## Section: New Results

### Estimation of Hölderian regularity using Genetic Programming

Participants : Leonardo Trujillo, Pierrick Legrand, Jacques Levy-Vehel, Gustavo Olague.

The concept of Hölderian regularity allows us to characterize the singular structures contained within a signal [32] . A quantitative measure of the regularity of a signal can be obtained from measuring Hölder exponents, either within a local region of the signal or at each point. In our work, we are interested in measuring the pointwise Hölder exponent which is defined below.

**Definition** Let f: , and x_{0} .
If , and a polynom P of degree <s
and a constant c exist such that

then the pointwise Hölder exponent of f at x_{0} is
.

However, this exponent is very costly to estimate in the common framework (oscillations or wavelets). Therefore, we will propose a new estimator of this exponent.

Over the past two decades, Genetic Programming (GP) has proven to be a powerful paradigm for the development of computer systems that that are able to solve complex problems without the need for substantial amounts of a priori knowledge. Indeed, GP solves two tasks simultaneously because it can search for the desired functionality and also determine the structure that the final solutions possess. The former is achieved by using the basic principles of artificial evolution, a stochastic search process that is guided by fitness and which uses simple variation operators. On the other hand, the latter is produced by eliminating strict structural restrictions on the evolving population that most evolutionary techniques require, as well as other black-box methods such as neural nets [30] , [31] . The overall flexibility of GP has allowed researchers to apply it to a very diverse set of problems from different disciplines [38] . Therefore, we have chosen GP as our search and optimization algorithm to produce novel estimators of Hölderian regularity for digital images.

We obtained some new results listed below.
1) We studied the problem of estimating a measure of Hölderian regularity using a GP-based search, which is the first such study. 2) We have successfully produced several operators that are capable of extracting good estimations of the pointwise Hölder exponent when compared with a canonical estimator. 3) The evolved estimators do not require parameter tuning or design choices, because this is implicitly carried out
during the optimization process. Hence, our evolved operators provide a *simpler* , in this sense, estimation method.4) We have applied our evolved operators to the problem of local image description, and we show that several of them
achieve a comparable performance when compared with a canonical estimation method [39] .