Kernel-Based Adaptive Image Retrieval Methods - Multimedia Database Retrieval: Technology and Applications - page 52

( , ) ( )

Regardless of the approach to estimating P c I and P I , there are two problems

( | )

with above estimation of P . First, the background information I may include

different numbers of indexes, which requires separate estimations of the model for
different sizes of I . Second, when collecting training data, we cannot guarantee
enough or even available samples for a certain configuration of c and I , where the
configuration refers to a particular instantiation of the number of random variables
of c

c I

∪

I and their values.
To deal with the estimation of the context model efficiently, the P

( | )

is
approximated using a distribution of a set of binary random variables estimated
based on the maximum entropy (ME) principle. In this approach, an image is
represented using a C -dimensional vector of binary random variables, denoted
Y

c I

t , where the value of each variable Y c is defined by

=( , ,..., )

Y 1 Y 2 Y C

1ift e c -th image is relevant to a query
0

Y c = (2.86)

otherwise

1 C 's, the data utilized by the
context modeling procedure belong to the set of vertices of a C -dimensional hyper-
cube. Given a set of N training samples, denoted Y 1 ,

Instead of being from the Cartesian product of | I | +

Y 2 ,..., Y N , we can estimate the

and then calculate the conditional probability P Y c |

, which is

P ( Y ) Y I 1 , Y I 2 ,..., Y I | I |

represented as P ( Y c | Y I ) in what follows. To approximate the P ( c | I ) in Eq. ( 2.82 ),

the following formula is utilized

P ( Y c | Y I )

P ( c | I )= (2.87)

C
v 1 P

( Y v | Y I )

∑

=

As the size of the concept ensemble, i.e. C , grows, the computational intensity
of the calculation of P

( |

Y I ) increases exponentially. Therefore, it would be more

efficient if we can directly estimate P

Y c

( |

Y I ) based on a set of training samples.

To this end, the ME approach demonstrated in [ 50 ] is employed, which estimates
a conditional distribution by maximizing its Rényi entropy. Essentially, the ME
principle states that the optimal model should only respect a certain set of statistics
induced from a given training set and otherwise be as uniform as possible. The
ME approach searches for the conditional distribution P

Y c

( |

Y I ) , with the maximum

entropy, among all the distributions which are consistent with a set of statistics
extracted from the training samples. Therefore, it can be considered as constrained
optimization, which is formulated as

Y c

P

2

] − y c , y I

max ( Y I = y I ) P ( Y c = y c | Y I = y I ) , (2.88)

P ( Y c | Y I ) ∈ [ 0 , 1