<s>
Geometric	B-Algorithm
and	I-Algorithm
Topological	I-Algorithm
Inference	I-Algorithm
is	O
a	O
monograph	O
in	O
computational	O
geometry	O
,	O
computational	B-Algorithm
topology	I-Algorithm
,	O
geometry	B-Algorithm
processing	I-Algorithm
,	O
and	O
topological	B-Algorithm
data	I-Algorithm
analysis	I-Algorithm
,	O
on	O
the	O
problem	O
of	O
inferring	O
properties	O
of	O
an	O
unknown	O
space	O
from	O
a	O
finite	O
point	B-Algorithm
cloud	I-Algorithm
of	O
noisy	O
samples	O
from	O
the	O
space	O
.	O
</s>
<s>
The	O
first	O
part	O
covers	O
basic	O
tools	O
from	O
topology	O
needed	O
in	O
the	O
study	O
,	O
including	O
simplicial	O
complexes	O
,	O
Čech	O
complexes	O
and	O
Vietoris	O
–	O
Rips	O
complex	O
,	O
homotopy	O
equivalence	O
of	O
topological	O
spaces	O
to	O
their	O
nerves	O
,	O
filtrations	O
of	O
complexes	O
,	O
and	O
the	O
data	B-General_Concept
structures	I-General_Concept
needed	O
to	O
represent	O
these	O
concepts	O
efficiently	O
in	O
computer	O
algorithms	O
.	O
</s>
<s>
A	O
second	O
introductory	O
part	O
concerns	O
material	O
of	O
a	O
more	O
geometric	O
nature	O
,	O
including	O
Delaunay	B-Algorithm
triangulations	I-Algorithm
and	O
Voronoi	B-Architecture
diagrams	I-Architecture
,	O
convex	O
polytopes	O
,	O
convex	O
hulls	O
and	O
convex	B-Algorithm
hull	I-Algorithm
algorithms	I-Algorithm
,	O
lower	B-Algorithm
envelopes	I-Algorithm
,	O
alpha	O
shapes	O
and	O
alpha	O
complexes	O
,	O
and	O
witness	O
complexes	O
.	O
</s>
<s>
The	O
fourth	O
part	O
shows	O
how	O
,	O
with	O
weaker	O
assumptions	O
about	O
the	O
samples	O
,	O
it	O
is	O
still	O
possible	O
to	O
recover	O
useful	O
information	O
about	O
the	O
space	O
,	O
such	O
as	O
its	O
homology	O
and	O
persistent	B-Algorithm
homology	I-Algorithm
.	O
</s>
