Biomedical Engineering Reference
In-Depth Information
As this estimator is convex in o (
x
), the global minimum for o (
x
) can be obtained
∇J obs ( o (
x
)) w.r.t o is null. This leads to solving the
at the point where the gradient
following Euler-Lagrange expression
i (
x
)
∗ h (
1
x
)=0 .
(4.17)
( h ∗ o )(
x
)+ b (
x
)
In an iterative form, the estimation can be written as:
)=
)
i (
x
)
o ( n +1) (
∗ h (
o ( n ) (
x
x
·
x
) .
(4.18)
( h ∗
o ( n ) )(
x
)+ b (
x
)
This RL algorithm given in Algorithm 5 , for the Poisson statistics, is a form of the
Expectation-maximization (EM) algorithm for Maximum likelihood (ML). The first
estimate in this iterative procedure is usually set to either the observed image/mean
of the observation or a smoothed version of the observation. We also notice in
Eq. ( 4.18 ) that if the initial estimate o (0) (
x
) is positive, the successive estimates
remain positive as well.
The earliest application of this algorithm was in astronomy and later it was used
in biological image processing as discussed by Holmes in [ 36 ]. As the iterative
algorithms are time consuming, Biggs et al. showed in [ 7 ] a simple way to accelerate
this. Like the previous direct inversion, this iterative algorithm does not allow
reconstruction of information outside the spatial frequency bandwidth. However as
it includes a positivity constraint, it is less ill-posed than the previous inversion
(in the Gaussian noise case). Moreover, as it is an iterative process, it can be
stopped before convergence which prevents noise amplification. Undeniably, the
RL algorithm is the most popular deconvolution algorithm for microscopy, but as
the inversion process is ill-posed, when n →∞
, the solution diverges due to noise
amplification (ringing artefact). An ad hoc approach to avoid the divergence of the
solution is to manually terminate the algorithm at a certain number of iterations.
Algorithm 5 The RL deconvolution algorithm
Input: Observation i ( x ) x ∈ Ω s , background b ( x ) 0 , criterion ε ∈ R
+ .
Output: Restored specimen o ( x ) .
1: Calculate PSF h ( x ) ∈O (Eq. 4.5 ),
2: Initialize: n ← 0 , o
( n ) ( x ) Mean( i ( x )) .
( n )
( n− 1)
( n )
3: while |o
− o
|/o
≥ ε do
( n +1) ( x ) by Eq. ( 4.18 ). }
4:
{ Deconvolve: o
5:
{ Sub-space projection (scale): o
( n +1) ( x ) for flux preservation Eq. ( 4.9 ) }
6:
{ Set: o
( n ) ( x ) ← o
( n +1) ( x ) and n ← ( n +1). }
7: end while
 
Search WWH ::




Custom Search