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This joint energy functional is then minimised by an alternating minimisation. In the PSF estimation
step,Rh(h) is representedasaquadraticform,Rh(h)= ∑
i,j,i′,j′Hi,j;i′,j′hi,jhi′,j′,with thecoefficient
matrix (Hessian)H=(Hi,j;i′,j′)i,j,i′,j′ givenby
H= sxsy∑
k=1 A vk(L(f))
mx,my TA vk(L(f))
mx,my
σk(L(f))2 , (5)
and this iscombinedwith thedata termfrom(3) toestablishaquadraticminimisationproblemforh.
In our re-implementation of the PSF estimation from [5], this quadratic minimisation problem is
solvedvia thecorresponding linearequationsystemandanLUdecomposition[9,pp.52p.], followed
by a projection step that eliminates negative entries inh and normalisesh to unit total weight. As
a refinement of the projection step, it turned out useful to cut off even small positive entries inhby
a threshold, thus additionally enforces sparsity of the PSF. Experiments indicate that the threshold is
best adaptedasamultipleof somequantile, e.g.0.1 times the95%-quantileof theentriesofh.
The image estimation step that alternates with PSF estimation comes down to a TV-regularised non-
blinddeconvolutionproblemforwhichseveral approachesexist. In [5] themethod from[4] isused.
3. Robust ImageandPSFEstimation
As demonstrated in e.g. [1, 8, 14, 13], robust data terms allow to achieve favourable deconvolution
results even with imprecise estimates of the PSF or slight deviations from the spatial invariant blur
model. While the latter is generally relevant in deconvolution of real-world images, robustness to
imprecise PSF estimates is particularly useful in blind deconvolution. This makes it attractive to
incorporate robustdata terms into the frameworkof [5],which isour goal in this section.
Dueto thealternatingminimisationstructureof themethod,weconsider the twostepsseparately. We
start with the image estimation, which is tantamount to non-blind deconvolution. Thus, we simply
have to replace theTVdeconvolutionmodelwithasuitable robustapproach. In thiswork,wechoose
RRRL[13] for thispurpose,which is afixed point iteration associated to theenergy function
E(u)= ∑
i,j Φ (
[u∗h]i,j−fi,j−fi,j ln [u∗h]i,j
fi,j )
dx+αRu(u) . (6)
We prefer this method for efficiency reasons; note that the non-blind deconvolution step is needed in
each iteration of the alternating minimisation. RRRL is known to evolve fast toward a good solution
during the first few iterations, see also [14], whereas methods based on approaches as in [1] tend to
require more iterations. Following [13], the data term penaliser in the RRRL method is chosen as
Φ(z)=2 √
z,whereas the image regulariserRu is chosen as totalvariationas inSection2.
For the PSF estimation, we insert a penaliser functionΦas mentioned above into the discretised data
term from (3), which is then combined with the unaltered regulariserRh from (4) to yield a (partial)
discreteenergy function for theestimationofh:
E(h)= ∑
i,j Φ
( (fi,j− [u∗h]i,j)2 )
dx+αRh(h) . (7)
Unlike itscounterpart inSection2., thisenergyfunction isno longerquadratic. Equating thegradient
(i.e. the derivatives w.r.t.hi,j) to zero now yields a system of nonlinear equations for the PSF entries.
56
Proceedings
OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“
- Title
- Proceedings
- Subtitle
- OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“
- Authors
- Peter M. Roth
- Kurt Niel
- Publisher
- Verlag der Technischen Universität Graz
- Location
- Wels
- Date
- 2017
- Language
- English
- License
- CC BY 4.0
- ISBN
- 978-3-85125-527-0
- Size
- 21.0 x 29.7 cm
- Pages
- 248
- Keywords
- Tagungsband
- Categories
- International
- Tagungsbände