Recognition I: Extended Gaussian Images

1
Recognition IExtended Gaussian Images

• Andrew Nashel
• COMP 290-075 Computer Vision
• http//www.cs.unc.edu/nashel/290-075/

2
Overview

• Motivation
• Gaussian Image
• Extended Gaussian Image
• Gaussian Curvature
• Extended Circular Image
• Complex EGI

3
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

4
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

5
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.

6
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

EGI of Block
Block
7
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.

8
Problems with the EGI

• EGIs only uniquely define convex objects.
• An infinite number of non-convex objects may have
  the same EGI

Areas abc def
e
c
9
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

10
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

• 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.

11
Defining Curvature Continued

• If we integrate over a patch O on the object we
  have
• 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.

??O K dO ??S dS S
12
Defining Curvature Continued

• Similarly, if we integrate over a patch S
• 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.

??S 1/K dS ??O dO O
13
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
• the inverse of the Gaussian curvature, where
  (?,?) is the
• point on the Gaussian sphere corresponding to the
  
• point (u,v) on the object.

14
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.

15
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

16
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

17
The Extended Circular Image

• The extended circular image is the 2-D equivalent
  of the extended Gaussian image.

18
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/eci
  morph/www/code/MMorph.html

19
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.

20
The Complex EGI

• We see that it is a simple modification to handle
  displacement determination

21
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?

22
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/ma
  noli/ecimorph/www/ecimorph.html
• Kang, S.B. and K. Ikeuchi. 1990. 3-D Object Pose
  Determination Using Complex EGI. tech. report
  CMU-RI-TR-90-18, Robotics Institute, Carnegie
  Mellon University.