Transcript Recognition I: Extended Gaussian Images Andrew Nashel COMP 290-075: Computer Vision

Recognition I:
Extended Gaussian Images
Andrew Nashel
COMP 290-075: Computer Vision
http://www.cs.unc.edu/~nashel/290-075/
Overview
Motivation
 Gaussian Image
 Extended Gaussian Image

– Gaussian Curvature
– Extended Circular Image

Complex EGI
Motivation


Object recognition is often one of the ultimate goals
for vision systems.
It is necessary for real world interactions such as:
– Navigation through environments
– Robotic handling of objects
– Object inspection

It brings together many components of computer
vision:
– Depth extraction
– Image segmentation
– Geometric modeling
The Gaussian Image

Surface normal information for any object may be
mapped onto a unit (Gaussian) sphere by finding the
point on the sphere with the same surface normal:
n̂
~
n̂
~
Properties of the Gaussian Image

This mapping is called the Gaussian image of the
object when the surface normals for each point on the
object are placed such that:
– tails lie at the center of the Gaussian sphere
– heads lie on the sphere at the matching normal point



In areas of convex objects with positive curvature, no
two points will have the same normal.
Patches on the surface with zero curvature (lines or
areas) may correspond to a single point on the sphere.
Rotations of the object correspond to rotations of the
sphere.
The Extended Gaussian Image

We can extend the Gaussian image by
– placing a mass at each point on the sphere equal to the area
of the surface having the given normal
– masses are represented by vectors parallel to the normals,
with length equal to the mass

An example:
Block
EGI of Block
Using the EGI



EGIs for different objects or object types may be
computed and stored in a model database as a
surface normal vector histogram.
Given a depth image, surface normals may be
extracted by plane fitting.
By comparing EGI histogram of the extracted
normals and those in the database, the identity and
orientation of the object may be found.
Problems with the EGI


EGIs only uniquely define convex objects.
An infinite number of non-convex objects may have
the same EGI:
Areas: a+b=c
d+e=f
b
d
b
d
e
a
e
a
e
c
Gaussian Curvature

Formally, we will develop the extended Gaussian
image based upon the Gaussian curvature of the
object.

Consider a patch of area
O on the object, and the
corresponding area S on
the Gaussian sphere:
Defining Gaussian Curvature

Given patches O and S, we define Gaussian
curvature K as the limit of the ratio of the two areas
as they approach zero:
dS
S
K = lim
=
O0 O
dO
– If the object surface is strongly curved, then the corresponding
points on the Gaussian sphere will be spread out.
– If the surface is planar, the normals will be parallel and will
map to a single point on the Gaussian sphere.
Defining Curvature Continued

If we integrate over a patch O on the object we have:
∫∫O K dO = ∫∫S dS = S
where S is the area of the patch on the Gaussian sphere.

We call the expression on the left the integral
curvature. This allows us to handle surfaces with
discontinuities in surface normals.
Defining Curvature Continued

Similarly, if we integrate over a patch S:
∫∫S 1/K dS = ∫∫O dO = O
where O is the area of the patch on the object.

This relationship suggests that the inverse of the
curvature will be used to define the extended
Gaussian image.
Defining EGI


Let u and v be used to specify points on the original
surface, and let  and  specify points on the
Gaussian sphere.
We now define the extended Gaussian image as:
G(,) =
1
K(u,v)
the inverse of the Gaussian curvature, where (,) is the
point on the Gaussian sphere corresponding to the
point (u,v) on the object.
The Discrete Case EGI

To represent the information of the Gaussian sphere
in a computer, the sphere is divided into cells:

For each image cell on the left, a surface orientation
is found and accumulated in the corresponding cell of
the sphere.
Discrete Approximation

In an actual implementation of a discrete EGI, we
start with a surface orientation map.

Shown here is a needle
diagram of an inclined
torus obtained by
photometric stereo:
Orientation Histogram

The discrete approximation of the EGI is called the
orientation histogram.

The needle diagram of the
torus is projected onto a
tessellated unit sphere to
create an orientation
histogram, displayed as a
set of spikes:
The Extended Circular Image

The extended circular image is the 2-D equivalent of
the extended Gaussian image.
Polygon Morphing with the ECI

An alternative to pixel-based morphing algorithms for
convex polygons:
– First compute the ECI representation of the source and
target polygons.
– Match source and target normals on the ECI circle to create
source-target pairs.
– Interpolate weights and angles between pairs to find the ECI
of intermediate steps.
– Reconstruct the convex polygon from the ECI.

Java implementation:
http://web.mit.edu/manoli/ecimorph/www/code/MMorph.html
The Complex EGI


Another problem with the EGI is that the weights in
the representation only encode area information and
not positional data, thus it is impossible to determine
translation.
The Complex EGI is an alternative formulation in
which the weight at each discrete cell is a complex
number:
– The magnitude at each cell is the surface area as in the
standard EGI.
– The phase is the signed distance of the surface patch from a
designated origin along the normal.
The Complex EGI

We see that it is a simple modification to handle
displacement determination:
Conclusions


The extended Gaussian image is a useful technique
for representing the shape of an object, and even its
position (complex EGI).
However, it is only a component tool to be used in a
shape recognition process which also includes:
–
–
–
–
Surface orientation determination
Image segmentation into objects
Prototypical models/object databases
System control - what object to handle/inspect?
References




Horn, B.K.P. 1984. Extended Gaussian images. In Proceedings
of the IEEE 72, 12 (Dec.), pp. 1656-1678.
Horn, B.K.P. 1986. Robot Vision. MIT Press, Cambridge, MA,
pp. 365-399.
Kamvysselis, M. 1997. 2D Polygon Morphing using the
Extended Gaussian Image.
http://web.mit.edu/manoli/ecimorph/www/ecimorph.html
Kang, S.B. and K. Ikeuchi. 1990. 3-D Object Pose
Determination Using Complex EGI. tech. report CMU-RI-TR-9018, Robotics Institute, Carnegie Mellon University.