Abstract
Image resampling is a widely used tool in image processing. The upsampling increases the number of pixels and introduces new information to the image which can have undesired effects, like ringing artifacts and oscillations, aliasing “jagged” lines effect, or introduces too much numerical diffusion. Histopolation upsampling methods produce much sharper images but are more prone to aliasing and ringing effect and oscillations which appear as spurious signals near sharp transitions in color intensity. In this paper, we propose an efficient and fast quasi-histopolation algorithm based on the Canonical Complete Chebyshev–Schoenberg operator approximations, applied dimension by dimension. These approximations, because of their shape preserving properties, avoid oscillations. Presented methods have several tunable parameters that control tension and shape properties of the approximation which are used to reduce the aliasing effect while keeping the image visually sharp. Numerical tests on real and artificial images demonstrate the effectiveness and show the computational efficiency of the proposed algorithm.
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References
Battiato, S., Gallo, G., Stanco, F.: A locally adaptive zooming algorithm for digital images. Image Vis. Comput. 20, 805–812 (2002)
Bosner, T.: Knot insertion algorithms for Chebyshev splines. Ph.D. thesis, Department of Mathematics, University of Zagreb (2006). https://web.math.pmf.unizg.hr/~tinab/TinaBosnerPhD.pdf
Bosner, T., Crković, B., Škifić, J.: Tension splines with application on image resampling. Math. Commun. 19(3), 517–529 (2014)
Bosner, T., Rogina, M.: Non-uniform exponential tension splines. Numer. Algorithms 46, 265–294 (2007)
Bosner, T., Rogina, M.: Variable degree polynomial splines are Chebyshev splines. Adv. Comput. Math. 38, 383–400 (2013)
Bosner, T., Rogina, M.: Quadratic convergence of approximations by CCC-Schoenberg operators. Numer. Math. 135, 1253–1287 (2017). https://doi.org/10.1007/s00211-016-0831-0
Costantini, P.: Shape-preserving interpolation with variable degree polynomial splines. In: Hoschek, J., Kaklis, P.D. (eds.) Advanced Course on FAIRSHAPE, pp. 87–114. Vieweg+Teubner Verlag, Wiesbaden (1996)
Costantini, P.: Variable degree polynomial splines. In: Méhauté, A.L., Rabut, C., Schumaker, L.L. (eds.) Curves and Surfaces with Applications in CAGD, pp. 85–94. Vanderbilt University Press, Nashville (1997)
Costantini, P.: Curve and surface construction using variable degree polynomial splines. Comput. Aided Geom. Des. 17, 419–446 (2000)
Costantini, P.: Properties and applications of new polynomial spaces. Int. J. Wavelets Multiresolut. Inf. Process. 4(3), 489–507 (2006)
Costantini, P., Lyche, T., Manni, C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005)
FFmpeg Developers: ffmpeg tool (version 3.3.4-2) (2017). http://ffmpeg.org/
Gao, R., Song, J., Tai, X.: Image zooming algorithm based on partial differential equations. Int. J. Numer. Anal. Model. 6(2), 284–292 (2009)
Getreuer, P.: Image interpolation with geometric contour stencils. Image Process. On Line 1, 98–116 (2011). https://doi.org/10.5201/ipol.2011.g_igcs
Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Addison-Wesley, New York (1992)
Goodman, T., Mazure, M.L.: Blossoming beyond extended Chebyshev spaces. J. Approx. Theory 109, 48–81 (2001)
Hou, H.S., Andrews, H.C.: Cubic splines for image interpolation and digital filtering. IEEE Trans. Acoust. Speech Signal Process. 26(6), 508–517 (1978)
ImageMagic Developers: Imagemagick tool (2018). http://www.imagemagick.org/
Kaklis, P.D., Pandelis, D.G.: Convexity preserving polynomial splines of non-uniform degree. IMA J. Numer. Anal. 10, 223–234 (1990)
Kaklis, P.D., Sapidis, N.S.: Convexity-preserving interpolatory parametric splines of non-uniform polynomial degree. Comput. Aided Geom. Des. 12, 1–26 (1995)
Keys, R.G.: Cubic convolution interpolation for digital image processing. IEEE Trans. Acoust. Speech Signal Process. 29(6), 1153–1160 (1981)
Kvasov, B.I.: Shape-Preserving Spline Approximation. World Scientific, Singapore (2000)
Mazure, M.L.: Quasi-Chebyshev splines with connection matrices: application to variable degree polynomial splines. Comput. Aided Geom. Des. 18, 287–298 (2001)
Mazure, M.L.: Chebyshev–Schoenberg operators. Constr. Approx. 34, 181–208 (2011)
Mazure, M.L.: Piecewise Chebyshev–Schoenberg operators: shape preservation, approximation and space embedding. J. Approx. Theory 166, 106–135 (2013)
Mitchell, D.P., Netravali, A.N.: Reconstruction filters in computer-graphics. Comput. Graph. 22(4), 221–228 (1988)
Pidatella, R.M., Stanco, F., Santaera, C.: ENO/WENO interpolation methods for zooming of digital images. In: Cutello, V., Fotia, G., Puccio, L. (eds.) Applied and Industrial Mathematics in Italy II. Series on Advances in Mathematics for Applied Sciences, vol. 75, pp. 480–491. World Scientific Publishing, Singapore (2007)
Robidoux, N., Gong, M., Cupitt, J., Turcotte, A., Martinez, K.: CPU, SMP and GPU implementations of Nohalo level 1, a fast co-convex antialiasing image resampler (2009)
Robidoux, N., Turcotte, A., Gong, M., Tousignant, A.: Fast Exact Area Image Upsampling with Natural Biquadratic Histosplines, pp. 85–96. Springer, Berlin, Heidelberg (2008)
Schoenberg, I.J.: Splines and histograms. In: Blanc, C., Ghizzetti, A., Ostrowski, A., Todd, J., van Wijngaarden, A. (eds.) Spline Function and Approximation Theory, ISNM, vol. 21, pp. 277–358. Birkhäuser Verlag, Basel und Stuttgart (1973)
Schumaker, L.L.: On Tchebycheffian spline functions. J. Approx. Theory 18, 278–303 (1976)
Schumaker, L.L.: Spline Functions: Basic Theory. Wiley, New York (1981)
Strong, D.M., Chan, T.F.: Edge-preserving and scale-dependent properties of total variation regularization. In: Inverse Problems, pp. 165–187 (2000)
Tian, Q., Wen, H., Zhou, C., Chen, W.: A fast edge-directed interpolation algorithm. In: Huang, T., Zeng, Z., Li, C., Leung, C.S. (eds.) Neural Information Processing, pp. 398–405. Springer, Berlin (2012)
Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)
Wang, Z., Simoncelli, E.P., Bovik, A.C.: Multi-scale structural similarity for image quality assessment. In: Proceedings of IEEE Asilomar Conference on Signals, Systems, and Computers, pp. 1398–1402 (2003)
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Work of the first author has been fully supported by the University of Zagreb, under the project “Numerical algorithms”, year 2018. Work of the second author has been fully supported by the University of Rijeka under the project number uniri-prirod-18-9.
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Bosner, T., Crnković, B. & Škifić, J. Application of CCC–Schoenberg operators on image resampling. Bit Numer Math 60, 129–155 (2020). https://doi.org/10.1007/s10543-019-00770-7
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DOI: https://doi.org/10.1007/s10543-019-00770-7