The Shepard method, introduced in [She68], is one of the best ways to solve scattered data approximation problems, i.e., to reconstruct unknown function values from some given scattered data.

Besides function approximation, the Shepard method appears in other areas where irregular data are given and there is a need to reconstruct a surface based on them, such as cartography, meteorology, geology, etc. Recently, a novel technique for neural networks based on the Shepard interpolation has been developed and successfully tested in solving different tasks like time series classification (see, e.g., [SmiWil18, SmiWil18-2]), image classification (see, e.g., [SmiWil18-3]), image recognition (see, e.g., [Wil16]), and inpainting (see, e.g., [RenXu15]).

In this paper, we focus on another application of a combined-type Shepard operator, specifically in image reconstruction. This problem has been intensively studied based on radial basis functions approaches (see, e.g., [Pae22, SavKojUnno02, Ska13, UhSka05, WWW06, ZapVanSka08, ZapVanSka09]), but few results have been reported for the Shepard operator. Here, we will reconstruct damaged images, both black-and-white and color, using the combined Shepard operator of inverse quadratic and inverse multiquadric type. Image reconstruction is typically required in cases involving inpainting or noise. Since our focus is on restoration rather than damage detection, we assume that the area to be reconstructed has already been identified. Test results show that this operator could be a powerful tool in this kind of restoration problem. This paper is organized as follows. Section II describes the mathematical method that will be used, specifically the Shepard operator of inverse quadratic and inverse multiquadric type, introduced in [CatMal22]. Using this operator, Section III presents the method for black-and-white image reconstruction, considering both global and local approaches. Section IV discusses the reconstruction of color images based on a multivalued Shepard-type operator. In these sections, the images are damaged by the Salt-and-Pepper noise. Finally, Section V presents numerical examples for the reconstruction of both black-and-white and color images, damaged by Gaussian and Poisson noise.

Several authors have worked over the years to improve the method proposed in [She68]. For example, different weight functions have been considered to overcome the disadvantages of the original operator (see, e.g., [FraNie80, Ren88]). Another improvement was made by combining the Shepard basis functions with different operators to increase the reproduction quality (see, e.g., [CaiDelTom12, Cat07, Cat05, CatMal21, CatMal22, DelTom16-2, Ren99]). In [CatMal22], the bivariate Shepard operator combined with inverse quadratic and inverse multiquadric radial basis functions (RBFs) was introduced. In the classical way, for $𝐱=(x,y)\in X\subseteq {ℝ}^2$, it is defined as

for a set of $N$ given interpolation nodes $𝒳=\{{𝐱}_i=(x_i,y_i)|i=1,\mathrm{\dots },N\}$ on which the values of a real-valued function $f$ are known, and with $A_{i,\mu }$ given by

considering the control parameter $\mu >0$ and $r_i(𝐱)=\sqrt{{(x-x_i)}^2+{(y-y_i)}^2}$.

The functions ${\varphi }_i^{\beta }$ are given by

with $ϵ$ being a shape parameter. To determine the coefficients ${\alpha }_j,a,b,c$, one has to solve a system of the following form

where ${\varphi }_{ij}={\left[1+{\left(ϵr_{ij}\right)}^2\right]}^{\beta }$, $r_{ij}=\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2}$ and $f_i=f({𝐱}_i)$, for $i,j=1,\mathrm{\dots },N$. The parameter $\beta$ is chosen as $\beta =-1$ to obtain the case of the inverse quadratic RBF and as $\beta =-1/2$ for the case of inverse multiquadric RBF. The value of the shape parameter $ϵ$ was the subject of discussion in many papers (see, e.g., [FasZha07, Fra82, Har71]). Here we followed the idea proposed in [Har71] and computed it as $ϵ=1/(0.815d),$ where $d$ is given as $d=\frac{1}{N}\sum _{i=1}^N{d}_i$, with $d_i$ being the distance from the node $i$ to its closest neighbor.

This kind of interpolant will represent the main tool in the reconstruction of both black-and-white and color images, following two approaches: a global and a local one.

Consider an original, uncorrupted black-and-white image whose matrix representation is denoted by $M$, where each entry $M(x,y)$ stores a pixel value of the image. Sometimes the image may contain defective pixels, which can be restored based on the correct pixel values that are available. Let us denote the matrix representation of the corrupted image by $\widehat{M}$. The coordinates of a valid pixel ${𝐟}_i$ are denoted by ${𝐱}_i=(x_i,y_i)$, i.e., ${𝐟}_i=\widehat{M}(x_i,y_i)$ and the coordinates of a defective pixel $\widehat{{𝐟}_i}$ are ${\widehat{𝐱}}_i=({\widehat{x}}_i,{\widehat{y}}_i)$, i.e., $\widehat{{𝐟}_i}=\widehat{M}({\widehat{x}}_i,{\widehat{y}}_i)$. Given the fact that the correct pixels are known, the goal is to reconstruct the values ${\widehat{𝐟}}_i$ for all ${\widehat{𝐱}}_i$. In our approach, we are going to deliberately corrupt several pixels using the Salt-and-Pepper noise. This consists of changing the values of a specific percentage of pixels into $0$ (black = pepper) or $255$ (white = salt), as in

Figure 1 shows the matrix form of a black-and-white image with resolution $8\times 8$ in both cases: as an uncorrupted image (Subfigure 1(a)) and as a corrupted image using the Salt-and-Pepper noise for $50\%$ of the pixels (Subfigure 1(b)).

Consider a multivalued function $f:X\subseteq {ℝ}^2\to {ℝ}^m,f=(f_1,\mathrm{\dots },f_m)$, and a set of $N$ interpolation nodes ${𝐱}_i=(x_i,y_i)\in X,i=1,\mathrm{\dots },N$. The Shepard-type multivalued interpolation operator was defined in [SomSoos16] as

for $x\in X$, $A_{i,\mu }$ given in (2), $\mu >0$. Based on this, we can define the multivalued Shepard operator combined with inverse quadratic and inverse multiquadric RBF in the bivariate case as

for $x\in X$, $A_{i,\mu }$ given in (2), $\mu >0$, and

The coefficients ${\alpha }_{j,k},a_k,b_k,c_k$ are found solving similar systems as the one in (4). A color image is represented as an $m\times n\times 3$ structure with each component defining the colors red, green, and blue (RGB) of every pixel. To reconstruct colored images, we use the multivalued Shepard operator introduced in (10) for $m=3$, applying similar methods as for black-and-white images. Computationally, we apply Algorithm 2 to each of the three color components: red, green, and blue. We consider only the case of local approximation and since the reconstructions of the three components are independent, we consider parallel computing, performed in Matlab. For numerical examples, we corrupted 50$\%$ of the pixels using the Salt-and-Pepper noise. We used the multivalued Shepard operator given in (10) for $\beta =-1$ (IQ) and $\beta =-1/2$ (IMQ) to reconstruct the previously mentioned images in color format. The images were processed with a resolution of $256\times 256$ pixels (Figure 5), using neighborhoods of radius $r=2$ and an initial tolerance $\epsilon =0.25$. The shape parameter $ϵ$ was computed following the previously mentioned ideas (see, e.g., [Har71]).

In the last part of this paper, we consider two additional types of noise, specifically Gaussian and Poisson noise, applied to corrupt $50\%$ of the pixels in the previously mentioned images. We consider a resolution of $128\times 128$ pixels, for both black-and-white and color images.

For Gaussian noise, the corrupted pixels are obtained as $\widehat{M}({\widehat{x}}_i,{\widehat{y}}_i)=M(x_i,y_i)+e(x_i,y_i)$, where $e(x_i,y_i)$ represents noise sampled from a Gaussian distribution. The visual representations are shown in Figure 6.

Poisson noise, rather than being added artificially, is generated from the data following a Poisson distribution. The graphical representations for this case are shown in Figure 7.

It can be observed that this operator is effective in reconstructing various types of damaged images. A more in-depth study of the latter two types of noise could be a subject for future research.

In this paper, we focused on the restoration of black-and-white and color images. The Shepard operator is well-known for its advantages in reconstructing function values from a limited number of known data points, and its practical benefits in image reconstruction have been demonstrated through the methods discussed above. We proposed two approaches: a global and a local one, with the latter providing better results, as expected.