17-12-2012, 05:42 PM
Polygonal Approximation of Closed Curves across Multiple Views
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Abstract
Polygon approximation is an important step in the recognition
of planar shapes. Traditional polygonal approximation
algorithms handle only images that are related by a similarity
transformation. The transformation of a planar shape as
the viewpoint changes with a perspective camera is a general
projective one. In this paper, we present a novel method
for polygonal approximation of closed curves that is invariant
to projective transformation. The polygons generated by
our algorithm from two images, related by a projective homography,
are isomorphic. We also describe an application
of this in the form of numeral recognition. We demonstrate
the importance of this algorithm for real-life applications
like number plate recognition and aircraft recognition.
Introduction
We recognize a large number of familiar and novel objects
every day with little effort. We can also recognize many
objects that may vary significantly in form when viewed
from different view points. Objects can be recognized even
when they are partially obstructed. The recognition of objects
from different views is a well studied problem in computer
vision. Most of the literature, however, is confined to
similarity transformations between different views, and not
the general case.
Planar objects form a subset of all objects but are of great
practical interest. We use several planar objects every day,
such as books, drawings, cloth, etc. Many non planar objects
can be approximated by planar ones when viewed from
sufficiently far. Many planar object recognition algorithms
represent the boundary object using a single parameter vector.
Polygonal approximation – i.e, approximating a given
closed curve as a 2D polygon – provides a simple representation
of the planar object boundary. Parameterizing the
boundary using a polyline representation makes recognition
easy. Polygonal approximation has also been used as an intermediate
step in various applications such as volume rendering
and multiresolution modeling [3, 12].
Preliminaries
Image-to-Image Homographies
When an object is imaged from multiple viewpoints, the
points on the planar object boundary (and also the points
inside) undergo a transformation. The transformation that
the coordinates of each point of a plane undergoes from one
image to the other can be mathematically described as a
general projective or linear operation in homogeneous coordinates.
Results
The results of the proposed algorithm on two planar boundaries
are shown in the Figure 2. The results were obtained
on images of size . We considered only the boundary
pixels for our algorithm. The original image is shown
on top with two projectively transformed versions below
it. The boundary pixels are drawn in black colour. The
red lines in the figure show the polygonal approximation of
the boundary using our algorithm. The polygon approximation
has twenty sides. The number of sides of the polygon
and their relative position with respect to the object remain
same across all projective transformations we have considered.
To quantitatively compare the results, we projectively
transformed the polygonal approximation of the original
image by the same projective transformations shown as
green lines. The green and red polygons in the other figures
are identical in all cases except one or two nodes which got
shifted by 1-2 pixels due to discretization. Another example
is shown on the right in Figure 2. Here an aircraft image is
polygonal approximated with a twenty five sided polygon.
The boundary of the aircraft is shown in black and the red
lines show the polygonal approximation of the boundary.
Our experience with other planar boundaries is also very
good.
Numeral Recognition
An important application of polygonal approximation is
planar object recognition [7, 10]. We demonstrate the applicability
of this algorithm for recognition of numerals across
multiple views. Numeral recognition has been conventionally
addressed among the document image processing community.
There are many other situations in image and video
processing where the numerals are to be recognized under
projective transformation. Conventional OCRs are not designed
to address this problem. We considered numeral images
of size for this experiment. Thirty four different
views of each numeral were considered for experimentation.
Thirty of these images used as inputs are shown
in Figure 4.
Application Domains
Number Plate Recognition: There are many number
plate detection algorithms for images and videos. Coupled
with this we need a module which can recognize alphanumerals
under projective transformation. In a number
plate recognition system, the input is a projectively transformed
image of numerals and alphabets, as shown in Figure
5 .It’s objective is to identify the number on the number
plate. To test the feasibility of number plate recognition using
our algorithm, we analyzed a few real images of number
plates. The test was limited to recognition of digits and can
be easily extended for alphabets as well. A sample polygonal
approximation is shown in Figure 6. We are yet to test
this system on a large database of number plates.
Conclusions and Future Work
We presented an approach to generate a projectively invariant
polygonal approximation using invariant properties of
the cross-ratio of areas. We demonstrated how planar shape
recognition can be achieved using this algorithm. This can
be applied to real-life problems such as number plate recognition
and aircraft recognition. We plan to use polygon approximation
for object recognition with occlusion. Also,
other applications have to be explored like shape from texture.
Repeated texture elements can be polygon approximated
and may be used for reconstruction of shape.