Elsevier

Information Sciences

Volume 209, 20 November 2012, Pages 61-79
Information Sciences

Evolving estimators of the pointwise Hölder exponent with Genetic Programming

https://doi.org/10.1016/j.ins.2012.04.043Get rights and content

Abstract

The regularity of a signal can be numerically expressed using Hölder exponents, which characterize the singular structures a signal contains. In particular, within the domains of image processing and image understanding, regularity-based analysis can be used to describe local image shape and appearance. However, estimating the Hölder exponent is not a trivial task, and current methods tend to be computationally slow and complex. This work presents an approach to automatically synthesize estimators of the pointwise Hölder exponent for digital images. This task is formulated as an optimization problem and Genetic Programming (GP) is used to search for operators that can approximate a traditional estimator, the oscillations method. Experimental results show that GP can generate estimators that achieve a low error and a high correlation with the ground truth estimation. Furthermore, most of the GP estimators are faster than traditional approaches, in some cases their runtime is orders of magnitude smaller. This result allowed us to implement a real-time estimation of the Hölder exponent on a live video signal, the first such implementation in current literature. Moreover, the evolved estimators are used to generate local descriptors of salient image regions, a task for which a stable and robust matching is achieved, comparable with state-of-the-art methods. In conclusion, the evolved estimators produced by GP could help expand the application domain of Hölder regularity within the fields of image analysis and signal processing.

Highlights

Genetic Programming is applied to the problem of Hölder regularity estimation for digital images. ► Evolved operators provide an accurate estimaton that can be computed in real-time. ► Hölder regularity is applied to local image description and matching.

Introduction

Image analysis and understanding entails the detection and extraction of meaningful descriptive features to carry out higher level tasks such as object recognition [47] or image indexing [61], two common examples. For most application, the most prominent and informative parts of an image correspond with regions that exhibit an irregular structure with a high amount of local variation. Therefore, many works have addressed the problem of detecting and describing salient image regions [38], [60], [57], [58]. One approach towards describing local shape and appearance within an image is through the concept of signal regularity, an approach that can characterize the singularities contained within non-differentiable signals [11], [35]. Regularity-based analysis has been used to describe local image patches [29], [58] and to detect salient regions [31].

In this paper, we focus on Hölder regularity, quantifies the amount of regularity any given point on a signal, using what is known as the pointwise Hölder exponent [11], [35] (see Section 3.1 for a formal definition). The Hölder exponent has proven to be a useful tool for image analysis [29], [58], [31], but computing the exponent is not a trivial task. In fact, closed form solutions only exist for a narrow class of functions, while for real-world signals the exponent must be estimated. Therefore, several estimation methods have been proposed, derived from a formal analysis of the Hölder exponent, using techniques from fractal theory and signal processing [53], [16], [31], [2]. However, some of these estimators are based on necessary assumptions regarding the underlying structure of the signal. Moreover, to use and develop practical implementations of current methods a system designer must make several parametric choices and ad hoc decisions. Furthermore, these estimators usually require a large run-time, which limits their use for systems that operate in real-time. Therefore, we pose the following research question: Can the pointwise Hölder exponent be estimated using an image operator that achieves an accurate estimation using a simple and fast algorithm? We believe that if such an operator exists, it can open new application domains for regularity-based image analysis.

Therefore, in this work the goal is to find operators that can provide a positive answer to the above question. To achieve this, we pose a search/optimization problem and solve it using Genetic Programming (GP). Over the past two decades, GP has proven to be a powerful paradigm for the development of computer algorithms that automatically synthesize solutions for complex tasks. Unlike black-box methods, when a search is properly framed, GP can produce solutions that are amenable to further analysis [57], [41], [18]. Moreover, GP is also quite flexible, with successful applications in various domains [21], [22], [19], including image analysis [13], [57], [39], [44], computer vision [25], [7], [40], [34] and applied mathematics [5], [15], [51]. The power and flexibility of GP comes from the fact that it solves two tasks simultaneously: (1) it searches for the desired functionality and (2) determines the structure of the solution [20], [26].

For these reasons, we use GP to search for specialized image operators that estimate the pointwise Hölder exponent for digital images. Experimental results show that GP is capable of evolving highly competitive estimators, that approximate the estimation produced by a traditional approach, the oscillations method, with a small error and a high correlation. Moreover, the GP estimators are efficient, in terms of computation time they achieve a 50% improvement with respect to one estimation method, and several orders of magnitude with respect to other approaches. In fact, a noteworthy result is that GP found a novel computational operator that extracts a measure of image regularity that can be implemented in real-time. The quality of the evolved estimators is verified using a common problem of modern computer vision, the description and matching of local image features [38]. In this task, the evolved estimators can be used to construct meaningful and discriminative local descriptors that compare favorably with the original Hölder-based descriptor [58], [55].

The remainder of this paper proceeds as follows. Section 2 contains a brief overview of related works and applications of GP. Then, Section 3 gives a brief introduction to Hölder regularity and GP. Section 4 poses the task of estimating the Hölder regularity of an image as an optimization problem, and presents a GP approach to solve it. Experimental results are detailed in Section 5, and qualitative and quantitative comparisons are made between the evolved estimators and traditional methods. Moreover, in Section 6 the evolved estimators are used to build local image descriptors and are compared with the local Hölder descriptor of [58]. Finally, a summary and concluding remarks are outlined in Section 7.

Section snippets

Related work

Image analysis encompasses a diverse group of complex problems, where the relationships between the input signal and desired output are poorly understood, closed-form analytical solutions normally do not exist, and the structure of the desired solution cannot de defined a priori. Conversely, a large amount of experimental data is usually available or easy to obtain. These characteristics makes problems in this field appropriate candidates for GP-based solutions [25], [7]. For instance, GP has

Theoretical background

The aim of this section is to present a concise introduction to Hölder regularity and the GP paradigm, the research areas that intersect in this paper. However, some details are omitted for brevity, but the interested reader should refer to [53], [43] for a detailed introduction to Hölder regularity and GP respectively.

Problem statement

Let I represent a digital 2D signal, or more specifically an image, and suppose that HI is a matrix that contains the value of the pointwise Hölder exponent for every pixel in I. Then, we can pose the problem of finding an optimal operator Ko as follows,Ko=argminK{Err[K(I),HI]},where Err[,] represents an error measure. In previous work [54], HI was set using a prescribed regularity function, and synthetic images of multifractional Brownian motions that share the same underlying regularity were

Experimental results

This section provides a detailed description of the GP system, the experimental results, and comparisons with traditional estimators.

Application to local image description

Despite the encouraging results presented above, a question remains: can the HGP estimators provide a useful estimation for higher level applications? Here, this question is addressed by applying the evolved operators on a difficult computer vision problem.

Recently, many computer vision systems are based on the detection and description of local and sparse image features. The approach was introduced in [45], [33] and consists of the following basic steps. First, small image regions centered

Summary and concluding remarks

In this paper, the task of developing a new estimator of image regularity is posed as an optimization problem and solved using Genetic Programming. The goal is to synthesize image operators that can approximate the oscillations method for Hölder exponent estimation. Additionally, the evolved estimators should also be simpler, easier to implement, and exhibit a lower runtime. Indeed, a standard implementation of GP was able to solve this problem and satisfy the desired criteria. This allowed us

Acknowledgements

The authors thank Bordeaux University, the Institut de Mathématiques de Bordeaux, INRIA ALEA team and all the people involved in the invited professor program who selected and supported the first author during the development of this research. Thanks are also given to the support provided by the Departamento en Ingeniería Eléctrica y Electrónica from the Instituto Tecnológico de Tijuana. Finally, this research was partially funded by CONACYT México, through Project 155045 – ”Evolución de

References (62)

  • R.M.A. Azad et al.

    Abstract functions and lifetime learning in genetic programming for symbolic regression

  • P. Balasubramaniam et al.

    Solution of matrix riccati differential equation for nonlinear singular system using genetic programming

    Genetic Programming and Evolvable Machines

    (2009)
  • H. Barnum et al.

    Quantum circuits for or and of ors

    Journal of Physics A: Mathematical and General

    (2000)
  • O. Barriere, Synthése et estimation de mouvements browniens multifractionnaires et autres processus á régularité...
  • S. Cagnoni, E. Lutton, G. Olague, (Eds.), Genetic and Evolutionary Computation for Image Processing and Analysis,...
  • K. De Jong

    Evolutionary Computation: A Unified Approach

    (2001)
  • M. Ebner

    A real-time evolutionary object recognition system

  • K. Falconer

    Fractal Geometry: Mathematical Foundations and Applications

    (1990)
  • S. Gustafson, E.K. Burke, N. Krasnogor, On improving genetic programming for symbolic regression, in: Proceedings of...
  • S. Jaffard, Wavelet techniques in multifractal analysis, in: Fractal Geometry and Applications: A Jubilee of Benoit...
  • M. Keijzer

    Scaled symbolic regression

    Genetic Programming and Evolvable Machines

    (2004)
  • M. Keijzer et al.

    Declarative and preferential bias in gp-based scientific discovery

    Genetic Programming and Evolvable Machines

    (2002)
  • J. Koza

    Human-competitive results produced by genetic programming

    Genetic Programming and Evolvable Machines

    (2010)
  • J.R. Koza

    Genetic Programming: On the Programming of Computers by Means of Natural Selection

    (1992)
  • J.R. Koza et al.

    Automatic creation of human-competitive programs and controllers by means of genetic programming

    Genetic Programming and Evolvable Machines

    (2000)
  • K. Krawiec

    Genetic programming-based construction of features for machine learning and knowledge discovery tasks

    Genetic Programming and Evolvable Machines

    (2002)
  • K. Krawiec et al.

    Visual learning by coevolutionary feature synthesis

    IEEE Transactions on Systems, Man, and Cybernetics, Part B

    (2005)
  • K. Krawiec, D. Howard, M. Zhang, Overview of object detection and image analysis by means of genetic programming...
  • W.B. Langdon et al.

    Foundations of Genetic Programming

    (2002)
  • P. Legrand, Debruitage et interpolation par analyse de la regularite hölderienne, Application a la modelisation du...
  • P. Legrand, J. Lévy-Véhel, Local regularity - based image denoising, ICIP03, Spain, IEEE International Conference on...
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