Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

epipolar planes meet the image planes at epipolar lines. From the epipolar
geometry of fig. 17.13, it is clear that for a given image point, there must be a
corresponding point on the second image. Thus if the image point in the first
image is shifted, the corresponding point will also be shifted on the epipolar
line of the second image.
Finding the corresponding points in images is generally referred to as
the correspondence problem [3]. There are many approaches to handle the
correspondence problem. In this book we will first determine the 3-D points of
an object from its 2-D image points and then for more than two images
determine the correspondence by measuring the shortest Euclidean distance
between the point sets of the two or more images. If the 2-D to 3-D mapping of
the points are satisfactory and the images consist of common points, then
determining the correspondence between the points of two or more images is
not a complex problem. For determining the 3-D points from their 2-D images
we, in this book, will use Kalman filtering [1]. However, before introducing
Kalman filtering, we briefly outline the minimal representation of 2-D lines, 3-
D lines and 3-D planes. With a minimal representation, we can extract 3-D
points from multiple 2-D points in different images by using Kalman filtering.
We use Kalman filtering because it has an advantage of recursively operating
on incoming data stream like 2-D points from n number of images. The more is
the value of n, the better will be the accuracy of the results. The other least
square estimators, unlike Kalman filtering, demand all the data set together; so
the user has no choice to control the level of accuracy at the cost of
computational time. In Kalman filtering, one can observe the improvement in
accuracy in the estimation of the parameter of lines or planes and accordingly
decide about the submission of new data points.
17.4.2.3 Minimal Representation
of Geometric Primitives
For estimation of parameters of 2-D lines, 3-D points, 3-D lines and 3-D
planes, we first represent them with minimal parameters. Further the
selected representation should be differentiable, so that we can employ the
principles of Linear Kalman filtering.
Representation of Affine lines in R 2 : A 2-D line can be represented by
at least two independent parameters. The simplest form of representation of a
2-D line is given by the following expressions.
Case 1: When the lines are not parallel to the Y-axis, they are represented
by
a x + y + p = 0 (17.15a)