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Signal Approximations Based on Nonlinear and Optimal Piecewise Affine Functions

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Abstract

In this work, we address the problem of piecewise affine approximations, that is, to find piecewise affine functions that well-approximate a given signal. The proposed approach is optimal in the sense of \({L}^2\) norm and formulated in a compact and explicit way; no fitting stage is needed. Also, affine parameters are obtained as closed formulas, and affine approximation functions are optimal in their corresponding subdomains. In addition, we state and prove a recursive formula for approximation errors, which makes the approach optimal and nonlinear, links also the subdomains and helps derive an algorithm of complexity of order \(O(MN^2)\), where M represents the number of piecewise affine approximants and N is the number of samples of the processed signal. Finally, obtained qualitative and quantitative results show that the presented method obtains good approximations and provides improvement over piecewise constant approximations.

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Notes

  1. Let \(u\in L^1(\Omega )\), and denote

    $$\begin{aligned} \int _\Omega \vert Du \vert = \sup \left\{ \int _\Omega u~\text {div}(\varphi )\textrm{d} x;~\varphi =(\varphi _1,\varphi _2) \in \mathbb {C}_0^1(\Omega ,\mathbb {R}^2) \text { and } \Vert \varphi \Vert \le 1\right\} , \end{aligned}$$

    where \(\mathbb {C}_0^1(\Omega )\) is the space of continuously differential functions with compact support in \(\Omega \), \(\text {div}(\varphi ) =\dfrac{\partial \varphi _1}{\partial x_1}+\dfrac{\partial \varphi _2}{\partial x_2}\) where derivatives are taken in a distributional sense, and \(\Vert \varphi \Vert =\Vert (\varphi _1^2+\varphi _2^2)^{1/2}\Vert _{L^\infty (\Omega )}\). The space of functions of bounded variation on \(\Omega \), denoted BV(\(\Omega \)), is the set of functions defined by

    $$\begin{aligned} BV(\Omega ) = \left\{ u\in L^1(\Omega ); ~\text {s.t.}~\int _\Omega \vert Du \vert <\infty \right\} . \end{aligned}$$

  2. \(W^{1,1}(\Omega ,\mathbb {R}^m)\) is the Sobolev space defined by all functions \(u\in L^1(\Omega ,\mathbb {R}^m)\) such that there exists \(g_1,\ldots , g_d\in L^1(\Omega ,\mathbb {R}^m)\) verifying:

    $$\begin{aligned} \int _\Omega u \dfrac{\partial \varphi }{\partial x_i} = -\int _\Omega g_i\varphi ,~\forall \varphi \in C_c^\infty (\Omega ) \text { and } \forall i=1,\ldots ,d \end{aligned}$$

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Authors and Affiliations

  1. NAGIP-Nonlinear Analysis and Geometric Information Processing Group, Department of Mathematics, University Iba Der Thiam of Thies, BP 967, Thies, Senegal

    El Hadji S. Diop & Ata Ngom

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

    V. B. Surya Prasath

  3. Department of Pediatrics, University of Cincinnati College of Medicine, Cincinnati, 45257, OH, USA

    V. B. Surya Prasath

  4. Department of Biomedical Informatics, University of Cincinnati, Cincinnati, 45267, OH, USA

    V. B. Surya Prasath

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

    V. B. Surya Prasath

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  1. El Hadji S. Diop
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Correspondence to V. B. Surya Prasath.

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Diop, E.H.S., Ngom, A. & Prasath, V.B.S. Signal Approximations Based on Nonlinear and Optimal Piecewise Affine Functions. Circuits Syst Signal Process 42, 2366–2384 (2023). https://doi.org/10.1007/s00034-022-02224-y

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