Interferometric Image Reconstruction using Closure Invariants and Machine
Learning–A

# Interferometric Image Reconstruction using Closure Invariants and Machine
Learning

Nithyanandan Thyagarajan  1 Lucas Hoefs E-mail:
<EMAIL_ADDRESS>1,2 and O. Ivy Wong1,3
1Commonwealth Scientific and Industrial Research Organisation (CSIRO)  Space &
Astronomy  P. O. Box 1130  Bentley  WA 6102  Australia
2Mechatronics Engineering Department  Curtin University  Bentley  WA 6102
Australia
3International Centre for Radio Astronomy Research  The University of Western
Australia  Crawley  WA 6009  Australia

(Accepted XXX. Received YYY; in original form ZZZ; 2023)

###### Abstract

Closure invariants in interferometry carry calibration-independent information
about the morphology of an observed object. Excepting simple cases, a mapping
between closure invariants and morphologies is not well established. We aim to
demonstrate that closure invariants can be used to classify the morphology and
estimate the morphological parameters using simple Machine Learning models. We
consider 6 morphological classes – point-like, uniform circular disc,
crescent, dual disc, crescent with elliptical accretion disc, and crescent
with double jet lobes – described by phenomenological parameters. Using simple
logistic regression, multi-layer perceptron (MLP), convolutional neural
network, and random forest models on closure invariants obtained from a sparse
aperture coverage, we find that all models except logistic regression are able
to classify the morphology with an $F_{1}$ score $\gtrsim 0.8$. The
classification accuracy notably improves with greater aperture coverage. We
also estimate morphological parameters of uniform circular disc, crescent, and
dual disc using simple MLP models, and perform a parametric image
reconstruction. The reconstructed images do not retain information about
absolute position or intensity scale. The estimated parameters and
reconstructed images are found to correspond well with the inputs. However,
the prediction accuracy worsens with increasing morphological complexity. This
proof-of-concept method opens an independent approach to interferometric
imaging under challenging observing conditions such as that faced by the Event
Horizon Telescope and Very Long Baseline Interferometry in general, and can
complement other methods to robustly constrain an object’s morphology.

###### keywords:

Algorithms – Machine Learning – methods: data analysis – techniques: image
processing – techniques: interferometric

##  1 Introduction

Image synthesis using radio interferometric measurements requires an array of
receiver elements sampling spatial correlations of the impinging radiation on
the aperture to infer the spatial intensity distribution on the sky within the
telescope’s field of view (Thompson et al., 2017; Taylor et al., 1999). As the
spatial resolution of the image scales inversely with the largest spacing
between the interferometer array elements, obtaining fine details in the image
requires a technique called Very Long Baseline Interferometry (VLBI) which
requires array elements widely separated from each other, with typical
separations spanning continental or even planet-sized scales. Due to the
sparseness of measurements and demanding requirements of accurate signal
calibration required in maintaining a high degree of spatial coherence while
combining signals from such large separations, VLBI imaging is extremely
challenging in general (Thompson et al., 2017; Walker, 1999).

There are certain invariants in interferometry like closure phases (Jennison,
1958) and closure amplitudes (Twiss et al., 1960) that are immune to
propagation and instrumental effects that are associable with individual array
elements, and are thus independent of calibration and errors therein. Being
true observables of the observed object’s morphology, they implicitly or
explicitly serve as useful anchors for inferring an object’s morphology. A few
examples of classic VLBI successes that used closure quantities include the
discovery of the double-lobed structures of Cygnus A (Jennison, 1957; Jennison
& Latham, 1959) and Centaurus A (Twiss et al., 1960), determination of the
core-jet morphology of quasar 3C 147 (Wilkinson et al., 1977), and providing
the first direct evidence for superluminal expansion of the relativistic jet
in quasar 3C 273 (Pearson et al., 1981). A recent example is the event
horizon-scale imaging using the Event Horizon Telescope (EHT) data of the
central supermassive black holes at the centres of M87 (Event Horizon
Telescope Collaboration et al., 2019a, b, c, d, e, f) and Sgr A⋆ (Event
Horizon Telescope Collaboration et al., 2021) by the EHT collaboration (EHTC).

The sparseness of data, low signal-to-noise conditions, choices of initial
models in deconvolution methods, even small miscalibrations, and other
unconscious biases in the analysis workflow can lead to artefacts and
diverging image morphologies, and thus misinterpretations of the results. For
instance, Carilli & Thyagarajan (2022) detailed the effects of the choice of
initial models in the iterative imaging process. The choice of introducing
unconstrained “boxes” in the CLEAN algorithm (Högbom, 1974; Schwab, 1984) can
lead to divergent results (Cornwell et al., 1999; Miyoshi et al., 2022). The
EHTC have acknowledged these challenges and followed a very detailed
verification process to mitigate these risks. Subsequent analysis by other
groups using different methods have found consistent results (Sun & Bouman,
2020; Carilli & Thyagarajan, 2022; Arras et al., 2022; Medeiros et al., 2023).
Nevertheless, independent confirmation by independent methods is paramount in
such important scientific studies.

While closure invariants are mathematically well-defined and have been
extensively used for decades, their physical interpretation has been unclear.
A first geometric understanding of closure phase was presented in Thyagarajan
& Carilli (2022). Closure invariants are adept at distinguishing point-like
and centrosymmetric objects (unit closure amplitudes and vanishing closure
phases) from non-symmetric morphologies, and corruptions that are element-
based rather than spatially correlated (Thompson et al., 2017). Similarly, in
parametric measurements, they can be used to infer the presence of
polarisation inherent to the object (Broderick & Pesce, 2020; Samuel et al.,
2022). However, a detailed mapping between the object’s morphology and closure
invariants has not been clearly established.

In this work, we demonstrate that closure invariants can be used to
distinguish between different image morphologies and provide a pathway for
parametric image reconstruction using machine learning (ML) methods that could
map such relationships. Use of closure invariants makes it largely immune to
the aforementioned risks present in traditional interferometric image
synthesis. Interferometric imaging based on closure invariants was studied in
Chael et al. (2018). They employed a minimisation scheme over closure
quantities supplemented by some a priori information and regularisers that
favour certain image features. In contrast, we don’t impose a priori
constraints or regularisers, and instead rely entirely on closure invariants
and ML to determine the morphological class and the respective parameters.
Thus, it can provide independent inputs to interferometric image
reconstruction.

This work is a first proof of concept of the ability to classify and
parametrically reconstruct radio interferometric images using closure
invariants and simple ML methods. We investigate the performance of these ML
methods across varying levels of aperture coverage on morphological classes of
varying complexities.

We introduce interferometric closure invariants in §2. We present a proof of
concept for morphological classification using simple ML methods in §3. The
effect of varying levels of aperture coverage on their performance is also
investigated. A proof of concept for estimating the morphological parameters
and reconstructing images using ML methods for some chosen morphologies is
presented in §4. §5 presents a discussion and summary.

##  2 Closure Invariants in Interferometry

The concepts of closure phases (Jennison, 1958) and closure amplitudes (Twiss
et al., 1960) have been in use in radio astronomy for many decades. They have
been integral to advances in calibration and interferometric synthesis imaging
(Cornwell & Fomalont, 1999; Thompson et al., 2017; Carilli et al., 2022). They
were extended to polarimetric measurements through the formalism of closure
traces (Broderick & Pesce, 2020). Recently, a unified and general theory of
interferometric closure invariants for co-polar and polarimetric measurements
has emerged from Thyagarajan et al. (2022) and Samuel et al. (2022),
respectively. We will primarily use the former because this work pertains to
co-polar measurements. Below is a brief outline of the theory of closure
invariants in co-polar interferometry relevant for this work.

In radio interferometry, the basic measurement units are spatial correlations,
known as visibilities, in the aperture plane corresponding to different array
element spacings. In an interferometer array with $N$ elements labelled by the
indices $a=0,\ldots,N-1$, each element, $a$, measures the amplitude and phase
of the stochastic electric field, $e_{a}$ (represented by a complex number),
incident on it. The true spatial correlation (visibility) between pairs of
array elements is $V_{ab}\coloneqq\langle e_{a}e_{b}^{\dagger}\rangle$, where,
$\dagger$ denotes complex conjugation.

The visibilities are related to the spatial distribution of intensities on the
sky or image plane (van Cittert, 1934; Zernike, 1938). Under certain
reasonable assumptions (Thompson et al., 2017; Clark, 1999), these
visibilities denote the Fourier components (in the aperture plane) of the
images that we are aiming to classify and reconstruct,

$\displaystyle V_{ab}\equiv V(\mathbf{u}_{ab})$
$\displaystyle=\iint_{\mathbb{S}}I(\hat{\mathbf{s}})\,e^{-2\pi
i\mathbf{u}_{ab}\cdot\hat{\mathbf{s}}}\,\mathrm{d}^{2}\hat{\mathbf{s}}\,,$ (1)

where, $\hat{\mathbf{s}}$ denotes a unit vector covering the sky surface,
$\mathbb{S}$. $\mathbf{u}_{ab}\equiv(u,v)$ denotes the spacial frequency and
is related to the spacing between elements, $\mathbf{x}_{ab}$, as
$\mathbf{u}_{ab}=\mathbf{x}_{ab}/\lambda$, where, $u$ and $v$ refer to $x$\-
and $y$-components of $\mathbf{u}_{ab}$ in the aperture (Fourier) plane, and
$\lambda$ is the wavelength. $I(\hat{\mathbf{s}})$ is the spatial distribution
of intensities on $\mathbb{S}$.

In real-world measurements, corruptions caused by propagation and instrumental
effects affect the array element’s measurement as $e_{a}^{\prime}=g_{a}e_{a}$,
where, $g_{a}$ is a complex corruption factor. Therefore, the measured
visibility is also corrupted as $V_{ab}^{\prime}=g_{a}V_{ab}g_{b}^{\dagger}$.
This necessitates calibration and elimination of these multiplicative
corrupting factors. There are several situations when accurate calibration can
be challenging and can lead to artefacts in the images (Cornwell & Fomalont,
1999).

Invariants like closure phases and amplitudes are special interferometric
quantities constructed in specific ways that eliminate the corrupting effects
caused by the $g_{a}$ terms partially or altogether (Thompson et al., 2017;
Cornwell & Fomalont, 1999). Because of this property, even when they are
constructed with corrupted data, they remain true observables of the physical
system under study independent of local corrupting factors.

In this work, we adopt the Abelian gauge theory formalism (Thyagarajan et al.,
2022) to obtain a complete and independent set of closure invariants in co-
polar interferometric measurements. We start by identifying a specific antenna
location as the reference vertex (indexed as element 0) and constructing all
possible closed triangular loops starting and ending with this vertex
(Thompson et al., 2017). For an $N$-element array, this gives us
$(N-1)(N-2)/2$ independent triangular loops. On each triangular loop, we
define an advariant as

$\displaystyle\mathcal{A}_{0ab}^{\prime}$
$\displaystyle=V_{0a}^{\prime}\widehat{V}_{ab}^{\prime}V_{b0}^{\prime}=|g_{0}|^{2}\mathcal{A}_{0ab}\,,$
(2)

where, $\widehat{Z}=(Z^{\dagger})^{-1}$ for a non-zero complex number, $Z$.
There are $(N-1)(N-2)/2$ complex advariants, each with a pair of real and
imaginary parts. This amounts to $(N-1)(N-2)$ real numbers, all with the same
unknown scaling factor, $|g_{0}|^{2}$, that is associated with the reference
vertex. In the final step, invariants are obtained from advariants by
eliminating the unknown scaling factor by dividing all of them by any one of
them that is non-zero such as the mean, maximum, or root-mean-square, etc.
This effectively yields $N^{2}-3N+1$ real-valued invariants after the loss of
one degree of freedom in the unknown scale elimination step.

##  3 Morphological Classification

The first goal of this paper is to explore the possibilities of classifying
image morphologies using simple ML methods entirely from interferometric
closure invariants rather than visibilities because the latter are not immune
to calibration errors. The fiducial morphological classes we have considered
are outlined below.

We consider six different morphological classes (0–5) along with the
parametrisations described below. Closure invariants are insensitive to some
parameters such as absolute location and absolute intensity scale. The
parameters that can be inferred for image reconstruction after the
morphological classification are italicised:

  1. 0 –

Point-like: location, intensity,

  2. 1 –

Uniform circular disc: radius ($R$), constant intensity throughout the disc,

  3. 2 –

Dual circular discs: radii of the two discs ($R_{1}$, $R_{2}$), offset of the
centre of the second disc from the first ($\delta x$, $\delta y$), intensity
of first disc ($I_{1}$), ratio of intensities of the two discs
($I_{1}/I_{2}$),

  4. 3 –

Crescent: radii of the outer and inner discs ($R_{1}$, $R_{2}$), offset of the
centre of the inner disc from the first ($\delta x$, $\delta y$), intensity of
outer disc ($I_{1}$), intensity of inner disc is constrained to be $-I_{1}$,

  5. 4 –

Crescent with elliptical accretion disc: crescent parameters as above, an
elliptical accretion disc parametrised by semi-major axis ($R_{a}$), semi-
minor axis fixed at $0.2R_{a}$, a position angle ($\theta_{a}$), and an
intensity ratio between the crescent and the accretion disc
($I_{\textrm{c}}/I_{\textrm{a}}$),

  6. 5 –

Crescent with jet lobes: crescent parameters as above, two diametrically
opposite jet lobes with offsets from the crescent centre ($\pm\delta x$,
$\pm\delta y$), and jet lobe intensities relative to the crescent intensity
($I_{1}/I_{\textrm{c}}$, $I_{2}/I_{\textrm{c}}$).

Figure 1 shows examples of these morphological classes obtained with random
instances of the respective parameters. Note that these parametrisations are
purely phenomenological and not physical.

(a)  (b)  (c)

(d)  (e)  (f)

Figure 1: Example images of six morphological classes that are classified
using closure invariants. Each pixel corresponds to 3.51 $\mu$as. (a) Point-
like objects are simulated at random locations with random intensities. (b)
Uniform circular disc morphology is parametrised by its radius. (c) Double
disc morphology is parametrised by the radii of the two discs, the $X$\- and
$y$-offsets between their centres, and the ratio of their intensities. (d) A
crescent morphology is parametrised by the inner and outer radii, and the
$X$\- and $y$-offsets of the inner hollow region from the centre. (e) A
crescent with accretion disc morphology is parametrised by the crescent
parameters in (d), an elliptical accretion disc parametrised by semi-major
axis with an ellipticity of 0.2 and a position angle, and an intensity ratio
between the crescent and the accretion disc. (f) A crescent with double jet
morphology consists of a crescent as parametrised above, in addition to two
diametrically opposite jets with $x$\- and $y$-offsets from the centre, and
two intensities relative to the crescent structure.

###  3.1 Input measurements

For sampling the Fourier (aperture) plane, we used frequencies of
$\nu_{1}=230$ GHz and $\nu_{2}=345$ GHz, using the eht-imaging software, which
contains the exact locations of the EHT array elements (Event Horizon
Telescope Collaboration et al., 2019b). Two fiducial times of
$t_{1}=\texttt{2017-04-05 04:46:05}$ UTC and $t_{2}=\texttt{2017-04-05
05:34:05}$ UTC were chosen to have seven locations of telescopes of the EHT
array for which an object at
RA=12${}^{\textrm{h}}$30${}^{\textrm{m}}$49.42${}^{\textrm{s}}$,
Dec=+12∘23′28.04′′ (J2000) would be visible. The seven locations correspond to
that of the Atacama Large Millimeter/submillimeter Array (ALMA; Matthews et
al., 2018) and the Atacama Pathfinder Experiment telescope (APEX; Wagner et
al., 2015) in Chile, the Large Millimeter Telescope Alfonso Serrano (LMT;
Ortiz-León et al., 2016) in Mexico, the IRAM 30 m telescope on Pico Veleta
(PV; Greve et al., 1995) in Spain, the Submillimeter Telescope Observatory
(SMT; Baars et al., 1999) in Arizona, the James Clerk Maxwell Telescope (JCMT)
and the Submillimeter Array (SMA; Young et al., 2016) in Hawai’i.

Fig. LABEL:fig:uv shows the aperture sampled in this study. The red and blue
symbols denote 230 GHz and 345 GHz, respectively, while the $+$ and $\times$
symbols denote the two times at which the aperture was sampled, thereby
yielding four possible combinations. We employ three sampling configurations
in this paper to study the effect of varying the level of aperture plane
sampling on the morphological classification accuracy. These aperture
configurations correspond to time $t_{1}$ at frequency $\nu_{1}$, times
$t_{1}$ and $t_{2}$ at frequency $\nu_{1}$, and times $t_{1}$ and $t_{2}$ at
frequencies $\nu_{1}$ and $\nu_{2}$. They are denoted by 1Fx1T, 1Fx2T, and
2Fx2T, respectively. The seven elements of the EHT array yielded 21 snapshot
visibilities at any given time and frequency. Therefore, the three sampling
configurations consist of 21, 42, and 84 visibilities, respectively.

(a)  (b)

Figure 2: (a) Aperture (Fourier) plane sampling (in units of wavelengths)
obtained using locations of seven telescopes in the EHT array at $\nu_{1}=230$
GHz (red) and $\nu_{2}=345$ GHz (blue) at times $t_{1}=\texttt{2017-04-05
04:46:05}$ UTC ($+$ symbols) and $t_{2}=\texttt{2017-04-05 05:34:05}$ UTC
($\times$ symbols). The inset shows a zoomed view of the shortest element
spacings. The three aperture sampling configurations in this paper correspond
to $\nu_{1}$ at $t_{1}$ (red $+$ denoted by 1Fx1T), $\nu_{1}$ at $t_{1}$ and
$t_{2}$ (red $+$ and $\times$ denoted by 1Fx2T), and $\nu_{1}$ and $\nu_{2}$
at $t_{1}$ and $t_{2}$ (red and blue $+$ and $\times$ denoted by 2Fx2T). (b)
Closure invariants for 10,000 random realisations of parameters under each of
the six morphological classes as labelled on the $y$-axis. The closure
invariants are indexed on the $x$-axis. The aperture models 1Fx1T, 1Fx2T, and
2Fx2T correspond to the first 30, 60, and 120 indices, respectively. The
values of closure invariants are indicated by the colour scale. Alternating
bands of 15 indices correspond to the real and imaginary values of the complex
advariants from which the real-valued co-polar closure invariants are derived
using Abelian gauge theory formalism (Thyagarajan et al., 2022), in which all
point-like morphologies have zero imaginary values and unit real values
regardless of their location or absolute intensities.

We generated 10,000 instances of $64\times 64$ images for each morphological
class using random realisations of their respective parameters (see Fig. 1).
The pixel resolution in angular units is 3.51 $\mu$as. Assuming all the array
elements were identical and had uniform power sensitivity across the entire
image, the visibilities were simulated using a direct Fourier Transform at the
antenna spacings for the various realisations of the images under each of the
morphological classes using equation (1). No noise was added to the simulated
visibilities.

The closure invariants were constructed from equation (2) using the Abelian
gauge theory formalism for co-polar invariants (Thyagarajan et al., 2022). In
this formalism, a 7-element array will yield 15 independent triads and a total
of 30 real numbers (from 15 complex-valued advariants) with one unknown but
common scale factor shared between them. Dividing them all by any one of them
(which is non-zero) will eliminate this common scale factor and provide 29
closure invariants with the thirtieth number being the trivial unity as a
result of this division as detailed in Thyagarajan et al. (2022). In this
study, we not only keep the 29 invariants but also the additional trivial
number – the unity – purely for bookkeeping convenience even though it
contributes no valuable information.

Fig. LABEL:fig:closure-invariants shows the closure invariants computed for
various realisations of each morphology. The classes are labelled on the left
with horizontal lines denoting class boundaries, each of which include 10,000
realisations. The closure invariants are numbered along the $x$-axis.
Alternating bands of 15 closure invariants correspond to the real and
imaginary parts of the advariants (Thyagarajan et al., 2022), respectively.
The three aperture sampling configurations, 1Fx1T, 1Fx2T, and 2Fx2T correspond
to the first 30, 60, and 120 closure invariants, respectively. The colour
scale represents the value of the closure invariants.

It is noted that for a point-like morphology, all the complex-valued
advariants are real and equal-valued. Therefore, the closure invariants
consist of equal numbers of ones and zeros. This is equivalent to having zero
closure phases and unit closure amplitudes independent of the location or the
absolute intensity, as expected from the centrosymmetry of a point-like object
(Thompson et al., 2017; Thyagarajan et al., 2022). Similarly, the symmetry of
a uniform circular disc is reflected in the vanishing of all the imaginary
parts. More complicated morphological classes have a correspondingly non-
trivial combination of real and imaginary parts.

The input for the training of our ML classifiers and parameter estimators
comprises of these closure invariants and the morphological class labels they
were generated from. The ML model so generated will then take an unknown set
of closure invariants as input and predict the morphological class and the
respective parameters. Note that neither the images nor the visibilities
themselves are used as inputs.

###  3.2 Machine Learning Models

We considered four simple ML methods for classifying the image morphologies,
whose details and performance are characterised below. We used PyTorch for
implementing most of the models except for the Random Forest classifier for
which we used scikit-learn. The ML models are visually illustrated in Fig. 7
in Appendix A. In each of these methods, we split the data randomly with 80%
used for training and 20% for validation. We randomised the composition of the
training and validation sets several times maintaining the same proportional
split, and found no significant change in the results. For each method, the
minimum possible number of training epochs was chosen to avoid over-fitting
and to preserve the method’s generalisation capabilities for unseen data.

####  3.2.1 Logistic Regression

Logistic regression (LR) estimates the probability of an event occurring, such
as the morphological class, based on a given data set of independent
variables, namely, the closure invariants. LR is considered a discriminative
model, which means that it attempts to distinguish between classes. Since the
dependent variable (morphological class) has six possible outcomes, and no
specified order, we employ multinomial LR, which is a simple network
consisting two layers – input (corresponding to the number of closure
invariants) and output (number of morphological classes).

The training was performed using Sigmoid activation, Adam optimiser (learning
rate $10^{-4}$), and 10 epochs. Fig. LABEL:fig:cmat-LR shows the confusion
matrix between true and predicted classes using LR for the 2Fx2T aperture
model. The presence of significant off-diagonal values indicates that LR’s
classification was not very accurate even though it is notably better than a
random classification. Even point-like morphologies got accurately classified
in only 63% of the instances. The classification performance on other
morphologies is only $\sim 33-38$%. LR is one of the simplest classification
schemes and is not expected to perform with high accuracy for non-binary
classifications.

(a)  (b)

(c)  (d)

(e)

Figure 3: Confusion matrices of (a) Logistic Regression (LR), (b) Multi-layer
Perceptron (MLP), (c) Convolutional Neural Network (CNN), and (d) Random
Forest from training for image morphology classification. (e) $F_{1}$ score
distributions for the four ML classification models from a randomised balance
of input morphologies for aperture coverage models 1Fx1T (orange), 1Fx2T
(cyan), and 2Fx2T (grey). The box-whisker plots show median, lower and upper
quartiles, 95% confidence limits, and outliers beyond.

####  3.2.2 Multi-layer Perceptron

Multi-layer Perceptron (MLP), a supervised learning algorithm, is a fully
connected feed-forward artificial neural network with at least three layers
(input, output, and at least one hidden layer) that can learn to classify non-
linearly separable patterns.

15-epoch training with an Adam optimiser (learning rate $10^{-3}$) was done
with Rectified Linear Unit (ReLU) non-linear activation in each of the hidden
layers. Fig. LABEL:fig:cmat-MLP shows the confusion matrix obtained for the
2Fx2T aperture model. Relative to LR’s performance, the confusion matrix is
much tighter around the diagonal. Point-like morphologies are perfectly
predicted. The uniform circular disc morphology is predicted accurately at the
rate of 91%. The prediction accuracy decreases with increasing complexity of
the underlying morphology. The worst performance is seen in the case of
morphology classes 4 (crescent with accretion disc) and 5 (crescent with jet
lobes) at a prediction accuracy of $\simeq 63-64$%, which is still better than
the best performance of LR in the case of point-like morphology.

####  3.2.3 Convolutional Neural Network

A convolution neural network (CNN) consists of three types of layers –
convolutional layer, pooling layer, and fully-connected layer. The
convolutional layer consists of filtering kernels that extract features. The
pooling layer effectively does downsampling thereby effecting a dimensionality
reduction. The fully-connected layer performs the classification. CNNs perform
well in scenarios where ordering present in input data can be exploited.

Our CNN model training used an Adam optimiser (learning rate $10^{-3}$) and 15
epochs with ReLU activation in the hidden layers. Fig. LABEL:fig:cmat-CNN
shows the confusion matrix after training on the 2Fx2T aperture model. The CNN
fares better for the circular disc (97% in contrast to MLP’s 91%) and worse
for the crescent (68% compared to MLP’s 75%), while remaining similar for the
rest of the morphological classes. Overall, the CNN’s performance is on a
scale similar to the MLP. We note that this is most likely because the input
closure invariants do not have any implicit ordering that the CNN can take
advantage of, which is one of its key strengths. The ordering of closure
invariants based on the triad’s shape, orientation, size or other
characteristics to improve the CNN’s performance could be a useful avenue for
future exploration.

####  3.2.4 Random Forest

A random forest classifier (RF) is an estimator that fits a number of decision
tree classifiers on various sub-samples of the data set and uses averaging to
improve the predictive accuracy and control over-fitting. We used 200 trees in
the forest. Fig. LABEL:fig:cmat-RF shows the confusion matrix obtained with
training on the 2Fx2T aperture model with an RF classifier. The point-like and
uniform circular disc morphologies are classified perfectly. The double disc
and crescent morphologies are classified at $\gtrsim 90$% accuracy, and the
crescent with accretion disc and jet lobes are classified with $\gtrsim 76$%
accuracy. Overall, the RF classifier performed better than all the other ML
classifiers used in this work.

###  3.3 Effect of aperture coverage on classifiers’ performance

We trained the aforementioned models for the three aperture models with
different levels of aperture coverage with a balanced input of morphological
classes. To test their performance on unknown test data, we used the $F_{1}$
score (Rijsbergen, 1979), a harmonic mean of precision and recall, and is a
measure of the test’s accuracy. During testing, to avoid skewing the results
based on the level of balance of input data, we used 1000 iterations each
containing 100 randomly drawn test inputs (excluding the training set) with no
regard for the balance of classes to determine the $F_{1}$ score statistics
even including the worst case of imbalance across classes.

Fig. LABEL:fig:F1 shows the distribution of $F_{1}$ scores for randomly drawn
test inputs. LR and RF are found to have the worst and best $F_{1}$ scores,
respectively. MLP and CNN have intermediate $F_{1}$ scores which are
comparable. In all cases, the $F_{1}$ scores show marked improvement as the
aperture coverage improves. This is because the morphological information
sampled by the successively improving aperture coverage helps build better
predictive ML classifiers.

Even with limited aperture coverage (Fig. LABEL:fig:uv), we find that the
accuracy in a multi-class classification can be $>80$% using simple ML models.
Increasing the aperture coverage is expected to improve the classification
accuracy further. Refinements to these models for improved classification
accuracy is left to future work.

##  4 Morphological Parametrisation

Here, we investigate the ability of ML to estimate parameters that are
retrievable using only closure invariants and implement a parametric image
reconstruction. Absolute position and intensity information is lost while
using closure invariants, and we don’t attempt to retrieve such information.
We approach this parametrisation not in sequence to the classification but as
a separate process. The morphological class is assumed to have been specified
independently. Our approach is to parametrically reconstruct an image once
that classification is specified. Here, we only consider the 2Fx2T aperture
model.

Point-like morphologies can only be detected and classified, but cannot be
reconstructed because they are parametrised by a single location and single
intensity, whose absolute values are not accessible by closure invariants. We
study the parametric image reconstruction of the following morphological
classes: uniform circular disc (1), dual disc (2), and crescent (3), noting
that we do not estimate an absolute location or an absolute intensity scale.
We primarily use MLP to determine the morphological parameters. The MLP models
used for the different morphologies are visually illustrated Fig. 8 in
Appendix A. The specific MLP model is briefly described below for each
morphology.

###  4.1 Uniform circular disc

The only parameter that we can estimate with closure invariants is the radius
of the circular disc. We used an an MLP model with ReLU activation and an Adam
optimiser (learning rate $10^{-3}$). After 10 epochs, the prediction accuracy
asymptotically approached 100%. Fig. 4 shows a histogram of the relative error
between the prediction and the true value in per cent. The prediction is
accurate to within 0.04% with 95% confidence.

Figure 4: Percentage error in the prediction of uniform circular disc radius
using MLP (see Fig. LABEL:fig:disc-graph). The box-whisker plot shows the
median, lower and upper quartiles, and the 95% confidence interval ($\simeq\pm
0.04$%) of the prediction error.

###  4.2 Crescent

In the morphologies we studied, the crescent model offers the next level of
complexity relative to the uniform circular disc. The parameters of the
crescent model we estimate are the radii of the outer and inner discs, $R_{1}$
and $R_{2}$, respectively, and the $x$\- and $y$-offsets of the centre of the
inner disc from the outer one, $\delta x$, and $\delta y$, respectively. We
used an an MLP model with ReLU activation, 20% dropout of neurons between
layers, and an Adam optimiser (learning rate $10^{-3}$). We performed a
10-epoch training with 8-fold cross-validation.

Figs. LABEL:fig:cr-R1–LABEL:fig:cr-dY show the predicted parameters against
the true values as “violin” plots. It is seen that the predictions of all the
four parameters are consistent with their respective input values to within
95% confidence over the range of values considered.

(a)  (b)

(c)  (d)

(e)  (f)  (g)

Figure 5: Estimated parameters of crescent using MLP (see Fig.
LABEL:fig:crescent-graph) and parametric image reconstruction example.
Predicted parameters against input parameters are shown for (a) outer disc
radius, $R_{1}$, (b inner disc radius, $R_{2}$, (c) $x$-offset, $\delta x$,
and (d) $y$-offset, $\delta y$ between the two discs. The diagonal lines
denote perfect prediction. The mean and central 95th percentile are marked on
the “violin” plots. The shade of the “violin” indicates the relative number of
input data in that bin. (e) Input crescent image example. (f) Parametrically
reconstructed crescent image from estimated parameters. (g) Difference between
input and reconstructed image.

As an example, Fig. LABEL:fig:cr-inp shows an input image of a crescent. With
the predicted parameters, Fig. LABEL:fig:cr-pred shows a parametrically
reconstructed image. The difference between the input and the parametrically
reconstructed image is shown in Fig. LABEL:fig:cr-diff. While the
reconstructed image resembles the input image reasonably, the prediction
accuracy is not as high as the uniform circular disc case. This is because the
same amount of input data is used to constrain a larger number of parameters,
thereby leading to a slight degradation in prediction accuracy.

###  4.3 Dual disc

A double disc is more complex than a crescent because in addition to the radii
of the two discs ($R_{1}$ and $R_{2}$) and the displacement of the centre of
one relative to the other ($\delta x$ and $\delta y$), the ratio of two
intensities ($I_{1}/I_{2}$) is an extra parameter. Again, we used an an MLP
model with ReLU activation, 20% dropout of neurons between layers, and an Adam
optimiser (learning rate $10^{-3}$). We performed a 10-epoch training with
8-fold cross-validation.

Figs. LABEL:fig:dd-R1–LABEL:fig:dd-dY show the input against the predicted
parameters as violin plots. The ideal prediction is enveloped by the central
95% of the distribution of predicted parameters in all cases except $R_{2}$.
The model suffers from an over-prediction of $R_{2}$ at smaller values and a
significant under-prediction at larger values.

(a)  (b)  (c)

(d)  (e)

(f)  (g)  (h)

Figure 6: Estimated parameters of dual using MLP (see Fig. LABEL:fig:dualdisc-
graph) and parametric image reconstruction example. Predicted parameters
against input parameters are shown for (a) larger disc radius, $R_{1}$, (b
inner disc radius, $R_{2}$, (c) ratio of two disc intensities, $I_{1}/I_{2}$,
(d) $x$-offset, $\delta x$, and (e) $y$-offset, $\delta y$ between the two
discs. The diagonal lines and violin plots have same meaning as in Fig. 5.
Panels (f), (g), and (h) show input, reconstructed, and difference images,
respectively, of a dual disc example.

Figs. LABEL:fig:dd-inp–LABEL:fig:dd-diff show an input, the parametric
reconstruction, and the difference image for the dual disc model. The
prediction accuracy appears to have worsened relative to the crescent case.
This is again attributed to an increase in the number of morphological
parameters for the same number of input closure invariants, which results in
inaccurate constraints on $R_{2}$. Refinement of ML models for improving the
prediction accuracy will be subject of future work.

##  5 Discussion and Summary

Extreme VLBI conditions such as requirement of high-precision calibration,
sparse data, and low signal-to-noise pose a big challenge to interferometric
imaging. In such cases, closure invariants, which are immune to calibration
errors, critically provide a reliable information anchor for estimating the
image morphology and preventing divergent solutions and artefacts, and have
indeed been successfully employed for decades. Despite their extensive use, a
mapping between closure invariants and morphological parameters has not been
straightforward. This, and the notion that closure invariants can provide
independent avenues for constraining image morphology are the primary
motivators of this study.

We demonstrate that Machine Learning methods applied on the closure invariants
can be used to classify the image morphology as well as determine the
morphological parameters. The latter can aid in parametric image
reconstruction. We studied the ability of simple methods like logistic
regression, multi-layer perceptron, convolutional neural network, and random
forest classifier to classify six different image morphologies – point-like,
uniform circular disc, dual disc, crescent, crescent with elliptical accretion
disc, and crescent with double jet lobes. Among them, all except logistic
regression yield $F_{1}$ scores $\gtrsim 0.8$ even with relatively sparse
aperture coverage. The classification accuracy of all ML models notably
improve with increasing aperture coverage.

We employed simple multi-layer perceptron models to estimate morphological
parameters such as radii, relative offsets, ratio of intensities, etc. for the
uniform disc, crescent, and dual disc morphologies. With these estimates, we
carried out parametric image reconstruction without information about absolute
position or intensity scale. The estimated parameters and reconstructed images
were largely found to be consistent and significantly correlated with the
corresponding truth values. Increasing the morphological complexity while
using a relatively sparse aperture coverage resulted in a corresponding loss
of accuracy in the estimated parameters. Increasing the aperture coverage and
the sophistication of the ML models is expected to improve the prediction
accuracy, which will be subject of future study.

Our proof-of-concept method demonstrates that closure invariants can be used
for constraining object morphologies and reconstructing images from
interferometric data. It offers an independent approach that may be
particularly suited for sparsely-sampled observations with challenging
calibration requirements such as the EHT. Machine learning methods can be
useful in mitigating the degeneracies imposed by local minima and sparsity of
information that may be encountered in existing methods when mapping from
closure invariants to image morphologies. The potential of this method
demonstrated herein warrants further exploration, the next steps of which will
be to characterise its performance in the presence of noise and compare to
other imaging methods. This approach could be used to complement, rather than
replace, traditional methods since closure invariants do not carry any
inherent information about absolute positions or intensities, and potentially
other information due to the lost degrees of freedom relative to having
reliably calibrated visibilities.

## Acknowledgements

We thank John Carilli, Foivos Diakogiannis, Ron Ekers, Timothy Galvin, Aidan
Hotan, Daniel Mitchell, John Morgan, Bojan Nikolic, Rajaram Nityananda, Cheng
Soon Ong, and Maxim Voronkov for their valuable inputs in the preparation of
this manuscript.

Software: eht-imaging (Chael, 2023), joblib (Joblib Development Team, 2020),
Matplotlib (Hunter, 2007), NumPy (Harris et al., 2020), PyTorch (Paszke et
al., 2019), Scikit-learn (Pedregosa et al., 2011), SciPy (Virtanen et al.,
2020), Torchview (Kurttutan, 2023).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A Visualisation of ML models

ML models used in the classification and parametrisation processes are
illustrated graphically in Fig. 7 and 8, respectively.

(a)  (b)  (c)

Figure 7: Machine learning models used for classification for the 2Fx2T
aperture model.

(a)  (b)  (c)

Figure 8: MLP models used for parametrisation of the (a) disc, (b) crescent,
and (c) dual disc morphologies for the 2Fx2T aperture model.