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FINAL EXAMINATION
FRIDAY DEC. 11
2-5PM
225NEB
November 9, 1998
1
THE REFLECTANCE MAP AND
SHAPE-FROM-SHADING
Part II
November 9, 1998
2
ASSUMPTIONS USED FOR
SHAPE FROM REFLECTANCE
•
•
•
•
Orthographic Projection
Lambertian Reflectance
Incident orientation of light source(s) are known.
Distance from light source to object is much
greater than the size of the object.
• Therefore reflectance is only dependent upon
relative surface orientation (i.e., no dependence
upon depth or 3-D position)
November 9, 1998
3
REFLECTANCE MAP IS A VIEWER-CENTERED
REPRESENTATION OF REFLECTANCE
(f x , f y , -1) = (p, q, -1)
p, q comprise a gradient or gradient space representation for
local surface orientation.
Reflectance map expresses the reflectance of a material directly
in terms of viewer-centered representation of local surface
orientation.
November 9, 1998
4
REFLECTANCE MAP IS A VIEWER-CENTERED
REPRESENTATION OF REFLECTANCE
(fx , fy , -1) = (0,1,f x)
(1,0,f y)
z=f(x,y)
Surface
Orientation
Z
(0,1,f x)
(1,0,f y)
(fx, fy , -1)
Depth
X
Y
November 9, 1998
dy
dx
IMAGE PLANE
5
LAMBERTIAN REFLECTANCE MAP
LAMBERTIAN MODEL
E = L r COS q
Y
Image
Plane
Z
(p,q,-1)
X
Optic
Axis
COSq 
November 9, 1998
q
(ps,qs,-1)
1  pps  qqs
1  p 2  q 2 1  ps 2  qs 2
6
November 9, 1998
7
LAMBERTIAN REFLECTANCE MAP
I0 
1  pps  qqs
1  p 2  q 2 1  ps 2  qs 2
I0 2 (1  p 2  q 2 )(1  ps 2  qs 2 )  (1  pps  qqs )2
ps  0, qs  0
1
p  q  2 1
I0
2
November 9, 1998
2
8
LAMBERTIAN REFLECTANCE MAP
ps=0.7 qs=0.3
I0 2 (1  p 2  q 2 )(1  ps 2  qs 2 )  (1  pps  qqs )2
ps= -2 qs= -1
November 9, 1998
9
PHOTOMETRIC STEREO
Derivation of local surface normal at each pixel
creates the derived normal map.
November 9, 1998
10
November 9, 1998
11
SHAPE FROM SHADING
(Calculus of Variations Approach)
• First Attempt: Minimize error in agreement
with Image Irradiance Equation over the
region of interest:
2
(
I
(
x
,
y
)

R
(
p
,
q
))
dxdy

object
November 9, 1998
12
SHAPE FROM SHADING
(Calculus of Variations Approach)
• Better Attempt: Regularize the Minimization of
error in agreement with Image Irradiance Equation
over the region of interest:
2
2
2
2
2
p

p

q

q


(
I
(
x
,
y
)

R
(
p
,
q
))
dxdy
x
y
 x y
object
November 9, 1998
13
TEXTURE CLASSIFICATION PROJECT
• Characterize each texture so as to
differentiate it from one another.
• Probably should use 2nd order cooccurrence matrices and their respective
properties. Examine what is a good
neighborhood size to work with and
how many grey levels for computation
of co-occurrence matrix.
• Might want to examine eigenvalue
invariants of symmetric co-occurrence
matrix but not absolutely necessary.
• May have to look at ‘ranges of values’
that differentiates the class for each
texture.
November 9, 1998
14
MEDIAL AXIS SKELETONIZATION
PROJECT
• Derive an algorithm for implementing
the medial axis skeleton in an image.
• Implement this on a segmentation of
labeled connected components in the
image on the right.
• Derive an algorithm for computing the
number of holes in a connected
component and implement it on the
image on the right.
November 9, 1998
15
PHOTOMETRIC STEREO PROJECT
• Compute the normal map for part of a
sphere with Phong reflectance model
from 4-light source photometric stereo
(use a=0.3, b=0.7, n=3.0).
• Need to compute each surface normal
from 3-light sources that mutually
illuminate a surface point with primarily
diffuse reflection.
• Arrange 4-light sources so as to
maximally cover as much of the sphere
as possible with mutual illuminaton
(WARNING: Placing light sources too
close together produces measurement
errors).
• Integrate the normal map to produce a
depth map.
November 9, 1998
PHONG MODEL
n
E = L (a COS q  b COS a)
Diffuse
albedo
Specular
albedo
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