Abstract
Finding optimal inpainting data plays a key role in the field of image compression with partial differential equations (PDEs). In this paper, we optimise the spatial as well as the tonal data such that an image can be reconstructed with minimised error by means of discrete homogeneous diffusion inpainting. To optimise the spatial distribution of the inpainting data, we apply a probabilistic data sparsification followed by a nonlocal pixel exchange. Afterwards we optimise the grey values in these inpainting points in an exact way using a least squares approach. The resulting method allows almost perfect reconstructions with only 5% of all pixels. This demonstrates that a thorough data optimisation can compensate for most deficiencies of a suboptimal PDE interpolant.
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References
Bae, E., Weickert, J.: Partial differential equations for interpolation and compression of surfaces. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, J.-L., Mørken, K., Schumaker, L.L. (eds.) MMCS 2008. LNCS, vol. 5862, pp. 1–14. Springer, Heidelberg (2010)
Belhachmi, Z., Bucur, D., Burgeth, B., Weickert, J.: How to choose interpolation data in images. SIAM Journal on Applied Mathematics 70(1), 333–352 (2009)
Bertalmío, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proc. SIGGRAPH 2000, New Orleans, LI, pp. 417–424 (July 2000)
Bruckstein, A.M., Holt, R.J., Netravali, A.N.: Holographic representations of images. IEEE Transactions on Image Processing 7(11), 1583–1597 (1998)
Carlsson, S.: Sketch based coding of grey level images. Signal Processing 15, 57–83 (1988)
Di Blasi, G., Francomano, E., Tortorici, A., Toscano, E.: A smoothed particle image reconstruction method. Calcolo 48(1), 61–74 (2011), http://dx.doi.org/10.1007/s10092-010-0028-3
Elder, J.H.: Are edges incomplete? International Journal of Computer Vision 34(2/3), 97–122 (1999)
Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.P.: Image compression with anisotropic diffusion. Journal of Mathematical Imaging and Vision 31(2–3), 255–269 (2008)
Hummel, R., Moniot, R.: Reconstructions from zero-crossings in scale space. IEEE Transactions on Acoustics, Speech, and Signal Processing 37, 2111–2130 (1989)
Johansen, P., Skelboe, S., Grue, K., Andersen, J.D.: Representing signals by their toppoints in scale space. In: Proc. Eighth International Conference on Pattern Recognition, Paris, France, pp. 215–217 (October 1986)
Kanters, F.M.W., Lillholm, M., Duits, R., Janssen, B.J.P., Platel, B., Florack, L.M.J., ter Haar Romeny, B.M.: On image reconstruction from multiscale top points. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 431–442. Springer, Heidelberg (2005)
Lillholm, M., Nielsen, M., Griffin, L.D.: Feature-based image analysis. International Journal of Computer Vision 52(2/3), 73–95 (2003)
Mainberger, M., Bruhn, A., Weickert, J., Forchhammer, S.: Edge-based image compression of cartoon-like images with homogeneous diffusion. Pattern Recognition 44(9), 1859–1873 (2011)
Masnou, S., Morel, J.M.: Level lines based disocclusion. In: Proc.1998 IEEE International Conference on Image Processing, Chicago, IL, vol. 3, pp. 259–263 (October 1998)
Morton, K.W., Mayers, L.M.: Numerical Solution of Partial Differential Equations, 2nd edn. Cambridge University Press, Cambridge (2005)
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
Schmaltz, C., Gwosdek, P., Bruhn, A., Weickert, J.: Electrostatic halftoning. Computer Graphics Forum 29(8), 2313–2327 (2010)
Schmaltz, C., Weickert, J., Bruhn, A.: Beating the quality of JPEG 2000 with anisotropic diffusion. In: Denzler, J., Notni, G., Süße, H. (eds.) DAGM 2009. LNCS, vol. 5748, pp. 452–461. Springer, Heidelberg (2009)
Uhlir, K., Skala, V.: Reconstruction of damaged images using radial basis functions. In: Proc.13th European Signal Processing Conference (EUSIPCO), Antalya, Turkey, pp. 160–163 (September 2005)
Zeevi, Y., Rotem, D.: Image reconstruction from zero-crossings. IEEE Transactions on Acoustics, Speech, and Signal Processing 34, 1269–1277 (1986)
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Mainberger, M. et al. (2012). Optimising Spatial and Tonal Data for Homogeneous Diffusion Inpainting. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24785-9_3
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DOI: https://doi.org/10.1007/978-3-642-24785-9_3
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