLEAN algorithms. Scaleinsensitive CLEAN algorithms decompose the true sky image as a collection of point sources or scaled delta functions. The original algorithm (Högbom 1974) has been proposed by Högbom in the 1970s. It is a matchedpursuit algorithm and employs iterations to approximate the true sky image. For pointlike emission, it is quite good and speedy enough. However, it needs a huge number of scaled delta functions to approximate diffuse emission and complex images, and it lacks a mechanism to introduce the dependence among the neighboring points. Thus the residuals of extended sources often include some significantly correlated structures.
Scalesensitive CLEAN algorithms (Bhatnagar & Cornwell 2004; Cornwell 2008) introduce a priori the knowledge that the true image is composed of different spatial scales. The pixels within one scale are dependent, which can constrain the unsampled spatial frequencies. Extended emission can be represented with a small number of components in scalesensitive CLEAN algorithms. Even though some papers (Zhang et al. 2016a,b) have improved scalesensitive CLEAN algorithms, they are still not computationally efficient for compact emission.
So far, these deconvolution algorithms are designed for either compact or diffuse emission. However, observations contain both compact and diffuse emission within the field of view, which requires an algorithm that can process both compact and diffuse emission well. To solve the problem, we propose an efficient algorithm that combines AspClean2016 and HgClean. This is not a direct combination of the two algorithms, but a more sophisticated way; it only combines in the phase of the component search in the minor cycle and uses the same major cycle.
The paper is structured as follows. In Sect. 2 we describe the imaging theory, the HgClean algorithm, and the AspClean2016 algorithm. In Sect. 3 we describe the fusedClean algorithm in detail. In Sect. 4 we give some examples and compare our results to other classical algorithms. In Sect. 5 we discuss this algorithm and conclude.
2. Imaging theory and CLEAN algorithms
To understand our algorithm, here we introduce the radio interferometric imaging theory and two algorithms that are used in the proposed algorithm.
2.1. Imaging theory
In the van Cittert–Zernike theorem (Thompson et al. 2017), the true visibility function and the sky brightness function are a pair of Fourier transformations. The true visibilities are V^{true} (1)
where F is the Fourier transformation and I^{true} is the sky brightness function. This is an ideal and continuous case. The true image can be recovered by directly employing inverse Fourier transformation. For real cases, however, the sample is incomplete and noisy. The measured visibilities V^{measured} (2)
where S is the sampling function and N is the noise in Fourier domain. Here we ignore these operations of weighting, convolutional interpolation, and resampling, and express the dirty image I^{dirty} as(3)
where F^{−1} is the inverse Fourier transform operation. In the convolution theorem of Fourier transform theory, the dirty image I^{dirty} can be expressed as(4)
where B^{dirty} is a Toeplitz matrix that is composed of the shifted dirty beams. Because of the incomplete sampling, the dirty beam often has many nonignorable widespread sidelobes. Deconvolution is a solver that removes the effect of these dirty beam sidelobes.
2.2. HgClean algorithm
The HgClean algorithm decomposes the true sky brightness as a set of scaled delta functions,(5)
where is the amplitude of the nth component and δ_{n} is the delta function in the position (x_{n}, y_{n}). The error of the component estimation in the minor cycle is corrected in the fusedClean algorithm by updating residual visibilities from the original visibility data in the major cycle. Very many components are needed to represent a large diffuse emission. However, the computational load of the algorithm for each component is small. This is effective for compact emission.
2.3. AspClean2016 algorithm
The AspClean2016 algorithm (Zhang et al. 2016a) is an efficient implementation of the AspClean algorithm (Bhatnagar & Cornwell 2004). It parameterizes the true sky image as a collection of circular Gaussian functions with different scales,(6)
where α_{n} is the amplitude of the nth component, σ_{n} is the scale of the Gaussian component, and x_{n} and y_{n} are the parameters of positions.
The AspClean algorithm finds the bestfit scale components with active sets by minimizing the objective function χ^{2} for each component(8)
where is the residual image in (n − 1)th iteration, is the current model component, and ∥ ⋅ ∥_{2} is the Euclidean norm. Since the convolution is in the componentfitting objective function, the AspClean algorithm is computationally expensive. It is removed to speed up the process by analytically computing model components in the AspClean2016 algorithm (Zhang et al. 2016a). The procedure of the AspClean2016 algorithm is similar to the AspClean algorithm, but it minimizes a different objective function of the fitting part,(9)
where is a Gaussian component that was fitted from the residual image. This is very efficient for diffuse emission (Zhang et al. 2016a). However, it is still timeconsuming for compact emission.
3. FusedClean algorithm
To achieve the goal of simultaneously processing both compact and diffuse emission well, the fusedClean algorithm combines the advantages of the different algorithms and can automatically trigger different algorithms in different situations to speed up and improve the performance of the algorithm. The basic procedure is as follows.

Smooth residual image (the dirty image for the first time) with s scales.

Find peaks from these smoothed residual images ; the global peak G_{l}(a_{l},x_{l},y_{l},σ_{l}) is used as the initial guess of the current parameters.

Trigger an algorithm according to the current situation.

Find new parameters of the current model component .

Update the model image .

Calculate the residual image .

Iterate until one of the termination criteria is satisfied.

Compute the restored image with the restored beam B^{clean} after m iterations.
All model components are found by equivalently minimizing the objective function in the major cycle,(10)
It can find the best fit to the measured visibilities.
In the beginning phase of deconvolution, the matchedfiltering technique is used to find the initial position and scale of the strongest emission in the same way as in the AspClean2016 algorithm. If the initial scale is larger than the scale of the dirty beam, then the minor cycle of the AspClean2016 algorithm is triggered once. The optimal scale and position are found by explicitly minimizing the objective function,(11)
where is a Gaussian function with parameters (amplitude α_{nb}, location x_{nb}, y_{nb}, and width ω_{nb}). The updated direction is estimated by computing the gradient of χ^{2} with respect to the parameters p_{nb} given by(12)
where p_{nb} ≡ {α_{nb}, x_{nb}, y_{nb}, ω_{nb}}. For the secondorder optimization method (e.g., the Levenberg–Marquardt algorithm Marquardt 1963), we also need to compute or approximate the Hessian matrix. After converging to the solution of , the parameterized component can be analytically computed as(13)
where ω_{n}, ω_{nb}, ω_{b} are the widths of the current model component , and the Gaussian beam approximated from the dirty beam, respectively. The amplitude α_{n} of is computed as(14)
where α_{b} and α_{nb} are the amplitudes of the Gaussian beam and , respectively.
Explicitly, minimization optimizes each component, which is a best fit for the current residuals. The AspClean2016 algorithm (Zhang et al. 2016a) has shown that the analytical computation of components is very efficient for diffuse emission.
When the initial scale estimated by matchedfiltering technique is smaller than a threshold (e.g., 1.2 times the width of the dirty beam) or very small scale components frequently appear in the last iterations (e.g., the FWHM of five components are smaller than 1 pixel in the last ten components), then the minor cycle of the HgClean algorithm will be triggered. Compared to scalesensitive CLEAN algorithms, the HgClean is more efficient for compact emission. It does not perform an explicit fitting, but determines the peak of the current residuals and then subtracts a scaled version of the dirty beam from the current residuals. In other words, the updated direction is estimated by finding the peak of the current residuals,(15)
where p_{n} = and is the peak point located at (x_{n}, y_{n}) in the nth iterations. The iterative search for and then the shiftandsubtract operation in the HgClean algorithm is equivalent to a fast implementation of the minimization of the objective function given by Eq. (8). When the HgClean is triggered, it will be ran for several times. In practice, more compact emission will appear in the residuals when the deconvolution reaches deeper, so that a monotonic function for the triggering number can process it well. We have found the following relation to work well:(16)t_{tn} is the times of executing HgClean when HgClean is triggered tnth times. The specific form of this function in Eq. (16) is not important, but with the increase of the triggering times for the HgClean algorithm, the function should be increasing.
In the fusedClean algorithm, the HgClean algorithm is used for compact emission and the AspClean2016 algorithm is used for diffuse emission. The scale adaptivity of the AspClean2016 algorithm can separate emission and noise, while the HgClean algorithm is efficient for compact emission. In the fusedClean algorithm, the two algorithms are triggered alternately, so that emission and noise are effectively separated.
Fig. 1. Results of simulated EVLA observations of the M 31 image deconvolved by the fusedClean algorithm. Panel a: original image; panel b: dirty image with Briggs weighting; panel c: model image; and panel d: residual image. 

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4. Numerical experiment and comparison
In this section, we apply the fusedClean algorithm to the EVLA^{1} simulated data to evaluate its performance and compare it to these frequently used CLEAN algorithms: the HgClean algorithm, the MsClean algorithm, and the AspClean2016 algorithm. The test M 31 image is shown in Fig. 1a, and its brightness ranges from 0 Jy pixel^{−1} to 0.1 Jy pixel^{−1}. We performed a simulated observation with the B configuration of EVLA using the CASA software^{2}. The observation was made in L band with a bandwidth of 1 GHz and 32 channels, and it lasted six hours. Gaussian white noise was added to the “measured” visibilities, which means that the dirty image has a noise level of RMS 5 × 10^{−5} Jy. The resolution of the images was 1^{″} and the width of the main lobe of the dirty beam was about 2^{″}. The corresponding dirty image with the robust (=0) weighting is shown in Fig. 1b, where the data range reaches from −0.039 Jy pixel^{−1} to 0.61 Jy pixel^{−1}.
The deconvolution results are shown in Fig. 1. The model image displayed in Fig. 1c is composed of 155 extended components and 20 196 compact components, which can represent the true emission well. No significant signal in the residual image displayed in Fig. 1d indicates that the fusedClean deconvolution can extract the signal fully, and signal and noise are effectively separated.
These scale choices of deconvolving the M 31 image and the change of the model flux with iterations are displayed in Fig. 2, which contains only the first 100 iterations and the last 10 000 iterations for effective visualization. The choices of algorithms can be known through the choices of the component scale sizes. If the scale size is greater than zero, then the AspClean2016 algorithm is selected; otherwise the HgClean algorithm is selected. We can know the deconvolution behaviors of the fusedClean algorithm from Fig. 2. 1) Most flux was recovered in the beginning phase of the deconvolution. The main emission is reconstructed in the first ∼50 iterations. AspClean2016 is more frequently executed in the beginning phase because AspClean2016 has more sparse representation capacity for diffuse emission. 2) Most iterations were used to reconstruct weak and smallscale emission. This compact emission corresponds to compact sources or broken emission from an inaccurate representation of diffuse emission. HgClean is required frequently to deconvolve this compact emission to approximate the latent true image. 3) After triggering the HgClean algorithm, some weak and largescale emission may appear and can be reconstructed effectively by the AspClean2016 algorithm. 4) We did not set the iteration number for the AspClean2016 and HgClean algorithms. The 155 AspClean2016 and 20 196 HgClean deconvolutions are completely determined by the scale complexity of the dirty image and by the parameters of the deconvolution algorithm. This also shows the adaptive capacity of calling these two subalgorithms.
In short, diffuse emission is recovered with the AspClean2016 algorithm, and compact emission is reconstructed with the HgClean algorithm. The AspClean2016 algorithm is almost always used preferentially when the residual image contains much signal. As the deconvolution continues, the signals in the residual image will decrease and much scaleless emission will appear. Then the HgClean algorithm dominates the deconvolution process.
To compare the performance of the fusedClean algorithm with other typical CLEAN deconvolution algorithms, the corresponding deconvolution results are displayed in Fig. 3 and listed in Table 1. The model images displayed in Fig. 3a0 from the HgClean algorithm are composed of 100 000 compact components, and the corresponding residual image in Fig. 3a1 contains many correlated features because delta function cannot physically represent diffuse emission well. The model image from the MsClean algorithm that uses enumeration scales has 2000 components and less signal in the residuals. The AspClean2016 and fusedClean algorithms use adaptive scales. They can represent an image more sparsely than the previous two algorithms. The fusedClean algorithm combines the AspClean2016 algorithm with the HgClean algorithm, which represents compact emission more effectively. This can separate signal and noise more effectively. No significant signal is in the residual image displayed in Fig. 3d1 from the fusedClean algorithm. The fusedClean deconvolution has the highest dynamic range (defined in Li et al. 2011) in this experiment from Table 1. All these results show that the performance of the fusedClean algorithm is excellent, which can be also proved by the numerical comparison in Table 1.
It is worth mentioning that the fusedClean algorithm is more robust and faster than the AspClean2016 algorithm. A combination of the HgClean algorithm equivalently introduces a new scheme to jump out of the possible local optimum in the AspClean2016 algorithm. In the M 31 simulation, the fusedClean deconvolution took about 4 min, which is about four times faster than the AspClean2016 algorithm. Experiments were ran on a typical graphics workstation. The speed increase arises because the HgClean algorithm was used to represent compact emission. In the AspClean2016 algorithm, finding a compact component from the current residual image needs to be explicitly fitted. This is achieved through iterative optimization, which is time consuming. However, finding a compact component through the HgClean algorithm only requires identifying the maximum value in the current residual image and some simple operations.
Fig. 2. Scale choices and model flux for deconvolving the dirty M 31 image with fusedClean. 

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Fig. 3. Deconvolution results of the M 31 image. From left to right, columns: HgClean, MsClean, AspClean2016, and fusedClean. From top to bottom, rows: model images and the residual images. 

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Numerical comparison of different deconvolution algorithms for the “M 31” simulation.
5. Discussion and summary
The fusedClean introduces a good algorithm framework and the thought of algorithm union. In other words, this is a general method that can be applied to more algorithmic combinations than a mere combination of the HgClean aglorithm and the AspClean2016 algorithm. Combined algorithms can combine the advantages of different algorithms without maintaining their disadvantages. It performs excellently well, which is difficult for a single algorithm. In addition, if a combined algorithm can reduce the total computation complexity, then it is very helpful that a deconvolution algorithm can be developed into software. For this purpose, many scaleinsensitive and scalesensitive algorithms can be combined to speed up the deconvolution process. For example, the MsClean (Cornwell 2008) or MTMFS (Rau & Cornwell 2011) can be combined with a scaleinsensitive algorithm such as the Clark CLEAN (Clark 1980).
An algorithm combination should consider intrinsic factors and relations of model decomposition among iterations properly. In other words, a combined algorithm should be not a simple mechanical concatenation. A simple mechanical concatenation may not be sufficient to improve the performance.
Compressivesensing based deconvolution algorithms make the obvious assumption that the signal is sparse in a certain domain. The CLEANbased algorithm does not make such an assumption. Therefore, the performance of CLEANbased algorithms (e.g., runtime and fidelity) is more stable when conditions change (see Li et al. 2011). At the same time, the fusedClean algorithm is naturally applicable to the CLEAN algorithm framework, that is, to minor cycle and major cycle (it is also implemented in the standard minor and major cycles) and also integrates well with some other typical synthesis imaging techniques, such as widefield corrections.
The minor cycle of the fusedClean contains the component estimation methods of the AspClean2016 and the HgClean algorithms. The AspClean2016 algorithm employs an analytical way to significantly reduce the computational load and at the same time keeps the excellent performance of adaptive scale deconvolution. The advantage of the HgClean algorithm is that it is excellent and fast for compact emission, but its disadvantage is that it is slow and difficult to fully represent for diffuse emission. The fusedClean algorithm combines the speed and excellent performance of the AspClean2016 algorithm for diffuse emission with the speed of the HgClean algorithm for compact emission, and it avoids the slow speed of the AspClean2016 algorithm for compact emission and the poor performance for diffuse emission of the HgClean algorithm.
Tests show that the performance of the fusedClean algorithm is better than these typical CLEANbased deconvolution algorithms. The algorithm is implemented with the CASA and Python language. The work to build it as an available deconvolution algorithm into the CASA software package is currently ongoing.
Acknowledgments
We thank the people who worked and are working on the Python and CASA projects, which provide an excellent development and simulation environment. The work is supported by the Open Research Program of the CAS Key Laboratory of Solar Activity (KLSA201805) and the Guizhou Science & Technology Cooperation Project–Talent Platform ([2017]5788).
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All Tables
Numerical comparison of different deconvolution algorithms for the “M 31” simulation.
All Figures
Fig. 1. Results of simulated EVLA observations of the M 31 image deconvolved by the fusedClean algorithm. Panel a: original image; panel b: dirty image with Briggs weighting; panel c: model image; and panel d: residual image. 

Open with DEXTER  
In the text 
Fig. 2. Scale choices and model flux for deconvolving the dirty M 31 image with fusedClean. 

Open with DEXTER  
In the text 
Fig. 3. Deconvolution results of the M 31 image. From left to right, columns: HgClean, MsClean, AspClean2016, and fusedClean. From top to bottom, rows: model images and the residual images. 

Open with DEXTER  
In the text 
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