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Learning from Measurements Alone

DIP reconstructs from a single measurement but requires per-image optimisation. Noise2Noise and SURE train amortised networks but need either paired data or Gaussian noise assumptions. Equivariant Imaging (EI) offers a fundamentally different approach: it learns a reconstruction network from unpaired measurements alone, using the symmetries of the signal class as self-supervision.

The key insight, due to Tachella, Chen, and Davies (2021), is that if the signal class is invariant under a group $\mathcal{G}$ of transformations, then measurement consistency plus equivariance uniquely determines the reconstruction --- even in the null space of the sensing operator $\mathbf{A}$ .

Definition:
Equivariant Imaging (EI)

Equivariant Imaging trains a reconstruction network using only unpaired measurements, by enforcing two losses:

1. Data consistency: $\mathcal{L}_{\text{DC}} = \mathbb{E}_\mathbf{y}\bigl[\|\mathbf{A}f_\theta(\mathbf{y}) - \mathbf{y}\|^2\bigr]$

2. Equivariance: $\mathcal{L}_{\text{EI}} = \mathbb{E}_{\mathbf{y}, g}\bigl[\|f_\theta(\mathbf{A}T_g f_\theta(\mathbf{y})) - T_g f_\theta(\mathbf{y})\|^2\bigr]$

where $T_g$ is a transformation from a group $\mathcal{G}$ under which the signal class is invariant.

The total loss is: $\mathcal{L} = \mathcal{L}_{\text{DC}} + \lambda\,\mathcal{L}_{\text{EI}}.$

EI requires no paired training data and no pretrained model. It learns to reconstruct from measurements alone, using the group symmetries as a self-supervisory signal. The key insight is that the symmetries provide information about the null space of $\mathbf{A}$ : the equivariance constraint fills in the missing information that measurements cannot provide.

Theorem: EI Recovers the Null Space

Let $\mathbf{A} \in \mathbb{R}^{M \times N}$ with $M < N$ (underdetermined), and let $\mathcal{G}$ be a group of transformations acting on $\mathbb{R}^N$ . If:

The signal class is invariant under $\mathcal{G}$ : $T_g \mathbf{x} \in \mathcal{X}$ for all $\mathbf{x} \in \mathcal{X}$ and $g \in \mathcal{G}$ .
The group action is transitive on the null space: for any $\mathbf{v} \in \text{null}(\mathbf{A})$ , there exist $g_1, \ldots, g_K \in \mathcal{G}$ and $\alpha_1, \ldots, \alpha_K$ such that $\mathbf{v} = \sum_k \alpha_k\, \mathbf{P}_{\text{null}}\, T_{g_k}\, \mathbf{P}_{\text{range}}\, \mathbf{u}$ for some $\mathbf{u}$ .

Then the EI loss $\mathcal{L}_{\text{DC}} + \lambda\,\mathcal{L}_{\text{EI}}$ has a unique minimiser $f_\theta^*$ satisfying:

$f_\theta^*(\mathbf{A}\mathbf{x} + \mathbf{w}) \to \mathbf{x} \qquad \text{as } \sigma^2 \to 0.$

Data consistency constrains the reconstruction in the range of $\mathbf{A}^H$ (the measured subspace). The equivariance loss constrains the null space: when we transform the reconstruction $T_g \hat{\mathbf{x}}$ , the null-space component changes, and requiring the network to reconstruct it correctly from new measurements provides indirect observations of the null space.

Proof

Range-null decomposition

Any signal decomposes as $\mathbf{x} = \mathbf{x}_R + \mathbf{x}_N$ where $\mathbf{x}_R \in \text{range}(\mathbf{A}^H)$ and $\mathbf{x}_N \in \text{null}(\mathbf{A})$ .

Data consistency uniquely determines $\mathbf{x}_R$ (in the noiseless case): $\mathbf{A}f_\theta(\mathbf{y}) = \mathbf{y}$ implies $f_\theta(\mathbf{y})_R = \mathbf{A}^\dagger\mathbf{y}$ .

Equivariance constrains the null space

For a transformation $T_g$ , the transformed signal $T_g\mathbf{x}$ has a different range-null decomposition. The requirement $f_\theta(\mathbf{A}T_g\hat{\mathbf{x}}) = T_g\hat{\mathbf{x}}$ constrains how the network maps range-space information to null-space content.

If $T_g$ mixes the range and null spaces (transitive action), then different transformations provide complementary constraints on $\mathbf{x}_N$ , collectively determining it uniquely. $\blacksquare$

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Equivariant Imaging: Effect of Symmetry Group

Visualise how equivariant imaging uses symmetries to learn reconstruction without ground truth. The plot shows the reconstruction PSNR as a function of the number of group transformations used in training.

More transformations provide more self-supervisory signal, improving reconstruction quality. Rotations are the most informative for isotropic scenes; shifts are best for systems with partial Fourier measurements (SAR, diffraction tomography).

Parameters

Number of transforms4

Transform type

M/N

(compression ratio)0.25

Example: Equivariant Imaging for Partial Fourier Measurements

Apply equivariant imaging to an RF imaging system with partial Fourier sensing $\mathbf{A} = \mathbf{P}_\Omega\mathbf{F}$ . Identify appropriate symmetries and explain why shifts are particularly effective.

Solution

Forward model

$\mathbf{A} = \mathbf{P}_\Omega\mathbf{F}$ : the DFT followed by selection of frequencies in $\Omega \subset \{0, \ldots, N-1\}$ . The null space consists of signals whose Fourier support lies entirely in $\Omega^c$ (unmeasured frequencies).

Why shifts work

By the Fourier shift theorem, a spatial shift by $\Delta$ pixels corresponds to phase modulation in frequency: $[\mathbf{F}T_\Delta\mathbf{x}]_k = e^{-j2\pi k\Delta/N}\hat{x}_k$ .

The EI loss for the shifted signal creates measurements at the same frequency locations $\Omega$ , but with different phase factors. This creates a system of equations that constrains the unmeasured frequency content through the reconstruction network.

Practical protocol

For each training measurement $\mathbf{y}_j = \mathbf{P}_\Omega\mathbf{F}\mathbf{x}_j$ :

Reconstruct: $\hat{\mathbf{x}}_j = f_\theta(\mathbf{y}_j)$
Sample random shift $\Delta \sim \text{Unif}\{0, \ldots, N-1\}$
Create virtual measurement: $\tilde{\mathbf{y}} = \mathbf{P}_\Omega\mathbf{F}(T_\Delta\hat{\mathbf{x}}_j)$
Re-reconstruct: $\hat{\mathbf{x}}'_j = f_\theta(\tilde{\mathbf{y}})$
EI loss: $\|\hat{\mathbf{x}}'_j - T_\Delta\hat{\mathbf{x}}_j\|^2$

Definition:
Measurement Splitting

Measurement splitting is a complementary self-supervised strategy that splits the measurements into two disjoint subsets:

$\mathbf{y}_1 = \mathbf{A}_1\mathbf{x} + \mathbf{w}_1, \qquad \mathbf{y}_2 = \mathbf{A}_2\mathbf{x} + \mathbf{w}_2$

where $\mathbf{A} = [\mathbf{A}_1; \mathbf{A}_2]$ . Train the network on $\mathbf{y}_1$ and validate on $\mathbf{y}_2$ :

$\mathcal{L}_{\text{split}} = \|\mathbf{A}_2 f_\theta(\mathbf{y}_1) - \mathbf{y}_2\|^2.$

This is a Noise2Noise-like loss applied to subsets of measurements.

Measurement splitting requires no symmetry assumptions but needs enough measurements in each split. It can be combined with EI for additional self-supervision.

Example: Equivariant Imaging for Multi-View RF Imaging

A multi-static RF imaging system with $I$ transmitters and $J$ receivers collects measurements from multiple viewpoints. How can equivariant imaging exploit the multi-view geometry?

Solution

Multi-view forward model

Each transmitter-receiver pair $(i, j)$ provides measurements $\mathbf{y}_{i,j} = \mathbf{A}_{i,j}\mathbf{c} + \mathbf{w}_{i,j}$ where $\mathbf{A}_{i,j}$ depends on the array geometry and the transmit/receive steering vectors.

The combined sensing matrix $\mathbf{A} = [\mathbf{A}_{1,1}; \ldots; \mathbf{A}_{I,J}]$ has a structured null space determined by the spatial frequency coverage.

Perspective equivariance

For scenes that are invariant under rotations about the vertical axis (common for urban/indoor environments), rotating the scene is equivalent to permuting the transmitter-receiver pairs.

This geometric equivariance provides a natural group $\mathcal{G}$ for EI: rotate the reconstructed scene, re-measure with the appropriate permuted sensing matrix, and require consistency.

Practical considerations

The rotation group for RF imaging is typically discrete (e.g., 4-fold or 8-fold symmetry for planar arrays). Additional symmetries may include:

Translations (shift-invariant scenes)
Reflections (symmetric array configurations)
Scale invariance (for multi-frequency measurements)

The effectiveness depends on how well these symmetries mix the null space of $\mathbf{A}$ .

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Self-Supervised Methods: Data Requirements and Assumptions

Method	Data Needed	Key Assumption	Null-Space Info	Best For
DIP	1 measurement	CNN spectral bias	Architecture bias	No data available
Noise2Noise	Noisy pairs	Zero-mean independent noise	None (denoising only)	Repeated measurements
SURE	Noisy images	Gaussian noise, known $\sigma^2$	None (denoising only)	Gaussian noise regime
EI	Unpaired measurements	Signal symmetries	Group action on null space	Structured forward models
Meas. splitting	Redundant measurements	Enough per split	Cross-validation	Over-sampled systems

Common Mistake: Wrong Symmetry Group Degrades EI Performance

Mistake:

Using rotation equivariance for RF scenes that are not rotationally invariant (e.g., corridor environments, elongated structures, or scenes with strong directional features).

Correction:

The EI loss penalises violations of the assumed symmetry. If the scene is not truly invariant under $\mathcal{G}$ , the equivariance constraint becomes a wrong prior that biases the reconstruction.

Best practice: Choose symmetries that match the scene statistics. For urban RF imaging, use discrete rotations (90-degree) and reflections. For general scenes, use shifts (which assume stationarity, a weaker assumption). When in doubt, use measurement splitting instead.

Quick Check

Why does data consistency alone (without equivariance) fail to produce good reconstructions for underdetermined inverse problems?

Data consistency is not a valid loss function

The null space of $\mathbf{A}$ is unconstrained --- infinitely many signals produce the same measurements

Neural networks cannot learn linear inverse problems

Gradient descent fails to converge for the data consistency loss

Correction:

The null space of

\mathbf{A}

is unconstrained --- infinitely many signals produce the same measurements

Correct. For $M < N$ , the system $\mathbf{A}\mathbf{x} = \mathbf{y}$ has infinitely many solutions differing by null-space components. Without additional constraints (symmetries, priors, regularisation), the network can produce any null-space content.

Historical Note: From Edinburgh to DeepInverse

2021

Equivariant imaging was developed by Julian Tachella, Dongdong Chen, and Mike Davies at the University of Edinburgh. Davies --- a leading figure in compressed sensing and computational imaging --- saw that the missing piece in self-supervised imaging was a principled way to handle the null space.

The EI framework drew on ideas from geometric deep learning (group equivariant networks) and classical sampling theory (the role of symmetries in Fourier analysis). The team subsequently developed the DeepInverse library, an open-source PyTorch toolkit that implements EI alongside DIP, Noise2Noise, SURE, and many other methods discussed in this chapter.

Mike Davies was also referenced by Caire as a key figure in the intersection of compressed sensing and RF imaging.

Equivariant Imaging (EI)

A self-supervised framework for inverse problems that learns reconstruction from unpaired measurements by exploiting known symmetries of the signal class, providing information about the null space of the sensing operator.

Related: Measurement Splitting

Measurement Splitting

A self-supervised strategy that splits measurements into disjoint subsets, training on one and validating on the other, providing a Noise2Noise-like loss for inverse problems.

Related: Equivariant Imaging (EI)

Key Takeaway

Equivariant imaging trains reconstruction networks from measurements alone, using signal symmetries as self-supervision. The equivariance loss constrains the null space of $\mathbf{A}$ , complementing data consistency that constrains the range. For RF imaging, translations and rotations are natural symmetries for scenes with spatial stationarity or isotropy. The choice of symmetry group must match the scene statistics --- wrong symmetries act as a harmful prior.

Equivariant Imaging