![](bg2.png)
frame HR methods attempt to magnify the image
without introducing blur. These last methods use
other parts of the low resolution images to make an
extimation on what the high resolution images
should look like. AFM imaging requires to work
using a single-frame approach: given a single image
of sample scanned at low resolution, return the
image that is mostly likely to be generated from a
noiseless high resolution scan of the same sample
portion. After HR equilavent image is reached, 3D
recostruction is made possible using a Markov
Random Field (MRF) approach. In a MRF method
we consider a couple (h
m
, N), where a stochastic
process is indexed by an augmented voxel h
m
for
which, for every couple (x,y) of the 2D image, any
augmented voxel depends only on its immediate
neighbours of the N set, where N is a parameter
space. The choice of N depends by the system
variables conditional probability distributions, where
system variables provide the basic tool for modelling
spatial continuity.
We apply our method to cells derived by stem
primitive cells and differentiated in osteocytes or
adipocytes, or fibroblast. Any differentiated stem
cell develops specific organs and functions, and
recognition such organs is not a trivial question. The
3D reconstruction is useful when does not lose
information of the primitive image and gives the
possibility to identify unambiguous such specific
organs.
2 INFERENTIAL GENERATIVE
MODEL
Despite the AFM ability to reach high spatial
resolution, the acquired surface topography image
can sometimes not correspond to the real surface
features due to the effect of the instrument on the
object producing artefacts. These artefacts can be
generally taken into consideration while
qualitatively interpreting the AFM results. However,
3D reconstruction tools require quantitative
estimation and reconstruction of sample true
geometry. During scanning, two major AFM
artefacts can appear: a profile broadening effect due
to the tip-sample convolution and the height
lowering effect due to the elastic deformation of
studied samples.
The first effect can be schematised as follows:
the tip moving across an object surface can be
approximated by a sphere of radius R moving along
a sphere of radius r surface, i.e., the tip describes arc
of radius R+r. The lateral dimension of the surface
objects is r
c
=2(R
⋅
r)
1/2
and the relative height of the
object
z=r[1-(1-(r
c
/(R+r))
2
]
(1)
The minimum separation between two asperities or
local pattern that can be detected is d= (8RΔz)
1/2
, that
is also the lateral resolution.
Before to build the 3D structures, the source
images are processed in order to improve their
resolution. To do this operation, we adopt a
Bayesian method. Hardie et al (Hardie et al, 1997)
demonstrated that low-resolution images can be
updated using super-resolution image estimate, and
that this improves a Maximum a Posteriori (MAP)
super-resolution image estimate. Pickup et al.
(Pickup et al., 2009) used a similar joint MAP
approach to learn more general geometric patterns,
configuring the correspondent super-resolution
images and valuing the prior parameters
simultaneously. Another remarkable result for the
inferential super-resolution has been reached by
Tipping and Bishop (Tipping and Bishop, 2003),
they used a Maximum Likelihood (ML) point
estimate of the image parameters found by
integrating the high-resolution image out the
registration problem and optimising the marginal
probability of the observed low-resolution images
directly.
We follow a generative model based on an idea
as proposed by Torres-Mendez et al. (Torres-
Mendez et al., 2007) carried out from single-frame
methods. The basic idea can be summarized as
follows: given a Low Resolution (LR) image α of
size h
α
×w
α
pixels, we want to estimate the
correspondent HR image ω of size h
ω
×w
ω
, with
equal or greater size of the input image α. From α,
we must generate L images of smaller size (scaled),
that we can call observable images l, with l=0…L.
Any point in the LR image is considered as a node in
a Markovian process, and a possible neighbourhood
node in the HR image is defined by a pairwise
potential. If we denote x
i
as a set of hidden nodes in
the output ω, and the y
i
as the observable nodes in α
image, and defining the pairwise potential between
the variables x
i
, and x
j
, by
Ψ
ij
and the local evidence
potential associated with the variables x
i
and y
j
by
Φ
i
, the joint probability correspondent to the
Markovian process can be written as
()
)
()
ii
i
iji
ji
ij
yxxx
Z
yxP ,,
1
,
,
∏∏
= Φ
ψ
(2)
where Z is a normalization constant. Our problem
consists to maximize P(x,y), maximization that
corresponds to find the most likely state for all
hidden nodes x
i
, given all the node y
i
. To remove
INFERENTIAL MINING FOR RECONSTRUCTION OF 3D CELL STRUCTURES IN ATOMIC FORCE
MICROSCOPY IMAGING
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