<s>
The	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
a	O
feature	B-Algorithm
extraction	I-Algorithm
technique	O
used	O
in	O
image	B-General_Concept
analysis	I-General_Concept
,	O
computer	B-Application
vision	I-Application
,	O
and	O
digital	B-Algorithm
image	I-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
This	O
voting	O
procedure	O
is	O
carried	O
out	O
in	O
a	O
parameter	O
space	O
,	O
from	O
which	O
object	O
candidates	O
are	O
obtained	O
as	O
local	O
maxima	O
in	O
a	O
so-called	O
accumulator	O
space	O
that	O
is	O
explicitly	O
constructed	O
by	O
the	O
algorithm	O
for	O
computing	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
classical	O
Hough	B-Algorithm
transform	I-Algorithm
was	O
concerned	O
with	O
the	O
identification	O
of	O
lines	O
in	O
the	O
image	O
,	O
but	O
later	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
has	O
been	O
extended	O
to	O
identifying	O
positions	O
of	O
arbitrary	O
shapes	O
,	O
most	O
commonly	O
circles	O
or	O
ellipses	O
.	O
</s>
<s>
The	O
Hough	B-Algorithm
transform	I-Algorithm
as	O
it	O
is	O
universally	O
used	O
today	O
was	O
invented	O
by	O
Richard	O
Duda	O
and	O
Peter	O
Hart	O
in	O
1972	O
,	O
who	O
called	O
it	O
a	O
"	O
generalized	B-Algorithm
Hough	I-Algorithm
transform	I-Algorithm
"	O
after	O
the	O
related	O
1962	O
patent	O
of	O
Paul	O
Hough	O
.	O
</s>
<s>
The	O
transform	O
was	O
popularized	O
in	O
the	O
computer	B-Application
vision	I-Application
community	O
by	O
Dana	O
H	O
.	O
Ballard	O
through	O
a	O
1981	O
journal	O
article	O
titled	O
"	O
Generalizing	B-Algorithm
the	I-Algorithm
Hough	I-Algorithm
transform	I-Algorithm
to	I-Algorithm
detect	I-Algorithm
arbitrary	I-Algorithm
shapes	I-Algorithm
"	O
.	O
</s>
<s>
It	O
was	O
initially	O
invented	O
for	O
machine	O
analysis	O
of	O
bubble	B-Algorithm
chamber	I-Algorithm
photographs	O
(	O
Hough	O
,	O
1959	O
)	O
.	O
</s>
<s>
The	O
Hough	B-Algorithm
transform	I-Algorithm
was	O
patented	O
as	O
in	O
1962	O
and	O
assigned	O
to	O
the	O
U.S.	O
Atomic	O
Energy	O
Commission	O
with	O
the	O
name	O
"	O
Method	O
and	O
Means	O
for	O
Recognizing	O
Complex	O
Patterns	O
"	O
.	O
</s>
<s>
although	O
it	O
was	O
already	O
standard	O
for	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
since	O
at	O
least	O
the	O
1930s	O
.	O
</s>
<s>
In	O
automated	O
analysis	O
of	O
digital	B-Algorithm
images	I-Algorithm
,	O
a	O
subproblem	O
often	O
arises	O
of	O
detecting	O
simple	O
shapes	O
,	O
such	O
as	O
straight	O
lines	O
,	O
circles	O
or	O
ellipses	O
.	O
</s>
<s>
In	O
many	O
cases	O
an	O
edge	B-Algorithm
detector	I-Algorithm
can	O
be	O
used	O
as	O
a	O
pre-processing	O
stage	O
to	O
obtain	O
image	O
points	O
or	O
image	O
pixels	O
that	O
are	O
on	O
the	O
desired	O
curve	O
in	O
the	O
image	O
space	O
.	O
</s>
<s>
Due	O
to	O
imperfections	O
in	O
either	O
the	O
image	O
data	O
or	O
the	O
edge	B-Algorithm
detector	I-Algorithm
,	O
however	O
,	O
there	O
may	O
be	O
missing	O
points	O
or	O
pixels	O
on	O
the	O
desired	O
curves	O
as	O
well	O
as	O
spatial	O
deviations	O
between	O
the	O
ideal	O
line/circle/ellipse	O
and	O
the	O
noisy	O
edge	O
points	O
as	O
they	O
are	O
obtained	O
from	O
the	O
edge	B-Algorithm
detector	I-Algorithm
.	O
</s>
<s>
The	O
purpose	O
of	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
to	O
address	O
this	O
problem	O
by	O
making	O
it	O
possible	O
to	O
perform	O
groupings	O
of	O
edge	O
points	O
into	O
object	O
candidates	O
by	O
performing	O
an	O
explicit	O
voting	O
procedure	O
over	O
a	O
set	O
of	O
parameterized	O
image	O
objects	O
(	O
Shapiro	O
and	O
Stockman	O
,	O
304	O
)	O
.	O
</s>
<s>
The	O
simplest	O
case	O
of	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
detecting	O
straight	O
lines	O
.	O
</s>
<s>
The	O
plane	O
is	O
sometimes	O
referred	O
to	O
as	O
Hough	B-Algorithm
space	I-Algorithm
for	O
the	O
set	O
of	O
straight	O
lines	O
in	O
two	O
dimensions	O
.	O
</s>
<s>
This	O
representation	O
makes	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
conceptually	O
very	O
close	O
to	O
the	O
two-dimensional	O
Radon	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
mathematically	O
equivalent	O
to	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
,	O
but	O
the	O
two	O
transformations	O
have	O
different	O
computational	O
interpretations	O
traditionally	O
associated	O
with	O
them	O
.	O
</s>
<s>
Given	O
a	O
shape	O
parametrized	O
by	O
,	O
taking	O
values	O
in	O
the	O
set	O
called	O
the	O
shape	O
space	O
,	O
one	O
can	O
interpret	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
as	O
the	O
inverse	O
transform	O
of	O
a	O
probability	O
distribution	O
on	O
the	O
image	O
space	O
to	O
the	O
shape	O
space	O
,	O
and	O
interpret	O
shape	O
detection	O
as	O
maximum	O
likelihood	O
estimation	O
.	O
</s>
<s>
Explicitly	O
,	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
performs	O
an	O
approximate	O
naive	B-General_Concept
Bayes	I-General_Concept
inference	O
.	O
</s>
<s>
The	O
assumption	O
of	O
naive	B-General_Concept
Bayes	I-General_Concept
means	O
that	O
all	O
pixels	O
in	O
the	O
image	O
space	O
provide	O
independent	O
evidence	O
,	O
so	O
that	O
their	O
likelihoods	O
multiply	O
,	O
that	O
is	O
,	O
their	O
log-likelihoods	O
add	O
.	O
</s>
<s>
The	O
linear	O
Hough	B-Algorithm
transform	I-Algorithm
algorithm	O
estimates	O
the	O
two	O
parameters	O
that	O
define	O
a	O
straight	O
line	O
.	O
</s>
<s>
For	O
each	O
pixel	O
at	O
and	O
its	O
neighborhood	O
,	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
algorithm	O
determines	O
whether	O
there	O
is	O
enough	O
evidence	O
of	O
a	O
straight	O
line	O
at	O
that	O
pixel	O
.	O
</s>
<s>
The	O
final	O
result	O
of	O
the	O
linear	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
a	O
two-dimensional	O
array	O
(	O
matrix	O
)	O
similar	O
to	O
the	O
accumulator	O
—	O
one	O
dimension	O
of	O
this	O
matrix	O
is	O
the	O
quantized	O
angle	O
,	O
and	O
the	O
other	O
dimension	O
is	O
the	O
quantized	O
distance	O
.	O
</s>
<s>
The	O
Hough	B-Algorithm
transform	I-Algorithm
accumulates	O
contributions	O
from	O
all	O
pixels	O
in	O
the	O
detected	O
edge	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
different	O
example	O
showing	O
the	O
results	O
of	O
a	O
Hough	B-Algorithm
transform	I-Algorithm
on	O
a	O
raster	O
image	O
containing	O
two	O
thick	O
lines	O
.	O
</s>
<s>
Since	O
edge	B-Algorithm
detection	I-Algorithm
generally	O
involves	O
computing	O
the	O
intensity	O
gradient	O
magnitude	O
,	O
the	O
gradient	O
direction	O
is	O
often	O
found	O
as	O
a	O
side	O
effect	O
.	O
</s>
<s>
Fernandes	O
and	O
Oliveira	O
suggested	O
an	O
improved	O
voting	O
scheme	O
for	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
that	O
allows	O
a	O
software	O
implementation	O
to	O
achieve	O
real-time	O
performance	O
even	O
on	O
relatively	O
large	O
images	O
(	O
e.g.	O
,	O
1280×960	O
)	O
.	O
</s>
<s>
The	O
Kernel-based	O
Hough	B-Algorithm
transform	I-Algorithm
uses	O
the	O
same	O
parameterization	O
proposed	O
by	O
Duda	O
and	O
Hart	O
but	O
operates	O
on	O
clusters	O
of	O
approximately	O
collinear	O
pixels	O
.	O
</s>
<s>
Limberger	O
and	O
Oliveira	O
suggested	O
a	O
deterministic	O
technique	O
for	O
plane	O
detection	O
in	O
unorganized	O
point	B-Algorithm
clouds	I-Algorithm
whose	O
cost	O
is	O
in	O
the	O
number	O
of	O
samples	O
,	O
achieving	O
real-time	O
performance	O
for	O
relatively	O
large	O
datasets	O
(	O
up	O
to	O
points	O
on	O
a	O
3.4GHz	O
CPU	O
)	O
.	O
</s>
<s>
It	O
is	O
based	O
on	O
a	O
fast	O
Hough-transform	O
voting	O
strategy	O
for	O
planar	O
regions	O
,	O
inspired	O
by	O
the	O
Kernel-based	O
Hough	B-Algorithm
transform	I-Algorithm
(	O
KHT	O
)	O
.	O
</s>
<s>
This	O
3D	O
kernel-based	O
Hough	B-Algorithm
transform	I-Algorithm
(	O
3DKHT	O
)	O
uses	O
a	O
fast	O
and	O
robust	O
algorithm	O
to	O
segment	O
clusters	O
of	O
approximately	O
co-planar	O
samples	O
,	O
and	O
casts	O
votes	O
for	O
individual	O
clusters	O
(	O
instead	O
of	O
for	O
individual	O
samples	O
)	O
on	O
a	O
(	O
)	O
spherical	O
accumulator	O
using	O
a	O
trivariate	O
Gaussian	O
kernel	O
.	O
</s>
<s>
The	O
approach	O
is	O
several	O
orders	O
of	O
magnitude	O
faster	O
than	O
existing	O
(	O
non-deterministic	O
)	O
techniques	O
for	O
plane	O
detection	O
in	O
point	B-Algorithm
clouds	I-Algorithm
,	O
such	O
as	O
RHT	B-Algorithm
and	O
RANSAC	B-Algorithm
,	O
and	O
scales	O
better	O
with	O
the	O
size	O
of	O
the	O
datasets	O
.	O
</s>
<s>
A	O
circle	O
,	O
for	O
instance	O
,	O
can	O
be	O
transformed	O
into	O
a	O
set	O
of	O
three	O
parameters	O
,	O
representing	O
its	O
center	O
and	O
radius	O
,	O
so	O
that	O
the	O
Hough	B-Algorithm
space	I-Algorithm
becomes	O
three	O
dimensional	O
.	O
</s>
<s>
The	O
generalization	O
of	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
for	O
detecting	O
analytical	O
shapes	O
in	O
spaces	O
having	O
any	O
dimensionality	O
was	O
proposed	O
by	O
Fernandes	O
and	O
Oliveira	O
.	O
</s>
<s>
For	O
more	O
complicated	O
shapes	O
in	O
the	O
plane	O
(	O
i.e.	O
,	O
shapes	O
that	O
cannot	O
be	O
represented	O
analytically	O
in	O
some	O
2D	O
space	O
)	O
,	O
the	O
Generalised	B-Algorithm
Hough	I-Algorithm
transform	I-Algorithm
is	O
used	O
,	O
which	O
allows	O
a	O
feature	B-Algorithm
to	O
vote	O
for	O
a	O
particular	O
position	O
,	O
orientation	O
and/or	O
scaling	O
of	O
the	O
shape	O
using	O
a	O
predefined	O
look-up	O
table.The	O
Hough	B-Algorithm
transform	I-Algorithm
accumulates	O
contributions	O
from	O
all	O
pixels	O
in	O
the	O
detected	O
edge	O
.	O
</s>
<s>
Hough	B-Algorithm
transform	I-Algorithm
can	O
also	O
be	O
used	O
for	O
the	O
detection	O
of	O
3D	O
objects	O
in	O
range	B-Algorithm
data	I-Algorithm
or	O
3D	O
point	B-Algorithm
clouds	I-Algorithm
.	O
</s>
<s>
The	O
extension	O
of	O
classical	O
Hough	B-Algorithm
transform	I-Algorithm
for	O
plane	O
detection	O
is	O
quite	O
straightforward	O
.	O
</s>
<s>
A	O
plane	O
is	O
represented	O
by	O
its	O
explicit	O
equation	O
for	O
which	O
we	O
can	O
use	O
a	O
3D	O
Hough	B-Algorithm
space	I-Algorithm
corresponding	O
to	O
,	O
and	O
.	O
</s>
<s>
This	O
formulation	O
of	O
the	O
plane	O
has	O
been	O
used	O
for	O
the	O
detection	O
of	O
planes	O
in	O
the	O
point	B-Algorithm
clouds	I-Algorithm
acquired	O
from	O
airborne	O
laser	O
scanning	O
and	O
works	O
very	O
well	O
because	O
in	O
that	O
domain	O
all	O
planes	O
are	O
nearly	O
horizontal	O
.	O
</s>
<s>
For	O
generalized	O
plane	O
detection	O
using	O
Hough	B-Algorithm
transform	I-Algorithm
,	O
the	O
plane	O
can	O
be	O
parametrized	O
by	O
its	O
normal	O
vector	O
(	O
using	O
spherical	O
coordinates	O
)	O
and	O
its	O
distance	O
from	O
the	O
origin	O
resulting	O
in	O
a	O
three	O
dimensional	O
Hough	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
This	O
results	O
in	O
each	O
point	O
in	O
the	O
input	O
data	O
voting	O
for	O
a	O
sinusoidal	O
surface	O
in	O
the	O
Hough	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Hough	B-Algorithm
transform	I-Algorithm
has	O
also	O
been	O
used	O
to	O
find	O
cylindrical	O
objects	O
in	O
point	B-Algorithm
clouds	I-Algorithm
using	O
a	O
two	O
step	O
approach	O
.	O
</s>
<s>
A	O
high-dimensional	O
parameter	O
space	O
for	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
not	O
only	O
slow	O
,	O
but	O
if	O
implemented	O
without	O
forethought	O
can	O
easily	O
overrun	O
the	O
available	O
memory	O
.	O
</s>
<s>
Even	O
if	O
the	O
programming	O
environment	O
allows	O
the	O
allocation	O
of	O
an	O
array	O
larger	O
than	O
the	O
available	O
memory	O
space	O
through	O
virtual	O
memory	O
,	O
the	O
number	O
of	O
page	B-Architecture
swaps	I-Architecture
required	O
for	O
this	O
will	O
be	O
very	O
demanding	O
because	O
the	O
accumulator	O
array	O
is	O
used	O
in	O
a	O
randomly	O
accessed	O
fashion	O
,	O
rarely	O
stopping	O
in	O
contiguous	O
memory	O
as	O
it	O
skips	O
from	O
index	O
to	O
index	O
.	O
</s>
<s>
Yonghong	O
Xie	O
and	O
Qiang	O
Ji	O
give	O
an	O
efficient	O
way	O
of	O
implementing	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
for	O
ellipse	O
detection	O
by	O
overcoming	O
the	O
memory	O
issues	O
.	O
</s>
<s>
The	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
only	O
efficient	O
if	O
a	O
high	O
number	O
of	O
votes	O
fall	O
in	O
the	O
right	O
bin	O
,	O
so	O
that	O
the	O
bin	O
can	O
be	O
easily	O
detected	O
amid	O
the	O
background	O
noise	O
.	O
</s>
<s>
Also	O
,	O
when	O
the	O
number	O
of	O
parameters	O
is	O
large	O
(	O
that	O
is	O
,	O
when	O
we	O
are	O
using	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
with	O
typically	O
more	O
than	O
three	O
parameters	O
)	O
,	O
the	O
average	O
number	O
of	O
votes	O
cast	O
in	O
a	O
single	O
bin	O
is	O
very	O
low	O
,	O
and	O
those	O
bins	O
corresponding	O
to	O
a	O
real	O
figure	O
in	O
the	O
image	O
do	O
not	O
necessarily	O
appear	O
to	O
have	O
a	O
much	O
higher	O
number	O
of	O
votes	O
than	O
their	O
neighbors	O
.	O
</s>
<s>
(	O
Shapiro	O
and	O
Stockman	O
,	O
310	O
)	O
Thus	O
,	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
must	O
be	O
used	O
with	O
great	O
care	O
to	O
detect	O
anything	O
other	O
than	O
lines	O
or	O
circles	O
.	O
</s>
<s>
Finally	O
,	O
much	O
of	O
the	O
efficiency	O
of	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
dependent	O
on	O
the	O
quality	O
of	O
the	O
input	O
data	O
:	O
the	O
edges	O
must	O
be	O
detected	O
well	O
for	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
to	O
be	O
efficient	O
.	O
</s>
<s>
Use	O
of	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
on	O
noisy	O
images	O
is	O
a	O
very	O
delicate	O
matter	O
and	O
generally	O
,	O
a	O
denoising	O
stage	O
must	O
be	O
used	O
before	O
.	O
</s>
<s>
In	O
the	O
case	O
where	O
the	O
image	O
is	O
corrupted	O
by	O
speckle	O
,	O
as	O
is	O
the	O
case	O
in	O
radar	O
images	O
,	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
sometimes	O
preferred	O
to	O
detect	O
lines	O
,	O
because	O
it	O
attenuates	O
the	O
noise	O
through	O
summation	O
.	O
</s>
