Cross-cumulants Super-resolution optical fluctuation imaging

principles of sofi cross-cumulant calculation , distance-factor: (a) 4th-order cross-cumulant calculation combinations repetitions . (b) distance-factor decay along arrows.

in more advanced approach cross-cumulants calculated taking information of several pixels account. cross-cumulants can described follows:

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{\displaystyle cc_{n}({\vec {r}},\tau _{1\ldots n-1}=0)=\prod _{j<l}^{n}u{\bigg (}{\frac {{\vec {r}}_{j}-{\vec {r}}_{l}}{\sqrt {n}}}{\bigg )}\cdot \sum _{i=1}^{n}u^{n}{\bigg (}{\vec {r}}_{i}-{\frac {\sum _{k}^{n}{\vec {r}}_{k}}{n}}{\bigg )}\varepsilon _{i}^{n}w_{i}(0)}

j, l , k indices contributing pixels whereas index current position. other values , indices used before. major difference in comparison of equation equation auto-cumulants appearance of weighting-factor



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{\displaystyle u(r_{j}-r_{l}/{\sqrt {n}})}

. weighting-factor (also termed distance-factor) psf-shaped , depends on distance of cross-correlated pixels in sense contribution of each pixels decays along distance in psf-shaped manner. in principle means distance-factor smaller pixels further apart. cross-cumulant approach can used create new, virtual pixels revealing true information labelled specimen reducing effective pixel size. these pixels carry more information pixels arise simple interpolation.

in addition cross-cumulant approach can used estimate psf of optical system making use of intensity differences of virtual pixels due loss in cross-correlation aforementioned. each virtual pixel can re-weighted inverse of distance-factor of pixel leading restoration of true cumulant value. @ last psf can used create resolution dependency of n nth-order cumulant re-weighting optical transfer function (otf). step can replaced using psf deconvolution associated less computational cost.

cross-cumulant calculation requires usage of computational more expensive formula comprises calculation of sums on partitions. of course owed combination of different pixels assign new value. hence no fast recursive approach usable @ point. calculation of cross-cumulants following equation can used:

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{\displaystyle k_{n}{\bigg (}{\vec {r}}={\frac {1}{n}}\sum _{i=1}^{n}{\vec {r_{i}}}{\bigg )}=\sum _{p}(-1)^{|p|-1}(|p|-1)!\prod _{p\in p}{\big \langle }\prod _{i\in p}f({\vec {r}})_{i}{\big \rangle }_{t}}

in equation p denotes amount of possible partitions, p denotes different parts of each partition. in addition index different pixel positions taken account during calculation f image stack of different contributing pixels. cross-cumulant approach facilitates generation of virtual pixels depending on order of cumulant mentioned. these virtual pixels can calculated in particular pattern original pixels 4th-order cross-cumulant image, depicted in lower image, part a. pattern arises simple calculation of possible combinations of original image pixels a, b, c , d. here done scheme of combinations repetitions . virtual pixels exhibit loss in intensity due correlation itself. part b of second image depicts general dependency of virtual pixels on cross-correlation. restore meaningful pixel values image smoothed routine defines distance-factor each pixel of virtual pixel grid in psf-shaped manner , applies inverse on image pixels related same distance-factor.