Multi-Scale CLEAN in Interferometric Imaging

• Multi-Scale CLEAN is a deconvolution algorithm that models radio and X-ray images using multiple spatial scales instead of point sources.
• It employs basis functions at various widths to efficiently represent extended or diffuse emissions and reduce imaging artifacts.
• Variants like MS-MFS and Autocorr-CLEAN optimize performance and accuracy while addressing challenges in reproducibility and computational efficiency.

Multi-Scale CLEAN, commonly written as msCLEAN or MS-CLEAN, is a deconvolution algorithm for interferometric imaging that extends Högbom CLEAN from a point-source model to a representation built from components at multiple spatial scales. In radio astronomy it is used to remove the point spread function (PSF) from dirty images produced by aperture-synthesis telescopes, and closely related formulations have also been developed for hard X-ray Fourier imaging. Rather than treating the sky exclusively as a sum of delta functions, Multi-Scale CLEAN models emission with a set of basis functions at different widths and iteratively subtracts the PSF at the scale and location that best matches the residual image, making it better suited than classical CLEAN to extended or diffuse structure (Wright et al., 2024, Müller, 18 Sep 2025).

1. Historical and inverse-problem context

The original CLEAN algorithm of Högbom is described as a matching pursuit, non-linear, iterative algorithm for deconvolving the PSF from radio astronomical images. Operationally, it identifies the maximum pixel in the dirty image, subtracts a gain-scaled PSF centered on that location, repeats until thresholding criteria are met, and restores the discovered point sources with an idealized clean beam (Wright et al., 2024). In this formulation, interferometric imaging begins from a dirty image related to the true sky by convolution with the PSF, and deconvolution aims to recover the actual sky brightness from that blurred measurement.

Multi-Scale CLEAN was introduced to address a specific weakness of point-source CLEAN: extended emission is inefficiently represented by sums of delta functions, which can lead to artifacts, poor convergence, and limited dynamic range when diffuse structure is present. The multiscale extension therefore fits flux at multiple, pre-defined angular scales, usually with circular Gaussians or tapered paraboloids, so that substantial portions of diffuse emission can be modeled in a single subtraction step rather than being fragmented into many point components (Müller, 18 Sep 2025).

This shift from a single-scale to a multi-scale representation is also the basis of later wideband and modality-specific extensions. In wideband synthesis imaging, MS-MFS extends MS-CLEAN by coupling spatial scales with spectral basis functions; in hard X-ray imaging, multi-scale CLEAN has been reformulated for instruments whose native data are sparse samples of the Fourier transform rather than direct images (Rau et al., 2011, Volpara et al., 2023).

2. Core model and iterative mechanics

The defining feature of Multi-Scale CLEAN is its image model. One formulation represents the brightness distribution as

$$I(x,y) = \sum_{i=1}^N \sum_{q_i=1}^{Q_i} I_{q_i} m_i(x-x_{q_i}, y-y_{q_i}) + B(x,y),$$

where $m_i(x,y)$ are normalized basis functions at scale $i$, $I_{q_i}$ are amplitudes, and $B(x,y)$ is background or residual structure (Volpara et al., 2023, Catalano et al., 8 Sep 2025). A radio-imaging summary presents the same principle as a sum of scale-dependent kernels convolved with the PSF,

$$I_{\mathrm{Dirty}}(x, y) \approx \sum_{j} \sum_{k} a_{jk} \cdot K_{j}(x - x_k, y - y_k) * \mathrm{PSF}(x, y),$$

with $a_{jk}$ the component amplitudes and $K_j$ the kernels associated with scale $j$ (Wright et al., 2024).

At each iteration, the algorithm evaluates candidate scales and locations and selects the best-matching component. A compact statement of the selection step is

$$(\hat{x}, \hat{s}) = \arg\max_{(x,s_i)} (I_{\mathrm{dirty} \ast P_{s_i}})(x),$$

where $m_i(x,y)$0 is the basis function at scale $m_i(x,y)$1. The corresponding scale-convolved PSF is then subtracted with a gain factor, and the model is updated (Müller, 18 Sep 2025).

Several implementation details are central to practical behavior. Scaling kernels are applied to generate multiple scaled versions of the image and the PSF; the global maximum is searched across all scales; and a bias term is used to permit fair comparisons across scales (Wright et al., 2024). In optimized variants, this scale bias is explicit. One such formulation defines

$m_i(x,y)$2

with default $m_i(x,y)$3, so that adjacent scales retain a constant preference ratio and bias settings become independent of the chosen maximum scale (Offringa et al., 2017).

The algorithm is usually embedded in the major/minor cycle structure common to CLEAN-based imaging. In that setting, the major cycle recomputes residuals using the measurement operator, while the minor cycle performs greedy image-plane deconvolution. This architectural pattern remains visible even in convex-optimization methods that later generalized CLEAN-like updates into parallel forward-backward iterations (Onose et al., 2016).

3. Software implementations and reproducibility issues

Although Multi-Scale CLEAN is often treated as a named algorithmic family, recent implementation comparisons show that software-specific realizations can produce materially different results. A direct comparison among WSCLEAN, CASA, and a prototype implementation for the Square Kilometer Array matched operational parameters such as gain, threshold, and iteration limit in order to compare the algorithms rather than tuning choices (Wright et al., 2024).

On a simulated point-source dataset deconvolved with Högbom CLEAN, none of the software packages detected all simulated point sources to within $m_i(x,y)$4 of the simulated values. The reported mean errors were $m_i(x,y)$5 for WSCLEAN, $m_i(x,y)$6 for CASA, and $m_i(x,y)$7 for the SKA prototype, with correspondingly different spreads and extrema (Wright et al., 2024). Although that experiment is not an msCLEAN benchmark, it frames the broader reproducibility problem for CLEAN implementations.

The msCLEAN comparison on the supernova remnant G055.7+3.4 is more directly relevant. Using VLA data and consistent parameters of 1000 iterations and a $m_i(x,y)$8 mJy threshold, the three packages produced different images for the same settings, with obvious and substantial differences in reconstructed structure and detailed morphology. Histogram analysis showed different amounts of flux were cleaned, and the SKA prototype performed a deeper clean than WSCLEAN or CASA, subtracting more flux and reconstructing more faint or extended emission (Wright et al., 2024).

These comparisons indicate that package-specific optimizations and implementation details such as scale biasing, kernel choice, and handling of residuals substantially affect the output. A practical implication, stated explicitly in the comparison study, is that direct comparison or combination of images derived from different packages should be approached with caution, because differences may reflect algorithmic discrepancies rather than only astrophysical structure (Wright et al., 2024).

4. Major extensions and algorithmic variants

Multi-Scale CLEAN has generated a large family of related methods that preserve the multiscale deconvolution idea while modifying the basis, optimization strategy, or data model.

Variant	Defining idea	Representative source
MS-MFS	Spatial and spectral basis functions	(Rau et al., 2011)
WSClean optimized multiscale	Subminor-loop optimization, scale-dependent masking, joined-channel cleaning	(Offringa et al., 2017)
DoB-CLEAN	Multi-scale, multi-directional wavelet dictionaries	(Müller et al., 2023)
Autocorr-CLEAN	Clusters of CLEAN components informed by residual autocorrelation	(Müller et al., 22 Apr 2025)

MS-MFS, described as a multi-scale multi-frequency deconvolution algorithm for wide-band synthesis imaging, models the sky as a linear combination of spatial basis functions and Taylor-polynomial spectral basis functions. It performs image reconstruction via a linear-least-squares approach with iterative $m_i(x,y)$9 minimization, and can reconstruct both spatial and spectral structure of compact and extended emission at the continuum sensitivity level and at the angular resolution allowed by the highest sampled frequency (Rau et al., 2011).

An important implementation-oriented development is the optimized multi-scale algorithm in WSClean. Rather than evaluating all scales at every minor iteration, it generates convolved images for several scales, selects the scale with the highest bias-adjusted peak, and then runs a subminor loop using only that chosen scale. The paper reports that in single-frequency mode the minor loop is over an order of magnitude faster than CASA’s multi-scale implementation, while in joined-channel multi-frequency mode it is $i$0-$i$1 orders of magnitude faster than CASA MSMFS in the reported tests (Offringa et al., 2017).

Other variants expand the shape dictionary beyond isotropic scale atoms. DoB-CLEAN approaches VLBI imaging with multi-scalar and multi-directional wavelet dictionaries, using a difference-of-elliptical-spherical-Bessel dictionary fitted to the $i$2-coverage for feature extraction and a difference-of-elliptical-Gaussian dictionary for the final clean representation. The method is reported to achieve super-resolution compared to CLEAN and to suppress artifacts associated with unsampled Fourier regions (Müller et al., 2023).

Autocorr-CLEAN proposes a different route to multiscale adaptivity: it fits a cluster of CLEAN components to the autocorrelation function of the residual in a subminor loop, thereby deriving continuously changing, potentially non-radially symmetric basis functions. The method is reported to achieve the superior reconstruction fidelity of modern multiscalar approaches and to be up to a magnitude faster than the classical CLEAN procedure in the reported experiments (Müller et al., 22 Apr 2025).

A broader review places these developments alongside Asp-CLEAN, DoB-CLEAN, and hybrid methods such as Autocorr-CLEAN, CG-CLEAN, and PolyCLEAN, and argues that multiscale and autodaptive variants reduce clean-bowl artifacts, improve extended-structure recovery, and remain the most pragmatic near-term choice for routine interferometric imaging (Müller, 18 Sep 2025).

5. Domain-specific adaptations and imaging workflows

Although Multi-Scale CLEAN is most closely associated with radio interferometry, it has also been adapted to hard X-ray Fourier imaging. For RHESSI, a multi-scale formulation decomposes the PSF into scale-specific components, maintains multiple dirty maps, and introduces a bias toward finer scales so that small, weak sources are not masked by brighter extended flux. In the reported synthetic and real-data tests, multi-scale CLEAN performed better than the original CLEAN in terms of accuracy, photometry, and regularization, and no final convolution with a CLEAN beam was required in the stated formulation (Volpara et al., 2023).

A closely related adaptation was developed for STIX on Solar Orbiter. There, the $i$3 plane is partitioned into scale subsets, the image is modeled as a sum over scale-dependent basis functions plus background, and the final map is obtained directly from the accumulated component map and estimated background, again without an additional convolution step. Using synthetic STIX data, the study reports that multi-scale CLEAN improves automation and resolving power relative to standard CLEAN, particularly for sources with different angular scales or closely spaced features (Catalano et al., 8 Sep 2025).

In radio practice, the algorithm is often embedded in detailed, instrument-specific workflows. The MAPS large program used CASA tclean with deconvolver="multiscale" and scales=[0, 5, 15, 25] pixels to synthesize molecular-line image cubes of protoplanetary disks. The workflow relied on Keplerian masks, typically cleaned to a threshold of $i$4 RMS, and applied the JvM correction to compensate for the mismatch between dirty-beam and clean-beam volumes; the paper states that if the residual scaling is not corrected, flux could be overestimated by up to $i$5 for faint emission (Czekala et al., 2021).

These cases show that Multi-Scale CLEAN is not a single invariant procedure. The choice of scales, masks, residual handling, restoration strategy, and spectral treatment can be integral to the scientific validity of the final image. This suggests that “Multi-Scale CLEAN” is best understood as a deconvolution class defined by multiscale component selection, rather than as a uniquely specified algorithm.

6. Relation to alternatives, scalability, and current role

Multi-Scale CLEAN occupies an intermediate position between classical CLEAN and more explicitly optimization-based methods. The comparison with MORESANE is illustrative. MORESANE combines synthesis and analysis approaches to sparse recovery, learns atoms from the data iteratively using analysis priors, and in simulated MeerKAT data was reported to provide competitive results relative to multi-scale CLEAN in terms of total flux or surface-brightness conservation and model fidelity, with particular strength for diffuse and extended sources (Dabbech et al., 2014).

Convex-optimization approaches push further in this direction. Scalable splitting methods for SKA-era imaging recast the inverse problem with explicit $i$6 regularization and proximal splitting, while preserving an analogy to the CLEAN major-minor cycle through forward and backward steps. In the summary provided for that work, these methods considerably outperform MS-CLEAN and related hybrid methods in SNR and residual artifacts on challenging simulated problems, while also supporting parallel and distributed computation (Onose et al., 2016).

At the same time, Multi-Scale CLEAN itself has been adapted to large-scale computation. A decentralized framework that partitions visibilities by spatial frequency applies the method independently on multiple nodes, exchanges filtered deconvolved residual images after each major cycle, and reconstructs pseudo full-resolution images locally. Across four datasets, the paper reports images of similar quality to the serial counterpart and close to the optimal $i$7 speedup for larger datasets with two partitions (Wang et al., 29 Jul 2025).

Current reviews therefore describe a mixed picture. CLEAN remains dominant because of its speed, robustness, and integration with calibration pipelines, while multiscale variants address the point-source limitations of standard CLEAN and are widely implemented in packages such as CASA and WSClean. At the same time, Bayesian inference, compressive sensing, regularized maximum likelihood, and AI-driven methods offer higher fidelity, more flexible priors, and uncertainty quantification at greater computational cost. The stated near-term expectation is that multiscale CLEAN and hybrid variants will remain sufficient for most routine imaging, while forward-modeling and AI-based frameworks are more likely to shape the long-term future of interferometric reconstruction (Müller, 18 Sep 2025).