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Weighted quasi-interpolant spline approximations: Properties and applications

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Continuous representations are fundamental for modeling sampled data and performing computations and numerical simulations directly on the model or its elements. To effectively and efficiently address the approximation of point clouds, we propose the weighted quasi-interpolant spline approximation method (wQISA). We provide global and local bounds of the method and discuss how it still preserves the shape properties of the classical quasi-interpolation scheme. This approach is particularly useful when the data noise can be represented as a probabilistic distribution: from the point of view of non-parametric regression, the wQISA estimator is robust to random perturbations, such as noise and outliers. Finally, we show the effectiveness of the method with several numerical simulations on real data, including curve fitting on images, surface approximation, and simulation of rainfall precipitations.

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The authors thank Dr. Bianca Falcidieno and Dr. Michela Spagnuolo for the fruitful discussions; Dr. Oliver J. D. Barrowclough, and Dr. Tor Dokken for their concern as supervisors; Dr. Georg Muntingh for his constructive suggestions that have enhanced the mathematical structure and the exposition of the article; the reviewers, for their positive suggestions, which have significantly contributed to extend the references.


This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 675789. This work has been co-financed by the “POR FSE, Programma Operativo Regione Liguria” 2014-2020, no. RLOF18ASSRIC/68/1, and partially developed in the CNR-IMATI activities DIT.AD021.080.001 and DIT.AD009.091.001.

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Appendix A. Univariate case

Appendix A. Univariate case

We will suppose—up to a rotation—that the point cloud \(\mathcal {P}\) can be locally represented by a function of the form \(f:[a,b]\subset \mathbb {R}\to \mathbb {R}\).

Definition 7

Let \(\mathcal {P}\subset \mathbb {R}^{2}\) be a point cloud, \(p\in \mathbb {N}^{\ast }\) and x = [x1,…,xn+p+ 1] a (p + 1)-regular (global) knot vector with fixed boundary knots xp+ 1 = a and xn+ 1 = b. The weighted quasi-interpolant spline approximation of degree p to the point cloud \(\mathcal {P}\) over the knot vector x is defined by

$$ f_{w}(x):=\sum\limits_{i=1}^{n}\hat{y}_{w}(\xi^{(i)})B[\mathbf{x}^{(i)}](x), $$

where ξ(i) := (xi + … + xi+p)/p are the knot averages and

$$ \hat{y}_{w}(t):=\frac{\sum\limits_{(x,y)\in\mathcal{P}}y\cdot w_{t}(x)}{\sum\limits_{(x,y)\in\mathcal{P}}w_{t}(x)} $$

are the control points estimators of weight functions \(w_{t}:\mathbb {R}\to [0,+\infty )\).

1.1 A.1 Properties

1.1.1 A.1.1 Global and local bounds

Proposition 2 (Global bounds)

Let \(\mathcal {P}\subset \mathbb {R}^{2}\) be a point cloud. Given \(y_{\min \limits }, y_{\max \limits }\in \mathbb {R}\) that satisfy

$$ y_{\min}\le y\le y_{\max}, \quad \text{ for all } (x,y)\in\mathcal{P}, $$

then the weighted quasi-interpolant spline approximation to \(\mathcal {P}\) from some spline space \(\mathbb {S}_{p,\mathbf {x}}\) and some weight function w has the same bounds

$$ y_{min}\le f_{w}(x)\le y_{max}, \quad \text{ for all } x\in\mathbb{R}. $$


From the partition of unity property of a B-spline basis, it follows that


where the inequalities and are a direct consequence of defining \(\hat {y}_{w}\) by means of a convex combination. □

The bounds of Proposition 2 can potentially lead to local bounds. We discuss this situation in Corollary 2.

Corollary 2 (Local bounds)

Let \(\mathcal {P}\subset \mathbb {R}^{2}\) be a point cloud. If x ∈ [xμ,xμ+ 1) for some μ in the range p + 1 ≤ μn, then

$$ \alpha(\mu)\le f_{w}(x)\le \beta(\mu), $$

for some α(μ),β(μ) which belong to [ymin,ymax].


By using the property of local support for B-splines, it follows that

$$ f_{w}(x)= \sum\limits_{i=\mu-p}^{\mu}\hat{y}_{w}(\xi^{(i)})B[\mathbf{x}^{(i)}](x) $$

over [xμ,xμ+ 1). Thus, we can re-write the chain of inequalities (22) as



$$\mathcal{P}_{i}:={\textstyle\bigcup\limits_{i=\mu-p,\ldots,\mu}} \left\{\text{supp}\left( w_{\xi^{(i)}}(\cdot)\right) \right\}\cap\mathcal{P}.$$

Notice that the set of points which are effectively used to compute the approximation, i.e.,

$$ \mathcal{P}^{\ast}:=\bigcup\limits_{i=p+1,\ldots,n}\mathcal{P}_{i} $$

may be a proper subset of \(\mathcal {P}\).

1.1.2 A.1.2 Preservation of monotonicity

Definition 8 (w-monotonicity)

Let \(w_{t}:\mathbb {R}\to [0,+\infty )\) be a family of weight functions, where \(t\in \mathbb {R}\). A point cloud \(\mathcal {P}\subset \mathbb {R}^{2}\) is said to be w-increasing if for all x1x2, \(\hat {y}_{w}(x_{1})\le \hat {y}_{w}(x_{2})\). \(\mathcal {P}\) is said to be w-decreasing if for all x1x2, \(\hat {y}_{w}(x_{1})\ge \hat {y}_{w}(x_{2})\).

The key ingredient to prove the preservation of monotonicity (Fig. 10) through our method is the following lemma.

Fig. 10
figure 10

w-monotonicity and its preservation. a shows an example of an estimator \(\hat {y}_{w}:\mathbb {R}\to \mathbb {R}\) (in red) for a given point cloud (in blue) with respect to a 3-NN weight function. b graphically compares the original point cloud (in blue) with its wQISA (in red)

Lemma 2

Let \(p\in \mathbb {N}^{\ast }\) and x = [x1,…,xn+p+ 1] be a (p + 1)-regular (global) knot vector with fixed boundary knots xp+ 1 = a and xn+ 1 = b. In addition, let \(f={\sum }_{i=1}^{n}c_{i}B[\mathbf {x}^{(i)}]\in \mathbb {S}_{p,\mathbf {x}}\). If the sequence of coefficients \(\{c_{i}\}_{i=1}^{n}\) is increasing (decreasing) then f is increasing (decreasing).


The Lemma is proven in [39], pp. 114–115. □

Proposition 3

Let \(\mathcal {P}\subset \mathbb {R}^{2}\) be a point cloud, \(p\in \mathbb {N}^{\ast }\) and x = [x1,…,xn+p+ 1] be a (p + 1)-regular (global) knot vector with fixed boundary knots xp+ 1 = a and xn+ 1 = b. If \(\mathcal {P}\) is w-increasing (decreasing) then fw is also increasing (decreasing).


By definition of w-increasing (decreasing) point cloud, the sequence of control points \(\{\hat {y}_{w}(\xi ^{(i)})\}_{i=1}^{n}\) is increasing (decreasing). By Lemma 2, this is sufficient to conclude that fw is increasing (decreasing). □

1.1.3 A.1.3 Preservation of convexity

Definition 9 (w-convexity)

Let \(w_{t}:\mathbb {R}\to [0,+\infty )\) be a family of weight functions, where \(t\in \mathbb {R}\). A point cloud \(\mathcal {P}\subset \mathbb {R}^{2}\) is said to be w-convex if for all x1x2 and for any λ ∈ [0, 1],

$$\hat{y}_{w}((1-\lambda)x_{1}+\lambda x_{2})\le(1-\lambda)\hat{y}_{w}(x_{1})+\lambda\hat{y}_{w}(x_{2}).$$

\(\mathcal {P}\) is said to be w-concave if \(\mathcal {P}_{-}:=\{(x,-y) | (x,y)\in \mathcal {P}\}\) is w-convex.

The preservation of convexity (Fig.11) is a consequence of the following lemma.

Fig. 11
figure 11

w-convexity and its preservation. a shows an example of an estimator \(\hat {y}_{w}:\mathbb {R}\to \mathbb {R}\) (in red) for a given point cloud (in blue) with respect to a 3-NN weight function. b graphically compares the original point cloud (in blue) with its wQISA (in red)

Lemma 3

Let \(p\in \mathbb {N}^{\ast }\) and x = [x1,…,xn+p+ 1] be a (p + 1)-regular (global) knot vector with fixed boundary knots xp+ 1 = a and xn+ 1 = b. Lastly, let \(f={\sum }_{i=1}^{n}c_{i}B[\mathbf {x}^{(i)}]\in \mathbb {S}_{p,\mathbf {x}}\). Define Δci by

$$ {\Delta} c_{i}:= \begin{cases} \frac{c_{i}-c_{i-1}}{x_{i+p}-x_{i}}, &\text{ if }x_{i}<x_{i+p} \\ {\Delta} c_{i-1} &\text{ if }x_{i}=x_{i+p} \end{cases} $$

for i = 2,…,n. Then, f is convex on [xp+ 1,xn+ 1] if it is continuous and if the sequence \(\{\Delta c_{i}\}_{i=2}^{n}\) is increasing.


See [39], p. 118. □

Proposition 4

Let \(\mathcal {P}\subset \mathbb {R}^{2}\) be a point cloud, \(p\in \mathbb {N}^{\ast }\) and x = [x1,…,xn+p+ 1] be a (p + 1)-regular (global) knot vector with fixed boundary knots xp+ 1 = a and xn+ 1 = b. If \(\mathcal {P}\) is w-convex (concave) then fw is also convex (concave).



$$ {\Delta} c_{i}:= \frac{\hat{y}_{w}(\xi^{(i)})-\hat{y}_{w}(\xi^{(i-1)})}{x_{i+p}-x_{i}}=\frac{\hat{y}_{w}(\xi^{(i)})-\hat{y}_{w}(\xi^{(i-1)})}{(\xi^{(i)}-\xi^{(i-1)})p} $$

with xi < xi+p. Since \(\mathcal {P}\) is w-convex then these differences must be increasing and consequently fw is convex by Lemma 3. □

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Raffo, A., Biasotti, S. Weighted quasi-interpolant spline approximations: Properties and applications. Numer Algor 87, 819–847 (2021).

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Mathematics Subject Classification 2010