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Relative Magnitude of Gaussian
Curvature from Shading Images
Using Neural Network
Yuji Iwahori
1
, Shinji Fukui
2
,
Chie Fujitani
1
, Yoshinori Adachi
1
and
Robert J. Woodham
3
1
Chubu University
, Japan
2
Aichi University of Education, Japan
3
University of British Columbia, Canada
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Introduction
Surface Curvature is the invariant
feature for the viewing direction.
It can be used to many
applications in the field of
computer vision.
Using a fixed camera, multiple
light sources, that is, multiple
shading images are used as
input.
Empirical implementation with
the neural network has been
performed to obtain the relative
magnitude of Gaussian
Curvature.
2D Shading
Images
Recovering
Gaussian
Curvature
for 3D Shape
description
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Previous Approaches (1)
[Woodham, 1994]
Look Up Table (LUT) based Photometric
Stereo:
Empirical estimation of surface gradient and
surface curvature under the different light
source directions.
[Iwahori, Woodham, 1995]
Neural network implementation of
Photometric Stereo (to obtain the surface
orientation)
with PCA (principal component analysis) to the co

linear light sources to remove the correlation
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Previous Approaches (2)
[Angelopoulou and Wolff, 1998]
Recovering the sign of Gaussian curvature
without knowing the surface gradient
[Okatani and Deguchi, 1998]
also formulated the method to recover the
sign of Gaussian curvature.
These approaches are applicable for the diffuse
reflectance. Also, the sign of GC is simple
but limited information.
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New Proposed Approach
Extension of [Iwahori, Fukui, Woodham, 1998]
which proposed the classification of surface
curvature using neural network.
1.
Empirical approach using Radial Basis
Function neural network (
RBF

NN
) to perform
the non

parametric functional approximation.
2.
In addition,
the relative magnitude of Gaussian
curvature
could be obtained for the general
surface reflectance (not limited to the diffuse
reflectance).
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Principle (Empirical Constraint)
Orthographic projection is assumed.
Test object and a calibration sphere with the
same reflectance property, are observed. As
far as the observed image irradiance
(intensity) vector is the same for both objects,
the corresponding surface normal vector
should be the same.
Three shading images with different light
sources are used to determine the curvature
sign and the relative magnitude.
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RBF neural network
The corresponding coordinate (x,y) on a
sphere which has the same triple of the
image irradiance for a test object and a
sphere can used to obtain the information of
the surface curvature.
RBF neural network is used to realize this
purpose.
Features:
Non

parametric functional approximation
(interpolation) in multidimensional spaces
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RBF neural network
E
1
E
2
E
3
S
S
x
sph
y
sph
f
(
E

c
m
)
f
(
E

c
1
)
f
(
E

c
2
)
RBF NN learns the mapping of (E1,E2,E3) to (x,y) of each
point on a sphere. After the learning, given the triple of
(E1,E2,E3) of local five points on a test object, we get the
corresponding (x,y) onto a sphere.
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Local Surface Curvature
(a)
Convex
(b) Concave
(d) plane
(e) convex
parabolic
(f) concave
parabolic
(c) hyperbolic
G>0
G>0
G<0
G=0
G=0
G=0
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Mapping onto Sphere by Neural Net
(1) Local convex
surface
(2) Local concave
surface
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Mapping onto Sphere by Neural Net
(4) Local plane
(3) Local hyperbolic
surface
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Mapping onto Sphere by Neural Net
(5) Convex parabolic
surface
(6) Concave parabolic
surface
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Relative Magnitude of Gaussian
Curvature
G
It can be estimated from the area value
surrounded by four mapped points onto
a sphere. Let be the
height,
This is the definition of G.
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Relative Magnitude of Gaussian
Curvature
G
The area value
S
surrounded by four mapped points is
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Experiments (Procedures)
Three shading images are used for a test
object and a sphere with the same
reflectance property,
Input images are taken under the same
illuminating conditions for both a test object
and a sphere.
Neural network learning is done for a sphere.
Generalization for a test object is done..
Classification and the Encoding the relative
magnitude are done.
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Experiments (Results)
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Evaluation (Simulation)
Normalized true distribution
of Gaussian curvature from
Hessian matrix of 2

D Sinc
Function
Normalized estimated
distribution of Gaussian
curvature based on the
proposed approach
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Evaluation (Accuracy)
Red curve is
true
, while
blue curve is
estimated
curve of the
average
cross section.
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Conclusion
A new method is proposed to recover the
relative magnitude of Gaussian curvature with
the neural network implementation.
Entire approach is empirical under the
condition that no explicit assumptions are not
used for the surface reflectance function nor
the illuminating directions.
Robust results are directly obtained in
addition to the classification of local surfaces.
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Further works
Extension without using any calibration
object.
For a test object with multiple color
textures.
For cast

shadow or inter

reflection.
Thank you
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