MROP: Modulated Rank-One Projections for compressive radio interferometric imaging

TL;DR

The paper tackles the challenge of explosive RI data volumes by introducing Modulated Rank-One Projections (MROP) as a data-acquisition compression scheme for RI imaging. It develops two acquisition paradigms—antenna-based MROP and visibility-based MROP with CROP/IROP variants—analyzes the resulting noise statistics to show preserved i.i.d. Gaussian noise under normalization, and provides a thorough cost/memory assessment that favors MROP over classical and BDA schemes. The approach is validated through extensive simulations with realistic ground-truths and through real VLA data, demonstrating that imaging quality can be preserved while reducing data to approximately the image size, enabling efficient, scalable RI imaging with the uSARA solver. These results have practical impact by enabling petabyte-scale RI imaging pipelines to operate within feasible storage and compute budgets, while maintaining calibration-friendly data models and compatibility with standard weighting schemes.

Abstract

The emerging generation of radio-interferometric (RI) arrays are set to form images of the sky with a new regime of sensitivity and resolution. This implies a significant increase in visibility data volumes, which for single-frequency observations will scale as $\mathcal{O}(Q^2B)$ for $Q$ antennas and $B$ short-time integration intervals (or batches), calling for efficient data dimensionality reduction techniques. This paper proposes a new approach to data compression during acquisition, coined modulated rank-one projection (MROP). MROP compresses the $Q\times Q$ batchwise covariance matrix into a smaller number $P$ of random rank-one projections and compresses across time by trading $B$ for a smaller number $M$ of random modulations of the ROP measurement vectors. Firstly, we introduce a dual perspective on the MROP acquisition, which can either be understood as random beamforming, or as a post-correlation compression. Secondly, we analyse the noise statistics of MROPs and demonstrate that the random projections induce a uniform noise level across measurements independently of the visibility-weighting scheme used. Thirdly, we propose a detailed analysis of the memory and computational cost requirements across the data acquisition and image reconstruction stages, with comparison to state-of-the-art dimensionality reduction approaches. Finally, the MROP model is validated for monochromatic intensity imaging both in simulation and from real data, with comparison to the classical and baseline-dependent averaging (BDA) models, and using the uSARA optimisation algorithm for image formation. Our results suggest that the data size necessary to preserve imaging quality using MROPs is reduced to the order of image size, well below the original and BDA data sizes.

Figures (9)

• Figure 1: Schematic of the radio interferometric sensing context. Far-away cosmic electric fields following a Gaussian random process, i.e., $s(\boldsymbol{l},t)\sim_{ \mathrm i.i.d.\xspace } \mathcal{C}\mathcal{N}(0,\sigma^2(\boldsymbol{l}))$ with an intensity distribution $\sigma^2(\boldsymbol{l})$, are received by $Q$ antennas. The antennas, with projected positions in $\Omega(t):=\{ \boldsymbol{p}_q^\perp(t) \}_{q=1}^Q$, have the same direction-dependent gain $g(\boldsymbol{l})$ focusing on a specific region $\mathcal{S}$ of the sky. Each antenna $q\in \llbracket Q \rrbracket$ integrates---with its own geometric delay $\boldsymbol{p}_q^\perp(t)^\top\boldsymbol{l}$ the electric field contributions from all direction into a noisy measurement $x_q(t)=\int_{\mathbb{R}^2} g(\boldsymbol{l})s(\boldsymbol{l},t)e^{\frac{\mathrm{i}\mkern1mu2\pi}{\lambda}\boldsymbol{p}_q^\perp(t)^\top\boldsymbol{l}} \mathrm{d}\boldsymbol{l} + n_q(t)$.
• Figure 2: Computation schemes with batchwise implementations. (a-c) computations at the acquisition: from the antenna signals $\mathcal{X}$ to the (compressed) visibilities. (a) Classical acquisition. For each batch $b$, the sample covariance matrix is computed as $\mathbf{C} _b = {\frac{1}{I}} \sum_{i=1}^I \boldsymbol{x}_b[i] \boldsymbol{x}_b[i]^*$. Selecting the upper triangular part of the covariance matrices provides the visibilities in $\boldsymbol{\mathcal{T}} \mathbf{C} _b$. The visibilities can be weighted as $\boldsymbol{\mathcal{T}}\widetilde{\mathbf{W}}_b \odot \boldsymbol{\mathcal{T}} \mathbf{C} _b = \boldsymbol{\mathcal{T}} \widetilde{ \mathbf{C} }_b$. An optionalrandom Gaussian compression of the (vectorised) visibilities $\boldsymbol{v}$ is possible using an i.i.d. i.i.d. random Gaussian matrix $\mathbf{A}$ as $\boldsymbol{y} =\mathbf{A}\boldsymbol{v}$. Its batchwise implementation (not shown here) implies aggregating the projections over batches as $\boldsymbol{y}=\sum_{b=1}^B \mathbf{A}_b \boldsymbol{v}_b$. (b) Antenna-based acquisition of MROP. For each batch $b$, $P$ (biased) ROPs of the covariance matrix are obtained as $y_{pb}' = {\frac{1}{I}} \sum_{i=1}^I \boldsymbol{\alpha}_{pb}^* \boldsymbol{x}_b[i] \boldsymbol{x}_b[i]^* \boldsymbol{\beta}_{pb}$, $\forall p\in\llbracket P \rrbracket$. The ROPs are debiased as $y_{pb} = y_{pb}'-\boldsymbol{\alpha}_p^* \mathbf{C} _b^\mathrm{d} \boldsymbol{\beta}_p$. $M$ copies of the ROP vector $\boldsymbol{y}_b$, with random signs encoded into $\boldsymbol{\gamma}_b = (\gamma_{mb})_{m=1}^M$ and represented by the black and white cells, are opened as $\boldsymbol{y}_b\times \boldsymbol{\gamma}_b$. These $PM$ MROPs are progressively summed over the $B$ batches as $\mathbf{Z}^{(b)} = \mathbf{Z}^{(b-1)} + \boldsymbol{y}_b \times \boldsymbol{\gamma}_b$ with $\mathbf{Z}^{(0)}=\mathbf{0}_{P\times M}$. There is no visibility weighting here. (c) Visibility-based acquisition of MROP. For each batch $b$, the sample covariance matrix is computed like in (a), giving direct access to the upper triangular covariance matrix $\boldsymbol{\mathcal{T}} \mathbf{C} _b$ and allowing for a weighting as $\boldsymbol{\mathcal{T}}\widetilde{\mathbf{W}}_b \odot \boldsymbol{\mathcal{T}} \mathbf{C} _b = \boldsymbol{\mathcal{T}}\widetilde{ \mathbf{C} }_b$. Then, symmetrising the weighted visibility matrix as $\boldsymbol{\mathcal{T}}\widetilde{ \mathbf{C} }_b + (\boldsymbol{\mathcal{T}}\widetilde{ \mathbf{C} }_b)^*=\widetilde{ \mathbf{C} }_b^\mathrm{h}$, $P$ random ROPs of the weighted interferometric matrix are obtained as $y_{pb} = \boldsymbol{\alpha}_{pb}^* \widetilde{ \mathbf{C} }_b^\mathrm{h} \boldsymbol{\beta}_{pb}$, $\forall p\in\llbracket P \rrbracket$. From that, the random modulations are the same as in (b). (d) Forward MROP imaging model. Starting from the FFT of the (zero-padded) discrete image $\mathbf{FZ}\boldsymbol{\sigma}$, the visibilities related to the batch $b$ are interpolated as $\bar{\boldsymbol{v}}_b = \mathbf{G}_b \mathbf{FZ}\boldsymbol{\sigma}$ then reshaped to get the upper triangular matrix $\boldsymbol{\mathcal{T}} \mathbf{C} _b$. The next operations are the same as in (c) with the weighting, ROPs, and random modulations. A key observation in Fig. \ref{['fig:acquisition']}(b-d) is that the number of measurements never exceeds $PM$ both during the acquisition and the imaging because the ROPs and modulations can be computed and integrated batchwise.
• Figure 3: Structure of the operators in the MROP imaging model. An abuse of notation is made to simplify the illustration by considering that the interpolation operator $\mathbf{G}$ directly outputs the hermitian visibility matrix $\mathbf{C} _b^\mathrm{h}$ instead of the upper triangular matrix $\boldsymbol{\mathcal{T}} \mathbf{C} _b$, i.e., $\mathbf{G}_b \mathbf{F}\mathbf{Z}\boldsymbol{\sigma}=\mathop{\mathrm{vec}}\nolimits( \mathbf{C} _b^\mathrm{h})$. The $VB$ visibilities are obtained by interpolating with $\mathbf{G}$, and at a cost $\mathcal{O}(JVB)$, between the on-grid frequencies of the image obtained from a (real) FFT of the zero-padded image $\mathbf{F}\mathbf{Z}\boldsymbol{\sigma}$ at a cost $\mathcal{O}(2N\log N)$. Each visibility is assigned a weight using the diagonal visibility weighting operator $\mathbf{W} \in \mathbb{R}^{VB\times VB}$ at a cost $\mathcal{O}(VB)$. The simplification of this illustration appears in the CROP operator $\mathbf{D}$ whose blocks $\{ \mathbf{R}_b \}_{b=1}^B$ have here rows that are the vectorisation of a rank-one matrix, i.e., $(\mathbf{R}_b)_p = \mathop{\mathrm{vec}}\nolimits(\boldsymbol{\alpha}_{pb} \boldsymbol{\beta}_{pb}^*)^\top$ instead of the more complicated expression in \ref{['eq:Rb']}. The computation of the CROP operator $\mathbf{D}$ requires $\mathcal{O}(2PVB)$ operations. The MROPs are finally obtained by applying the modulation operator $\mathbf{M}$ at a negligible cost $\mathcal{O}(PMB)$.
• Figure 4: Ground-truth images made of $256\times 256$ pixels and shown in logarithmic scale. (a) 3c353, (b) Abell 2034, (c) Messier 106, (d) PSZ2 G165.68+44.01.
• Figure 5: One of the $uv$-coverages of simulated MeerKAT observations used within our experiments, with a super-resolution factor of 1.5. (a) classical, (b) averaged $uv$ points with BDA, (c) subsampled $uv$ points with $B_{\rm bda} = 2$, (d) subsampled $uv$ points with $B_{\rm bda}=32$. Each panel is accompanied by a zoom-in of the $uv$-coverage, showing baselines shorter than 1 km in its top left corner. For BDA, the central batch's colour is selected from the available averaged colours. Furthermore, concentric circles appear around the centre, depicting different averaging regimes. Each subplot lives in $[0,1) \times [0,1)$. The $uv$-coverage $\mathcal{V}_b$ associated with batch $b$ has a unique colour, and the colours are faded from blue for $b=1$ to red for $b=B$.