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Institute
of
Technology
Department
of
Computer
Science
and
Electrical
Engineering
6.801/6.866
Machine
Vision
Quiz
I
Handed
out:
2004
Oct.
21st
Due
on:
2003
Oct.
28th
Problem
1:
Uniform
reﬂecting
properties
are
a
prerequisite
for
the
usual
shape
from
shading
methods.
Consider
now
a
surface
covered
by
a
material
of
spatially
varying
reﬂectance.
Suppose
that
the
brightness
can
be
treated
as
the
product
of
a
spatially
varying
‘reﬂectance’
or
‘albedo’
ρ(x,
y)
,
and
a
‘geometric
factor’
R(p,
q)
that
depends
only
on
surface
orientation.
There
are
two
unknowns—
z(x,
y)
and
ρ(x,
y)
—at
every
position
on
the
surface,
so
a
single
image
will
not
provide
enough
information
to
recover
both
(consider,
for
example,
a
photographic
print
of
a
rounded
object
where
the
bright
ness
variations
could
either
be
from
a
rounded
object
of
uniform
albedo
or
from
a
ﬂat
object
of
varying
albedo).
Now
suppose
we
take
two
images
under
different
lighting
conditions.
(a)
Combine
the
two
resulting
image
irradiance
equations
in
such
a
way
as
to
eliminate
ρ(x,
y)
.
Suppose
the
new—now
‘
ρ
free’—equation
can
be
written
in
the
form
E (x,
y)
=
R (p,
q).
(b)
Show
that
if
the
underlying
surface
actually
is
a
Lambertian
reﬂector.
then
the
isophotes
in
gradient
space
are
straight
lines.
When
will
they
be
parallel
straight
lines?
(c)
Show
that
the
isophotes
all
go
through
a
common
point
in
gradient
space
in
the
case
that
they
are
not
parallel.
Where
in
gradient
space
would
you
expect
the
highest
accuracy
in
recovering
surface
orientation?
That
is,
where
is
‘brightness’
(in
the
ρ
free
equation)
most
affected
by
small
changes
in
surface
orientation?
Relate
this
back
to
the
original
imaging
situation.
How
should
the
lighting
be
arranged
to
obtain
high
accuracy?
Problem
2:
An
edge
detection
method
starts
by
ﬁnding
the
brightness
gradient
(E
x
,E
y
)
at
each
picture
cell
using
some
local
estimator
of
the
partial
deriva
tives
of
image
brightness
E(x,
y)
.
The
magnitude
of
the
brightness
gradient
is
then
computed.
Next,
“nonmaximum
suppression’’
keeps
for
further
consider
ation
only
pixels
where
the
magnitude
of
the
gradient
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 Spring '04
 BertholdHorn

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