Abstract
We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the RL integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one.
Highlights
The theory of multifunctions is an important field of research
Since interval arithmetic, introduced by Moore in [1], it appears a natural option for handling the uncertainty in data and in sensor measurements, particular attention was addressed to the study of interval-valued multifunctions and multimeasures because of their applications in statistics, biology, theory of games, economics, social sciences and software, to keep track of rounding errors in calculations and of uncertainties in the knowledge of the exact values of physical and technical parameters
The advantage of using the notion of interval-valued multifunction in signal analysis is that this formalism allows to include in a unique framework possible uncertainty or the noise on the evaluation of an image at any given pixel
Summary
The theory of multifunctions is an important field of research. Since interval arithmetic, introduced by Moore in [1], it appears a natural option for handling the uncertainty in data and in sensor measurements, particular attention was addressed to the study of interval-valued multifunctions and multimeasures because of their applications in statistics, biology, theory of games, economics, social sciences and software, to keep track of rounding errors in calculations and of uncertainties in the knowledge of the exact values of physical and technical parameters (see for example [2,3,4,5]). In the literature several methods of integration for functions and multifunctions have been studied extending the Riemann and Lebesgue integrals In this framework a generalization of Riemann sums was given in [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] while another generalization is due to Kadets and Tseytlin [38], who introduced the absolute Riemann-Lebesgue | RL| and unconditional Riemann-Lebesgue RL integrability, for Banach valued functions with respect to countably additive measures. The advantage of using the notion of interval-valued multifunction in signal analysis is that this formalism allows to include in a unique framework possible uncertainty or the noise on the evaluation of an image at any given pixel
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