Super-resolution capacity of variance-based fluorescence microscopies, from Random Illumination Microscopy (RIM) to Super-resolved Optical Fluctuation Imaging (SOFI)

Simon Labouesse (1), Jérôme Idier (2), Marc Allain (3), Guillaume

Giroussens (3), Thomas Mangeat (1) and Anne Sentenac (3)*

(1) LITC Core Facility, Centre de Biologie Intégrative, Université de Toulouse, CNRS, UPS, 31062, Toulouse, France.

(2) Nantes Université, École Centrale Nantes, CNRS, LS2N, Nantes, France. and

(3) Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France.

*anne.sentenac@fresnel.fr

Improving the resolution of fluorescence microscopy beyond the diffraction limit can be achieved by acquiring and processing multiple images of the sample under different illumination patterns (periodic grids, focused beams, or more generally speckles). When the illuminations are known, the super-resolved reconstruction is generally formed from a linear combination of the multiple diffraction-limited images, and the resolution gain is easily determined. On the other hand, when the illuminations are unknown, the resolution gain is seldom well-defined. In this work, we consider the recent Random Illumination Microscopy (RIM) technique where the illuminations are unknown speckles and the reconstructions are formed from the variance of the images. We show that an unambiguous two-fold resolution gain can be obtained only when the speckle correlation length coincides with the width of the observation point spread function. Last, we analyze the difference between the variance-based techniques using random speckled illuminations (as in RIM) and those obtained using random fluorophore activation (as in Super-resolution Optical Fluctuation Imaging, SOFI).

The light intensity recorded by the camera of a fluorescence microscope cannot exhibit spatial frequencies above $2/\lambda$ where $\lambda$ is the wavelength of the emitted light. This low-pass filtering, due to the loss of the evanescent waves at the detector plane, cannot be circumvented. Therefore the challenge of super-resolution imaging is to recover spatial frequencies of the sample fluorescence density beyond $2/\lambda$ from data that are frequency-limited to $2/\lambda$. A widespread solution consists in processing multiple images obtained by changing the illumination, like translating focused spots [1-4] or rotating and translating periodic light patterns [5, 6]. The data processing of most techniques using structured illuminations requires the knowledge of the illumination patterns, either explicitly as in Structured Illumination Microscopy [5, 6] or implicitly as in confocal or Image Scanning microscopy [1]. In this context, Random Illumination Microscopy [7] stands out as an exception as it does not require knowledge of the illumination patterns: the super-resolved image is formed from the variance of multiple diffraction-limited images recorded under different random speckled illuminations. While attractive because of its simplicity and significant image improvement [7], RIM variance-based processing lacks a rigorous analysis of its resolution potential, the non-linearity of the variance being a significant obstacle to its derivation. In this work, we study the sample information that can be extracted from the variance of speckled images as a function of the statistical properties of the random illumination and we derive the condition under which the variance can provide a resolution gain.

# I. DEFINING THE RESOLUTION IN THE SPATIAL FREQUENCY SPACE

To model the data provided by a fluorescence microscope under an inhomogeneous illumination, we introduce the point spread function of the microscope $h$ and the illumination intensity function $E$. Importantly, these two functions are defined at a macroscopic scale inside the sample, through the averaging over regions large enough to contain thousands of atoms (typically of the order of a thousand nm³), to wash out the microscopic fluctuations. In this context, we define the macroscopic fluorescence density $\rho$ such that $V\rho(\mathbf{r})E(\mathbf{r})$ is the energy (detected by the camera) of the fluorescent light emitted by a macroscopic volume $V$ centered about $\mathbf{r}$. Hereafter we neglect the Poisson noise. The fluorescence density depends on the fluorophore concentration and the molecular brightness.

With these definitions, the microscope image can be written as,

$$I(\mathbf{r}) = \int \rho(\mathbf{r}')E(\mathbf{r}')h(\mathbf{r} - \mathbf{r}')d\mathbf{r}' \quad (1)$$

where $\mathbf{r}$ indicates a position in the image domain that is conjugated to a point in the object domain. This model can be applied to two- or three-dimensional imaging configurations. In the spatial frequency-space (Fourier space), Eq. (1) reads,

$$\tilde{I}(\boldsymbol{\nu}) = \tilde{h}(\boldsymbol{\nu}) \int \tilde{\rho}(\boldsymbol{\nu} - \boldsymbol{\nu}') \tilde{E}(\boldsymbol{\nu}') d\boldsymbol{\nu}' \quad (2)$$
where $\tilde{f}(\nu) = \int f(\mathbf{r})e^{-i2\pi\nu \cdot \mathbf{r}}d\mathbf{r}$ stands for the Fourier transform of $f$.

If $E$ is a constant, as in a standard fluorescence microscope, the recorded image depends only on the sample spatial frequencies belonging to the support of the Optical Transfer Function (OTF) $\tilde{h}$, noted $W_h$, which is at best a disk of radius $2/\lambda$ (in the 2D case) or exhibits a torus-like shape in the 3D case [8].

We now consider cases where the illumination $E$ is inhomogeneous. In periodic Structured Illumination Microscopy (SIM), the illumination is a periodic light grid generally formed from the interference of two or three collimated laser beams that is translated and rotated [6]. In point-scanning microscopy, the illumination is a focused beam that is scanned across the sample [4]. In Random Illumination Microscopy, the illumination is a random speckle obtained, for example, by passing a laser beam through a diffuser. Noting $W_E$ the support of $\tilde{E}$, the recorded image depends on the sample spatial frequencies in the domain

$$W_{hE} = \{\nu - \mu | \nu \in W_h, \mu \in W_E\}, \quad (3)$$

which is no longer limited by $2/\lambda$ and corresponds to the frequency support of $hE$.

Yet, sensitivity to spatial frequencies of the sample outside $W_h$ is a necessary but not a sufficient condition for being able to form a super-resolved image. One also needs a technique for extracting the high spatial frequencies of the sample from the diffraction-limited images $I_n$ obtained for various illumination intensities $E_n$. Hereafter, the gain of resolution of a super-resolved technique will be measured through the ability to recover $\tilde{\rho}$ beyond $W_h$.

## II. RESOLUTION OF TECHNIQUES USING KNOWN INHOMOGENEOUS ILLUMINATIONS

When the illumination function is well known, as in periodic SIM or point-scanning microscopy, the super-resolved image is obtained through a linear combination of the recorded data, and the super-resolution capacity of the technique is easily determined.

In periodic SIM, the sample is illuminated successively by $N$ different light grids $E_{n=1...N}$. The resolution of a system of linear equations permits the recovery of the object spatial frequencies in $\bigcup_{n=1}^N W_{hE_n}$ [6].

In point-scanning microscopy, one records images for different positions of a focused illumination. We note $E(\mathbf{r})$ the illumination intensity at $\mathbf{r}$ when the beam is focused at the origin. The image that is recorded when the illumination is focused at $\mathbf{r}_o$ reads,

$$I(\mathbf{r}, \mathbf{r}_o) = \int \rho(\mathbf{r}')E(\mathbf{r}' - \mathbf{r}_o)h(\mathbf{r} - \mathbf{r}')d\mathbf{r}'. \quad (4)$$

Taking the Fourier transform of $I(\mathbf{r}, \mathbf{r}_o)$ with respect to

($\mathbf{r}, \mathbf{r}_o$) yields,

$$\tilde{I}(\nu, \nu_o) = \tilde{h}(\nu)\tilde{\rho}(\nu - \nu_o)\tilde{E}(\nu_o). \quad (5)$$

In theory, it is possible to recover the sample frequencies $\tilde{\rho}(\mu)$ for any $\mu \in W_{hE}$ whatever the shape of the translated illumination. Yet, in practice, this reconstruction scheme is never used. In confocal microscopy, for example, the super-resolved image is obtained without numerical processing, by simply recording the signal at the (conjugated) position of the focused beam, $I_{SR}(\mathbf{r}) = I(\mathbf{r}, \mathbf{r}) = \int \rho(\mathbf{r}')E(\mathbf{r}' - \mathbf{r})h(\mathbf{r} - \mathbf{r}')d\mathbf{r}'$. The frequency support of the function $E(-\mathbf{u})h(\mathbf{u})$ being $W_{hE}$, the confocal approach directly recovers the sample frequencies over the whole accessible super-resolved domain. Yet, this ideal resolution requires the use of an infinitely small pinhole which is impossible in practice.

We now turn to microscopy configurations where the inhomogeneous illuminations are unknown. These approaches ease the experimental implementation, as the control of the illuminations becomes minimal, but require more complex reconstruction schemes.

## III. RESOLUTION OF TECHNIQUES USING RANDOM ILLUMINATIONS (RIM)

In the last fifteen years, it was observed that super-resolved images of the sample could be built from multiple low-resolution images acquired with random speckled illuminations. The first reconstruction techniques [9–12] estimated both the sample and the illuminations using advanced regularization techniques, such as sparsity or binarity. The complexity of the reconstruction procedures and the influence of the regularization in the final result, prevented any rigorous determination of the super-resolution capacity of these approaches.

Recently [13], it was proposed to reconstruct the sample from the second-order statistics of the speckled images. The major interest of this statistical approach (known as RIM for Random Illumination Microscopy) is that it does not require the knowledge (nor the estimation) of the illuminations. It only requires the knowledge of the speckle statistics (namely the mean and auto-covariance) which are theoretically well-defined and very robust to misalignments or aberrations [14]. More precisely, RIM consists in recording multiple images of a sample under different fully developed speckled illuminations $E_n$. The speckled patterns can be considered different realizations of a second-order stationary random process $E$, with constant mean $\langle E \rangle$ (where $\langle \cdot \rangle$ indicates the ensemble average) and auto-covariance, $C(\mathbf{r}-\mathbf{r}') = \langle E(\mathbf{r})E(\mathbf{r}') \rangle - \langle E \rangle^2$. The frequency support of $C$ is the same as the frequency support of each speckled illumination and is noted $W_E$ [14, 15]. It is demonstrated mathematically in [13] that, if $W_E = W_h$, the sample frequencies in the enlarged domain $W_{h2}$ can be recovered from the square root of the
covariance matrix of the speckled images defined as,

$$ \mathrm{Cov}(\mathbf{r}, \mathbf{r}') = \langle I(\mathbf{r})I(\mathbf{r}') \rangle - \langle I(\mathbf{r}) \rangle \langle I(\mathbf{r}') \rangle. \quad (6) $$

Yet, forming the covariance matrix, let alone its square root, is numerically untractable. In practice, RIM reconstruction method, named algoRIM, processes only the variance $V_\rho(\mathbf{r}) = \mathrm{Cov}(\mathbf{r}, \mathbf{r})$ of the speckled images. The sample is estimated iteratively so as to minimize a distance between the experimental variance image $V^{\mathrm{exp}}$ and the simulated variance image for a given $\rho$, $V_\rho$ [7, 16]. The minimization procedure is not a simple task as the variance image is quadratically linked to the sample $\rho$ through [7, 17]

$$ V_\rho(\mathbf{r}) = \int d\mathbf{r}_1 d\mathbf{r}_2 \\ h(\mathbf{r} - \mathbf{r}_1)\rho(\mathbf{r}_1)C(\mathbf{r}_1 - \mathbf{r}_2)\rho(\mathbf{r}_2)h(\mathbf{r} - \mathbf{r}_2). \quad (7) $$

On many calibrated and biological samples, RIM exhibited a resolution twice better than that of a classical fluorescence microscope, which indicated that the sample estimated by algoRIM presented spatial frequencies in the enlarged domain $W_{h2}$ [7, 17]. Now, to be fully confident in RIM results, it is necessary to show that the solution of the variance-matching procedure (which ideally provides the same variance image as the experimental one, $V_\rho = V^{\mathrm{exp}}$) has the same spectrum $\tilde{\rho}$ as the actual sample in a domain about $W_{h2}$. The question is: if two samples yield the same variance image, do they have the same spectrum in a domain larger than $W_h$? If yes, the size of this domain will define RIM super-resolution capacity.

We recall that the variance is formed from the square of raw images that are frequency-limited to $W_h$ but depend on the sample spectrum in $W_{hE}$. The variance is sensitive to the spatial frequencies of the sample in $W_{hE}$ (we show on a specific example in appendix B that this is indeed the case), but its frequency support is $W_{h2}$. In other terms, the number of unknowns (the sample frequencies in $W_{hE}$) is, in general, different from the number of data (the variance frequencies in $W_{h2}$). Thus, it is likely that the answer to the question raised above will be different when $W_{hE}$ is the same as, included in, or bigger than, $W_{h2}$.

In the classical RIM configuration, the illumination is performed through the same objective as the observation. If the Stokes shift can be neglected [7, 14] the frequency support of the speckled illumination $W_E$ is the same as that of the point spread function, $W_h$. Hence, the frequency support of the variance $W_{h2}$ matches the support of the sample spectrum it depends on, $W_{hE}$. In this case, we demonstrate in Appendix A that if two samples have the same variance image, they have necessarily the same sample spectrum within $W_{h2}$. This result ensures the unicity of the variance-matching solution and determines the super-resolution capacity of RIM: RIM achieves the same resolution as an ideal confocal microscope in which the focused beam and the point spread function share the same frequency support.

We now consider the case where the speckle correlation length is larger than the width of the point spread function, i.e. $W_E \subset W_h$. In this case, it is always possible to filter the raw images to transform $h$ into $C$ and the variance of the modified images gives access to the sample spatial frequencies in $W_{E2}$ at least. Yet, this result is not totally satisfactory. Indeed, in this case, the frequency support of the variance $W_{h2}$ is larger than the frequency support of the sample spectrum it depends on, $W_{hE}$. In other terms, we have more data than unknowns. It is likely that, with some approximations, the resolution can be improved further (maybe up to $W_{hE}$), but this remains to be demonstrated.

Last, we study the configurations where the speckle correlation length is smaller than the width of the point spread function, $W_h \subset W_E$. In this case, we have fewer data than unknowns ($W_{h2} \subset W_{hE}$), and we foresee major difficulties. Indeed, we show in appendix B, that there is a loss of information in the variance image that prevents the reconstruction of the sample spectrum in $W_{hE}$ and even in $W_{h2}$. In particular, two samples with different spectra in $W_{h2}$ can provide the same variance image. These issues can be reduced with a priori information on the sample and regularization techniques, but the resolution gain, if any, will not be universal. The ambiguity of the variance image when $W_h \subset W_E$ applies in particular to configurations where the speckles are assumed to be spatially uncorrelated. This assumption amounts to considering that the observation point spread function and the fluorescence density vary slowly over the speckle grain size. This is the case when near-field speckles are used together with far-field detection [18] or optical speckles with acoustic detection [19]. In these techniques, hereafter called speckle-SOFI, the expression of the variance simplifies to,

$$ V_{\mathrm{speckle-SOFI}}(\mathbf{r}) \approx C_0 \int h^2(\mathbf{r} - \mathbf{r}')\rho^2(\mathbf{r}')d\mathbf{r}'. \quad (8) $$

where $C_0 = \int C(\mathbf{r})d\mathbf{r}$. We observe that the variance is now linearly linked to the square of the sought parameter (optical absorption or fluorescence density) that is filtered over $W_{h2}$. Now, knowing the Fourier transform of $\rho^2$ in $W_{h2}$ does not mean that $\tilde{\rho}$ can be retrieved over $W_{h2}$ (except if $\rho$ is binary). We show in appendix B that, when the speckled illumination is spatially uncorrelated, it is possible to find samples with different fluorescence density spectra in $W_{h2}$ that have the same variance image.

## IV. RESOLUTION OF TECHNIQUES USING RANDOM ACTIVATION OF THE FLUOROPHORES (SOFI)

In this last section, we differentiate fluctuation imaging using quasi-uncorrelated speckled illuminations from Super-resolution Optical Fluctuation Imaging (SOFI). In SOFI, the intensity fluctuations observed in the recorded
images come from the random activation of the fluo-
rophores and not from the illumination (which is kept
homogeneous and equal to E₀ during the whole exper-
iment). To account for this phenomenon, one needs
to explicit further the characteristics of the fluorescence
density ρ which is related to the fluorophore concentra-
tion and the molecular brightness. We define the fluo-
rophore concentration g at the macroscopic scale such
that, Vg(r) is the number of fluorophores contained in
a macroscopic volume V centered about r. Next, we
introduce the mean molecular brightness b which ac-
counts for the fluorophores' quantum yield and for the
environment-dependent ability of the incident/emitted
photons to reach the fluorophore/detector. If all the flu-
orophores are activated in V, the mean brightness b is
defined such that Vg(r)b(r)E₀ is the energy measured
by the camera of the photons emitted from V. In other
terms, if all the fluorophores are activated, the fluores-
cence density is the product of the fluorophore concen-
tration times the mean brightness, ρ = g × b.

In SOFI, only a few fluorophores of V are activated
during the image recording and they change at each novel
image. Let us assume that they follow a Poisson point
process of intensity proportional to the total number of
fluorophores in V. Then, the number of activated flu-
orophores in V observed when recording one image be-
comes a Poisson variable of parameter Vg(r)p(r) where
p is the mean percentage of activation. Under this as-
sumption, we show in appendix D that the variance of
SOFI images reads,

$$
V_{\text{SOFI}}(\mathbf{r}) = E_0^2 \int h^2(\mathbf{r} - \mathbf{r}')b^2(\mathbf{r}')g(\mathbf{r}')p(\mathbf{r}')d\mathbf{r}'. \quad (9)
$$

While RIM is able to recover the fluorescence density
ρ = g × b over W<sub>h2</sub>, SOFI has a similar super-resolution
capacity, but the latter applies to a distinct density g ×
b<sup>2</sup> × p = ρ × b × p.

It is worth noting that if the mean brightness *b* is homogeneous, RIM and SOFI are able to restore the fluorophore concentration *g* over *W*<sub>*h*2</sub>, (provided the mean activation percentage *p* in SOFI is also homogeneous). On the contrary, even if *b* is homogeneous, fluctuation imaging using quasi-uncorrelated speckled illuminations (speckle-SOFI) can only restore the square of the fluorophore concentration, *g*<sup>2</sup>, over *W*<sub>*h*2</sub>. Thus, SOFI and speckle-SOFI yield a *priori* different results and their umbrella denomination as ‘fluctuation imaging’ can be misleading.

V. CONCLUSION

There exist three main super-resolved microscopy tech-
niques that form the super-resolved images from the vari-
ance of multiple diffraction-limited images. RIM uses
speckled illuminations that are correlated over a distance
comparable to the width of the observation point spread
function, speckle-SOFI uses 'quasi uncorrelated' speckled

illuminations and SOFI takes advantage of the random
activation of fluorophores. In this work, we have shown
that while these three techniques can offer a doubling of
resolution, this is not applied to the same sample param-
eters. SOFI recovers the fluorophore concentration times
the square of the brightness while Speckle-SOFI recov-
ers the square of the fluorophore concentration times the
square of the brightness. For its part, RIM recovers the
fluorescence density, i.e. the fluorophore concentration
times the brightness, in the same way as an ideal confocal
microscope. Our demonstration provides a solid theoret-
ical ground for the two-fold resolution gain, the optical
sectioning, and the linearity to fluorescence observed in
the last RIM experiments [7, 17].

Appendix A: Super-resolution capacity of RIM
when $W_E = W_h$

In this appendix, we consider the classical RIM config-
uration where the speckle auto-covariance C is similar to
the observation point spread function h. Experimentally,
this condition is fulfilled in epi-fluorescence microscopy
when the speckled illumination fills the pupil of the col-
lection objective and the Stokes shift is negligible. In
this case, we show that if two samples ρ₁ and ρ₂ have
the same variance image, they have the same spectrum
in W<sub>h2</sub>.

a. Demonstration of the bijection between the variance image and the sample spectrum in W<sub>h2</sub>

First of all, we recall that the variance is formed from
the square of images acquired under random illumina-
tions whose frequency support W<sub>E</sub> is equal to W<sub>h</sub>. From
Eq. (3) we deduce that the variance is only sensitive to
the sample spectrum in W<sub>hE</sub> = W<sub>h2</sub>. Thus, two samples
with the same spectrum in W<sub>h2</sub> will provide the same
variance image. The reciprocal is significantly more dif-
ficult to demonstrate due to the quadratic link between
the variance and the sample. It requires the introduction
of a bilinear symmetric operator B<sub>U,V</sub> acting on real in-
tegral functions (U, V),

$$
\begin{equation}
\begin{split}
B_{U,V}(\mathbf{r}) = \int & \mathrm{d}\mathbf{r}_1 \mathrm{d}\mathbf{r}_2 \\
& h(\mathbf{r} - \mathbf{r}_1)U(\mathbf{r}_1)C(\mathbf{r}_1 - \mathbf{r}_2)V(\mathbf{r}_2)h(\mathbf{r} - \mathbf{r}_2).
\end{split}
\tag{A1}
\end{equation}
$$

such that the variance $V_\rho$ is equal to $B_{\rho,\rho}$.

The bilinear operator $B_{U,V}$ can be cast as the product
of linear operators acting on $U$ and $V$. To this aim,
we define $h_E$ such that $\tilde{h}_E = \sqrt{\tilde{C}}$ (we recall that $\tilde{C}$ is
always positive as $C$ is an autocovariance function) which
satisfies

$$
\int h_E(\mathbf{r}_1 - \mathbf{x}) h_E(\mathbf{r}_2 - \mathbf{x}) d\mathbf{x} = C(\mathbf{r}_1 - \mathbf{r}_2). \quad (A2)
$$
Introducing $h_E$ in Eq. (A1), we obtain,

$$B_{U,V}(\mathbf{r}) = \int M_U(\mathbf{r}, \mathbf{x}) M_V(\mathbf{r}, \mathbf{x}) d\mathbf{x} \quad (A3)$$

with

$$M_V(\mathbf{r}, \mathbf{x}) = \int h(\mathbf{r} - \mathbf{r}_1) V(\mathbf{r}_1) h_E(\mathbf{r}_1 - \mathbf{x}) d\mathbf{r}_1. \quad (A4)$$

At this point, we note that the Fourier transform of $M_V$ with respect to $(\mathbf{r}, \mathbf{x})$,

$$\tilde{M}_V(\boldsymbol{\nu}, \boldsymbol{\mu}) = \tilde{h}(\boldsymbol{\nu}) \tilde{V}(\boldsymbol{\mu} + \boldsymbol{\nu}) \tilde{h}_E(\boldsymbol{\mu}), \quad (A5)$$

is bounded, so $M_V$ is an analytic function. In addition, if $h = C$, we demonstrate (at the end of this section) that,

$$\int B_{U,V}(\mathbf{r})V(\mathbf{r})d\mathbf{r} = \int |M_V(\mathbf{r}, \mathbf{x})|^2 U(\mathbf{r})d\mathbf{r}d\mathbf{x}. \quad (A6)$$

We now consider two fluorescence densities, $\rho_1(\mathbf{r}) \ge 0$ and $\rho_2(\mathbf{r}) \ge 0$ that have the same RIM variance, $B_{\rho_1, \rho_1}(\mathbf{r}) = B_{\rho_2, \rho_2}(\mathbf{r})$. Using the bilinearity and symmetry of $B_{U,V}$, we can show that $B_{\rho_1, \rho_1} - B_{\rho_2, \rho_2} = B_{\rho_1+\rho_2, \rho_1-\rho_2} = 0$. This last property implies, in particular, that,

$$\int B_{\rho_1+\rho_2, \rho_1-\rho_2}(\mathbf{r})[\rho_1(\mathbf{r}) - \rho_2(\mathbf{r})]d\mathbf{r} = 0. \quad (A7)$$

which, using Eq. (A6), can be cast as,

$$\int |M_{\rho_1-\rho_2}(\mathbf{r}, \mathbf{x})|^2 [\rho_1(\mathbf{r}) + \rho_2(\mathbf{r})] d\mathbf{r} d\mathbf{x} = 0. \quad (A8)$$

We now assume that $\rho_1 + \rho_2$ stays strictly positive in a non-empty open set $\Omega$. In this case, Eq. (A8) is satisfied if and only if $M_{\rho_1-\rho_2}(\mathbf{r}, \mathbf{x}) = 0$ for $\mathbf{r} \in \Omega$ and for all $\mathbf{x}$. Since $M_{\rho_1-\rho_2}$ is analytic, $M_{\rho_1-\rho_2}(\mathbf{r}, \mathbf{x}) = 0$ for all $\mathbf{x}$ and for $\mathbf{r} \in \Omega$ implies that $M_{\rho_1-\rho_2}(\mathbf{r}, \mathbf{x}) = 0$ for all $\mathbf{x}$ and all $\mathbf{r}$, thus $\tilde{M}_{\rho_1-\rho_2}(\boldsymbol{\nu}, \boldsymbol{\mu}) = 0$ for all $\boldsymbol{\nu}$ and $\boldsymbol{\mu}$. From Eq. (A5), the nullity of $\tilde{M}_{\rho_1-\rho_2}$ is obtained only if $\tilde{\rho}_1(\boldsymbol{\eta}) - \tilde{\rho}_2(\boldsymbol{\eta}) = 0$ for $\boldsymbol{\eta} \in W_{h^2}$. Hence, if $\rho_1$ and $\rho_2$ have the same RIM variance, they have the same spatial frequencies in $W_{h^2}$.

We have thus demonstrated that there is a one-to-one correspondence between the spatial frequencies of the variance of diffraction-limited speckled images and the spatial frequencies of the sample fluorescence density in the enlarged frequency domain $W_{h^2}$ provided $C = h$.

b. *Proof of Equation (A6)*

The demonstration of Equation (A6) requires that $h = C$. In this case, Eq. (A1) can be rewritten as,

$$B_{U,V}(\mathbf{r}) = \int d\mathbf{r}_1 d\mathbf{r}_2 \\ h(\mathbf{r} - \mathbf{r}_1) U(\mathbf{r}_1) h(\mathbf{r} - \mathbf{r}_2) V(\mathbf{r}_2) h(\mathbf{r}_1 - \mathbf{r}_2).$$

Using $h(\mathbf{r} - \mathbf{r}_2) = \int h_E(\mathbf{r} - \mathbf{x}) h_E(\mathbf{r}_2 - \mathbf{x}) d\mathbf{x}$, one obtains

$$\int B_{U,V}(\mathbf{r}) V(\mathbf{r}) d\mathbf{r} = \int d\mathbf{r} d\mathbf{r}_1 d\mathbf{r}_2 d\mathbf{x} \\ h(\mathbf{r} - \mathbf{r}_1) V(\mathbf{r}) h_E(\mathbf{r} - \mathbf{x}) h(\mathbf{r}_1 - \mathbf{r}_2) V(\mathbf{r}_2) h_E(\mathbf{r}_2 - \mathbf{x}) U(\mathbf{r}_1).$$

Recalling the expression of $M_V$ in Eq. (A4) and the symmetry of $h$ and $h_E$, we get

$$\int B_{U,V}(\mathbf{r}) V(\mathbf{r}) d\mathbf{r} = \int |M_V(\mathbf{r}_1, \mathbf{x})|^2 U(\mathbf{r}_1) d\mathbf{x} d\mathbf{r}_1.$$

Appendix B: What happens when $W_h \subset W_E$?

We now study the configurations where $W_h \subset W_E$, namely the speckle correlation length is smaller than the width of the point spread function. This situation is encountered with near-field speckles or when the illumination pupil is larger than the collection pupil. In this case, we show that even if the variance is sensitive to the sample spectrum in $W_{h_E}$, there is a loss of information that prevents its recovery. In addition, we show that two samples with different spectra in $W_{h^2}$ can provide the same variance image.

To simplify the discussion, we assume that $(W_h, W_E)$ are centered plain disks with frequency cut-offs $\nu_h$ and $\nu_E$ respectively, with $\nu_h < \nu_E$. We further assume that the point spread function $h$ is symmetric so that $\tilde{h}$ is a real positive symmetric function, like $\tilde{C}$. We consider a sample whose spectrum is restricted to the null frequency and a high frequency $\pm \mathbf{k}$, $\rho(\mathbf{r}) = A + B \cos(2\pi \mathbf{k} \cdot \mathbf{r} + \varphi)$ with $(A, B)$ real positive such that $\rho$ is real positive. The variance of the raw images, given by Eq. (7), obtained with such sample reads,

$$V_{\rho}(\mathbf{r}) = A^2 \alpha + B^2 \beta(\mathbf{k}) + 2AB\gamma(\mathbf{k}) \cos(2\pi\mathbf{k} \cdot \mathbf{r} + \varphi) \\ + B^2 \eta(\mathbf{k}) \cos(4\pi\mathbf{k} \cdot \mathbf{r} + 2\varphi), \quad (B1)$$

with

$$\alpha = \int |\tilde{h}|^2(\boldsymbol{\nu}) \tilde{C}(\boldsymbol{\nu}) d\boldsymbol{\nu}, \\ \beta(\mathbf{k}) = \int |\tilde{h}|^2(\boldsymbol{\nu} + \mathbf{k}) \tilde{C}(\boldsymbol{\nu}) d\boldsymbol{\nu}, \\ \gamma(\mathbf{k}) = \int \tilde{h}(\mathbf{k} - \boldsymbol{\nu}) \tilde{h}(\boldsymbol{\nu}) \tilde{C}(\boldsymbol{\nu}) d\boldsymbol{\nu}, \\ \eta(\mathbf{k}) = \int \tilde{h}(\boldsymbol{\nu}) \tilde{h}(2\mathbf{k} - \boldsymbol{\nu}) \tilde{C}(\mathbf{k} - \boldsymbol{\nu}) d\boldsymbol{\nu}.$$

We observe that, as long as $k < \nu_h + \nu_E$, $\beta(\mathbf{k}) \neq 0$ and the variance depends on the high spatial frequency of the sample, $B$. This result confirms the sensitivity of the variance to sample spatial frequencies in $W_{h_E}$. However, if $2\nu_h < k \le \nu_h + \nu_E$, $\gamma(\mathbf{k}) = \eta(\mathbf{k}) = 0$ so that $V_{\rho}(\mathbf{r}) = \alpha A^2 + \beta(\mathbf{k})B^2$. In this case, the variance is sensitive to the amplitudes of the null and high frequencies of the sample, $(A, B)$, but it has lost the information about the phase
of the high frequency, $\varphi$. Worse, this example shows that a uniform sample defined by $\rho_1(\mathbf{r}) = (A^2 + \beta(\mathbf{k})B^2)^{\frac{1}{2}}/\alpha$ will have the same variance as the inhomogeneous sample defined by $\rho(\mathbf{r}) = A+B\cos(2\pi\mathbf{k}\cdot\mathbf{r}+\varphi)$. Thus, when $\nu_h < \nu_E$, the identifiability of the sample spatial frequencies from the variance is lost, even for frequencies belonging to $W_{h^2}$. This assertion is particularly counter-intuitive as it shows that decreasing the size of the speckle grains below the width of the observation point spread function is a priori detrimental to the sample reconstruction.

## Appendix C: Imaging with quasi-uncorrelated speckles, speckle-SOFI

When the point spread function and fluorescence density vary slowly over the width of the speckle autocovariance function, the variance of the diffraction-limited images is linearly linked to the square of $\rho$ filtered over $W_{h^2}$. In this section, we provide an example of two positive functions with different spectra in the super-resolved domain $W_{h^2}$ which, when squared, have exactly the same spectra in $W_{h^2}$.

We consider $g$ the sum of a constant and a one-dimensional cosine along the x-axis with a frequency $k$ laying in $W_{h^2}$ but not in $W_h$, and $f$ the sum of a constant and two cosines with period $k$ and $2k$. Note that $2k$ lays outside $W_{h^2}$. We adapt the constant and the cosine amplitudes so that $f$ and $g$ are positive and $f^2$ and $g^2$ are equal in $W_{h^2}$. A possible solution is,

$$ 
\begin{aligned}
f(x) &= 6 + \sqrt{2} \cos(2\pi kx) + \cos(4\pi kx) \\
g(x) &= \frac{\sqrt{101} + 7}{2\sqrt{2}} + \frac{\sqrt{101} - 7}{2} \cos(2\pi kx)
\end{aligned}
\quad (C1) 
$$

Noting $\mathcal{F}$ the low pass filter operator that removes all the frequencies outside $W_{h^2}$, we find,

$$ \mathcal{F}[f^2](x) = \mathcal{F}[g^2](x) = 75 + 13\sqrt{2} \cos(2\pi kx). $$

## Appendix D: Modeling SOFI at the macroscopic scale

Generally, SOFI data are modeled with a discrete sum that depends on the fluorophore positions [20]. However,

it is clearly impossible to recover the fluorophores positions from SOFI (second-order) image, except if a constraint of sparsity is assumed. It may thus be interesting to relate SOFI images to sample characteristics that are defined at a macroscopic scale, such as the fluorophore concentration.

In structured illumination microscopy, one assumes that all the fluorophores are activated. The fluorescence density $\rho$ is written as the product of the fluorophore concentration $g$ with a mean brightness $b$. The intensity recorded by the camera is modeled as,

$$ I(\mathbf{r}) = \int h(\mathbf{r} - \mathbf{r}')E(\mathbf{r}')g(\mathbf{r}')b(\mathbf{r}')d\mathbf{r}, \quad (D1) $$

where $E$ is the inhomogeneous illumination intensity and $h$ the microscope point spread function.

In SOFI, the illumination $E_0$ is homogeneous but the fluorophores oscillate between an activated and non-activated state. Thus, only a subset of the fluorophores present in the sample contributes to the image intensity at a given time $t$. The activated fluorophores in the (macroscopic) volume $V$ centered about $\mathbf{r}$ can be seen as points popping up at random and independently of one another. This process is conveniently modeled with a Poisson point process of intensity proportional to the number of fluorophores in $V$. Under this assumption, the number of activated fluorophores in $V$ at time $t$, written as $Vq(\mathbf{r}, t)$, where $q$ is the activated fluorophore concentration, is a Poisson variable of parameter $Vg(\mathbf{r})p(\mathbf{r})$ with $p$ the mean percentage of activation. With this definition, the intensity of the image recorded at $t$ can be written as,

$$ I(\mathbf{r}, t) = E_0 \int h(\mathbf{r} - \mathbf{r}')q(\mathbf{r}', t)b(\mathbf{r}')d\mathbf{r}'. \quad (D2) $$

It is thus a filtered Poisson variable whose time variance reads [21, Chap. 5],

$$ V_{\text{SOFI}}(\mathbf{r}) = E_0^2 \int h^2(\mathbf{r} - \mathbf{r}')b^2(\mathbf{r}')g(\mathbf{r}')p(\mathbf{r}')d\mathbf{r}'. $$

Note that this result recovers the expression given in the original SOFI paper [20] if one expresses the fluorophore concentration distribution as $\sum_{k=1}^K \delta(\mathbf{r} - \mathbf{r}_k)$ where $\mathbf{r}_k$ is the position of the $k^{th}$ fluorophore.

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