### Theory and algorithm

SOFI is based on the computation of temporal cumulants or spatio-temporal cross-cumulants. Cumulants are a statistical measure related to moments. Because cumulants are additive, the cumulant of a sum of independently fluctuating fluorophores corresponds to the sum of the cumulant of each individual fluorophore. This leads to a point-spread function raised to the power of the cumulant order *n* and therefore a resolution improvement of
\sqrt{n}, respectively almost *n* with subsequent Fourier filtering (Dertinger et al.
2010). So far, SOFI has been used exclusively to improve structural details in an image (Dertinger et al.
2009; Dertinger et al.
2010). Information about the on-time ratio, the molecular brightness and the concentration has to our knowledge never been exploited before and therefore represents a new potential for super-resolved imaging.

In the most general sense, the cumulant of order *n* of a pixel set
\mathbb{P}=\{{\overrightarrow{r}}_{1},{\overrightarrow{r}}_{2},\dots ,{\overrightarrow{r}}_{n}\} with time lags
\overrightarrow{\tau}=\{{\tau}_{1},{\tau}_{2},\dots ,{\tau}_{n}\} can be calculated as (Leonov and Shiryaev
1959)

\begin{array}{l}{\kappa}_{n}\left(\overrightarrow{r}=\frac{1}{n}\sum _{i=1}^{n}{\overrightarrow{r}}_{i};\overrightarrow{\tau}\right)=\sum _{P}{(-1)}^{\left|P\right|-1}(\left|P\right|-1)\\ \phantom{\rule{1em}{0ex}}\prod _{p\in P}{\u3008\prod _{i\in p}I(\phantom{\rule{0.3em}{0ex}}{\overrightarrow{r}}_{i},t-{\tau}_{i})\u3009}_{t},\end{array}

(1)

where
{\left.\right)"\; close=">">\dots}_{}\n \n t\n \n stands for averaging over the time *t*. *P* runs over all partitions of a set
\mathbb{S}=\{1,2,\dots ,n\}, which means all possible divisions of
\mathbb{S} into non-overlapping and non-empty subsets or parts that cover all elements of
\mathbb{S}. |*P*| denotes the number of parts of partition *P* and *p* enumerates these parts.
I({\overrightarrow{r}}_{i},t) is the intensity distribution measured over time on a detector pixel
{\overrightarrow{r}}_{i}. We used the cross-cumulant approach without repetitions to increase the pixel grid density and eliminate any bias arising from noise contributions in auto-cumulants (Geissbuehler et al.
2011). A 4x4 neighborhood around every pixel was considered to compute all possible *n*-pixel combinations excluding pixel repetitions. For computational reasons, we kept only a single combination featuring the shortest sum of distances with respect to the corresponding output pixel
\overrightarrow{r}=\frac{1}{n}\sum _{i=1}^{n}{\overrightarrow{r}}_{i}. For even better signal-to-noise ratios, it would be beneficial to average over multiple combinations per output pixel. The heterogeneity in output pixel weighting arising from the different pixel combinations has been accounted for by the distance factor as described in (Dertinger et al.
2010).

Considering a sample composed of *M* independently fluctuating fluorophores and assuming a simple two-state blinking model (with characteristic lifetimes *τ*
_{on},*τ*
_{off}) with slowly varying molecular parameters compared to the size of the point-spread function (PSF), the cumulant of order *n* without time-lags can be interpreted as

\begin{array}{l}{\kappa}_{n}\left(\phantom{\rule{0.3em}{0ex}}\overrightarrow{r}\right)\propto \sum _{k=1}^{M}{\epsilon}_{k}^{n}{U}^{n}(\overrightarrow{r}-{\overrightarrow{r}}_{k}){f}_{n}\left({\rho}_{\mathrm{on},k}\right)\\ \phantom{\rule{3em}{0ex}}\approx {\epsilon}^{n}\left(\overrightarrow{r}\right){f}_{n}({\rho}_{\mathrm{on}};\overrightarrow{r})\sum _{k=1}^{M}{U}^{n}(\overrightarrow{r}-{\overrightarrow{r}}_{k})\end{array}

(2)

with
\epsilon \left(\overrightarrow{r}\right) the spatial distribution of the molecular brightness and
{\rho}_{\mathrm{on}}\left(\overrightarrow{r}\right)=\frac{{\tau}_{\mathrm{on}}\left(\overrightarrow{r}\right)}{{\tau}_{\mathrm{on}}\left(\overrightarrow{r}\right)+{\tau}_{\text{off}}\left(\overrightarrow{r}\right)} the on-time ratio.
U\left(\overrightarrow{r}\right) is the system’s PSF and
{f}_{n}({\rho}_{\mathrm{on}};\overrightarrow{r}) is the *n*-th order cumulant of a Bernoulli distribution with probability *ρ*
_{on}:

\begin{array}{l}{f}_{1}({\rho}_{\mathrm{on}};\overrightarrow{r})={\rho}_{\mathrm{on}}\\ {f}_{2}({\rho}_{\mathrm{on}};\overrightarrow{r})={\rho}_{\mathrm{on}}(1-{\rho}_{\mathrm{on}})\\ \phantom{\rule{3em}{0ex}}\phantom{\rule{1.25em}{0ex}}\vdots \\ {f}_{n}({\rho}_{\mathrm{on}};\overrightarrow{r})={\rho}_{\mathrm{on}}(1-{\rho}_{\mathrm{on}})\frac{\partial {f}_{n-1}}{\partial {\rho}_{\mathrm{on}}}\end{array}

(3)

Assuming a uniform spatial distribution of molecules inside a detection volume *V* centered at
\overrightarrow{r}, we may further approximate

\sum _{k=1}^{M}{U}^{n}(\overrightarrow{r}-{\overrightarrow{r}}_{k})\approx {\mathcal{\mathcal{E}}}_{V}\left\{{U}^{n}\left(\overrightarrow{x}\right)\right\}N\left(\overrightarrow{r}\right),

(4)

where
{\mathcal{\mathcal{E}}}_{V}\left\{{U}^{n}\left(\overrightarrow{x}\right)\right\}=1/V\underset{V}{\int}{U}^{n}\left(\overrightarrow{x}\right)d\overrightarrow{x} is the expectation value of
{U}^{n}\left(\overrightarrow{x}\right) or the *n*-th moment of
U\left(\overrightarrow{x}\right) (see (Kask et al.
1997) for some examples) and
N\left(\overrightarrow{r}\right) denotes the number of molecules within the detection volume *V* . Finally, we can write

{\kappa}_{n}\left(\overrightarrow{r}\right)\approx {\mathcal{\mathcal{E}}}_{V}\left\{{U}^{n}\left(\overrightarrow{x}\right)\right\}N\left(\overrightarrow{r}\right){\epsilon}^{n}\left(\overrightarrow{r}\right){f}_{n}({\rho}_{\mathrm{on}};\overrightarrow{r}).

(5)

Based on at least three different cumulant orders and approximation (5), it is possible to extract the molecular parameter maps
\epsilon \left(\overrightarrow{r}\right),
N\left(\overrightarrow{r}\right) and
{\rho}_{\mathrm{on}}\left(\overrightarrow{r}\right) by solving an equation system, or by using a fitting procedure. For example, the cumulant orders two to four can be used to build the ratios

\begin{array}{ll}{K}_{1}\left(\overrightarrow{r}\right)& =\frac{{\mathcal{\mathcal{E}}}_{V}\left\{{U}^{2}\left(\overrightarrow{x}\right)\right\}{\kappa}_{3}}{{\mathcal{\mathcal{E}}}_{V}\left\{{U}^{3}\left(\overrightarrow{x}\right)\right\}{\kappa}_{2}}\left(\overrightarrow{r}\right)\\ =\epsilon \left(\overrightarrow{r}\right)\left(1-2{\rho}_{\mathrm{on}}\left(\overrightarrow{r}\right)\right)\\ {K}_{2}\left(\overrightarrow{r}\right)& =\frac{{\mathcal{\mathcal{E}}}_{V}\left\{{U}^{2}\left(\overrightarrow{x}\right)\right\}{\kappa}_{4}}{{\mathcal{\mathcal{E}}}_{V}\left\{{U}^{4}\left(\overrightarrow{x}\right)\right\}{\kappa}_{2}}\left(\overrightarrow{r}\right)\\ ={\epsilon}^{2}\left(\overrightarrow{r}\right)\left(1-6{\rho}_{\mathrm{on}}\left(\overrightarrow{r}\right)+6{\rho}_{\mathrm{on}}^{2}\left(\overrightarrow{r}\right)\right)\end{array}

(6)

and to solve for the molecular brightness

\epsilon \left(\overrightarrow{r}\right)=\sqrt{3{K}_{1}^{2}\left(\overrightarrow{r}\right)-2{K}_{2}\left(\overrightarrow{r}\right)},

(7)

the on-time ratio

{\rho}_{\text{on}}\left(\overrightarrow{r}\right)=\frac{1}{2}-\frac{{K}_{1}\left(\overrightarrow{r}\right)}{2\epsilon \left(\overrightarrow{r}\right)},

(8)

and the molecular density

N\left(\overrightarrow{r}\right)=\frac{{\kappa}_{2}\left(\overrightarrow{r}\right)}{{\epsilon}^{2}\left(\overrightarrow{r}\right){\rho}_{\text{on}}\left(\overrightarrow{r}\right)\left(1-{\rho}_{\text{on}}\left(\overrightarrow{r}\right)\right)}.

(9)

The spatial resolution of the estimation is limited by the lowest order cumulant, i.e. the second order in this case. However, the presented solution is not unique. Basically any three distinct cumulant orders could have provided a solution. Furthermore, the method is not limited to a two-state system; it can be extended to more states as long as the differences in fluorescence emission are detectable. Additional details on the analytical developments as well as a theoretical investigation of the estimation accuracy of the different parameters under different conditions are given in the Additional file
1.

In order to correct for the amplified brightness and blinking heterogeneities without compromising the resolution, the cumulants have to be deconvolved first. For this purpose, we used a Lucy-Richardson algorithm (Lucy
1974; Richardson
1972) implemented in MATLAB (deconvlucy, The MathWorks, Inc.), which is an iterative deconvolution without regularization that computes the most likely object representation given an image with a known PSF and assuming Poisson distributed noise. Apart from the estimate of the cumulant PSF and the specification of a maximum of 100 iterations, all arguments have been left at their default values. Assuming a perfect deconvolution without regularization, the result could be interpreted as

{\stackrel{\u0306}{\kappa}}_{n}\left(\overrightarrow{r}\right)\approx {\epsilon}^{n}\left(\overrightarrow{r}\right){f}_{n}({\rho}_{\mathrm{on}};\overrightarrow{r})\sum _{k=1}^{M}\delta (\overrightarrow{r}-{\overrightarrow{r}}_{k}).

(10)

Taking then the *n*-th root linearizes the brightness response without cancelling the resolution improvement of the cumulant. To reduce the amplified noise and masking residual deconvolution artefacts, small values (typically 1-5% of the maximum value) are truncated and the image is reconvolved with
U\left(n\overrightarrow{r}\right). This leads to a final resolution improvement of almost *n*-fold compared to the diffraction-limited image, which is physically reasonable since the frequency support of the cumulant-equivalent optical transfer function (OTF) is *n*-times the support of the system’s OTF (cf. (Dertinger et al.
2010)). In contrast to Fourier reweighting (Dertinger et al.
2010), which is equivalent to a Wiener filter deconvolution and reconvolution with
U\left(n\overrightarrow{r}\right), we explicitly split these two steps and use an improved but computationally more expensive deconvolution algorithm that is appropriate for the subsequent linearization.

Since the cumulants are proportional to
{f}_{n}({\rho}_{\mathrm{on}};\overrightarrow{r}), which contains *n* roots for
{\rho}_{\mathrm{on}}\in \left[0,1\right], there might still be hidden image features in these brightness-linearized cumulants (result after deconvolution, *n*-th root and reconvolution with
U\left(n\overrightarrow{r}\right)). However, using the on-ratio map
{\rho}_{\mathrm{on}}\left(\overrightarrow{r}\right), it is straightforward to identify the structural gaps around the roots of *f*
_{
n
} and fill them in with the brightness-linearized cumulant of order n-1. To this end, the locations where *f*
_{
n
} approaches zero are translated into a weighting mask with smoothed edges around these locations. The thresholds have been defined by computing the crossing points of
{\left|{f}_{n}\right|}^{1/n} and
{\left|{f}_{n-1}\right|}^{1/(n-1)}. This mask is then applied on the *n*-th order brightness-linearized cumulant and its negation is applied on order n-1 (see Additional file
1 for further details). The result is a balanced cumulant image. It should be noted that the cancellation of
{f}_{n}\left(\overrightarrow{r}\right) by division is possible but not recommended, because it amplifies noisy structures in the vicinity of the roots of *f*
_{
n
}. The combination of multiple cumulant orders in a balanced cumulant image results in a better overall image quality. However, it is also possible that the on-ratio varies only slightly throughout the field of view, such that a combination of multiple cumulant orders is not necessary. Figure
1 illustrates the different steps of the algorithm based on a simulation of randomly blinking fluorophores, arranged in a grid of increasing density from right to left, increasing brightness from left to right and increasing on-time ratio from top to bottom.

### Experiments

In order to verify the concept experimentally, we used a custom-designed objective-type total internal reflection (TIR) fluorescence microscope with a high-NA oil-immersion objective (Olympus, APON 60XOTIRFM, NA 1.49, used at 100x magnification), blue (490nm, 8mW, epi-illumination) and red (632nm, 30mW, TIR illumination) laser excitation and an EMCCD detector (Andor Luca S). We imaged fixed HeLa cells with Alexa647-labelled alpha tubulin and used a chemical buffer containing cysteamine and an oxygen-scavenging system (Heilemann et al.
2008) (see Additional file
1 for further details) to generate reversible blinking and an increased bleaching stability. The blue laser excitation was used to accelerate the reactivation of dark fluorophores and to reduce the acquisition time. For data processing, 5000 images acquired at 69 frames per second (fps) were divided into 10 subsequences significantly shorter than the bleaching lifetime to avoid correlated dynamics among the fluorophores (Dertinger et al.
2010). The final bSOFI images are obtained by averaging over the processed subsequences.