Christoph Florian Schaller - FU Berlin, FB MI

Christoph Schaller - STORMicroscopy 10
we can write
f ′ (x n ) = ∑
N
√ (p i+ − p i− ) 2 + ∑ (C i − Ne i+ + Ne i− ) 1 (
2πσ σ 2 (i + 1 2 − x n)p i+ − (i − 1 )
2 − x n)p i−
and
f(x n ) = ∑ (C i − Ne i+ + Ne i− )(p i+ − p i− ).
Finally we need some way of calculating the values e i± = 1 2 erf( i± 1 2 −x0
σ √ ). This can be done by one of
2
several approximations for the error function according to the needed accuracy. In MATLAB the buildin
function erf(x) performs eciently with relative errors of order 10 −19 as it is an implementation of
the algorithm given by W. Cody in [12]. As we obtain an analogous relation g(y n ) in y-direction one
can iterate
x n+1 = x n − f(x n)
f ′ (x n ) , y n+1 = y n − g(y n)
g ′ (y n ) ,
starting o with (x 0 , y 0 ) located in the pixel center.
But as f(x n ), f ′ (x n ) and g(y n ), g ′ (y n ) are still N-dependant, this needs to be done in turn with
the weighted sum
N =
∑
Sij P ij (x n , y n )
∑
Pij (x n , y n ) 2 ,
where P ij (x n , y n ) is the probability to hit pixel (i, j) from center (x n , y n ) and consequently
P ij =
ˆ i+ 1
2
i− 1 2
ˆ j+ 1
2
j− 1 2
p G (x, y)dy dx = 1 4
[
erf( x − x ] i+ 1
n
σ √ 2 ) 2
i− 1 2
[
erf( y − y ] j+ 1
n
σ √ 2 ) 2
j− 1 2
= (e i+,x −e i−,x )(e i+,y −e i−,y ).
Nonetheless the required values of the error function are the same ones required for the iterations of
the pixel center and thus only need to be calculated once.
All in all this provides a method using no approximations apart from the calculation of the error
functions which can be done to whatever precision needed.
2.4 Poissonian background tting
After avoiding approximations in the tting algorithm itself we now want to have a closer look at the
background noise and how it is treated. We recall that background noise was estimated in a rather
simple fashion so far. Obviously subtracting a constant value from every pixel does not represent the
reality as the background noise is a random process following a certain distribution. As stated in [10]
and universally accepted, the background noise can be seen as a Poisson process and therefore every
pixel value should stem from the same Poisson distribution. Thus we want to include tting a noise
value for every pixel within a spot.
Assume a matrix K ij of observed photon counts including background noise is given. Now we
introduce a matrix b ij , which is meant to contain the number of photons most probably steming from
the background. Finally G ij denotes the matrix of the currently tted Gaussian distribution and b
the average background noise value.
Thence the probability of an observation is
P ij = GKij−bij
e −Gij
(K ij − b ij )!
} {{ }
P P SF
· bbij e −b
.
b ij !
} {{ }
P back
Here P P SF is the probability of observing K ij − b ij photons from the distribution G ij and P back