Abstract
The concepts of convolution and deconvolution are well known in the field of physical measurement. In particular, they are of interest in the field of metrology, since they can positively influence the performance of the measurement. Numerous mathematical models and computer developments dedicated to convolution and deconvolution have emerged, enabling a more efficient use of experimental data; this in sectors as different as biology, astronomy, manufacturing and energy industries. The subject finds today a new topicality because it has been made accessible to a large public for applications such as processing photographic images. The purpose of this paper is to take into account some recent evolutions such as the introduction of convolution methods in international test standards. Thus, its first part delivers a few reminders of some associated definitions. They concern linear systems properties, and integral transforms. If convolution, in most cases, does not create major calculation problems, deconvolution on the contrary is an inverse problem, and as such needs more attention. The principles of some of the methods available today are exposed. In the third part, illustrations are given on recent examples of applications, belonging to the domain of electrical energy networks and photographic enhancement.
Highlights
The purpose of this paper is to discuss the two mathematical concepts of convolution and deconvolution. these already appear as well-known, we observe today some reasons to reconsider them in the light of recent changes.We shall focus on the application of these concepts in the context of technical laboratories, practising either research, or accredited conformity tests.A significant evolution to take into account is the apparition, in international tests standards, of recommended calculation methods, which make use either of convolution or deconvolution.the applicability and efficiency of these methods, generally working on time signals, have to be questioned with respect to other currently used methods based on Fourier analysis.We wish here to introduce the physical concept of convolution, before presenting the formal mathematical definition which covers it
We immediately note a certain number of characteristics of these transforms: – a formal analogy with the Laplace transform, despite the different integration limits – the analogy between the two transforms themselves – the possibility of limiting the calculation, in general to the one of real integrals for the Fourier transform. All these properties are used on the level of numerical computation, to render the algorithms efficient (Fast Fourier Transform or F.F.T.); to re-use them in multiple ways, and to apply them to the problem of deconvolution
The Laplace transform has been widely used by electricians for the study of transient circuit conditions; electronics developed, and Fourier transform became generalized, for several reasons: – ease of calculation by discretized elementary operations ⇒ Discrete Fourier Transform (D.F.T.), or even by accelerated algorithms Fast Fourier Transform (F.F.T.) – the great faculty of data representation that it authorizes, for the properties of signals and systems: by switching to the auto-spectrum or to the power cross-spectrum and plotting in module/phase form (Bode diagram, Nyquist diagram), possible use in
Summary
The purpose of this paper is to discuss the two mathematical concepts of convolution and deconvolution These already appear as well-known, we observe today some reasons to reconsider them in the light of recent changes. Electronic, mechanical observation instrument is used, there is an almost systematic deterioration of the input information, whether it is mono or multidimensional: signal, image,. This distortion reflects the necessarily non-ideal nature of physical systems: the finite nature of these makes it impossible to transmit infinite information, limitations with regard to frequency; the same cause creates similar limitations in the domain of amplitudes [1,2,3]. Initial information is not necessarily impractical, but all the more complex as there are intermediate phenomena, often of random nature such as noise
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