Normalized Weighting Schemes for Image Interpolation Algorithms
1 Abstract —This paper presents and evaluates four weighting schemes for image interpolation algorithms. The first scheme is based on the normalized area of the circle, whose diameter is equal to the minimum side of a tetragon. The second scheme is based on the normalized area of the circle, whose radius is equal to the hypotenuse. The third scheme is based on the normalized area of the triangle, whose base and height are equal to the hypotenuse and virtual pixel length, respectively. The fourth weighting scheme is based on the normalized area of the circle, whose radius is equal to the virtual pixel length-based hypotenuse. Experiments demonstrated debatable algorithm performances and the need for further research.
Index Terms —Image interpolation; hypotenuse; circle; triangle; tetragon; normalization I. I NTRODUCTION HE commonly used normalization methods are based on interval arithmetic and fuzzy arithmetic [1]. Usually, normalization means rescaling variables, on the range between zero and one [2]. While this meaning holds in this work, it can vary from problem to problem in other works [3],[4]. Here, the news is normalization of areas, whose sum exceeds a unit square size, especially areas based on the Pythagorean theorem equation. In mathematics, the Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle [5]. Here, it can be written as equation Eq.1 or Eq.2 relating the lengths of the sides a , b , and c , of a right-angled triangle [5]. The Pythagorean equation has been widely used in many computer sciences and engineering techniques, including in [6], [7], but, with very few times for image interpolation purposes [8]. In mathematics, interpolation is an estimation method used to construct a new data value within the range of a set of known data [9]. It is widely and routinely used in digital zooming [10]. Digital zoom is one of image processing techniques used to get a closer view of image objects, but it produces more visual artefacts than optical zooming. One of its advantages is that it does not require the mechanical device of lens elements such as the one used in optical zoom [10]. Artefact-free digital image zooming remains very challenging to achieve due to many reasons, including the inaccuracy of interpolation algorithms. There are currently many approaches to image interpolation problems, particularly techniques developed to efficiently reduce visual artefacts in Olivier Rukundo, Department of Clinical Physiology, Lund University, Lund, Sweden. e-mail: [email protected] interpolated images [10], [11], [12]. A recent work, presented in [13], examined the origin of image pixels and divided image interpolation approaches into two major categories of non-extra-pixel and extra-pixel interpolation. Unlike, the extra-pixel approach, the non-extra pixel approach only depends on original or source image pixels [13]. It is important to note that, unlike the non-extra pixel, the extra-pixel approach category attracted many researchers, until now. In the extra pixel category, there are adaptive and non-adaptive interpolation techniques [8], [14], [15]. Also, among adaptive and non-adaptive, some techniques depend on traditional techniques, such as bilinear, to achieve improved outcomes relevant to the targeted problem or application of interest. Some of the related works, recently reviewed, show that there are techniques that specifically focus on minimizing damages to image edge structures . For example, in [16], the authors tackled the challenge of preserving edge structures and developed weighting strategies encompassing the bilinear and bicubic interpolators. In [17], the authors proposed a hybrid approach of switching between bilinear interpolation and covariance-based adaptive interpolation to reduce the overall computational complexity and achieve the efficient edge-directed interpolation algorithm. In [18], authors locally applied the ant colony algorithm-based weighting to optimize the bilinear interpolation algorithm in the efforts to reduce the edge blurriness artefacts. In [19], authors globally applied the ant colony algorithm-based weighting to optimize the bilinear interpolation algorithm in the efforts to further reduce the edge blurriness artefacts. Also, some techniques focus on generally preserving the texture of image content . For example, in [20], the authors proposed an automated minimum difference-directed weighting scheme based on a bilinear interpolation algorithm to tackle visual artefacts. In [21], the author studied the effects of rounding functions on the accuracy of the bilinear interpolation and proposed a rounding strategy that relatively improved the interpolation outcomes. In [14], the authors studied and presented the effects of rescaling the bilinear interpolator on image quality interpolation to reduce the blurriness artefacts and improve interpolation image quality. In [15], the authors applied the linear extrapolation concept and bilinear interpolation weighting scheme to achieve improved image interpolation quality outcomes. In [22], the authors proposed an image resolution enhancement method based on bicubic and extrapolation iterations for high
Normalized Weighting Schemes for Image Interpolation Algorithms
Olivier Rukundo T
2 resolution digital holographic reconstruction. In [23], the authors proposed a resolution enhancement algorithm based on a gradient-based interpolation technique for 1080p to 4k. In [24], the authors proposed a novel image interpolation method by using Gaussian-sinc automatic interpolators and achieved improvements in terms of objective image quality assessment metrics. In [25], authors dealt with the application of a new rational bi-cubic Ball function with six parameters in image interpolation and achieved improved objective image quality assessment metrics-based outcomes on grayscale images. Other techniques are dedicated to improving the quality of image and processing speed of interpolation algorithms . For example, in [11], the authors proposed a fast edge-directed interpolation algorithm to smooth edges and speed up the interpolation process, using three steps including the bilinear interpolation for non-edge pixels. In [26], the authors proposed a non-automated minimum difference-directed weighting scheme based on bilinear interpolation algorithm in the efforts to tackle visual artefacts and computational complexity of the image interpolation algorithm. In [27], the authors used the nearest neighbor algorithm to optimize the interpolation speed of the bilinear image interpolation algorithm. In [12], authors introduced an image interpolation method, in which the smooth and edge pixels are interpolated using simple line averaging and gradient-based interpolation, respectively, to reduce the complexity of the algorithm and preserve image edges. Another subcategory of techniques includes those using linear interpolation algorithms for specific tasks. For example, in [28], the authors proposed a capsule endoscopy image quality enhancement algorithm using a combination of two algorithms that use the bilinear interpolators. In [29], authors presented a preliminary study of the nearest neighbor interpolation-based full-reference metric for objective assessment of aliasing artefacts in interpolated images, in which bilinear interpolation method was one of the selected test algorithms. In [30], the author proposed a novel capsule endoscopy image quality enhancement algorithm based exclusively on the bilinear interpolation algorithm. In [31], authors developed and implemented the optimized version of bicubic interpolation algorithm for improved interpolated scan conversion outcomes in echocardiography. In [32], the author developed a bilinear weighting-based strategy that selectively minimizes empty bins in the low-resolution image before further processing. In [33], in a ground-penetrating radar survey, authors supported many conclusions including one suggesting the success of employing simple linear interpolation, bilinear interpolation, and inverse distance weighting interpolation techniques for maps improvements. In [34], the authors proposed an efficient data hiding scheme that uses image interpolation to keep a relative balance between hiding capacity and image quality. In [35], the authors proposed a correction method for color artefacts that combines the object information with weighted bilinear interpolation and keeps the continuity of the image while restoring the real color. In [36], the authors proposed an interpolation-based reversible data hiding scheme which includes the operation of stretching the size of the original image using the enhanced neighbor mean interpolation technique. In [37], authors proposed a method that combines an error hiding algorithm with image super-resolution reconstruction to achieve better visual effects, in which bilinear interpolation was used, in case of empty sets, to directly recover edge pixels. In [38], the authors proposed a 2-dimensional optimization interpolation scheme, derived from the broad class of Kalman filter or Bayesian estimation theory, for infrared products from hyperspectral satellite imagers and sounders. In [39], the authors developed and evaluated a resampling method based on the three-dimensional Lanczos kernel to achieve improved medical imaging outcomes. The last subcategory includes techniques that use other methods to achieve interpolation for different applications. For example, in [40], the authors proposed an interpolation technique based on mathematical morphology and shaping regularization for seismic image interpolation. In [41], the authors presented a novel adaptive rational fractal magnification algorithm that selects different interpolation forms according to regional characteristics. In [42], the authors proposed a novel fast interpolated compressed sensing technique based on a 2D variable density under-sampling scheme due to the advantages that interpolated compressed sensing provides in terms of scan time reduction. In [43], the authors proposed a deep learning concept based multi-scale attention-aware inception network to achieve accurate image interpolation outcomes. In brief, bilinear dependence is real in many techniques, therefore, its weighting scheme is examined, here, focusing on the geometric structure of its weights so that new relevant geometric weighting schemes can be proposed. The rest of paper is organized as follows. Section-II recaps the Pythagorean theorem and normalization. Section-III presents the normalized weighting schemes. Section-IV presents the virtual pixel length-based normalized weighting schemes. Section-V presents results and discussions. Section-VI gives conclusions. II. P YTHAGOREAN THEOREM AND NORMALIZATION
Named after the Greek mathematician Pythagoras, the theorem can be reduced to Eq.1 and Eq.2. a b c + = (1) a c b − (2) where a and b are called legs or catheti of the right triangle and c is called hypotenuse, as shown in Figure 1. Figure 1: c = hypotenuse, b and a = legs (or base and height)
3 Now, referring to [2], the normalization of an n -tuple of general weights to the corresponding n -tuple of normalization weights is described by the real-vector-valued function : n n n W S → defined for all ( ) ,... n n w w W in the following way shown in Eq. 3. ( )
11 1 1 ,..., : ,..., nn n ni ii i wwn w w w w = = = (3) Note that the normalization given by Eq. 3 provides weights whose sum is equal to one. III.
NORMALIZED WEIGHTING SCHEMES A. Tetragonal area
A tetragon is another name for a quadrilateral [44],[45]. It has properties such as having four sides or edges, four vertices or corners, and interior angles that add to 360 degrees [44]. Figure 2 shows that the unit square (P1, P2, P3, P4) is composed of four tetragons with areas depending on the x-y coordinates of P. Such areas, for example ( 2 ) ( 1) W x x y y = − − , are used as pixel weights and P value is obtained using Eq. (4). ( )
4( , ) 1 x y i ii
P W P = = (4) Here, it is important to note that the sum of all weights must be equal to one. Figure 2: Tetragon
However, given that the four tetragons areas sum up to one, in this case, there is no need for normalization of the area or weight. B. Minimum side-based diameter
Here, the minimum side length of any tetragons (if it is in the unit square), is used as the diameter of a circle, as shown in Figure 3. When the diameter is known, the area of the circle is calculated using Eq. 5. A diameter = (5) Here, it is important to note that this Eq.5-based area replaces the pixel weights, in Eq. 4. However, given that the four circular areas do not sum up to one, in this case, there is need for weight normalization, using Eq.3.
Figure 3: Minimum side-based diameter C. Hypotenuse-based radius
In the right-angled triangle, the hypotenuse is the longest side of the triangle. As shown in Figure 4, the hypotenuse is used as the radius of the circle. When the radius is known, the area of the circle is calculated using Eq. (6). A radius = (6) Also, here, it is important to note that this Eq.6-based area replaces the pixel weights, in Eq. 4. However, given that the four circular areas do not sum up to one, there is need for weight normalization, using Eq.3.
Figure 4: Hypotenuse-based radius D. Preliminary experiments
Figure 5 shows the preliminary interpolation outcomes from the traditional bilinear or tetragon-based algorithm (TB), minimum side-based diameter algorithm (MD), and hypotenuse -based radius algorithm (HR). As can be seen, the basic structures and features, in the red square test image, are relatively recovered after image interpolation, using the scaling ratio = 4. 4
Figure 5: TB (top-right), MD (bottom-left), HR (bottom-right)
IV.
VIRTUAL PIXEL LENGTH - BASED NORMALIZED WEIGHTING SCHEMES
It is important to note that, in this part, the virtual pixel length is equivalent to the grayscale pixel value. A. Virtual pixel length-based height
In this scheme, we refer to Figure 6, where the virtual pixel length (A) is used as the height of the triangle. Now, with the hypotenuse (B) as the base of the triangle, the area of the triangle can be calculated using Eq. 7. A Base Height = (7) It is important to note that the Eq.6-based area replaces the pixel weights, in Eq. 4. And, given that the four triangular areas do not sum up to one, there is need for weight normalization, using Eq.3.
Figure 6: Virtual pixel length-based height B. Virtual pixel length for hypotenuse-based radius
Here, Figure 7 shows the virtual pixel length (A), the base (B), and the hypotenuse (C). In this case, the hypotenuse (C) is used as the radius to find the area of a circle, using Eq. (6).
Figure 7: Virtual pixel length for hypotenuse-based radius
In Figure 7 case, the Eq.6-based area replaces the pixel weights, in Eq. 4. And, given that the four circular areas do not sum up to one, there is need for weight normalization, using Eq.3. Apart from that, it is important to note that this is inspired by one of the works presented in [36], in which the Pythagorean theorem equation was preliminary used, to tackle the image interpolation algorithmic efficiency, as the main concern. V.
RESULTS AND DISCUSSION
In this part, the popular datasets, and image quality assessment (IQA) metrics are briefly discussed. Image interpolation algorithms to be evaluated include the traditional nearest neighbor (TN), bicubic (TC), bilinear (TB), normalized minimum side diameter-based (MD), normalized hypotenuse radius-based (HR), normalized virtual pixel length-based height (AT), and normalized virtual pixel length for the hypotenuse-based radius(AC). The mentioned interpolation algorithms were manually implemented in MATLAB software (R2020a). Objective and subjective image quality assessments are also presented and discussed. A. Dataset
Here, the image dataset used contained 210 Textures, Aerials, Miscellaneous, and Sequences images downloaded from the USC-SIPI Database [46]. These images were resized to 512 × 512, 256 × 256, and 128 ×128, and converted to 8bits, using R2020a MATLAB software. The resized versions are accessible via GitHub.com/orukundo [47] – and each size category contains 210 different images. B. IQA Metrics
Here, only full reference IQA metrics were chosen to quantify the closeness or similarity of interpolated images against their corresponding pristine images [15]. Those chosen are the mean-squared error (MSE), structural similarity index (SSIM), and peak signal to noise ratio (PSNR). Given that these metrics are widely used, more details can be found in the literature and software, therefore not included in this part. It is important to note that, MATLAB’s TIC and TOC functions were used to measure the speed of different interpolation algorithms, at different scaling ratios, in terms of the elapsed time in seconds. Also, note that, here, the average means the average score or 5 performance achieved by each mentioned algorithm on 210 test images. C. Objective image quality assessment
As can be seen, in Figure 8, the TN is the fastest of all algorithms, mentioned, because, in each scaling ratio case, it used the smallest average time, among other methods mentioned. The elapsed time showed that TB is faster than MD, HR, AT, and AC. This can be explained by the new weighting schemes which included more computations thus increasing their MATLAB lines reading time. It is important to note that the software built-in versions may perform better than the manually implemented versions, presented here.
Figure 8: Average time in seconds
In Figure 9, the average results of each interpolation algorithm in terms of MSE are presented. As can be seen, the MSE errors of each method can be generalized, even if there are some exceptions with TC as well as the TN which normally produces heavy jagged artefacts. The TN exception was caused by the type of images used since it is likely to perform badly with only interpolating grain texture images [19]. Note that, in each case, TC produced the smallest average MSE error, among other methods mentioned thus achieved the best objective image quality. Also, referring to MSE error, note that the AC achieved a better objective image quality than TB, MD, HR, and AT.
Figure 9: Average MSE error
In Figure 10, the TB did not outperform the MD, HR, AT, and AC, in each scaling ratio case. Only TC seemed to perform better than all the remaining. Here, it is important to note that TC produced the largest average SSIM score, among other methods mentioned thus achieved the best objective image quality. Also, referring to the SSIM score, note that the TB achieved the better objective image quality than AC, MD, HR, and AT, at the ratio = 4 but tied with MD at the ratio = 2.
Figure 10: Average SSIM score
A similar situation is noticed with the PSNR outcome in Figure 11. Here, the PSNR values, produced by each interpolation algorithm, were almost equal in most cases, except in the TN and TC. Despite that, it is important to note that TC produced the largest average PSNR score, among other methods mentioned thus achieved the best objective image quality. Also, referring to the PSNR score, note that the AC achieved a better objective image quality than TB, MD, HR, and AT.
Figure 11: Average PSNR D. Subjective image quality assessment
As can be seen in Figure 12, Figure 13, and Figure 14, on the top or first row, there are: Boat, Man and Clock test images (left) and TN (right) interpolated images. In the second row, there are: TC (left) and TB (right) interpolated images. On the third row, there are: MD (left) and HR (right) interpolated images. On the fourth row, there are: AT (left) and AC (right) interpolated images. In Figure 12, with the TN image, some edges are heavily jagged while others are clearly defined, especially vertical and horizontal edges. However, the TN algorithm did not produce blurred edges. And there is no significant degradation of the TN interpolated image contrast. In the TC image, the oblique, vertical and horizontal edges are smoothed to some extent and slightly blurred. As can be seen, there is a slight degradation of the TB interpolated image contrast when compared to the source image (in the red square). In the TB image, regardless of the edge direction, the edges are blurred but not jagged. As can also be seen, there is a degradation of the TB interpolated image contrast when compared to the source image. 6 Figure 12: Boat test image In the MD image, regardless of the edge direction, the edges are slightly blurred and jagged to some extent. But there is no degradation of the MD interpolated image contrast as same as in TB or TC interpolated images. The MD interpolated image contrast looks better than TB image contrast, at the expense of slightly jagged artefacts. In the HR image, regardless of the edge direction, the edges are blurred and slightly jagged. As can be seen, there is a degradation of the HR interpolated image contract to some extent. In the AT image, the edges are blurred and jagged, especially at the oblique edges. As can be seen, such artefacts negatively affected the overall AC interpolated image contrast. In the AC image, the edges are slightly blurred and slightly jagged, especially at the oblique edges, but not the same way as in TC. As can be seen, such artefacts slightly affected the overall AC interpolated image contrast. This situation is quasi-repeated in Figure 13 and Figure 14, which involved different images. In some of the presented figures' cases, edges are clearly defined, which may lead to different conclusions, as until now the performance of image interpolation algorithms remain image dependent. Figure 13: Man test image Note that the quality of images presented here is affected, to some extent, by compression artefacts and viewing them directly from the paper format. 7 Figure 14: Clock test image VI.
CONCLUSION
Four image interpolation algorithms based on the normalized weighting schemes-based are presented in this paper. Experiments demonstrated that the MD-based, HR-based, AT-based, and AC-based took more time than the TB weighting scheme-based algorithm. Unlike HR, AT, and AC, the MD produced the MSE error greater than TB at scaling ratio = 4. But, at scaling ratio = 2, AC produced the smallest MSR error among BT, MD, HR AT, and AC. Like, HR, AT, and AC, the MD also achieved the SSIM score smaller than the TB at the scaling ratio = 4. But, at scaling ratio = 2, TB and MD tied SSIM score, and both achieved the best SSIM score among AT and AC. Like, HR, AT, and AC, the MD also achieved the PSNR score smaller than the TB at the scaling ratio = 4. But, at scaling ratio = 2, HR achieved the best PSNR score among, TB, MD, AT, and AC. In general, referring to objective and subjective evaluations, the overall performance of TC, AC, TB, MD, HR, and AT remains debatable, even if AC remains encouraging in this direction. It is important to note that many image processing software still rely on TB for different image processing operations which encouraged re-examinations and contributions. Further efforts may be devoted to the image dependence issues in digital image interpolation algorithms. A
CKNOWLEDGMENT
Olivier Rukundo would like to thank the reviewers and editors for constructive comments.
COMPETING INTERESTS
The author declares that he has no competing interest. R
EFERENCES [1]
Wang, Y.M., Elhag, T.M.S., On the normalization of interval and fuzzy weights, Fuzzy Sets and Systems, 157, 2006, pp. 2456 – 2471 [2]
Pavlacka, O., On various approaches to normalization of interval and fuzzy weights, Fuzzy Sets and Systems, 243, 2014, pp. 110 – 130 [3]
González, R.C., Woods, R.E., Digital Image Processing, Prentice Hall., 2007, p. 85 [4]
Daniel, F., Julia, K., A Student's Guide to the Mathematics of Astronomy, Cambridge University Press, 2013, p. 35 [5]
Sally, J.D., Sally, P., Chapter 3: Pythagorean triples. Roots to research: a vertical development of mathematical problems. American Mathematical Society Bookstore. 2007, page 63 [6]
Sadiq, A. S., Almohammad, T. Z., et al., An Energy-Efficient Cross-Layer approach for cloud wireless green communications, 2017 Second International Conference on Fog and Mobile Edge Computing (FMEC), Valencia, 2017, pp. 230-234 [7]
Fu, L., Yang, H.G., Zhou, C.C., A computer-aided geometric approach to inverse kinematics, Journal of Robotic Systems, 15(3), 1998, pp. 131-143 [8]
Olivier, R., Optimal Methods Research on Grayscale Image Interpolation, CNKI, TP391.41, 2012 [9]
W.F., Sheppard, Interpolation, In Chisholm, Hugh (ed.). Encyclopædia Britannica. 14 (11th ed.), Cambridge University Press., 1911, pp. 706–710 [10]
Rukundo, O., Evaluation of Rounding Functions in Nearest-Neighbour Interpolation, arXiv:2003.06885, 2020, p. 1-8 [11]
Tian, Q. C., Wen, H., et al.: A fast edge-directed interpolation algorithm. In: Huang, T.W., Zeng, Z.G., Li, C.D., Lueng, C.S. (eds.) International Conference on Neural Information Processing, LNCS, vol. 7665, 2012, pp. 398–405 [12]
Khan, S., Lee, D.H., et al., "Image Interpolation via Gradient Correlation-Based Edge Direction Estimation", Scientific Programming, vol. 2020, Article ID 5763837, 12 pages, 2020 [13]
Rukundo, O., Non-extra Pixel Interpolation, International Journal of Image and Graphics, Vol. 20, Issue 4, 2050031, 2020, p. 1-14 [14]
Rukundo, O., Schmidt, S., Effects of Rescaling Bilinear Interpolant on Image Interpolation Quality, Proc. SPIE 10817, Optoelectronic Imaging and Multimedia Technology V, 1081715, 2018 [15]
Rukundo, O., Schmidt, S., Extrapolation for Image Interpolation, Proc. SPIE 10817, Optoelectronic Imaging and Multimedia Technology V, 108171F, 2018 [16]
Zhang, L., Wu, X.: An edge-guided image interpolation algorithm via directional filtering and data fusion. IEEE Transactions on Image Processing, 15(8), 2006, pp. 2226–2238 [17]
Li, X., Orchard, M. T.: New edge-directed interpolation. IEEE Transactions on Image Processing, 10(10), 2001, pp. 1521–1527 [18]
Rukundo, O., Cao, H.Q., Huang, M.H., Optimization of Bilinear Interpolation Based on Ant Colony Algorithm”, Proc. 2nd Int. Conf. Electrical and Electronics Engineering, Macao, Dec.1-2, 2011. pp. 571-580 [19]
Rukundo, O., Cao, H.Q., Advances on Image Interpolation Based on Ant Colony Algorithm, SpringerPlus, 5:403, 2016 [20] Rukundo, O., Cao, H.Q., Nearest Neighbor Value Interpolation, International Journal of Advanced Computer Science and Applications (IJACSA), 3(4), 25 - 30, May 2012 [21]
Rukundo, O., Effects of Improved-Floor Function on the Accuracy of Bilinear Interpolation Algorithm, Computer and Information Science, Vol.8, No.4, 2015, pp.1–25 [22]
Huang, Z.Z., Cao, L.C., Bicubic interpolation and extrapolation iteration method for high resolution digital holographic reconstruction, Optics and Lasers in Engineering, Volume 130, 2020, 106090 [23]
Lee, Y.H., Yu, N.A., Tsai, C.Y., an image-upscaling engine for 1080p to 4k using gradient-based interpolation, International Journal of Electronics, 107:9,2020, pp.1386-1405 [24]
Xu, G., Ling, R., Deng, L., Wu, Q., Ma, W., Image interpolation via gaussian-sinc interpolators with partition of unity, Computers, Materials & Continua, vol. 62, no.1, 2020, pp. 309–319 [25]
Zulkifli, N.A.B., Karim, S.A.A., Shafie, A.B., Sarfraz, M., Ghaffar, A., Nisar, K.S., Image Interpolation Using a Rational Bi-Cubic Ball. Mathematics. 2019; 7(11):1045 [26]
Rukundo, O., Wu, K.N., Cao, H.Q., Image Interpolation Based on The Pixel Value Corresponding to The Smallest Absolute Difference, in Proc. 4th Int. Workshop. on Advanced Computational Intelligence, Wuhan, 2011, pp. 434-437 [27]
Rukundo, O., Maharaj, B.T., Optimization of Image Interpolation based on Nearest Neighbour Algorithm. 9th Int. Conf. on Computer Vision Theory and Applications (VISAPP 2014), Lisbon, 2014, pp. 641–647 [28]
Rukundo, O., Pedersen, M., Hovde, Ø., Advanced Image Enhancement Method for Distant Vessels and Structures in Capsule Endoscopy, Computational and Mathematical Methods in Medicine, vol. 2017, Article ID 9813165, 13 pages, 2017 [29]
Rukundo, O., Schmidt, S., Aliasing Artefact Index for Image Interpolation Quality Assessment, Proc. SPIE 10817, Optoelectronic Imaging and Multimedia Technology V, 108171E, 2018 [30]
Rukundo, O., Half-Unit Weighted Bilinear Algorithm for Image Contrast Enhancement in Capsule Endoscopy, Proc. SPIE 10615, Ninth International Conference on Graphic and Image Processing (ICGIP 2017), 106152Q, 2018 [31]
Rukundo, O., Schmidt, S.E., Von Ramm, O.T., Software Implementation of Optimized Bicubic Interpolated Scan Conversion in Echocardiography, arXiv:2005.11269, 2020, p. 1-10 [32]
Rukundo, O., Effects of Empty Bins on Image Upscaling in Capsule Endoscopy, Proc. SPIE 10420, Ninth International Conference on Digital Image Processing (ICDIP 2017), 104202P, July 21, 2017 [33]
Rucka, M., Wojtczak, E., Zielińska, M., Interpolation methods in GPR tomographic imaging of linear and volume anomalies for cultural heritage diagnostics, Measurement, Volume 154, 2020, 107494 [34]
Chen, Y.Q., Sun, W.J., Li, L.Y., et al., An efficient general data hiding scheme based on image interpolation, Journal of Information Security and Applications, Volume 54, 2020, 102584 [35]
Wang, X.H., Jia, X.Y., Zhou, W., et al., Correction for color artifacts using the RGB intersection and the weighted bilinear interpolation, Appl. Opt. 58, 2019, pp. 8083-8091 [36]
Hassan, F.S., Gutub, A., Efficient reversible data hiding multimedia technique based on smart image interpolation, Multimedia Tools and Applications (2020) 79:30087–30109 [37]
Jiang, C.J., Li, H.T., Zhou, S.B., et al., Image interpolation model based on packet losing network, Multimedia Tools and Applications (2020) 79:25785–25800 [38]
De Feis, I., Masiello, G., Cersosimo, A., Optimal Interpolation for Infrared Products from Hyperspectral Satellite Imagers and Sounders. Sensors. 2020; 20(8): 2352 [39]
Moraes, T., Amorim, P., Vicente Da Silva, J., Pedrini, H., Medical image interpolation based on 3D Lanczos filtering, Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 8:3, 2020, 294-300 [40]
Huang, W.L., Liu, J.X., Robust Seismic Image Interpolation with Mathematical Morphological Constraint, IEEE Transactions on Image Processing, Vol.29, 2020, pp. 819-829 [41]
Song, G., Qin, C., Zhang, K., Yao, X., Bao, F., Zhang, Y., Adaptive Interpolation Scheme for Image Magnification Based on Local Fractal Analysis, in IEEE Access, vol. 8, 2020, pp. 34326-34338 [42]
Murad, M., Bilal, M., Jalil, A., et al., Efficient Reconstruction Technique for Multi-Slice CS-MRI Using Novel Interpolation and 2D Sampling Scheme, in IEEE Access, vol. 8, 2020, pp. 117452-117466 [43]
Ji, J., Zhong, B., Ma, K.K., Image Interpolation Using Multi-Scale Attention-Aware Inception Network, in IEEE Transactions on Image Processing, vol. 29, 2020, pp. 9413-9428 [44]
List of Geometry and Trigonometry Symbols, Math Vault,