Christoph Florian Schaller - FU Berlin, FB MI

Christoph Schaller - STORMicroscopy 26
Here M rot is an arbitrary rotation matrix and as already mentioned −→ x n does not denote the
localization of a spot in frame n, but a mean localization of the surrounding frames. Now every
assigned pair ( −→ x n , x
−−→
n+1 ) provides two equations to estimate the six parameters
−→ v =
(
)
v 1
v 2
and M rot =
(
)
m 1 m 2
.
m 3 m 4
of the drift −−→ x n+1 − −→ x n . Hence we need to locate at least three beads in the surrounding frames. In case
of more localizations we calculate the linear least squares t since the drift is no longer parameterdependant
in a non-linear way.
At this point we only have to describe how to obtain such a t. Assume i ∈ N equations
a i,1 λ 1 + a i,2 λ 2 + ... + a i,k λ k = b i
for known right hand sides b i and factors a i,k are given and we want to estimate the k ∈ N unknowns
λ k ∈ R. This can be accomplished by minimizing the error in the Euclidean norm, i.e.
||b − Aλ|| 2 ≤ ||b − Av|| 2 ∀v ∈ R k .
Now according to [18], Theorem 2.14, λ ∈ R k is a minimizer, if and only if
A T Aλ = A T b (5.2)
Furthermore it is unique if A is injective.
In our case the right hand sides are the occuring drifts, the unknowns are the free parameters and
the respective factors follow from (5.1):
(
) (
b 2i
:=
b 2i+1
) (
x n+1,i
−
y n+1,i
λ := (v 1 , v 2 , m 1 , m 2 , m 3 , m 4 ),
a 2i := (1, 0, x n,i , y n,i , 0, 0) and
a 2i+1 := (0, 1, 0, 0, x n,i , y n,i ).
)
x n,i
,
y n,i
As we will not locate two spots in the exact same place A is obviously injective and thus a unique
minimizer is given by (5.2), which can be evaluated using basic linear algebra.