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Optimal Nonlinear Signal Approximations Based on Piecewise Constant Functions

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Abstract

We provide here an optimal method of approximating a signal by piecewise constant functions. To this end, we minimize over the signal subdomains a fidelity term between the signal and its corresponding piecewise approximations; subdomains being determined by the number of approximations samples used for. An optimal recursive relationship is then obtained and proven, which helps us to derive the proposed approximation algorithm. The complexity of the algorithm is O(\(\hbox {MN}^2\)), where N is the number of samples of the processed signal and M is the number of piecewise constant approximation functions. There are different techniques to approximate a signal using piecewise constant functions, wavelet decomposition is one of them by means of a Haar wavelet. Our approach is then compared to linear and nonlinear wavelet-based approximations, and both qualitative and quantitative results are provided on various tested signals, showing the efficiency of the proposed approach.

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Notes

  1. For \(f\in L^1(\varOmega )\), denote \(\int _\varOmega |Df| = \sup \left\{ \int _\varOmega f~\text {div}(\varphi )\mathrm {d}x;~\varphi =(\varphi _1,\varphi _2) \in C_0^1(\varOmega ,\mathbb {R}^2) \text { and } ||\varphi ||\le 1\right\} \), where \(C_0^1(\varOmega )\) is the space of continuously differential functions with compact support in \(\varOmega \), \(\text {div}(\varphi )=\dfrac{\partial \varphi _1}{\partial x_1}+\dfrac{\partial \varphi _2}{\partial x_2}\) and derivatives are taken in a distributional sense, \(||\varphi ||=||(\varphi _1^2+\varphi _2^2)^{1/2}||_{L^\infty (\varOmega )}\). BV(\(\varOmega \)) is the set of functions defined by \(BV(\varOmega ) = \left\{ f\in L^1(\varOmega ); ~\textit{s.t.}~\int _\varOmega |Df|<\infty \right\} \). See [1, 21, 26] for more details on BV spaces.

  2. \(W^s[0;~1]\) is the space of functions which are s times differentiable is the sense of Sobolev over the domain [0;  1]. A good review of Sobolev spaces can be found in [6].

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Authors and Affiliations

  1. Department of Mathematics, University of Thies, BP 967, Thies, Senegal

    El Hadji S. Diop

  2. Information Modeling and Processing of Maritime Data Group, French Naval Academy IRENav, 29240, Brest, France

    Abdel-Ouahab Boudraa

  3. Division of Biomedical Informatics, Cincinnati Children’s Hospital Medical Center, Cincinnati, OH, 45229, USA

    V. B. Surya Prasath

  4. Department of Pediatrics, College of Medicine, University of Cincinnati, Cincinnati, USA

    V. B. Surya Prasath

  5. Department of Biomedical Informatics, College of Medicine, University of Cincinnati, Cincinnati, OH, 45267, USA

    V. B. Surya Prasath

  6. Department of Electrical Engineering and Computer Science, University of Cincinnati, Cincinnati, OH, 45221, USA

    V. B. Surya Prasath

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  1. El Hadji S. Diop
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Diop, E.H.S., Boudraa, AO. & Prasath, V.B.S. Optimal Nonlinear Signal Approximations Based on Piecewise Constant Functions. Circuits Syst Signal Process 39, 2673–2694 (2020). https://doi.org/10.1007/s00034-019-01285-w

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