An application of the Shepard operator in image reconstruction

Abstract

We propose an application of the Shepard operator combined with two radial basis functions in image processing. This method aims to reconstruct damaged black-and-white or color images, considering both a global and a local approach. In the construction of the Shepard operator, we use the inverse quadratic and the inverse multiquadric radial basis functions.

Authors

Andra Malina
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Keywords

Shepard operator, image reconstruction, inverse quadratic, inverse multiquadric.

Paper coordinates

A. Malina, An Application of the Shepard Operator in Image Reconstruction, 2024 26th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), Timisoara, Romania, 2024, pp. 61–67, https://doi.org/10.1109/SYNASC65383.2024.00023.

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An application of the Shepard operator in image reconstruction

An application of the Shepard operator in image reconstruction

Andra Malina Babeş-Bolyai University, Faculty of Mathematics and Computer Science
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Cluj-Napoca, Romania
andra.malina@ubbcluj.ro
Abstract

We propose an application of the Shepard operator combined with two radial basis functions in image processing. This method aims to reconstruct damaged black-and-white or color images, considering both a global and a local approach. In the construction of the Shepard operator, we use the inverse quadratic and the inverse multiquadric radial basis functions.

Keywords:
Shepard operator, image reconstruction, inverse quadratic, inverse multiquadric.

I Introduction

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.

II Combined Shepard operator of inverse quadratic and inverse multiquadric type

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)X2, it is defined as

Sμβf(𝐱)=i=1NAi,μ(𝐱)ϕiβ(𝐱), (1)

for a set of N given interpolation nodes 𝒳={𝐱i=(xi,yi)|i=1,,N} on which the values of a real-valued function f are known, and with Ai,μ given by

Ai,μ(𝐱)=j=1jiNrjμ(𝐱)k=1Nj=1jkNrjμ(𝐱), (2)

considering the control parameter μ>0 and ri(𝐱)=(xxi)2+(yyi)2.

The functions ϕiβ are given by

ϕiβ(𝐱)=j=1iαj[1+(ϵrj(𝐱))2]β+ax+by+c,i=1,,N, (3)

with ϵ being a shape parameter. To determine the coefficients αj,a,b,c, one has to solve a system of the following form

(ϕ11ϕ12ϕ1Nx1y11ϕ21ϕ22ϕ2Nx2y21ϕN1ϕN2ϕNNxNyN1x1x2xN000y1y2yN000111000)(α1α2αNabc)=(f1f2fN000), (4)

where ϕij=[1+(ϵrij)2]β, rij=(xixj)2+(yiyj)2 and fi=f(𝐱i), for i,j=1,,N. The parameter β is chosen as β=1 to obtain the case of the inverse quadratic RBF and as β=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=1Ni=1Ndi, with di 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.

III Reconstruction of damaged black-and-white images using the bivariate Shepard operator

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 M^. The coordinates of a valid pixel 𝐟i are denoted by 𝐱i=(xi,yi), i.e., 𝐟i=M^(xi,yi) and the coordinates of a defective pixel 𝐟i^ are 𝐱^i=(x^i,y^i), i.e., 𝐟i^=M^(x^i,y^i). Given the fact that the correct pixels are known, the goal is to reconstruct the values 𝐟^i for all 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

𝐟i=M(xi,yi){0,1,,255}replaced by
{0,black = pepper255,white = saltobtaining𝐟^i=M^(x^i,y^i). (5)

Figure 1 shows the matrix form of a black-and-white image with resolution 8×8 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)).

Refer to caption
(a) Matrix with all correct pixels.
Refer to caption
(b) Matrix with 50% corrupted pixels by the Salt-and-Pepper noise.
Figure 1: Matrix representation of an image with 8×8 resolution.

III-A Global case

Paewpolsong et al. [Pae22] proposed an algorithm to reconstruct images that were damaged by Salt-and-Pepper noise. Their reconstruction method is based on using shapefree radial basis functions neural networks. Following some ideas proposed in this paper, we consider the reconstruction of a damaged black-and-white image using the bivariate Shepard operator Sμβf introduced in (1). For the reconstruction, we will use a global approach, where the value of a defective pixel is computed based on all the information provided by the set of correct pixels. Consider the matrix M^ associated with an image of resolution m×n that contains a percentage p% of damaged pixels. Let N be the number of valid pixels and N^ be the number of defective pixels.

Let 𝒳 be the set of interpolation nodes, 𝒳={𝐱i=(xi,yi),i=1,,N} where xi and yi represent the matrix coordinates of the correct pixels 𝐟i, i=1,,N. We apply the Shepard operator given in (1) to reconstruct the set of defective pixels, 𝒳^={𝐱^i=(x^i,y^i),i=1,,N^}. The reconstructed values of the damaged pixels are obtained as 𝐟^i=(Sμβf)(𝐱^i),i=1,,N^. The new matrix M corresponding to the reconstructed image will have the form:

M(x,y)={M^(x,y),(x,y) node(Sμβf)(x,y),(x,y) damaged.

The pseudo-code for this approach is given in Algorithm 1.

Data: damaged matrix M^
Result: reconstructed matrix M
𝒳={𝐱i=(xi,yi),i=1,,N} ;
  /* correct pixels’ coordinates */
𝒳^={𝐱^i=(x^i,y^i),i=1,,N^} ;
  /* defective pixels’ coordinates */
𝐟={M^(𝐱i),𝐱i𝒳,i=1,,N} ;
  /* correct pixels’ values */
M=M^ ;
for i=1N^ do
       M(𝐱^i)=Sμβ𝐟(𝐱^i) ;
        /* reconstruction of damaged pixels */
      
end for
Algorithm 1 Global reconstruction of black-and-white images.

For numerical experiments we considered three images downloaded from the TAMPERE17 noise-free image database \urlhttps://webpages.tuni.fi/imaging/tampere17/ with a resolution of 64×64 pixels. We corrupted 50% of the pixels with the Salt-and-Pepper noise and reconstructed them using the Shepard operator (1) combined with both RBFs, inverse quadratic (IQ) (β=1) and inverse multiquadric (IMQ) (β=1/2). The experiments were performed in Matlab, and the results are shown in Figure 2.

Refer to caption
Figure 2: 64×64 pixels images, 50% corrupted, reconstructed in the cases of IQ and IMQ.

III-B Local case

In the global method, an incorrect pixel’s value is reconstructed based on all the information provided by the correct pixels. However, this approach does not always produce the best results for the reconstructed image, since the pixels of an image have strongly local properties, as emphasized in [Ska13]. It may be more efficient to approximate the value of an incorrect pixel based on a local procedure. In [UhSka05], the focus was on reconstructing black-and-white images using the radial basis function approach in some neighborhoods of the defective pixels. The neighborhood is defined by a radius r that moves in both directions of the axes x and y, creating a window of dimensions (2r+1)×(2r+1) (see, e.g., [ZapVanSka08]).

The radius size could be chosen according to the image that is processed, but its choice is not an easy task. In Figure 3, a pixel’s neighborhood of radius r=4 is represented.

Refer to caption
Figure 3: Neighborhood of a pixel of radius r=4.

This approach leads to a smaller computational time compared to the global case, especially for high-resolution images because the systems (4) have a significantly reduced size.

The reconstruction of the damaged pixels is based on the combined Shepard operator given in (1) used locally, in each neighborhood. Besides the local behavior, some other differences appear in the algorithm, in contrast to the global case. First, a pixel is reconstructed only if a certain percentage of correct pixel exists within its neighborhood, determined by an initial desired tolerance ε. If this condition is not met, the value is not restored, the algorithm continues and the procedure is retried in the next iteration. The shape parameter ϵ of the functions ϕiβ defined in (3) is updated at each iteration based on the new set of correct pixels that is constructed. Finally, to avoid the case when the algorithm gets stuck due to the impossibility of reconstructing new pixels because of the unattained criteria, the tolerance can be increased after each iteration. This adjustment allows the reconstruction of a pixel even if the initial desired number of correct pixels in its neighborhood was not achieved. The pseudo-code is presented in Algorithm 2.

Data: damaged matrix M^, initial tolerance ε
Result: reconstructed matrix M
𝒳^={𝐱^i=(x^i,y^i),i=1,,N^} ;
  /* defective pixels’ coordinates */
M=M^ ;
while 𝒳^ not empty do
       N^ = size(𝒳^) ;
       for i=1N^ do
             Define 𝒳i = neighborhood of 𝐱^i ;
             Xi = {𝐱j𝒳i,j=1,,Ni} ;
              /* correct pixels in 𝒳i */
             if nr_wrong_pixels_neighborhood / nr_pixels_neighborhood < ε then
                   𝐟i={M^(𝐱j),𝐱jXi,j=1,,Ni} ;
                   M(𝐱^i)=Sμβ𝐟i(𝐱^i) ;
                    /* reconstruction of damaged pixels */
                  
             end if
            
       end for
      Update 𝒳^ ;
       ε=ε+0.01 ;
      
end while
Algorithm 2 Local reconstruction of black-and-white images.

For the numerical experiments in this case, we used the previously mentioned images, considering three different resolutions: 64×64, 128×128, and 256×256 pixels, respectively. The pixels corrupted with the Salt-and-Pepper noise were reconstructed using the Shepard operators (1) combined with the inverse quadratic (IQ) (β=1) and inverse multiquadric (IMQ) (β=1/2) radial basis functions. The radius of the neighborhood considered was r=4, and the initial tolerance was set to ε=0.25. We computed the Mean Absolute Errors (MAEs) and the Mean Square Errors (MSEs), defined as

MAE: 1Nk[i=1mj=1n|M(i,j)M(i,j)|],k=1,2, (6)
MSE: 1Nk[i=1mj=1n(M(i,j)M(i,j))2],k=1,2, (7)

where m×n is the resolution of the image, M is the matrix corresponding to the original image (without the damaged pixels), M is the reconstructed image, and Nk,k=1,2, is the number of all the pixels (N1) and the number of all the damaged pixels (N2), respectively. Table I presents the numerical results obtained in the case of 50% defective pixels. The graphical representations for the case of 256×256 pixels resolution are shown in Figure 4(a). Table II presents the MAEs and MSEs for the case of 75% defective pixels. The graphical representations in the case of 256×256 pixels are shown in Figure 4(b). In both cases, we compared the results with the ones obtained using the classical Shepard operator for image restoration (SC). It is defined as

Sμf(𝐱)=i=1NAi,μ(𝐱)f(𝐱i), (8)

for some known values of a target function f to be reconstructed on a set of N sample points 𝒳={𝐱i=(xi,yi)|i=1,,N} with Ai,μ given in (2), considering μ>0. Moreover, we performed the same tests using the radial basis function method as a reconstruction technique, as proposed, for example, in [UhSka05]. We considered the same RBFs used for the Shepard method: the inverse quadratic (RBF-IQ) and the inverse multiquadric (RBF-IMQ). We can observe that the results are comparable, emphasizing that the choice of the operator used depends on the available data sets. All the computations were performed in Matlab.

Image Size IQ IMQ RBF-IQ RBF-IMQ SC
(pixels) N1 N2 N1 N2 N1 N2 N1 N2 N1 N2
64 MAE 3.0488 6.4537 3.8503 7.7881 3.8838 7.9580 1.9785 4.0765 4.0989 7.5126
MSE 29.2734 61.9659 35.3613 71.2664 34.9431 71.5993 17.4766 36.0080 36.1318 67.0200
Squirrel 128 MAE 2.6635 5.4466 2.7916 5.8035 3.0629 6.3066 2.6510 5.4130 2.9958 5.5836
MSE 24.8775 50.8728 25.2861 52.5679 27.7273 57.0924 24.2982 49.6138 25.5072 52.2310
256 MAE 2.0421 4.2144 2.3256 4.7664 1.3852 2.8604 2.7158 5.3699 2.6661 4.4314
MSE 19.1413 39.5037 21.5356 44.1379 12.4294 25.6664 25.6967 50.8104 19.4090 41.7361
64 MAE 3.3040 6.6144 3.6333 7.2524 4.3599 8.6146 3.6831 7.2739 3.4768 5.9488
MSE 34.0203 68.1070 36.5508 72.9591 42.5686 84.1105 37.0300 73.1316 34.6287 59.2093
Duck 128 MAE 3.7920 7.6654 3.9347 7.8752 4.4873 8.8782 2.9347 5.9141 3.3517 6.6608
MSE 37.4204 75.6442 38.4586 76.9736 42.2457 83.5833 28.6941 57.8258 32.4424 74.4776
256 MAE 2.8885 6.7603 3.3504 7.7502 2.9630 5.9133 4.0513 8.1227 2.7705 5.5314
MSE 30.3921 62.6268 30.3486 62.1441 28.1477 56.1738 37.3027 74.7903 24.5686 49.0519
64 MAE 7.8159 15.7317 8.6021 17.1873 5.5688 11.2921 8.7522 17.3100 7.2290 14.2836
MSE 52.0884 104.8422 52.8828 105.6624 42.6194 86.4203 55.7437 110.2492 47.0215 92.9088
Lilac 128 MAE 6.4549 12.9971 7.3460 14.6562 7.5515 15.2032 3.2698 6.5605 5.6881 11.3499
MSE 45.0393 90.6875 46. 7339 93.2403 46.9867 94.5970 27.0303 54.2327 39.1420 78.1028
256 MAE 4.5035 9.0432 5.1046 10.1667 5.1958 10.4333 4.3784 8.7962 3.4615 6.9085
MSE 34.7859 69.6511 37.3420 74.3730 37.3673 75.0345 33.6234 67.5497 26.8552 53.5976
TABLE I: MAEs and MSEs for black-and-white images, 50% corrupted pixels, r=4.
Image Size IQ IMQ RBF-IQ RBF-IMQ SC
(pixels) N1 N2 N1 N2 N1 N2 N1 N2 N1 N2
64 MAE 5.7886 7.9165 6.1357 8.3941 6.6062 9.0771 3.8699 5.3370 9.1814 12.6623
MSE 52.8901 72.3332 54.7422 74.8911 57.0402 78.3753 35.1724 48.5071 71.0435 97.9778
Squirrel 128 MAE 4.5025 6.1633 4.6560 6.3485 4.5613 6.2387 3.0633 4.1810 5.9568 8.1412
MSE 41.3348 56.5820 42.4871 57.9318 40.3003 55.1198 27.3839 37.3757 48.6407 66.4772
256 MAE 3.5675 4.8870 3.7372 5.1113 3.6951 5.0583 3.4109 4.6693 4.9520 6.6118
MSE 32.8042 44.9376 34.3066 46.9211 33.8474 46.3345 31.5244 43.1546 46.2881 61.3837
64 MAE 5.8584 7.7833 5.3125 6.9543 7.0669 9.3465 5.4187 7.2485 5.6411 7.2084
MSE 64.2800 85.0149 65.9153 87.1776 66.4197 87.8447 53.3684 71.3903 55.8879 71.3837
Duck 128 MAE 6.3166 8.4290 6.5463 8.7577 6.3074 8.3798 6.3390 8.4990 5.3077 7.0592
MSE 60.6191 80.8913 61.3469 82.0697 60.1147 79.8670 60.4590 81.0606 48.8678 64.9931
256 MAE 5.5304 7.6872 5.6804 7.8754 5.3715 7.1759 6.5068 8.6925 5.0627 6.7633
MSE 49.2334 58.7967 49.7614 59.3976 49.5005 66.1285 58.4281 78.0550 43.7449 58.4395
64 MAE 13.9375 18.5290 14.3804 19.1178 10.8867 14.4732 9.8218 13.1042 15.0703 19.7726
MSE 84.0781 111.7767 85.8694 114.1581 74.5144 99.0623 65.7292 87.6961 80.7622 113.7531
Lilac 128 MAE 11.1888 15.0015 10.3228 14.4397 11.9753 16.0100 10.8036 14.4827 11.1688 14.9257
MSE 71.2964 95.5908 71.1661 95.1823 74.6075 99.7446 69.7742 93.5346 77.6111 99.3511
256 MAE 7.9741 10.6209 7.1637 9.8769 5.0419 6.7100 7.4306 9.9266 8.2623 10.6803
MSE 57.1398 76.1058 55.4276 77.8465 40.9014 54.4333 54.5833 72.9187 56.1876 78.8978
TABLE II: MAEs and MSEs for black-and-white images, 75% corrupted pixels, r=4.
Refer to caption
(a) 50% corrupted.
Refer to caption
(b) 75% corrupted.
Figure 4: 256×256 pixels images, reconstructed in the cases of IQ and IMQ.

IV Reconstruction of damaged color images using the bivariate multivalued Shepard operator

Consider a multivalued function f:X2m,f=(f1,,fm), and a set of N interpolation nodes 𝐱i=(xi,yi)X,i=1,,N. The Shepard-type multivalued interpolation operator was defined in [SomSoos16] as

k=1mSμ,kf(𝐱)=k=1mi=1NAi,μ(𝐱)fk(𝐱i), (9)

for xX, Ai,μ given in (2), μ>0. Based on this, we can define the multivalued Shepard operator combined with inverse quadratic and inverse multiquadric RBF in the bivariate case as

k=1mSμ,kβf(𝐱)=k=1mi=1NAi,μ(𝐱)ϕi,kβ(𝐱), (10)

for xX, Ai,μ given in (2), μ>0, and

ϕi,kβ(𝐱) =j=1iαj,k[1+(ϵrj)2]β+akx+bky+ck,
k=1,,m. (11)

The coefficients αj,k,ak,bk,ck are found solving similar systems as the one in (4). A color image is represented as an m×n×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 β=1 (IQ) and β=1/2 (IMQ) to reconstruct the previously mentioned images in color format. The images were processed with a resolution of 256×256 pixels (Figure 5), using neighborhoods of radius r=2 and an initial tolerance ε=0.25. The shape parameter ϵ was computed following the previously mentioned ideas (see, e.g., [Har71]).

Refer to caption
Figure 5: 256×256 pixels images, 50% corrupted, reconstructed in the cases of IQ and IMQ.

V Reconstruction of images damaged with Gaussian and Poisson noises

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 128×128 pixels, for both black-and-white and color images.

For Gaussian noise, the corrupted pixels are obtained as M^(x^i,y^i)=M(xi,yi)+e(xi,yi), where e(xi,yi) represents noise sampled from a Gaussian distribution. The visual representations are shown in Figure 6.

Refer to caption
(a) Black-and-white images.
Refer to caption
(b) Color images.
Figure 6: 128×128 pixels images, 50% corrupted by Gaussian noise.

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.

Refer to caption
(a) Black-and-white images.
Refer to caption
(b) Color images.
Figure 7: 128×128 pixels images, 50% corrupted by Poisson noise.

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.

VI Conclusions

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.

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