Abstract
We present Neural Space-filling Curves (SFCs), a data-driven approach to infer a context-based scan order for a set of images. Linear ordering of pixels forms the basis for many applications such as video scrambling, compression, and auto-regressive models that are used in generative modeling for images. Existing algorithms resort to a fixed scanning algorithm such as Raster scan or Hilbert scan. Instead, our work learns a spatially coherent linear ordering of pixels from the dataset of images using a graph-based neural network. The resulting Neural SFC is optimized for an objective suitable for the downstream task when the image is traversed along with the scan line order. We show the advantage of using Neural SFCs in downstream applications such as image compression. Project page: https://hywang66.github.io/publication/neuralsfc.
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References
Alexandrov, V., Alexeev, A., Gorsky, N.: A recursive algorithm for pattern recognition. In: Proceedings of IEEE International Conference Pattern Recognition, pp. 431–433 (1982)
Ansari, A., Fineberg, A.: Image data compression and ordering using Peano scan and lot. IEEE Trans. Consumer Electron. 38(3), 436–445 (1992)
Bader, M.: Space-filling curves: an introduction with applications in scientific computing, vol. 9. Springer Science & Business Media (2012). https://doi.org/10.1007/978-3-642-31046-1
Chen, H., He, B., Wang, H., Ren, Y., Lim, S.N., Shrivastava, A.: NeRV: neural representations for videos. Adv. Neural Inf. Process. Syst. 34, 21557–21568 (2021)
Dafner, R., Cohen-Or, D., Matias, Y.: Context-based space filling curves. In: Computer Graphics Forum, vol. 19, pp. 209–218. Wiley Online Library (2000)
Dai, H., Khalil, E.B., Zhang, Y., Dilkina, B., Song, L.: Learning combinatorial optimization algorithms over graphs. NIPS (2017)
Deudon, M., Cournut, P., Lacoste, A., Adulyasak, Y., Rousseau, L.-M.: Learning heuristics for the TSP by policy gradient. In: van Hoeve, W.-J. (ed.) CPAIOR 2018. LNCS, vol. 10848, pp. 170–181. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-93031-2_12
Drozdowski, M.: Scheduling for parallel processing, vol. 18. Springer (2009). https://doi.org/10.1007/978-1-84882-310-5
Ehrlich, M., et al.: Leveraging bitstream metadata for fast and accurate video compression correction. arXiv preprint arXiv:2202.00011 (2022)
Harary, F., Norman, R.Z.: Some properties of line digraphs. Rendiconti del Circolo Matematico di Palermo 9(2), 161–168 (1960)
Hilbert, D.: Über die stetige abbildung einer linie auf ein flächenstück. In: Dritter Band: Analysis\(\cdot \) Grundlagen der Mathematik\(\cdot \) Physik Verschiedenes, pp. 1–2. Springer (1935). https://doi.org/10.1007/978-3-662-38452-7_1
Hopfield, J.J., Tank, D.W.: “neural” computation of decisions in optimization problems. Biol. Cybern. 52(3), 141–152 (1985). https://doi.org/10.1007/BF00339943
Huffman, D.A.: A method for the construction of minimum-redundancy codes. Proc. IRE 40(9), 1098–1101 (1952)
Kamata, S., Eason, R.O., Kawaguchi, E.: An implementation of the Hilbert scanning algorithm and its application to data compression. IEICE Trans. Inf. Syst. 76(4), 420–428 (1993)
Karras, T., Laine, S., Aila, T.: A style-based generator architecture for generative adversarial networks. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 4401–4410 (2019)
Kipf, T.N., Welling, M.: Semi-supervised classification with graph convolutional networks. ICLR (2016)
Kool, W., Van Hoof, H., Welling, M.: Attention, learn to solve routing problems! ICLR (2019)
Lawder, J.K.: Calculation of mappings between one and n-dimensional values using the Hilbert space-filling curve. School of Computer Science and Information Systems, Birkbeck College, University of London, London Research Report BBKCS-00-01 August (2000)
LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998)
Lee, J.H., Hsueh, Y.C.: Texture classification method using multiple space filling curves. Patt. Recogn. Lett. 15(12), 1241–1244 (1994)
Lempel, A., Ziv, J.: Compression of two-dimensional data. IEEE Trans. Inf. Theory 32(1), 2–8 (1986)
Li, Y., et al.: TGIF: a new dataset and benchmark on animated gif description. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4641–4650 (2016)
Lieberman-Aiden, E., et al.: Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science 326(5950), 289–293 (2009)
Liu, Z., Luo, P., Wang, X., Tang, X.: Deep learning face attributes in the wild. In: Proceedings of International Conference on Computer Vision (ICCV) (December 2015)
Matias, Y., Shamir, A.: A video scrambling technique based on space filling curves. In: Conference on the Theory and Application of Cryptographic Techniques, pp. 398–417. Springer (1987). https://doi.org/10.1007/3-540-48184-2_35
Moon, B., Jagadish, H.V., Faloutsos, C., Saltz, J.H.: Analysis of the clustering properties of the Hilbert space-filling curve. IEEE Trans. knowl. Data Eng. 13(1), 124–141 (2001)
Moore, E.H.: On certain crinkly curves. Trans. Am. Math. Soc. 1(1), 72–90 (1900)
Munroe, R.: xkcd: Map of the internet. https://xkcd.com/195 (2006-12-11). Accessed 16 Nov 2021
Oord, A.V.D., Vinyals, O., Kavukcuoglu, K.: Neural discrete representation learning. NIPS (2017)
Ouni, T., Lassoued, A., Abid, M.: Gradient-based space filling curves: application to lossless image compression. In: 2011 IEEE International Conference on Computer Applications and Industrial Electronics (ICCAIE), pp. 437–442. IEEE (2011)
Peano, G.: Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen 36(1), 157–160 (1890)
Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36(6), 1389–1401 (1957)
Ramesh, A., et al.: Zero-shot text-to-image generation. ICML (2021)
Razavi, A., Oord, A.V.D., Vinyals, O.: Generating diverse high-fidelity images with VQ-VAE-2. NeurIPS (2019)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948)
Sierpínski, W.: Sur une nouvelle courbe continue qui remplit toute une aire plane. Bull. Acad. Sci. Cracovie (Sci. math. et nat. Serie A), pp. 462–478 (1912)
Thyagarajan, K., Chatterjee, S.: Fractal scanning for image compression. In: Conference Record of the Twenty-Fifth Asilomar Conference on Signals, Systems & Computers, pp. 467–468. IEEE Computer Society (1991)
Veličkovič, P., Cucurull, G., Casanova, A., Romero, A., Lió, P., Bengio, Y.: Graph attention networks. In: International Conference on Learning Representations (2018). https://openreview.net/forum?id=rJXMpikCZ
Vinyals, O., Bengio, S., Kudlur, M.: Order matters: sequence to sequence for sets. ICLR (2015)
Vinyals, O., Fortunato, M., Jaitly, N.: Pointer networks. NIPS (2015)
Welch, T.A.: Technique for high-performance data compression. Computer (1984)
Witten, I.H., Neal, R.M., Cleary, J.G.: Arithmetic coding for data compression. Commun. ACM 30(6), 520–540 (1987)
Xiao, H., Rasul, K., Vollgraf, R.: Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747 (2017)
Yoo, A.B., Jette, M.A., Grondona, M.: SLURM: simple Linux utility for resource management. In: Feitelson, D., Rudolph, L., Schwiegelshohn, U. (eds.) JSSPP 2003. LNCS, vol. 2862, pp. 44–60. Springer, Heidelberg (2003). https://doi.org/10.1007/10968987_3
Zhou, L., Johnson, C.R., Weiskopf, D.: Data-driven space-filling curves. IEEE Trans. Visual. Comput. Graph. 27(2), 1591–1600 (2020)
Zhu, J., Hoorfar, A., Engheta, N.: Bandwidth, cross-polarization, and feed-point characteristics of matched Hilbert antennas. IEEE Antennas Wireless Propag. Lett. 2, 2–5 (2003)
Ziv, J., Lempel, A.: Compression of individual sequences via variable-rate coding. IEEE Trans. Inf. Theory 24(5), 530–536 (1978)
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This work was partially supported by the Amazon Research Award to AS.
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Wang, H., Gupta, K., Davis, L., Shrivastava, A. (2022). Neural Space-Filling Curves. In: Avidan, S., Brostow, G., Cissé, M., Farinella, G.M., Hassner, T. (eds) Computer Vision – ECCV 2022. ECCV 2022. Lecture Notes in Computer Science, vol 13667. Springer, Cham. https://doi.org/10.1007/978-3-031-20071-7_25
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