Super-resolution multi-reference alignment

Abstract

We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in ${ℝ}^M$ is uniquely determined when the number L of samples per observation is of the order of the square root of the signal’s length ($L=O(\sqrt{M})$). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to 1/SNR³. In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled (L = M). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes.

1. Introduction

Model.

We study the problem of estimating a signal from its circularly shifted, sampled and noisy copies. More precisely, we consider N independent observations sampled from the model

$$y=P{R}_{s}x+\epsilon ,s~\text{Uniform}[0,\dots ,M-1],\epsilon ~\mathcal{N}\left(0,{\sigma }^{2}I\right),$$ (1.1)

where R_s denotes an operator that shifts the target signal $x\in {ℝ}^M$ circularly by s entries, that is, (R_sx)[n] = x[(n − s) mod M], and P denotes a fixed sampling operator that collects L ≤ M equally-spaced samples. We assume that the random variable s is distributed uniformly over [0, …, M −1], and the noise $\epsilon \in {ℝ}^L$ is i.i.d. Gaussian. Explicitly, the i-th observation reads:

$$\begin{gathered} y_i[\ell ]=P\left(R_{s_i}x\right)[\ell ]+{\epsilon }_i[\ell ] \\ =x\left[\ell K-s_i\right]+{\epsilon }_i[\ell ], \end{gathered}$$ (1.2)

where ℓ = 0, …, L − 1 and K ≔ M/L is assumed to be an integer. Importantly, the shifts s_i are all unknown, and thus (1.1) is a special case of the multi-reference alignment (MRA) model, which we review in Section 2. Figure 1 presents an example of two observations with signal-to-noise ratio (SNR) equal to one (namely, the expected squared norm of the noise equals the squared norm of the signal).

Fig. 1.

Two shifted copies of a signal of length M = 120 (the high-resolution signal) are presented in blue. The red squares display L = 15 noisy samples with SNR equal to one. The goal is to estimate the high-resolution signal from multiple noisy observations.

Our goal is to estimate x from N observations sampled from (1.1). In contrast to previous works on MRA, the individual observations are down-sampled, and therefore recovering the full signal x is also a special case of the super-resolution problem. Accordingly, we refer to x as the ‘high-resolution signal,’ whereas y₁, …, y_N are the ‘low-resolution observations.’ The parameter K can be thought of as a ‘super-resolution factor.’ The difficulty in estimating x resides chiefly in three factors: the additive noise, the unknown circular shifts (the nuisance variables of the problem) and the sampling operator.

The statistical model (1.1) suffers from an intrinsic symmetry: it is invariant under a global circular shift since p(y|x) = p(y|R_ix) for all i = 1, …, M − 1. In this case, we say that the goal is to recover the signal up to a global circular shift. More formally, the goal is to recover the orbit of x:

$$Gx:=\left\{gx\mid g\in G\right\},$$ (1.3)

where G ≔ {R₀, R₁, …, R_M−1} is the group of cyclic shifts ${\mathbb{Z}}_M$. However, as will be shown in Section 3, without prior information on the signal, even the orbit Gx is not identifiable from the observations, and thus prior information on x is necessary for its identification.

Connection with sampling theory:

We think of the discrete signal $x\in {ℝ}^M$ as Nyquist-rate samples of a continuous bandlimited signal. Specifically, let us define a real signal with bandlimit B as

$$x_c(t)=\sum _{k=-B}^B\widehat{x}[k]e^{2\pi \iota kt},t\in [0,1),$$ (1.4)

where $\iota =\sqrt{-1}$, and $\widehat{x}$ denotes the Fourier series coefficients of x. Since x_c is real, it follows that $\widehat{x}[k]=\overline{\widehat{x}[-k]}$. According to the well-known Shannon–Nyquist sampling theorem, the samples

$${\tilde{x}}_c[m]:=x_c(m/M)=\sum _{k=-B}^B\widehat{x}[k]e^{2\pi \iota km/M},m=0,\dots ,M-1,$$ (1.5)

characterize x_c uniquely when M ≥ 2B + 1. Model (1.1) is identical to rotating the discrete signal ${\tilde{x}}_c$ on the M-point grid ${\left\{m/M\right\}}_{m=0}^{M-1}$, sampling it L times, and adding noise.

With the above interpretation in mind, we identify the length of the signal in (1.1) with twice the signal’s bandwidth, namely, M = 2B + 1 ≈ 2B. Thus, if L < M, we say that each observation is sampled below the signal’s Nyquist rate, and thus the recovery process should compensate for an aliasing distortion. To avoid aliasing, the standard signal processing approach in many applications is to remove the high-frequency components before sampling [26,53]—namely, low-passfiltering the signal— and then estimate a down-sampled, smooth approximation of x. Although this strategy is generally optimal for a single observation, this is not necessarily true when multiple observations are available. In this work, we show that if sufficiently many observations are acquired (as a function of the noise level), then in principle it suffices to acquire only $L=O(\sqrt{M})$ samples at each observation to recover high-resolution details, even if the circular shifts are unknown and the noise level might be high.

The analogy between (1.1) and rotating a continuous bandlimited signal (1.4) holds only when the rotations are restricted to the grid ${\left\{m/M\right\}}_{m=0}^{M-1}$. In Section 6 we discuss potential extensions to more intricate models that permit rotations over a continuous interval.

Super-resolution:

The model (1.1) is an instance of the super-resolution from multiple observations problem: the task of estimating the fine details of a signal from its low-resolution observations. This problem has attracted the attention of numerous researchers in the last couple of decades in a variety of fields, such as computer vision, image processing and medical imaging; see for instance [28,32,46] and references therein. In particular, the statistical model in some of these works is akin to (1.1), see for example [49,50,61]. Nevertheless, as far as we know, previous works on super-resolution did not aim to derive and quantify the achievable super-resolution in the low SNR regime. To avoid confusion, we mention that there exists a different thread of research, which is not directly related to this work, that studies super-resolution from a single image based on prior knowledge (such as sparsity [15,19]), or machine learning techniques [36,41].

Main contributions:

In this paper we provide a detailed analysis of model (1.1) and derive fundamental conditions permitting an accurate estimate of x. In particular, we characterize the interplay between the number of observations N, the noise level σ, the signal’s length M, and the number of samples per observation L, in the low SNR regime.

The following theorem summarizes (informally) the theoretical contribution of this paper. Precise formulations and technical details are provided in Section 3.

THEOREM 1.1 (informal). Suppose that N observations from (1.1) are collected and σ → ∞. If N/σ⁶ → ∞ and $L\ge C\sqrt{M}$ for some constant C, then the maximum of the likelihood function p(y₁, …, y_N|x) is almost surely attained by a finite set of signals that includes the target signal x. If in addition x was drawn from a Gaussian prior, then almost surely there exists a single signal that achieves the maximum of the posterior distribution p(x|y₁, …, y_N).

Expectation–maximization:

As a computational scheme, we propose to retrieve the high-resolution signal x from the low-resolution observations y₁, …, y_N using an expectation-maximization (EM) algorithm; a detailed description is given in Section 4. Figure 2 shows a numerical example. A high-resolution signal of length M = 120 is estimated from N = 10⁴ observations in a noisy environment, where the SNR is equal to one and each observation is sampled at L = 15 points. The bandwidth of the signal is B = L, so that the sampling rate is half of the Nyquist rate. If we were to follow the Shannon–Nyquist sampling scheme of filtering out the L/2 high frequencies, the two peaks in the center of the signal would have been blurred into one, even with known circular shifts and in the absence of noise. In contrast, the EM algorithm resolves the two adjacent peaks and estimates the signal accurately. A detailed description of this simulation, and additional numerical experiments, are provided in Section 5. We note, however, that although the theoretical analysis guarantees identifiability in the regime M ≈ L², in our experiments the EM algorithm fails to estimate the high-resolution signal even when L ≈ M^2/3. Following [6,17,60], we postulate that this inadequate performance reflects a fundamental statistical-computational gap in the super-resolution problem, rather than a shortcoming of the EM framework.

Fig. 2.

An example of an accurate estimate when the SNR is equal to 1. In the experiment, N = 10⁴ observations were generated from a signal of length M = 120 (plotted in dashed blue; the same one as in Fig. 1). The bandwidth of the signal is B = 15 and it was sampled L = 15 times at each observation—half of the Nyquist sampling rate. The classical signal processing approach suggests to remove all frequencies beyond L/2 and then process the low-resolution data. This low-passed version of the signal is presented in red. Notably, the two peaks in the center of the signal are blurred and merged into one. In contrast, the EM algorithm resolves the two peaks and estimates the high-resolution signal accurately (in green).

Remark on terminology and notation:

We refer to each realization of the model (1.1) as an observation, and to the entries of each observation as samples. Namely, y_i[ℓ] denotes the ℓ-th sample of the i-th observation. In addition, in the sequel all indices should be considered as modulo M or L, depending on the context. When writing P ≳ Q^d we mean that Q^d is the leading-order term of the right-hand side expansion in Q. For example, M ≳ L²/6 implies that M is greater than L²/6 plus a linear polynomial in L.

2. Background on multi-reference alignment and invariants

The model (1.1) is a special case of the multi-reference alignment (MRA) problem. This problem entails estimating a signal from multiple noisy observations; in each observation the signal is acted upon by an unknown element of a known group G. In its most general form, the MRA model reads

$$y=T(g\circ x)+\epsilon ,g\in G,x\in \mathcal{X},$$ (2.1)

where T is a known linear operator, with the group G acting on a vector space χ [8]. Specifically, if $x\in {ℝ}^M$, G is identified with the group of circular shifts ℤ_M, and T is the sampling operator P, then the general MRA model (2.1) reduces to (1.1).

Similarly to many MRA models in the literature [1,2,4,7,14,17,42,47,51], this work is inspired by single-particle reconstruction problems using cryo-electron microscopy (cryo-EM) and X-ray free electron lasers (XFEL)—high-resolution structural methods for biological macromolecules [10,30,31,44,56]. In particular, this work is a first step towards understanding the resolution limits of these modalities; see further discussion in Section 6.

Suppose we collect N observations from (2.1). If the noise level is low, the standard approach is to estimate the group element g₁, …, g_N. For example, in (1.1) the unknown circular shifts s₁, …, s_N can be estimated by simultaneous clustering and synchronization (see Section 3.1). This can be done, for instance, using the Non-Unique Games framework [39]. However, in the low SNR regime—which is the main interest of this work—the group elements cannot be recovered reliably by any method [3,13]. Therefore, we consider two techniques that circumvent shift determination: estimation based on shift-invariant features, and the EM algorithm. In particular, we formulate EM in detail in Section 4, and present numerical experiments in Section 5.

For the theoretical analysis, we use features that are invariant under circular shifts. Specifically, the q-th order circular-shift invariant feature of a signal $z\in {ℝ}^L$ is simply its auto-correlation:

$$M_q(z)\left[{\ell }_1,\dots ,{\ell }_{q-1}\right]=\sum _{i=0}^{L-1}z[i]z\left[i+{\ell }_1\right]\dots z\left[i+{\ell }_{q-1}\right].$$ (2.2)

It is readily seen that this quantity remains unchanged under any circular shift of z, namely, $M_q(z)=M_q\left(R_{\overline{s}}z\right)$ for any fixed $\overline{s}$. These invariants can also be presented in Fourier domain. Specifically, let $\widehat{z}[k]$ denote the k-th Fourier coefficient of z. Then the polynomials

$${\widehat{M}}_q(z)\left[k_1,\dots ,k_{q-1}\right]=\widehat{z}\left[k_1\right]\cdots \widehat{z}\left[k_{q-1}\right]\widehat{z}\left[-k_1-\cdots -k_{q-1}\right],$$ (2.3)

are also invariant under circular shift. Throughout the paper, we use the terms auto-correlations, invariants, and invariant features interchangeably. Using these invariants, a variety of algorithms were proposed under different MRA setups [1,14,17,21,42,47], as well as for cryo-EM and XFEL [12,34,37,40,45,55,58].

In this work, we harness the first three invariants. The first invariant is the zero frequency ${\widehat{M}}_1(z)=\widehat{z}[0]$ (equivalently, the mean of the signal). The second invariant is the power spectrum of the signal ${\widehat{M}}_2(z)[k]=|z[k]{|}^2$ for k = 0, …, L − 1. Unfortunately, the mean and the power spectrum do not determine a general signal uniquely (see for example [11]). Thus, we need the third-order invariant, the bispectrum, which determines almost all signals uniquely [52,57]:

$${\widehat{M}}_3(z)\left[k_1,k_2\right]=\widehat{z}\left[k_1\right]\widehat{z}\left[k_2\right]\widehat{z}\left[-k_1-k_2\right],k_1,k_2=0,\dots ,L-1.$$ (2.4)

The bispectrum is a useful tool in many data processing applications, such as separating Gaussian and non-Gaussian processes [18], studying the cosmic background radiation, seismic, radar and EEG signals [23,43,59], MIMO systems [20] and classification [63].

For large σ, the variance of estimating the q-th order auto-correlation (either M_q or ${\widehat{M}}_q$) is proportional to σ^2q since the estimator involves the product of q noisy terms. Thus, reliable estimation requires an order of σ^2q observations. For the problem under consideration, it implies that we need to record N/σ⁶ ≫ 1 observations to obtain an accurate estimate of the bispectrum. Interestingly, it was shown that for the MRA model (2.1), the invariant features approach is optimal in the following sense. Let $\overline{q}$ be the lowest-order auto-correlation that identifies a generic signal (in our case, $\overline{q}=3$). Then, in the asymptotic regime where N and σ diverge (while L is fixed), the estimation error of any method is bounded away from zero if $N/{\sigma }^{2\overline{q}}$ is bounded from above [2,9]. In other words, $\overline{q}$ determines the minimal number of observations required for an accurate estimate in the low SNR regime. Remarkably, we show that for (1.1) and for a certain range of L, at the same estimation rate (i.e., N scales with σ⁶) one can reduce the sampling rate significantly below the Nyquist rate and still achieve an accurate estimate of the signal. In Section 6 we discuss the potential of super-resolution in case higher-order auto-correlations could be computed—that is, if more observations are available.

3. Analysis

The analysis is carried out in the asymptotic regime of N → ∞, while the dimension M remains fixed. Therefore, we assume, without of loss of generality, that ∥x∥² = M so that SNR=1/σ². By the term ‘accurate recovery’ we mean that the recovery error drops to zero almost surely. For example, the condition N/σ⁶ → ∞ ensures that we can almost surely estimate the bispectrum accurately.

In Section 3.1, we show that (1.1) can be interpreted as the heterogeneous multi-reference alignment (hMRA) model applied to K subsets of x, and formulate the likelihood function of (1.1). This, in turn, immediately implies that the signal is not determined uniquely from the likelihood function (a result implicit in earlier works, such as [61]):

Theorem 3.1. The likelihood function p(y₁, …, y_N|x) does not determine x uniquely, neither its orbit under ${\mathbb{Z}}_M$ (1.3).

Nevertheless, the likelihood function allows us to identify a family of signals which can be described as the orbit of x under a parameterized sub-group of the permutation group; we denote this orbit by G_Π,Lx from reasons that will be explained later. Our analysis consists of two stages: identifying the orbit G_Π,Lx from the observations, and finding a unique signal in G_Π,Lx that maximizes the posterior distribution. In particular, in Section 3.2 we use auto-correlation analysis to show that for any L ≤ 192 satisfying $L\gtrsim \sqrt{6M}$ (more accurately, any pair (L, M) satisfying (3.5) and (3.6)), the orbit G_Π,Lx can be computed from the first three auto-correlations of y; we conjecture it remains true for any L > 192. These auto-correlations can be estimated from the data if N/σ⁶ → ∞ in any SNR regime. Finally, in Section 3.3 we show that if x was drawn from almost any Gaussian prior on the signal, then there is a unique signal in G_Π,Lx that maximizes the posterior distribution.

The following summarizes the main results of this section:

Theorem 3.2. Suppose that N → ∞ observations from (1.1) are collected, N/σ⁶ → ∞, and that x was drawn from almost any Gaussian prior. Then, for L ≤ 192 and any K that satisfies (3.5), there exists a single signal that achieves the maximum of the posterior distribution p(x|y₁, …, y_N).

Conjecture 3.3. Suppose that N → ∞ observations from (1.1) are collected, N/σ⁶ → ∞, and that x was drawn from almost any Gaussian prior. Then, for any fixed M, there exists a single signal that achieves the maximum of the posterior distribution p(x|y₁, …, y_N) as long as $M\le L\cdot \mathcal{P}(L)$, where $\mathcal{P}(L)$ is given in (3.5).

3.1. Reduction to heterogeneous MRA and the likelihood function

Consider two realizations y_i, y_j generated, respectively, after shifting x by s_i and s_j, and recall that K = M/L is an integer. If s_i − s_j = cK for some integer c, then y_i is equal to a circular shift of y_j, with a different noise realization. It follows that any observation y_i is a noisy and circularly-shifted realization of one of the following K signals,

$$x_k:=[x[k],x[k+K],x[k+2K],\dots ,x[k+(L-1)K]],k=0,\dots ,K-1.$$ (3.1)

Namely, x_k [ℓ] = x[k + Kℓ] for ℓ = 0, …, L − 1. We refer to $x_0,\dots ,x_{K-1}\in {ℝ}^L$ as sub-signals. Using this notation, the model (1.1) can be written as

$$y=R_{\ell }{x}_k+\epsilon ,$$ (3.2)

where k is drawn uniformly at random from {0, …, K − 1}, R_ℓ is a circular shift on an L-point grid [0, 1, …, L − 1], and ℓ is distributed uniformly. The model (1.1) is thus equivalent to the hMRA model, recently studied in [6,17,42,47], applied to the sub-signals x₀, …, x_K−1.

Observations from the hMRA model (3.2) enable the recovery of x₀, …, x_K−1 up to a circular shift of each sub-signal and a permutation across signals. This can be seen by considering the marginalized likelihood of a single observation y:

$$p(y\mid x)=\frac{1}{M}\sum _{\ell =0}^{L-1}\sum _{k=0}^{K-1}p(y\mid x,\ell ,k)=\frac{1}{{\left(2\pi {\sigma }^2\right)}^{L/2}M}\sum _{\ell =0}^{L-1}\sum _{k=0}^{K-1}{e}^{-\frac{1}{2{\sigma }^2}{\Vert y-R_{\ell }{x}_k\Vert }_2^2}.$$ (3.3)

Plainly, p(y|x) is invariant under any permutation π (overall K! permutations)

$$\left(x_0,\dots ,x_{K-1}\right)\mapsto \left(x_{\pi (0)},\dots ,x_{\pi (K-1)}\right),$$

or circular shifts ℓ₀, …, ℓ_{K – 1} (overall L^K permutations)

$$\left(x_0,\dots ,x_{K-1}\right)\mapsto \left(R_{{\ell }_0}{x}_0,\dots ,R_{{\ell }_{K-1}}{x}_{K-1}\right).$$

This set of permutations, denoted by G_Π,L, includes K!L^K elements and constitutes a subgroup of the permutation group of M elements. The orbit of x under G_Π,L is illustrated in Fig. 3.

Fig. 3.

An illustration of the orbit G_Π,Lx; all four signals have the same likelihood function. (a) A signal of length M = 12 consists of K = 3 sub-signals (drawn in different colors). (b) Permuting the sub-signal (x₀, x₁, x₂) ↦ (x₁, x₂, x₀). (c) Shifting the sub-signals (x₀, x₁, x₂) ↦ (R₋₁x₀, R₋₂x₁, x₂). (d) Permuting and shifting the sub-signals (x₀, x₁, x₂) ↦ (R₋₂x₁, x₂, R₋₁x₀).

Importantly, previous works on hMRA aimed to retrieve the orbit G_Π,Lx. In this work we further wish to recover the high-resolution signal x by ordering the sub-signals x₀, …, x_K−1 properly, which is impossible based solely on the likelihood. Thus, we must impose some additional constraints on the signal. In particular, we show in Section 3.3 that for almost any Gaussian prior there is a single element of G_Π,Lx that achieves the maximum of the posterior distribution.

3.2. Identifying the orbit G_Π,Lx

3.2.1. The noiseless case

In the absence of noise, if we have observed each one of the K sub-signals x₀, …, x_K−1, we can determine the orbit G_Π,Lx immediately by considering all of their circular shifts and permutations. Therefore, the only question is how many observations from (1.1) are required to see each sub-signal x_k at least once; this problem is known in the combinatorics literature as the coupon collector’s problem (see, for instance, [29]). In expectation, it is known that KH_K observations are required to see all K signals, where H_K is the harmonic sum

$$H_K=\frac{1}{1}+\frac{1}{2}+\cdots +\frac{1}{K}=\text{log}K+\gamma +{ϵ}_K,$$

where γ ≈ 0.57721 is the Euler–Mascheroni constant and ϵ_K ~ 1/(2K) for large K. If K is large enough, the harmonic sum can be bounded by H_K ≤ C log K for some small constant C. For example, H_K ≤ 2 log K for any K ≥ 3. Thus, we say that in expectation N ≈ K log K = M/L log(M/L) observations suffice to characterize the orbit G_Π,Lx from noiseless observations. Yet, even in the absence of noise, Theorem 3.1 suggests that finding x itself from G_Π,Lx is a non-trivial task, requiring additional assumptions; we address this in Section 3.3.

3.2.2. Auto-correlation analysis

In the low SNR regime, we propose to estimate the orbit G_Π,Lx using the first three auto-correlations of the observations, or, equivalently, their Fourier counterparts: the mean, power spectrum, and bispectrum. Assuming N/σ⁶ → ∞ and considering (3.2), the invariants of the data converge to the average of the invariants of the K sub-signals, up to bias terms:

$$\begin{gathered} \frac{1}{N}\sum _{i=1}^N{\widehat{M}}_1\left(y_i\right)\stackrel{N\to \infty }{\to }\frac{1}{K}\sum _{k=0}^{K-1}{\widehat{M}}_1\left(x_k\right), \\ \frac{1}{N}\sum _{i=1}^N{\widehat{M}}_2\left(y_i\right)[\ell ]\stackrel{N\to \infty }{\to }\frac{1}{K}\sum _{k=0}^{K-1}{\widehat{M}}_2\left(x_i\right)[\ell ]+B_2, \\ \frac{1}{N}\sum _{i=1}^N{\widehat{M}}_3\left(y_i\right)\left[{\ell }_1,{\ell }_2\right]\stackrel{N\to \infty }{\to }\frac{1}{K}\sum _{k=0}^{K-1}{\widehat{M}}_3\left(x_i\right)\left[{\ell }_1,{\ell }_2\right]+B_3\left[{\ell }_1,{\ell }_2\right], \end{gathered}$$ (3.4)

where B₂ = σ² L1, $B_3\left[{\ell }_1,{\ell }_2\right]=\overline{x}{\sigma }^2{L}^{2}D\left[{\ell }_1,{\ell }_2\right]$, $1\in {ℝ}^L$ is a vector of ones, $\overline{x}$ is the average of x $D\in {ℝ}^{L\times L}$ is a zero matrix except D[0, 0] = 3 and D[i, 0] = D[0, i] = D[i, i] = 1 for i = 1, …, L − 1, and ℓ₁, ℓ₂, ℓ₃ = 0, …, L − 1. We note that if σ² is known, one can easily remove the bias factors from the second- and third-order invariants. As N → ∞, the left-hand side equals the right-hand side almost surely.

The reason we require N/σ⁶ → ∞ is that the third-order auto-correlation requires taking triple products of three noise terms (thus tripling the effective noise level), and thus for large σ the number of observations needs to scale at least as σ⁶ to keep the variance of the estimator under control; more precisely, only when N/σ⁶ → ∞ one can estimate the invariants accurately. Therefore, if N/σ⁶ → ∞, one can estimate the first three auto-correlations at any SNR levels. If σ is fixed while N → ∞, then one can estimate all auto-correlations at any SNR level. If σ → ∞ and N does not scale with σ⁶, namely, N/σ⁶ < C for some finite constant C, then the third-order auto-correlation cannot be consistently estimated from the observations.

3.2.3. Identifiability conditions for the orbit G_Π,Lx from the auto-correlations

As discussed in Section 2, it is well-known that the first three invariants determine a single generic signal uniquely [14,33,52]. Using tools from invariant theory and algebraic geometry, this result was recently extended to demixing of K ≥ 1 invariants as in (3.4) [6]. The framework of [6] is based on checking computationally the rank of the Hessian matrix of the map between a generic¹ signal and the invariants. If the rank is sufficiently high (depending on the algebraic structure of the problem), we say that the Hessian test is passed, implying that the orbit G_Π,Lx can be identified for generic x. The Hessian test is executed on pairs of parameters (K, L); if (K, L) passes the test, it implies identifiability for all pairs (K′, L′), K′ ≤ K. In particular, it was verified² for all L ≤ 192 that a set of generic signals is determined uniquely from (3.4), up to the symmetries that form the group G_Π,L, as long as

$$K<\mathcal{P}(L)≔\frac{L+3+\lfloor L/2\rfloor +\lceil (L-1)(L-2)/6\rceil }{L+1}\approx L/6.$$ (3.5)

This immediately implies that the number of required samples is bounded by

$$K=\frac{M}{L}\lesssim L/6\Rightarrow L\gtrsim \sqrt{6M}.$$ (3.6)

This bound is conjectured to hold true for any pair (L, K) that satisfies (3.5); see [6,17]. We note that the bound of (3.5) is tight in the sense that it agrees with a simple upper bound based on parameter counting: on the one hand, the bispectrum of a generic signal contains $\mathcal{P}(L)\cdot L\approx L^2/6$ distinct entries (out of L² entries in total), and on the other hand K signals consists of KL parameters.

The following proposition and conjecture summarize the result:

Proposition 3.4. Suppose that we acquire an average of the mean, power spectrum and bispectrum of K signals as in the right-hand side (3.4). Then, for L ≤ 192 and any K that satisfies (3.5), one can identify the orbit G_Π,Lx for generic x.

Conjecture 3.5. Suppose that we acquire an average of the mean, power spectrum and bispectrum of K signals as in the right-hand side of (3.4). Then, for any pair (K, L) that satisfies (3.5), one can identify the orbit G_Π,Lx for generic x.

3.2.4. A note on computational considerations

It is important to note that Proposition 1 does not claim that the bound (3.5) can be achieved using a computationally efficient (e.g., polynomial time) algorithm. In the context of the hMRA model, numerical evidence suggests that for i.i.d. standard Gaussian signals, one can estimate the orbit G_Π,Lx from a mix of bispectra using non-convex least squares only in the regime $K\le \sqrt{L}$—substantially below the identifiability regime K ≲ L/6 [17]. Recently, it was proven that i.i.d. standard Gaussian signals can be disentangled, with high probability, using a sum-of-squares algorithm as long as $K\le \sqrt{L}/\text{polylog }(L)$ [60]. In [6,17], it was conjectured that the $\sqrt{L}$ bound reflects a fundamental statistical-computational gap—namely, although it is statistically possible to recover approximately L/6 signals, any efficient (polynomial-time) algorithm can estimate at most $\sqrt{L}$ signals.

If indeed one can recover only up to $\sqrt{L}$ signals efficiently from (3.4), it implies that the orbit G_Π,Lx can be estimated efficiently in the regime $K=\frac{M}{L}\le \sqrt{L}\Rightarrow M\le L^{3/2}$ for i.i.d. Gaussian entries. Having said that, in contrast to the hMRA model, the goal of the super-resolution problem is not to recover the orbit G_Π,Lx, but rather the signal x itself; the latter task seems to be a significantly more challenging computational problem. In addition, the main interest of this work is in smooth signals (e.g., signals with decaying power spectrum). For such signals, the achievable performance of hMRA deteriorates [9], and even recovering $\sqrt{L}$ signals seem to be unreachable. Indeed, numerical experiments in Section 5 suggest that recovery is not attainable even in the regime M ≈ L^3/2—at least not with the EM algorithm.

3.3. Identifying a unique high-resolution signal from the orbit G_Π,Lx

Until now, we have shown that one can identify the orbit G_Π,Lx for generic x if $L\gtrsim \sqrt{6M}$, L < 192, and the first three auto-correlations can be estimated from the observations. Next, we wish to show how to determine a single signal out of G_Π,Lx.

Recall that the posterior distribution p(x|y₁, …, y_N) is proportional to the likelihood function p(y₁, …, y_N|x) times a prior on the signal p(x). According to (3.3), the likelihood is constant over the orbit G_Π,Lx. Consequently, we choose the signal in the orbit that best fits the prior. Importantly, this part is independent of the observations.

Many priors can be used. In this section, we focus on Gaussian signals with zero mean and covariance Σ, that is, $p(x)=\frac{1}{\sqrt{{(2\pi )}^M\left|\Sigma \right|}}{e}^{-\frac{1}{2}{x}^T{\Sigma }^{-1}x}$. Priors of this form are ubiquitous in signal processing, and have been considered in previous works, such as [49,50,61]. In particular, we wish to show that among all signals in G_Π,Lx, there is a single signal that maximizes p(x), or, equivalently, minimizes x^TΣ⁻¹x for a positive-definite matrix, Σ⁻¹. The next lemma shows that permuting a signal usually changes this quadratic form. To this end, we define the Σ⁻¹ norm by $\Vert z{\Vert }_{{\Sigma }^{-1}}^2:=z^T{\Sigma }^{-1}z$ for a positive-definite matrix Σ⁻¹.

Lemma 3.6. Let $\Omega \subset {ℝ}^n$ and denote by R₁ and R₂ two permutation matrices and their ratio by $R=R_2{R}_1^T$. Then the set

$${\mathcal{Z}}_{R_1,R_2}=\left\{z\in \Omega \mid {\Vert R_{1}z\Vert }_{{\Sigma }^{-1}}^2={\Vert R_{2}z\Vert }_{{\Sigma }^{-1}}^2\right\},$$

is a subset of Ω of measure zero if and only if Σ⁻¹ and R do not commute.

Proof. Let $A:=R_1^T{\Sigma }^{-1}{R}_1-R_2^T{\Sigma }^{-1}{R}_2$. Observe that A = 0 if and only if R and Σ⁻¹ commute. The condition ${\Vert R_{1}z\Vert }_{{\Sigma }^{-1}}^2={\Vert R_{2}z\Vert }_{{\Sigma }^{-1}}^2$ is equivalent to z^TAz = 0. Thus, if A = 0 then ${\mathcal{Z}}_{R_1,R_2}=\Omega$. Otherwise, A ≠ 0 and symmetric, i.e., there exists a basis Q of orthogonal eigenvectors such that ${(Qy)}^{T}AQy={\sum }_{i=1}^n{\lambda }_i{y}_i^2$, with λ₁ ≠ 0. Therefore, the condition (Qy)^TAQy = 0 means that $y_1^2={\sum }_{i=2}^n\frac{{\lambda }_i}{-{\lambda }_1}{y}_i^2$. Denote the indicator functions χ_Z and χ_Y for the two sets {z ∈ Ω | z^TAz = 0} and $\left\{y\in {ℝ}^n\mid y^T\left(Q^{T}AQ\right)y=0\right\}$, respectively. Since orthogonal transformations preserve integrals and $\Omega \subset {ℝ}^n$, we have

$$\left|{\mathcal{Z}}_{R_1,R_2}\right|={\int }_{\Omega }{\chi }_Z(z)dz\le {\int }_{{ℝ}^n}{\chi }_Y(y)dy={\int }_{{ℝ}^{n-1}}\left({\int }_{ℝ}{\chi }_Y\left(y_1,y_2,\dots ,y_n\right)d{y}_1\right)d{y}_2\cdots d{y}_n.$$

Recall that (Qy)^TAQy = 0 implies that $y_1=\pm {\left({\sum }_{i=2}^n\frac{{\lambda }_i}{-{\lambda }_1}{y}_i^2\right)}^{\frac{1}{2}}$. Thus, for any fixed y₂, …, y_n, the indicator function χ_Y is nonzero on only two points, implying ${\int }_{ℝ}{\chi }_Y\left(y_1,y_2,\dots ,y_n\right)d{y}_1=0$. Consequently, $\left|{\mathcal{Z}}_{R_1,R_2}\right|=0$. □

Any group of permutations over a finite set is finite, and thus there are finitely many pairs R₁ and R₂ of permutation matrices. Consequently, the set of signals for which ${\Vert R_{1}z\Vert }_{{\Sigma }^{-1}}^2={\Vert R_{2}z\Vert }_{{\Sigma }^{-1}}^2$ for some pairs of permutations is also of measure zero.

Corollary 3.7. Assume the conditions of Lemma 1 are met, that is, Σ⁻¹ and $R_2{R}_1^T$ do not commute for all pairs R₁ ≠ R₂ in the permutation group. Then for almost every x in Ω the minimum of the quadratic form y^TΣ⁻¹y is unique among all signals y in G_Π,Lx.

The case when Σ⁻¹ is a circulant matrix is of particular importance. Such a prior, reflecting a prior on the signal’s power spectrum, is popular in many signal processing tasks, as well as in cryo-EM; see for instance [54]. In this case, both the prior and the likelihood function (and thus the posterior) are shift-invariant, and therefore any signal is indistinguishable from its cyclic shifts. To account for this symmetry, we derive a distinct uniqueness result—up to a circular shift—for circulant matrices.

Proposition 3.8. Assume Σ⁻¹ is a circulant, positive-definite matrix of size n > 2. Then:

1. The Σ⁻¹ norm $\Vert \cdot {\Vert }_{{\Sigma }^{-1}}$ is invariant under cyclic shifts. Consequently, we may consider the quadratic form y^TΣ⁻¹y as a function over equivalence classes in G_Π,Lx, where two vectors are equivalent if one is a cyclic shift of the other.

2. If all eigenvalues of Σ⁻¹ are distinct, then for almost every signal x the minimum of the quadratic form y^TΣ⁻¹y is unique over the equivalence classes of G_Π,Lx.

The proposition is proved in Appendix A.

4. An expectation-maximization algorithm

Our theoretical study is based on invariant features. Conceptually, it suggests a two-stage procedure: it begins by identifying the orbit G_Π,Lx, and then choosing a unique signal according to the prior. Although identifying the orbit G_Π,Lx efficiently using bispectrum demixing is possible [17,60], it is unclear how to devise a tractable algorithm for the second step. As an alternative, we formulate an EM algorithm, described below, which aims to achieve the maximum of the posterior distribution by maximizing the likelihood function and the prior simultaneously [25]. EM is known to work quite well in many practical scenarios; see for instance its application to cryo-EM experimental datasets [48,54]. In what follows, we formulate EM for MRA with a general linear operator

$$y=T{R}_{s}x+\epsilon ,$$ (4.1)

where s is drawn from a uniform distribution on a discrete grid S_M with M points, and T is a general linear operator (not necessarily a sampling matrix). For the special case of super-resolution, a similar algorithm was already derived by [61].

EM is a common framework to compute the maximum aposteriori estimator (MAP). Hereafter, we formulate EM for the general model. Given a set of N independent observations y₁, …, y_N, the log-posterior distribution log p(x|y₁, …, y_N) is proportional to log p(y₁, …, y_N|x)p(x), where

$$\text{log}p\left(y_1,\dots ,y_N\mid x\right)=\sum _{i=1}^N\text{log}\sum _{s_{\ell }\in S_M}{e}^{-\frac{1}{2{\sigma }^2}{\Vert y_i-T{R}_{s_{\ell }}x\Vert }_2^2}+\text{ constant,}$$ (4.2)

is the log-likelihood function, and p(x) is a prior on the signal. We assume that the signal is drawn from a Gaussian prior with zero mean and known covariance Σ so that $p(x)~\mathcal{N}(0,\Sigma )$. EM aims to maximize the posterior iteratively, where each iteration consists of two steps. The first step, called E-step, computes the expected value of the likelihood x (note, not the marginalized likelihood) with respect to the circular shifts (i.e., the nuisance variables), given the current estimate of the signal x_t and the data y₁, …, y_N:

$$\begin{gathered} Q\left(x\mid x_t\right)={\mathbb{E}}_{s_1,\dots ,s_N\mid y_1,\dots ,y_N,x_t}\left\{\text{log}p\left(y_1,\dots ,y_N,s_1,\dots ,s_N\mid x\right)+\text{log}p(x)\right\} \\ =-\frac{1}{2{\sigma }^2}{\mathbb{E}}_{s_1,\dots ,s_N\mid y_1,\dots ,y_N,x_t}\left\{\sum _{i=1}^N\Vert y_i-T{R}_{s_i}x\Vert \right\}-\frac{1}{2}{x}^T{\Sigma }^{-1}x+\text{constant} \\ =-\frac{1}{2{\sigma }^2}\sum _{i=1}^N\sum _{s_{\ell }\in S_M}{w}_{i,\ell }\Vert y_i-T{R}_{s_{\ell }}x\Vert -\frac{1}{2}{x}^T{\Sigma }^{-1}x+\text{constant}, \end{gathered}$$ (4.3)

where

$$w_{i,\ell }=\frac{e^{\frac{-1}{2{\sigma }^2}{\Vert y_i-T{R}_{s_{\ell }}{x}_t\Vert }^2}}{{\sum }_{s_{\ell }\in S_M}{e}^{\frac{-1}{2{\sigma }^2}{\Vert y_i-T{R}_{s_{\ell }}{x}_t\Vert }^2}}.$$ (4.4)

The second step, called M-step, maximizes Q with respect to x. The solution is obtained by solving the linear system of equations

$$Ax=b,$$ (4.5)

where

$$A≔{\Sigma }^{-1}+\frac{1}{{\sigma }^2}\sum _{i,\ell }{\omega }_{i,\ell }\left(R_{\ell }^{-1}{T}^{T}T{R}_{\ell }\right),$$ (4.6)

$$b≔\frac{1}{{\sigma }^2}\sum _{i,\ell }{\omega }_{i,\ell }{R}_{\ell }^{-1}{T}^T{y}_i.$$ (4.7)

The EM algorithm iterates between computing the weights (4.4) and solving the linear system (4.5) until convergences. In our implementation, the algorithm halts when the relative difference between the posterior of two consecutive iterations falls below 10⁻⁵. In general, EM does not converge to the global maximum of the posterior distribution; however, each iteration is guaranteed not to decrease the posterior [25]. In addition, several recent works derived intriguing theoretical results for EM under specific statistical models. See for example [5,24], and in particular [27] that analyzes EM for the MRA model.

In the special case in which the linear operator T is the sampling operator (1.1), computing the weights and constructing A and b reduces to computing a set of correlations; this can be executed efficiently using FFT. For EM implementation when a blurring kernel is included, see [61].

On the connection between maximum likelihood estimation and the invariant approach.

A recent paper by Katsevich and Bandeira [35] studies Gaussian mixture models, for which heterogeneous MRA is a special case, in the parametric setup considered in this work: N → ∞, SNR→ 0, and fixed M. In particular, they show that log-likelihood maximization is equivalent to an asymptotic series of successively higher moment matching problems. In this sense, a method based on the bispectrum (a third-order moment), as we use for the analysis, can be thought of as a third-order approximation of the likelihood function.

5. Numerical results

We conducted three experiments to examine the performance of the EM algorithm. The first experiment demonstrates resolving two adjacent peaks from low-resolution observations. The next two experiments study the performance of the algorithm as a function of noise level and the number of samples. The code for all experiments and the EM algorithm is publicly available at https://github.com/TamirBendory/MRA-SR.

In all experiments, we set a prior on the signal’s power spectrum, and therefore, to account for the circular shift symmetry, the relative recovery error is defined as

$$\text{relative error}=\underset{\ell \in {\mathbb{Z}}^M}{\text{min}}\frac{\Vert R_{\ell }{x}_{\text{est}}-x\Vert }{\Vert x\Vert },$$ (5.1)

where $x\in {ℝ}^M$ is the underlying signal, and x_est is the output of the EM algorithm. The SNR is defined as SNR = ∥x∥²/(Mσ²). The EM iterations terminate when either a maximal number of iterations is reached, or the relative absolute difference of the posterior between two consecutive iterations drop below a tolerance parameter. In all experiments, the maximal number of EM iterations was set to be 100 iterations, and the tolerance parameters was 10⁻⁵. The EM may be initialized from multiple random points and thus produce different estimators. Among those estimators, we choose the one with the largest posterior. The posterior is computed at each EM iteration.

Experiment 1.

The signal in this experiment is of length M = 120 with bandwidth (the largest non-zero frequency, see (1.5)) B = 15. We generated N = 10⁴ observations by shifting the signal and sampling it at L = 15 points, corresponding to half of the Nyquist sampling rate. Then, an i.i.d. Gaussian noise was added, corresponding to SNR = 1. We ran the EM algorithm from five random initial points; each trial required 13–17 iterations to converge. Since the signal (only in this experiment) is bandlimited, in each iteration the current estimate is projected onto the low B = 15 frequencies. The target and estimated signals are presented in Fig. 2; the relative recovery error is 0.0614. Figure 4 displays the relative error per frequency. As can be seen, the relative error of frequencies above the Nyquist rate is still quite low, indicating that the EM algorithm resolves high frequencies accurately.

Fig. 4.

Recovery error per frequency of the experiment presented in Figure 2. The figure indicates that the EM algorithm succeeds to resolve frequencies beyond the largest frequency determined by the Nyquist sampling rate L/2 (vertical red line).

Experiment 2.

Figure 5 presents the error curve as a function of the SNR, in the high and low SNR regimes. For each SNR value, 50 trials were conducted and we present the median error. A signal of length M = 64 was drawn from a Gaussian distribution with zero mean, and a circulant covariance matrix, corresponding to power spectrum decaying linearly, that is, as 1/f. Following the circular shift, each observation was sampled at L = 32 equally-spaced points—corresponding to half of the Nyquist sampling rate.

Fig. 5.

Relative estimation error as a function of the SNR. In the high SNR regime, the relative error scales as SNR^−1/2, which is the same estimation rate as if there were no shifts (namely, the estimation rate of averaging independent Gaussian variables). In the low SNR regime, the error decays faster than SNR⁻¹, demonstrating a sharp transition from the high SNR regime.

Figure 5b presents the relative error as a function of the SNR, for 30 SNR values sampled uniformly on a logarithmic scale between 10^0.2 and 10², and N = 10² observations; this reflects the high SNR regime. Unfortunately, the EM seems to suffer from a flaw: it should be initialized from many points (among them we choose the one with the largest posterior value) in order to result in a consistent recovery. In this experiment, we initialized the algorithm from 1000 points; the computational load is still quite cheap since each trial requires only a few iterations (around 5). Yet, it suggests that EM may not be the optimal computational scheme in the high SNR regime. The slope of the error curve is approximately −1/2. Since the SNR is proportional to 1/σ², this indicates that the error scales as σ—the optimal estimation rate even if the circular shifts were known.

Figure 5a shows a similar experiment for SNR values ranging between 10^−0.6 and 1 and N = 10⁵ observations. In this low SNR regime, the EM algorithm seems to be more consistent, and thus we initialized it from merely 20 random points. In this regime, the slope of the error curve becomes steeper and the error slope is smaller than −1, implying that the error scales faster than σ⁴. This indicates, in line with previous works on MRA (with finite M), that the estimation rate in the high and low SNR regimes is drastically different [1,2,9,47]. In fact, our analysis predicts that as SNR→ 0, the slope of the error curve would tend to SNR^−3/2; see Section 3.

Experiment 3.

Figure 6 examines the recovery error for different values of M and L. For each M, we chose values of L so that M/L is an integer. The signals were generated as in Experiment 2 with SNR = 5 and N = 1000, and the EM was initialized from 50 random locations in each trial. For each pair (M,L), the mean error over 50 trials was recorded. The red vertical dashed line indicates L = M^2/3; this is the conjectured computational recovery limit for hMRA, namely, for recovering the orbit G_Π,Lx (see discussion in Section 3.2.4). Notwithstanding, we get relatively small recovery error only for much larger values of L, suggesting that the super-resolution problem is computationally more challenging than hMRA. In particular, our theoretical analysis is split into two stages: recovering the orbit G_Π,Lx, and recovering x from the orbit; the latter depends only on the prior, and not on the data. In contrary, the EM algorithm aims to implicitly carry out both stages simultaneously. We believe that the second stage, together with the smoothness of the signals (see Section 3.2.4), is the reason the performance of EM for super-resolution is inferior to what was demonstrated in previous MRA setups [1,14,42].

Fig. 6.

Relative recovery error as a function of L for different values of M. The red dashed line indicates L = M^2/3. The results suggest that the super-resolution problem is significantly harder than hMRA.

6. Discussion

Super-resolution limits:

This work analyzes the super-resolution from multiple observations problem in a noisy environment using the third-order auto-correlation. To use higher-order auto-correlations, more observations should be collected: the number of observations needs to scale as σ^2q to estimate the q-th order auto-correlation accurately. The q-th auto-correlation provides O(L^q−1) polynomial equations of the sought signals. Based on our analysis and the reduction of the super-resolution problem to the hMRA model (3.2), we expect that the q ≥ 3 auto-correlation would identify M = O(L^q−1) grid points. Such a result will follow directly from a generalization of [6] to higher-order auto-correlations. This leads us to the following conjecture:

Conjecture 6.1. Suppose that N observations from (1.1) are collected and each observation is sampled at L equally-spaced locations. Then, in the low SNR regime σ → ∞, if N/σ^2q → ∞ for some q ≥ 3, one can identify up to M = O(L^q−1) grid points. In other words, only L = O(M^1/(q−1)) samples per observation suffice for signal identification. In particular, for N → ∞ and any fixed noise level (that might be arbitrarily high), there is no theoretical limit on the achievable resolution.

Continuous super-resolution:

A natural generalization of the model considered in this work is the following. Let $x:S^1\to ℝ$ be a band-limited signal on the circle (1.4), and let R_θ denote a rotation, that is, (R_θx)(t) = x(t − θ), where θ is distributed uniformly on the circle. Together with i.i.d. Gaussian noise $\epsilon \in {ℝ}^L$, the data generative model reads

$$y=P\left(R_{\theta }x\right)+\epsilon ,\theta ~\text{Uniform }[0,1),\epsilon ~\mathcal{N}\left(0,{\sigma }^{2}I\right),$$ (6.1)

where P denotes a sampling operator that collects L equally-spaced point-wise samples. The goal is to estimate x from N observations sampled from (6.1). This setup is interesting in the sub-Nyquist regime, where P samples x below its Nyquist sampling rate. Although this model shares many similarities with (1.1), it poses some additional challenges that are beyond the scope of this work; we intend to address them in a follow-up work.

Super-resolution of images:

Although this paper deals with 1-D signals, the methodology can be extended to higher-dimensions. For example, an interesting MRA setup that was studied in [42] considers rotating 2-D ‘bandlimited’ images. Specifically, suppose that an image X belongs to the vector space of images that can expanded by finitely many coefficients in a steerable basis (such as Fourier-Bessel [62] or prolate spheroidal wave functions [38]):

$$X(r,\varphi )=\sum _{k=1}^{K_{\mathit{\text{max}}}}\sum _{q=1}^{Q_{\mathit{\text{max}}}}{a}_{k,q}{u}_{k,q}(r,\varphi ),$$ (6.2)

where (r, ϕ) are polar coordinates, $a_{k,q}$ are the expansion coefficients, and $u_{k,q}(r,\varphi )$ are the basis functions of the steerable basis. The images are acted upon by unknown elements of the group of inplane rotations SO(2). The steerability property implies that rotating an image by an angle α amounts to multiplying the expansion coefficients by e^ιkα:

$$X(r,\varphi -\alpha )=\sum _{k=1}^{K_{\mathit{\text{max}}}}\sum _{q=1}^{Q_{\mathit{\text{max}}}}{a}_{k,q}{e}^{-ık\alpha }{u}_{k,q}(r,\varphi ).$$ (6.3)

Accordingly, it is easy to see that the triple products $a_{k_1,q_1}{a}_{k_2,q_2}{a}_{-k_1-k_2,q_3}$ are invariant under rotations—these products form the bispectrum [42,62]. This in turn implies that for an image expanded by M coefficients, there are O(M^5/2) bispectrum entries. In this case, our framework suggests that, perhaps, one can identify an image from sufficiently many observations with merely L = O(M^−2/5) samples per observation.

Super-resolution in high dimensions:

This work studies the finite-dimensional regime (finite M) in which invariant features achieve the optimal estimation rate as SNR → 0. A recent work [51] uncovered that this is not the case in the high-dimensional regime M → ∞. In particular, it was shown that the parameter that controls the ‘hardness’of the model is α = M/(logMσ²): when α < 2 the samples complexity of the problem rapidly increases, whereas for α > 2 the effect of the unknown shifts is minor. Nevertheless, the authors of [51] only investigated random sub-sampling operators that do not include the super-resolution setup (1.1). The high-dimensional regime is of particular interest since it seems as a good model for modern cryo-EM datasets, where the dimensionality and the number of samples are of the same order of a few millions. In fact, high-dimensional statistical analysis has been already proven effective for cryo-EM data processing. For example, a covariance estimation technique based on high-dimensional analysis (the so-called spiked model) has significantly improved image denoising [16].

Cryo-EM and XFEL:

The main motivation of this work arises from cryo-EM and XFEL. The measurements in these applications (under simplifying assumptions, see for example [10]) agree with the general MRA model (1.1), where g ∈ SO(3) (the group of 3-D rotations), the 3-D Fourier transform of the signal x is assumed to be bounded in a ball (‘bandlimited’ volume), and the linear operator T collects samples of the 2-D tomographic projection of the rotated volume. The question then would be whether the maximal resolution of a 3-D reconstruction algorithm can surpass the resolution dictated by the detectors acquiring the data—that is, the resolution of the 2-D tomographic projection images. A recent proof of concept (on simulated data) promises an affirmative answer [22]. Extending our analysis to this case requires sophisticated tools and we leave it for a future research.

Appendix A. Proof of Proposition 3.8

The first lemma refines the condition of Lemma 1 in terms of invariant subspaces. Recall that a subspace V is R-invariant if R(V) ⊂ V.

Lemma A.1. Let S be a symmetric matrix. Then, R and S commute if and only if every eigenspace is R-invariant.

Proof. Since S is symmetric, we can decompose the space into real eigenspaces. First direction: if R and S commute then for any eigenvector v of S with eigenvalue λ, we have

$$S(Rv)=RSv=\lambda (Rv).$$

Namely, Rv is also in the eigenspace. Second direction: if any eigenspace V_λ is R-invariant then for any v ∈ V_λ we can write Rv in terms of the basis of V_λ, and to have

$$Rv=\sum _i{\alpha }_i{v}_i\Rightarrow S(Rv)=S\left(\sum _i{\alpha }_i{v}_i\right)=\sum _i{\alpha }_i\lambda v_i=\lambda Rv=R(Sv).$$

□

Lemma A.2. Let Σ⁻¹ be a circulant matrix. Then

1. Σ⁻¹ commutes with any cyclic permutation matrix R.

2. Conversely, if each eigenvalue of Σ⁻¹ has multiplicity 1 and Σ⁻¹ commutes with a permutation matrix R, then R must be a cyclic permutation.

Proof. Let W denote the DFT matrix, with entries $W_{j,\ell }=\frac{1}{\sqrt{n}}{\omega }^{(\ell -1)(j-1)},\omega =e^{-2\pi ı/n},1\le j,\ell \le n$. Then the columns w₀, …, w_n−1 of W are the eigenvectors of Σ⁻¹. Furthermore, if R is the cyclic permutation matrix corresponding to the permutation j ↦ j + k mod n, then Rw_ℓ = ω^(ℓ−1)kw_ℓ. Consequently, R preserves each eigenspace of Σ⁻¹, and so by Lemma 2 R and Σ⁻¹ commute. This proves the first statement.

For the converse, suppose each eigenvalue of Σ⁻¹ has multiplicity 1, and take any permutation matrix R that commutes with Σ⁻¹. From Lemma 2, R must leave each eigenspace of Σ⁻¹ fixed; consequently, for each eigenvector w_ℓ of Σ⁻¹ there is a scalar α_ℓ so that Rw_ℓ = α_ℓw_ℓ. Suppose the permutation corresponding to R sends index 1 to index k + 1; then

$$\frac{1}{\sqrt{n}}=W_{1,\ell }={\left(R{w}_{\ell }\right)}_{k+1}={\alpha }_{\ell }{W}_{k+1,\ell }={\alpha }_{\ell }\frac{1}{\sqrt{n}}{\omega }^{(\ell -1)k},$$ (A.1)

and so α_ℓ = ω^{−(ℓ−1)k}. Since Rw_ℓ = α_ℓw_ℓ, for any index j,

$${\left(R{w}_{\ell }\right)}_j={\omega }^{-(\ell -1)k}{W}_{j,\ell }={\omega }^{-(\ell -1)k}\frac{1}{\sqrt{n}}{\omega }^{(\ell -1)(j-1)}=\frac{1}{\sqrt{n}}{\omega }^{(\ell -1)(j-k-1)}=W_{j-k,\ell },$$ (A.2)

meaning that R cyclically shifts the entries of w_ℓ by k. Since this holds for all basis vectors w_ℓ, R is a cyclic shift. □

We may now prove Proposition 2. The first statement of Proposition 2 is identical to the first statement of Lemma 3. For the second statement, Lemma 1 tells us that for almost every signal x, the quadratic form y^TΣ⁻¹y takes on distinct values on each equivalence class of vectors in the orbit G_Π,Lx (where two vectors are equivalent if one is a cyclic shift of the other). Indeed, for any two permutation matrices R₁ and R₂, the set $\left\{x\mid x^T{R}_1^T{\Sigma }^{-1}{R}_{1}x=x^T{R}_2^T{\Sigma }^{-1}{R}_{2}x\right\}$ has measure zero if and only if $R_2{R}_1^T$ does not commute with Σ⁻¹; from Lemma 3, this latter condition is equivalent to $R_2{R}_1^T$ not being a cyclic permutation. Since there are only finitely many permutation matrices, the set

$$\left\{x\mid \exists R_1,R_2\text{ s}.\text{t}.R_2{R}_1^T\text{ not cyclic and }{x}^T{R}_1^T{\Sigma }^{-1}{R}_{1}x=x^T{R}_2^T{\Sigma }^{-1}{R}_{2}x\right\}$$ (A.3)

also has measure 0. Consequently, for almost every signal x, the equality $x^T{R}_1^T{\Sigma }^{-1}{R}_{1}x=x^T{R}_2^T{\Sigma }^{-1}{R}_{2}x$ can hold only when $R_2{R}_1^T$ is cyclic, i.e. when R₁x and R₂x are in the same equivalence class of G_Π,Lx.

Because the quadratic form y^TΣ⁻¹y takes on distinct values on each equivalence class in G_Π,Lx for almost every x, it immediately follows that for almost every x the minimum of y^TΣ⁻¹y over equivalence classes in G_Π,Lx is unique. This is the desired result.