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Stochastic
geometry and
image analysis
BY EVA B. VEDEL JENSEN
Eva B. Vedel Jensen (born 1951), DSc 1987, professor at Aarhus Univer-
sity 2003. Director of the Thiele Centre 2004-2009 and the research pro-
ject Histo Informatics 2007-2010 at Aarhus University. Coordinator of the
Marie Curie training site Advanced Medical Imaging and Spatial Statistics
(MISS) 2002-2006. Member of the Danish Natural Science Research Council
2001-2007. Since 2010 director of CSGB – a VKR Centre of Excellence.
In 2010 the VILLUM FOUNDATION donated DKK 25,000,000
to a VKR Centre of Excellence: Centre for Stochastic Geometry and
Advanced Bioimaging – CSGB. The Centre has been established as
an interdisciplinary collaboration between Aarhus University (AU),
Aalborg University (AAU) and University of Copenhagen (KU), in-
volving Department of Mathematical Sciences (AU), Clinical Institute
(AU), Department of Mathematical Sciences (AAU) and Department
of Computer Science (KU). CSGB will develop new mathematical and
statistical methods of analysing advanced image data from biological
tissue (bioimaging). A particular focus is the analysis of image data
at the molecular level, using state-of-the-art microscopy techniques.
Many of the methods will utilise the latest advances in stochastic ge-
ometry, a discipline on the borderline between mathematics and statis-
tics. The research takes place in a combined Danish and international
research environment. The centre is expected to contribute to added
knowledge in public health-care and the biomedical industry.
Mathematics and statistics play a key role in the biosciences. Dur-
ing the last 25 years mathematical and statistical models have been
developed for the analysis of cell populations, based on data from
traditional light microscopy. With the development of, e.g., laser
scanning and so-called cryo-electron microscopy it is, however,
now possible to acquire images of biological tissue (bioimaging)
at the molecular level and this requires a corresponding develop-
ment of mathematical and statistical methods.
Stochastic geometry
Stochastic geometry is an important prerequisite for this develop-
ment of methods. Stochastic geometry is used in modelling and
analysis of spatial structures such as curves and surfaces. Stereology
is an important subfield of stochastic geometry. As the name indi-
cates, stereology provides information about a spatial structure
(three dimensions) from a number of sections (two di-
mensions) through the structure (see fig. 1).
One of the principal contributions of stochastic
geometry is a number of geometric calcu-
lation formulae (functionals) to describe the
quantitative properties of a spatial structure, for in-
stance, volume, surface area, length, number and curva-
ture. These functionals have been used in numerous stereo-
logical studies of biological tissue at the light microscopy level.
Local stereology
In stochastic geometry, there has recently been a lot of focus on the
study of a number of new functionals, so-called volume and surface
tensors, that are, so to speak, able to ‘zoom in’ and provide more
detailed local information about a spatial structure than the more
general global information in the previous functionals. At the cen-
tre we want to utilise these advances in enhancing analysis of image
data from biological tissue.
For example, earlier stereological analyses of the cerebral cortex
in humans with dementia or schizophrenia do not show changes
in the total number of neurons. We expect that there are instead
changes in the spatial arrangement of neurons in certain regions
of the cortex. Likewise, disease-caused changes in tissue do not
Fig.1. Local stereology makes it possible
to calculate estimates of parameters
of a spatial structure, such as volume
and surface area, from observations
along lines or in planes passing
through a reference point. At CSGB,
local stereological methods of determining
so-called volume and surface
tensors are developed. These tensors
can provide new information about
shape and orientation distributions in
cell populations. Illustration: Markus
Kiderlen
44 STOCHASTIC GEOMETRY AND IMAGE ANALYSIS VILLUM FOUNDATION 45
TECHNICAL AND NATURAL SCIENCES