1
CSE252A
Illumination Cones
and
Uncalibrated Photometric Stereo
CS252A, Fall 2010
Computer Vision I
Photometric stereo
•
Single viewpoint, multiple images under
different lighting.
1.
Arbitrary known BRDF, known lighting
2.
Lambertian BRDF, known lighting
3.
Lambertian BRDF, unknown lighting.
CS252A, Fall 2010
Computer Vision I
Three Source Photometric stereo:
Step1
Offline:
Using source directions & BRDF, construct reflectance map
for each light source direction. R
1
(p,q), R
2
(p,q), R
3
(p,q)
Online:
1.
Acquire three images with known light source directions.
E
1
(x,y), E
2
(x,y), E
3
(x,y)
2.
For each pixel location (x,y), find (p,q) as the intersection
of the three curves
R
1
(p,q)=E
1
(x,y)
R
2
(p,q)=E
2
(x,y)
R
3
(p,q)=E
3
(x,y)
3.
This is the surface normal at pixel (x,y).
Over image, the
normal field is estimated
CS252A, Fall 2010
Computer Vision I
Reflectance Map of Lambertian Surface
What does the intensity
(Irradiance) of one pixel in one
image tell us? (e.
.g, let’s say the
Then, the normal lies on
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
R(p,q)
CS252A, Fall 2010
Computer Vision I
One viewpoint, two images, two light sources
Two super imposed reflectance maps
E
measured
E
measured
1
A third image would disambiguate between two possible n
R
1
(p,q)
R
2
(p,q)
CS252A, Fall 2010
Computer Vision I
Recovering the surface f(x,y)
Many methods: Simplest approach
1.
From estimate
n
=(n
x
,n
y
,n
z
), p=n
x
/n
z
, q=n
y
/n
z
2.
Integrate p=df/dx along a row (x,0) to get f(x,0)
3.
Then integrate q=df/dy along each column
starting with value of the first row
f(x,0)
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CS252A, Fall 2010
Computer Vision I
Lambertian Surface
At image location (u,v), the intensity of a pixel x(u,v) is:
e(u,v) = [a(u,v)
n
(u,v)]
·
[s
0
s
]
=
b
(u,v) ·
s
where
•
a(u,v) is the albedo of the surface projecting to (u,v).
•
n
(u,v) is the direction of the surface normal.
•
s
0
is the light source intensity.
•
s
is the direction to the light source.
^
n
^
s
^
^
a
e(u,v)
CS252A, Fall 2010
Computer Vision I
Lambertian Photometric stereo
• If the light sources
s
1
,
s
2
, and
s
3
are
known
, then
we
can
recover
b
at each pixel from as few as
three images. (Photometric Stereo: Silver 80,
Woodham81).
[e
1
e
2
e
3
] =
b
T
[
s
1
s
2
s
3
]
• i.e., we measure e
1
, e
2
, and e
3
and we know
s
1
,
s
2
,
and
s
3
. We can then solve for
b
by solving a linear
system.
• Surface normal is:
n
=
b
/
b
, albedo is: 
b

CSE252A
What is the set of images of an object
under all possible lighting conditions?
In answering this question, we’ll
arrive at a photometric setereo
method for reconstructing surface
shape w/ unknown lighting.
CSE252A
The Space of Images
• Consider an npixel image to be a point in an n
dimensional space,
x
∈
R
n
.
• Each pixel value is a coordinate of
x
.
x
1
x
2
x
n
• Many results will apply to linear transformations of
image space (e.g. filtered images)
• Other image representations (e.g. CayleyKlein
spaces, See Koenderink’s “pixel
f#@king paper”)
x
1
x
n
x
2
CSE252A
Assumptions
For discussion, we assume:
– Lambertian reflectance functions.
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 Fall '08
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 Singular value decomposition, light source, Illumination Cone

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