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Scattered Data Histopolation in Averaging Kernel Hilbert Spaces

Ludovico Bruni Bruno, Giacomo Cappellazzo, Wolfgang Erb, Mohammad Karimnejad Esfahani

TL;DR

The paper develops averaging kernel Hilbert spaces (AKHS) to enable histopolation from averaged data, replacing point samples with mean values over subdomains. It establishes a rigorous isometric link between AKHS and an associated RKHS, turning histopolation into RKHS interpolation and enabling unisolvence and error analysis via a history of kernel methods. It provides explicit AKHS constructions (including L2-based, RKHS-based, uniform translates, B-splines, and radial kernels), Fourier-Plancherel and Hankel-transform characterizations, and practical quadrature schemes for computing histopolation matrices. The work demonstrates convergence behavior and practical usefulness through 1D simulations and a 2D biomedical-imaging example, showing how histopolation can enhance image upscaling and compression when only averaged data are available. Overall, AKHS offer a robust, flexible framework for reconstructing functions from integral averages with provable guarantees and applicable numerical strategies.

Abstract

Kernel-based methods offer a powerful and flexible mathematical framework for addressing histopolation problems. In histopolation, the available input data does not consist of pointwise function samples but of averages taken over intervals or higher-dimensional regions, and these mean values serve as a basis for reconstructing or approximating the target function. While classical interpolation requires continuity of the underlying function, histopolation can be performed in larger function spaces. In the framework of kernel methods, we will introduce and study the so-called averaging kernel Hilbert spaces (AKHS's) for this purpose. Within this setting, we develop systematic construction principles for averaging kernels and provide characterizations based on the Fourier-Plancherel transform. In addition, we analyze several representative histopolation scenarios in order to highlight properties of this approximation method, including conditions for unisolvence and possible error estimates. Finally, we present numerical experiments that shed some light on the convergence behavior of the presented approach and demonstrate its practical effectiveness.

Scattered Data Histopolation in Averaging Kernel Hilbert Spaces

TL;DR

The paper develops averaging kernel Hilbert spaces (AKHS) to enable histopolation from averaged data, replacing point samples with mean values over subdomains. It establishes a rigorous isometric link between AKHS and an associated RKHS, turning histopolation into RKHS interpolation and enabling unisolvence and error analysis via a history of kernel methods. It provides explicit AKHS constructions (including L2-based, RKHS-based, uniform translates, B-splines, and radial kernels), Fourier-Plancherel and Hankel-transform characterizations, and practical quadrature schemes for computing histopolation matrices. The work demonstrates convergence behavior and practical usefulness through 1D simulations and a 2D biomedical-imaging example, showing how histopolation can enhance image upscaling and compression when only averaged data are available. Overall, AKHS offer a robust, flexible framework for reconstructing functions from integral averages with provable guarantees and applicable numerical strategies.

Abstract

Kernel-based methods offer a powerful and flexible mathematical framework for addressing histopolation problems. In histopolation, the available input data does not consist of pointwise function samples but of averages taken over intervals or higher-dimensional regions, and these mean values serve as a basis for reconstructing or approximating the target function. While classical interpolation requires continuity of the underlying function, histopolation can be performed in larger function spaces. In the framework of kernel methods, we will introduce and study the so-called averaging kernel Hilbert spaces (AKHS's) for this purpose. Within this setting, we develop systematic construction principles for averaging kernels and provide characterizations based on the Fourier-Plancherel transform. In addition, we analyze several representative histopolation scenarios in order to highlight properties of this approximation method, including conditions for unisolvence and possible error estimates. Finally, we present numerical experiments that shed some light on the convergence behavior of the presented approach and demonstrate its practical effectiveness.
Paper Structure (16 sections, 10 theorems, 92 equations, 8 figures, 1 table)

This paper contains 16 sections, 10 theorems, 92 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Averaging Kernel Hilbert Spaces
  3. Construction and examples of AKHS
  4. Square integrable functions
  5. Averaging kernels generated in reproducing kernel Hilbert spaces
  6. Uniform averaging kernels based on translates of a single domain
  7. B-splines in the univariate case
  8. Radial uniform averaging kernels for $d = 1$
  9. Radial uniform averaging kernels in higher dimensions $d > 1$
  10. Characterization of AKHS in terms of the Fourier transform
  11. Solution of histopolation problems
  12. Uniform error estimates for average values
  13. Approximate averages by quadrature formulas
  14. Numerical Experiments
  15. Convergence in the univariate case: $d = 1$
  16. ...and 1 more sections

Key Result

theorem 1

Assume that the functionals $\{ \lambda_{\tau} \ : \ \tau \in \mathcal{T}\}$ are continuous in the space $\mathcal{H}$. Then, the linear operator $\Lambda : \mathcal{N}_A(\Omega) \to \mathcal{N}_K(\mathcal{T})$ given as $(\Lambda f)(\tau) = \lambda_{\tau}(f)$ provides an isometric isomorphism betwee

Figures (8)

  • Figure 1: Histopolation of average values of the function $f(x) = \frac{1}{1+(x-0.4)^2}$ over $5$ interval segments with length $a \in \{0.05, 0.25, 1\}$ using shifts of averaged Matérn functions as kernels, see Example \ref{['ex:RUAK']} (i).
  • Figure 2: Histopolation of average values of the function $f(x) = \frac{1}{1+(x-0.4)^2}$ over $n \in \{5,25,100\}$ uniform intervals of length $a = 0.5$ using the indicator functions in \ref{['eq:indicatorkernel1D']} as averaging kernels.
  • Figure 3: The averaging and reproducing kernel based on the Matérn function. Left: the Matérn function $\phi(x) = e^{-\lambda |x|}$ compared to the averaging kernels $\alpha(x)/\kappa(0)$ given in Example \ref{['ex:RUAK']} (i) for three interval lengths $a \in \{0.1, 0.4, 0.8\}$. Right: the respective normalized reproducing kernels $\kappa(x)/\kappa(0)$.
  • Figure 4: Lagrange basis functions for histopolation on $5$ segments (red) with length $a = 0.05$ (left) and $a = 0.25$ (right) using the one-dimensional averaged Matérn function in Example \ref{['ex:RUAK']} (i).
  • Figure 5: Uniform errors (left) and uniform mean errors (right) of histopolation with the indicator kernel compared to the errors of the Matérn interpolation. While we do not have uniform convergence for fixed interval length, the errors of the mean values do converge uniformly also in this case.
  • ...and 3 more figures

Theorems & Definitions (21)

  • theorem 1
  • proof
  • lemma 1
  • lemma 2
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof
  • remark 1
  • ...and 11 more