<s>
The	O
Mumford	B-Algorithm
–	I-Algorithm
Shah	I-Algorithm
functional	I-Algorithm
is	O
a	O
functional	O
that	O
is	O
used	O
to	O
establish	O
an	O
optimality	O
criterion	O
for	O
segmenting	O
an	O
image	O
into	O
sub-regions	O
.	O
</s>
<s>
Ambrosio	O
and	O
Tortorelli	O
showed	O
that	O
Mumford	B-Algorithm
–	I-Algorithm
Shah	I-Algorithm
functional	I-Algorithm
E[ 	O
J	O
,	O
B	O
]	O
can	O
be	O
obtained	O
as	O
the	O
limit	O
of	O
a	O
family	O
of	O
energy	O
functionals	O
E[ 	O
J	O
,	O
z	O
,	O
ε	O
]	O
where	O
the	O
boundary	O
B	O
is	O
replaced	O
by	O
continuous	O
function	O
z	O
whose	O
magnitude	O
indicates	O
the	O
presence	O
of	O
a	O
boundary	O
.	O
</s>
<s>
Their	O
analysis	O
show	O
that	O
the	O
Mumford	B-Algorithm
–	I-Algorithm
Shah	I-Algorithm
functional	I-Algorithm
has	O
a	O
well-defined	O
minimum	O
.	O
</s>
<s>
the	O
last	O
integral	B-Algorithm
term	O
of	O
the	O
energy	O
functional	O
)	O
converge	O
to	O
the	O
edge	O
set	O
integral	B-Algorithm
∫Bds	O
.	O
</s>
<s>
The	O
energy	O
functional	O
E[ 	O
J	O
,	O
z	O
,	O
ε	O
]	O
can	O
be	O
minimized	O
by	O
gradient	B-Algorithm
descent	I-Algorithm
methods	I-Algorithm
,	O
assuring	O
the	O
convergence	O
to	O
a	O
local	O
minimum	O
.	O
</s>
<s>
The	O
Mumford-Shah	B-Algorithm
functional	I-Algorithm
can	O
be	O
split	O
into	O
coupled	O
one-dimensional	O
subproblems	O
.	O
</s>
