Bayesian Statistics Approach to Imaging of Aperture Synthesis Data: RESOLVE Meets ALMA ^†

Abstract

The Atacama Large Millimeter/submillimeter Array (ALMA) is currently revolutionizing observational astrophysics. The aperture synthesis technique provides angular resolution otherwise unachievable with the conventional single-aperture telescope. However, recovering the image from inherently undersampled data is a challenging task. The clean algorithm has proven successful and reliable and is commonly used in imaging interferometric observations. It is not, however, free of limitations. Point-source assumption, central to the clean is not optimal for the extended structures of molecular gas recovered by ALMA. Additionally, negative fluxes recovered with clean are not physical. This begs the search for alternatives that would be better suited for specific scientific cases. We present recent developments in imaging ALMA data using Bayesian inference techniques, namely the resolve algorithm. This algorithm, based on information field theory, has already been successfully applied to image the Very Large Array data. We compare the capability of both clean and resolve to recover known sky signal, convoluted with the simulator of ALMA observation data, and we investigate the problem with a set of actual ALMA observations.

1. Introduction

1.1. Aperture Synthesis

The Atacama Large Millimeter/submillimeter Array (ALMA) is revolutionizing observational astrophysics. With its 66 antennas located on the Atacama desert it provided the sharpest ever images of the submillimeter sky, for example, images of the protoplanetary disks at 1 au resolution [1]. In order to obtain such a resolution at a distance to a nearby star-forming region at 1.3 mm a telescope diameter of ∼15 km are needed. Since the construction challenges of such an antenna, especially if one would like to make it steerable, are far beyond current technical capabilities, in radio astronomy domain we often turn to aperture synthesis techniques, where instead of a single dish, a combination of smaller antennas is used, and with interference of signal between each antennas a resolution compared to the a telescope of a size of the greatest distance between the two antennas in an array (i.e., baseline) is achieved.

This does not come without a cost: sampling the baselines is never complete compared with a single dish telescope. This means that we do not receive complete information at all baselines, and thus to create an image of the sky we are operating with missing information.

A direct measurement of an interferometer is the interference pattern between two given antennas. This pattern is related to the sky brightness observed by the antennas. The recorded complex value, referred to as visibility, is then a Fourier transform of the sky brightness, the quantity which observations aim to recover. Therefore, a simplified imaging process consists of a (reverse) Fourier transformation of the measured visibilities (while the non-measured visibilities are set to 0) to obtain first approximation of the sky brightness, the so-called dirty image [2,3]. Once this is achieved, it becomes apparent that even with a modern interferometer such as ALMA with many baselines sampling the so-called UV plane we achieve a rather poor quality image. Refining this image presents the topic of this work.

1.2. CLEAN as a Standard Approach to the Imaging of the Interferometric Data

A standard technique to improve the image quality of the interferometric observations was developed by Hogbom in 1974 [4] and is called clean. Clean exploits the well-defined point-spread function of a given antenna configuration. The algorithm identifies the point sources in an initial dirty image, where the point sources are then approximated with a delta function, convolved with a dirty beam (i.e., the assumed pattern that the point-source would create on the dirty image), scaled with brightness of the suspected point source and the point-source patter is subtracted from the dirty image. This is an iterative process that ideally ends when all point sources are removed from the dirty image until it consists only of noise [2].

One issue of clean that can be easily identified is in the assumption that the sky is composed of point sources. This results in clean struggling with imaging of extended sky brightness structures. Several modifications to the original clean algorithm have been implemented to mitigate this issue, such as multi-scale clean which allows to set a point-source that would be a Gaussian rather than a delta function [5,6].

There are several steps in the process of cleaning which can be modified to improve the process. Masking is a method to restrict an area where the algorithm will search for point sources to a selected region in the sky. Weighting allows us to attribute different weights to different u, v scales, allowing for a trade-off between resolution and sensitivity.

Although clean became the gold standard of interferometric data imaging, producing many stunning images of astronomical objects, its limitation beg the search for alternatives, especially in cases where its assumptions are not met.

1.3. Resolve Algorithm and IFT

Imaging of the interferometric data can be presented as an inference problem, since we operate on an incomplete measurement problem, trying to find the true sky emission from the received data. Radio Extended SOurces Lognormal deconVolution Estimator resolve (https://gitlab.mpcdf.mpg.de/ift/resolve) [5,7] is designed in the Information Field Theory (IFT) framework [8]. IFT enables to use Bayesian inference methods in the context of mathematical framework of field theory. This is well fitted to the issue of imaging the sky brightness. IFT algorithms are implemented in resolve through Python package NIFTy (https://gitlab.mpcdf.mpg.de/ift/NIFTy) [9,10].

The measurement equation can be presented as: $d=R\left(s\right)+n$, where R is a response of the instrument to the original physical signal (s), and n is noise. In the IFT framework, obtained data d is analyzed in order to find the most probable form of s, which is signal from the field, in this case, the sky. This takes the form of Bayes’ theorem:

$$P\left(s\right|d)=\frac{P\left(d\right|s\left)P\right(s)}{P\left(d\right)},$$

(1)

$P\left(d\right|s)$—likelihood that a given data (d) has been produced from a signal (s), $P\left(s\right)$—is a prior knowledge about the signal, and P(d) is a normalization factor.

Resolve is developed in order to optimize imaging of extended and diffuse radio sources and to provide reliable noise estimation [5]. It has been successfully used for example in imaging of Cygnus A [11] observed with the VLA and the M87* black hole environment with VLBI [12]. In this work we want to explore resolve capabilities for ALMA interferometer, specifically compared with the most commonly used clean algorithm.

2. Test Datasets

In this section, we present data used to test interferometric imaging methods: resolve and clean. We use a simple simulated dataset in order to have a complete control over the input information, as well as the real ALMA observations.

2.1. Simulated Data

In this section, we describe the preparation of the simulated data. The major advantage in using simulated datasets to analyze the imaging procedures is a complete control of the input parameters. We control perfectly the shape and brightness of the sources, as well as configuration and behaviour of the telescope array. On the downside of this approach is the difficulty to realistically model the noise acquired during observation and simplistic assumptions about the sky brightness.

We generate a simple 2D array consisting of five Gaussian components of different brightness, size and position angle. This model is input into the Common Astronomy Software Applications package (CASA; [13]) task simalma. This task converts image to the sky model, imposing physical properties to the sky, such as spherical coordinates, pixel dimensions, field-of-view and brightness in physical units. Figure 1 (left) shows the sky model generated with simalma task in CASA. Afterwards, the task is simulating observations of the given sky model with ALMA observatory. For this specific case we simulate observations with ALMA configuration C-3 at 230 GHz (ALMA band 6), which results in effective resolution of 0.7″, determined approximately by largest available baseline (distance between two antennas). In the case of C-3 configuration this is ∼500 m.

The simalma task returns a calibrated Measurement Set (MS), that consists of complex visibilities. Those visibilities are Fourier transform of the sky brightness, therefore reverse Fourier transformation provides a dirty image of the observed sky. Further on we describe the process of imaging with resolve and tclean.

2.2. ALMA Data

Here, we describe archival ALMA observations used to test application of resolve on real-life example. It is important to compare it with a well-understood case, in which the calibration and imaging examples with other tools are already available. We select a protoplanetary disk Sz114 observed within DSHARP ALMA Large Program [14]. This represented a milestone program for the ALMA observatory, showcasing the richness of substructures within disks, often associated with ongoing planet formation. It is therefore especially interesting to search for innovative methods of imaging those data in order to verify current conclusions, as well as open avenues for new studies.

The Sz114 disk shows relatively smooth structure compared with other extreme cases and therefore it serves well as a test case to try to identify underlying structure with resolve. Figure 1 (right) presents image of the disk obtained with tclean as delivered by the Large Program team [14].

2.3. Imaging of the Simulated Dataset

First, in order to image the simulated measurement set, we use tclean task in CASA software which implements CLEAN algorithm as in Hogbom 1974 [4]. We ran tclean without any constraint on where to look for point sources in the image (i.e., without any masking), for 20,000 iterations, or until threshold of 0.3 mJy/beam was reached. Pixel size of the reconstructed image is set to 0.1″ and image size is set to 512 × 512 pixels.

This threshold is set based on the earlier, quick tclean, so that the algorithm does not attempt to find sources from the residual image consisting purely of noise. In the default settings, weighting of baselines is set to ’natural’, which means it associate the baseline with weight proportional to the sampling density (i.e., the most covered baselines have the highest weight). Since there is much more baselines sampling the larger scales, this results in putting more weight to lower resolution, which results in achieving lower resolution than the sky model image. Therefore we also attempt an imaging with Briggs weighting with robust parameter 0.5, which moves the balance of weighting toward longer baselines increasing the resolution but decreasing signal-to-noise ratio. The result of the tclean run is presented in Figure 2 (bottom).

For resolve image of the simulated data we run 30 iterations, using Newton optimizer. We did not make any assumptions on the presence of the point sources in the data. Parameters of the resolve run are summarized in

Table 1. Parameters of the resolve run on simulated dataset (left) and Sz114 ALMA data (right).

Parameter	Mean	Parameter	Mean
Offset	26	Offset	20
Zero mode	1 ± 0.1	Zero mode	1 ± 0.2
Fluctuations	5 ± 1	Fluctuations	3 ± 1
Power spectrum slope	−2 ± 0.5	Power spectrum slope	−4 ± 2
Flexibility	1.2 ± 0.4	Flexibility	4 ± 0.8
Asperity	0.2 ± 0.2	Asperity	2 ± 0.8

. For detailed explanation of the parameters see Arras et al. [11].

2.4. Imaging of the Real ALMA Observations

For the Sz114 protoplanetary disk observed within DSHARP ALMA Large Program [14], we have created an image from a single spectral window, in order for a direct comparison with resolve, which currently does not have the capability to create images of multiple spectral windows. From the publicly available MS file, we extracted spectral window 9 with split task in CASA and binned it into a single channel.

For this tclean run we implemented both standard and multiscale imaging in order to compare the outcomes, since the observed disk presents large variety of spatial scales. We create images with 0.005″ and 1024 × 1024 pixels and run 20,000 iterations up to noise threshold of 0.05 mJy is reached. In the case of multiscale clean, we specified three scales: at 0, 7, and 28 pixels, which means the tclean algorithm iterate in order to find three types of sources in the data: either a point source, which corresponds to scale 0, and extended components with Gaussian shape of 7 and 28 pixels of FWHM. Results of the imaging are presented in Figure 3 (bottom).

A resolve imaging followed the same procedure as in the simulated data, however, since the weights were obtained with from real observation, this results in more realistic treatment of the uncertainties. Parameters of this run are shown in Table 1.

3. Comparison between tclean and Resolve

In this initial study of performance of the resolve algorithm in operating with ALMA data, we focus on key aspects of the imaging process: flux recovery—how much flux observed on the sky is recovered by the imaging process and its accuracy; image quality—effective resolution and dynamic range.

3.1. Simulations

We compare the quality of resolve and tclean imaging of the simple simulated ALMA observation, as described in Section 2.1. Figure 3 presents resolve image and associated uncertainty map, and tclean image with two different weighting schemes: natural and Briggs weighting with robust parameter of 0.5.

We note that for tclean the imaging resulted with large areas with a negative flux, which do not occur in resolve—positive-only emission is one of the assumptions in the prior. On the uncertainty map produced by resolve, we observe much higher confidence associated with the source positions, compared with positions where no significant emission was detected.

We measure integrated fluxes of each of the Gaussian components in both imaging methods as well as in the input sky model. Result is presented in

Table 2. Peak flux (in Jy arcsec${}^{-2}$) of different components on the simulated image.

Comp	Model	tclean	tclean	Resolve
		Natural	Briggs
1	34.11	24.67	24.96	26.68 ± 5.76
2	23.31	21.68	22.19	23.59 ± 6.72
3	16.69	16.31	16.57	17.82 ± 3.86

. We can note that resolve and tclean both underestimate the flux of the brightest source (i.e., the Gaussian associated with the largest integrated flux). In case of tclean, this can be attributed to imperfect recovery of the signal from the point-spread function, but it is unclear why resolve underpredicts the flux. More extensive runs are needed to investigate this issue.

On the other hand, resolve reaches good accuracy on the other two measured components. We are able to provide much more reliable error bars on resolve by taking the standard deviation of the measured value from all samples in the final iteration.

3.2. ALMA Data

In the case of real ALMA data, it is more difficult to assess total recovered flux since we do not have information on true flux. We compare the peak and total flux integrating over different areas of the disk

Table 3. Peak (in Jy arcsec${}^{-2}$) and integrated flux at given radii of the Sz114 disk.

Radius	Best	tclean	tclean	Resolve
		Hogbom	Multiscale
peak	1.57	1.55	1.49	5.8 ± 3.82
0.06	8.63	8.85	8.95	9.48 ± 1.07
0.15	22.79	22.82	22.53	23.21 ± 0.68
0.35	47.45	47.07	47.24	47.24 ± 2.79

. First of all, it can be noted that all tclean images have comparable fluxes, and therefore the multiscale did not affect significantly the flux measurements.

In a comparison between resolve and tclean, we can see that peak flux measured from resolve image is three-times higher than the peak in tclean images. This can be purely a gridding effect as the pixel scale of the resolve image is much smaller.

At the same time, we note slightly comparable values of the integrated flux measured at from different radii of the disk.

At first sight, it appears that resolve is able to create a super-resolution image of the disk. We can use the uncertainty map to understand what confidence can be associated with those structures. We note on the uncertainty map that we typically achieve better than 20% confidence on the results; however, the map also reaches very low confidence in some areas.

4. Conclusions

In this work, we apply Bayesian inference and field theory in the framework of Information Field Theory (IFT) using the resolve algorithm, to create images from radio intereferometric observations obtained with ALMA array. The key conclusions are:

• The imaging of simulated ALMA Measurement Set with resolve results in successful recovery of the flux of different components in the simulated dataset.
• In one of the first attempts to apply resolve on real ALMA data we obtain a high-fidelity image of protoplanetary disk Sz 114, highlighting the potential of resolve to create super-resolution images.
• For both test cases, we obtain a robust estimation of the uncertainties on the measured fluxes, which presents one of the major advantages of resolve compared with tclean.

With these encouraging results, we further highlight areas where exciting developments can be made: resolve can be used to create spectral cubes, using correlation between channels as additional prior information; create combined maps of different antenna configuration, a particularly challenging case for tclean algorithm and where accurate assessment of the confidence in obtained data is especially necessary.