A Probabilistic Approach to Geometric Hashing using Line Features

CHAPTER 2. PRIOR AND RELATED WORK 21
is, the measured values are assumed to be distributed according to a Gaussian distribution
centered at the true values and having standard deviation ç. More precisely, let èx i ;y i èbe
the ëtrue" location of the i-th feature point in the scene. Let also èX i ;Y i è be the continuous
random variables denoting the coordinates of the i-th feature. The joint probability density
function of X i and Y i is then given by:
fèX i ;Y i è= 1
2çç 2 expè,èX i , x i è 2 +èY i , y i è 2
2ç 2 è:
Using this error model, he formulates a weighted voting scheme for the evidence accumulation
in geometric hashing.
He uses a weighted contribution to the modelèbasis
hypothesis of an entry ç in the hash space based on a scene point's hash ç in the same
space, using a formula of the form
logë1 , c + A expè, 1 2 èç , çèæ,1 èç , çè t èë;
where c and A are constants that depend on the scene density and æ is the covariance
matrix in hash space coordinates of the expected distribution of ç based on the error
model for èx; yè, the feature point in the scene.
He shows that when weighted voting is used according to the above formula, the evidence
accumulation can be interpreted as a Bayesian aposteriori classiæcation scheme. He
shows that the accumulations are related to
logëProbèH k jE 1 ; æææ;E n èë
where the hypothesis H k represents the proposition that a certain modelèbasis occurs in
the scene and matches the chosen basis, and E i 's are pieces of evidence given by scene
invariants.
The above formulation is implemented on a 8K-processor Connection Machine èCM-
2è and can recognize objects that have undergone a similarity transformation, from a
library of 32 models. The models used for experiments are military aircraft and production
automobiles. Features are extracted by using the Boie-Cox edge detector ë9ë and locating
the points of high curvature. The features are the coordinate pairs of these points ë44ë.
In this thesis, we also make use of a Bayesian aposteriori maximum likelihood recognition
scheme based on geometric hashing. We use somewhat simpliæed versions of the
formulae given by Rigoutsos. We make use of line features as opposed to point features.