<s>
In	O
photogrammetry	B-Application
and	O
computer	B-Algorithm
stereo	I-Algorithm
vision	I-Algorithm
,	O
bundle	B-General_Concept
adjustment	I-General_Concept
is	O
simultaneous	O
refining	O
of	O
the	O
3D	O
coordinates	O
describing	O
the	O
scene	O
geometry	O
,	O
the	O
parameters	O
of	O
the	O
relative	O
motion	O
,	O
and	O
the	O
optical	O
characteristics	O
of	O
the	O
camera(s )	O
employed	O
to	O
acquire	O
the	O
images	O
,	O
given	O
a	O
set	O
of	O
images	O
depicting	O
a	O
number	O
of	O
3D	O
points	O
from	O
different	B-Algorithm
viewpoints	I-Algorithm
.	O
</s>
<s>
Its	O
name	O
refers	O
to	O
the	O
geometrical	O
bundles	O
of	O
light	B-Device
rays	I-Device
originating	O
from	O
each	O
3D	O
feature	O
and	O
converging	O
on	O
each	O
camera	B-Algorithm
's	I-Algorithm
optical	O
center	O
,	O
which	O
are	O
adjusted	O
optimally	O
according	O
to	O
an	O
optimality	O
criterion	O
involving	O
the	O
corresponding	B-Algorithm
image	O
projections	O
of	O
all	O
points	O
.	O
</s>
<s>
Bundle	B-General_Concept
adjustment	I-General_Concept
is	O
almost	O
always	O
used	O
as	O
the	O
last	O
step	O
of	O
every	O
feature-based	O
3D	B-Algorithm
reconstruction	I-Algorithm
algorithm	O
.	O
</s>
<s>
It	O
amounts	O
to	O
an	O
optimization	O
problem	O
on	O
the	O
3D	O
structure	O
and	O
viewing	O
parameters	O
(	O
i.e.	O
,	O
camera	B-Architecture
pose	I-Architecture
and	O
possibly	O
intrinsic	O
calibration	O
and	O
radial	O
distortion	O
)	O
,	O
to	O
obtain	O
a	O
reconstruction	O
which	O
is	O
optimal	O
under	O
certain	O
assumptions	O
regarding	O
the	O
noise	O
pertaining	O
to	O
the	O
observed	O
image	O
features	O
:	O
If	O
the	O
image	O
error	O
is	O
zero-mean	O
Gaussian	O
,	O
then	O
bundle	B-General_Concept
adjustment	I-General_Concept
is	O
the	O
Maximum	O
Likelihood	O
Estimator	O
.	O
</s>
<s>
Bundle	B-General_Concept
adjustment	I-General_Concept
was	O
originally	O
conceived	O
in	O
the	O
field	O
of	O
photogrammetry	B-Application
during	O
the	O
1950s	O
and	O
has	O
increasingly	O
been	O
used	O
by	O
computer	B-Application
vision	I-Application
researchers	O
during	O
recent	O
years	O
.	O
</s>
<s>
Thus	O
,	O
the	O
minimization	O
is	O
achieved	O
using	O
nonlinear	O
least-squares	B-Algorithm
algorithms	O
.	O
</s>
<s>
Of	O
these	O
,	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
has	O
proven	O
to	O
be	O
one	O
of	O
the	O
most	O
successful	O
due	O
to	O
its	O
ease	O
of	O
implementation	O
and	O
its	O
use	O
of	O
an	O
effective	O
damping	O
strategy	O
that	O
lends	O
it	O
the	O
ability	O
to	O
converge	O
quickly	O
from	O
a	O
wide	O
range	O
of	O
initial	O
guesses	O
.	O
</s>
<s>
By	O
iteratively	O
linearizing	O
the	O
function	O
to	O
be	O
minimized	O
in	O
the	O
neighborhood	O
of	O
the	O
current	O
estimate	O
,	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
involves	O
the	O
solution	O
of	O
linear	O
systems	O
termed	O
the	O
normal	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
bundle	B-General_Concept
adjustment	I-General_Concept
,	O
the	O
normal	B-Algorithm
equations	I-Algorithm
have	O
a	O
sparse	B-Algorithm
block	O
structure	O
owing	O
to	O
the	O
lack	O
of	O
interaction	O
among	O
parameters	O
for	O
different	O
3D	O
points	O
and	O
cameras	O
.	O
</s>
<s>
This	O
can	O
be	O
exploited	O
to	O
gain	O
tremendous	O
computational	O
benefits	O
by	O
employing	O
a	O
sparse	B-Algorithm
variant	O
of	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
which	O
explicitly	O
takes	O
advantage	O
of	O
the	O
normal	B-Algorithm
equations	I-Algorithm
zeros	O
pattern	O
,	O
avoiding	O
storing	O
and	O
operating	O
on	O
zero-elements	O
.	O
</s>
<s>
Bundle	B-General_Concept
adjustment	I-General_Concept
amounts	O
to	O
jointly	O
refining	O
a	O
set	O
of	O
initial	O
camera	O
and	O
structure	O
parameter	O
estimates	O
for	O
finding	O
the	O
set	O
of	O
parameters	O
that	O
most	O
accurately	O
predict	O
the	O
locations	O
of	O
the	O
observed	O
points	O
in	O
the	O
set	O
of	O
available	O
images	O
.	O
</s>
<s>
More	O
formally	O
,	O
assume	O
that	O
3D	O
points	O
are	O
seen	O
in	O
views	O
and	O
let	O
be	O
the	O
projection	B-General_Concept
of	O
the	O
th	O
point	O
on	O
image	O
.	O
</s>
<s>
where	O
is	O
the	O
predicted	O
projection	B-General_Concept
of	O
point	O
on	O
image	O
and	O
denotes	O
the	O
Euclidean	O
distance	O
between	O
the	O
image	O
points	O
represented	O
by	O
vectors	O
and	O
.	O
</s>
<s>
Because	O
the	O
minimum	O
is	O
computed	O
over	O
many	O
points	O
and	O
many	O
images	O
,	O
bundle	B-General_Concept
adjustment	I-General_Concept
is	O
by	O
definition	O
tolerant	O
to	O
missing	O
image	O
projections	O
,	O
and	O
if	O
the	O
distance	O
metric	O
is	O
chosen	O
reasonably	O
(	O
e.g.	O
,	O
Euclidean	O
distance	O
)	O
,	O
bundle	B-General_Concept
adjustment	I-General_Concept
will	O
also	O
minimize	O
a	O
physically	O
meaningful	O
criterion	O
.	O
</s>
