Abstract
Formal Concept Analysis (FCA) decomposes a matrix into a set of sparse matrices capturing its underlying structure. A similar task for real-valued data, transform coding, arises in image compression. Existing cosine transform coding for JPEG image compression uses a fixed, decorrelating transform; however, compression is limited as images rarely consist of pure cosines. The question remains whether an FCA adaptive transform can be applied to image compression. We propose a multi-layer FCA (MFCA) adaptive ordered transform and Sequentially Sifted Linear Programming (SSLP) encoding pair for adaptive image compression. Our hypothesis is that MFCA’s sparse linear codes (closures) for natural scenes, are a complete family of ordered, localized, oriented, bandpass receptive fields, predicted by models of the primary visual cortex. Results on real data demonstrate that adaptive compression is feasible. These initial results may play a role in improving compression rates and extending the applicability of FCA to real-valued data.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chen, S.S., Donoho, D., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Scientific Computing 20, 33–61 (1998)
Mallat, S.: A Wavelet Tour of Signal Processing, The Sparse Way, 3rd edn. Academic Press (2008)
Lewicki, M.S., Sejnowski, T.J., Hughes, H.: Learning overcomplete representations. Neural Computation 12, 337–365 (1998)
Lewicki, M.S., Olshausen, B.A.: A probabilistic framework for the adaptation and comparison of image codes. J. Opt. Soc. Am. A 16, 1587–1601 (1999)
Olshausen, B.A., Field, D.J.: Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381(6583), 607–609 (1996)
Olshausen, B.O., Fieldt, D.J.: Sparse coding with an overcomplete basis set: a strategy employed by V1. Vision Research 37, 3311–3325 (1997)
Rubinstein, R., Bruckstein, A.M., Elad, M.: Dictionaries for Sparse Representation Modeling. Proceedings of the IEEE 98(6), 1045–1057 (2010)
Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1999)
Horev, I., Bryt, O., Rubinstein, R.: Adaptive image compression using sparse dictionaries. In: 19th Int. Conf. Sys., Sig. and Im. Process., pp. 592–595 (2012)
Pennebacker, W.B., Mitchell, J.L.: JPEG still image data compression standard. Springer, New York (1993)
Taubman, D.S., Marcellin, M.: JPEG2000: image compression fundamentals, standards and practice. Kluwer Academic Publishers, Norwell (2001)
Ahmed, N., Natarajan, T., Rao, K.R.: Discrete Cosine Transform. IEEE Trans. Computers C-32, 90–93 (1974)
Jolliffe, I.T.: Principal Component Analysis, 2nd edn. Springer Series in Statistics (October 2002)
de Fréin, R.: Ghostbusters: A Parts-based NMF Algorithm. In: 24th IET Irish Signals and Systems Conference, pp. 1–8 (June 2013)
de Fréin, R.: Formal concept analysis via atomic priming. In: Cellier, P., Distel, F., Ganter, B. (eds.) ICFCA 2013. LNCS, vol. 7880, pp. 92–108. Springer, Heidelberg (2013)
Bryt, O., Elad, M.: Compression of facial images using the K-SVD algorithm. Journal of Visual Communication and Image Representation 19(4), 270–283 (2008)
Howard, P.G., Vitter, J.S.: Arithmetic coding for data compression. Proceedings of the IEEE 82(6) (June 1994)
de Fréin, R., Rickard, S.T.: The synchronized short-time-Fourier-transform: Properties and definitions for multichannel source separation. IEEE Trans. Sig. Proc. 59(1), 91–103 (2011)
Lee, D.D., Seung, H.S.: Algorithms for non-negative matrix factorization. In: NIPS, pp. 556–562. MIT Press (2000)
Elad, M.: Sparse and redundant representations - from theory to applications in signal and image processing. Springer (2010)
Mallat, S., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Sig. Proc. 41(12) (1993)
Ganter, B.: Two Basic Algorithms in Concept Analysis. In: Kwuida, L., Sertkaya, B. (eds.) ICFCA 2010. LNCS, vol. 5986, pp. 312–340. Springer, Heidelberg (2010)
Lindig, C.: Fast Concept Analysis. Working with Conceptual Structures-Contributions to ICCS, pp. 235–248 (2000)
Kuznetsov, S.O.: A Fast Algorithm for Computing All Intersections of Objects in a Finite Semi-Lattice. Auto. Doc. and Math. Linguistics 27(5), 11–21 (1993)
Andrews, S.: In-Close, a Fast Algorithm for Computing Formal Concepts. In: The Seventeenth International Conference on Conceptual Structures (2009)
Vychodil, V.: A New Algorithm for Computing Formal Concepts. In: Cybernetics and Systems, pp. 15–21 (2008)
Krajca, P., Outrata, J., Vychodil, V.: Parallel Recursive Algorithm for FCA. In: CLA 2008, vol. 433, pp. 71–82 (2008)
Kuznetsov, S.O., Obiedkov, S.A.: Comparing Performance of Algorithms for Generating Concept Lattices. J. Exper. & Th. Artif. Intell. 14, 189–216 (2002)
Bordat, J.-P.: Calcul pratique du treillis de Galois d’une correspondance. Mathématiques et Sciences Humaines 96, 31–47 (1986)
Berry, A., Bordat, J.-P., Sigayret, A.: A Local Approach to Concept Generation. Annals of Mathematics and Artificial Intelligence 49(1), 117–136 (2006)
Norris, E.M.: An Algorithm for Computing the Maximal Rectangles in a Binary Relation. Rev. Roum. Math. Pures et Appl. 23(2), 243–250 (1978)
Dowling, C.E.: On the Irredundant Generation of Knowledge Spaces. J. Math. Psychol. 37, 49–62 (1993)
Godin, R., Missaoui, R., Alaoui, H.: Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices. Computational Intelligence 11, 246–267 (1995)
Carpineto, C., Romano, G.: A Lattice Conceptual Clustering System and Its Application to Browsing Retrieval. Machine Learning, 95–122 (1996)
Valtchev, P., Missaoui, R., Lebrun, P.: A Partition-based Approach Towards Constructing Galois (concept) Lattices. Discrete Math., 801–829 (2002)
Yu, Y., Qian, X., Zhong, F., Li, X.R.: An Improved Incremental Algorithm for Constructing Concept Lattices. In: Soft. Eng., World Congress, vol. 4, pp. 401–405 (2009)
Krajca, P., Vychodil, V.: Distributed Algorithm for Computing Formal Concepts Using Map-Reduce Framework. In: Adams, N.M., Robardet, C., Siebes, A., Boulicaut, J.-F. (eds.) IDA 2009. LNCS, vol. 5772, pp. 333–344. Springer, Heidelberg (2009)
Dean, J., Ghemawat, S.: MapReduce: Simplified Data Processing on Large Clusters. In: OSDI, p. 13 (2004)
Xu, B., de Fréin, R., Robson, E., Ó Foghlú, M.: Distributed Formal Concept Analysis Algorithms Based on an Iterative MapReduce Framework. In: Domenach, F., Ignatov, D.I., Poelmans, J. (eds.) ICFCA 2012. LNCS, vol. 7278, pp. 292–308. Springer, Heidelberg (2012)
Ekanayake, J., Li, H., Zhang, B., Gunarathne, T., Bae, S.H., Qiu, J., Fox, G.: Twister: a Runtime for Iterative MapReduce. In: HPDC 2010, pp. 810–818. ACM (2010)
Belohlavek, R., Glodeanu, C., Vychodil, V.: Optimal factorization of three-way binary data using triadic concepts. Order 30(2), 437–454 (2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
de Fréin, R. (2014). Multilayered, Blocked Formal Concept Analyses for Adaptive Image Compression. In: Glodeanu, C.V., Kaytoue, M., Sacarea, C. (eds) Formal Concept Analysis. ICFCA 2014. Lecture Notes in Computer Science(), vol 8478. Springer, Cham. https://doi.org/10.1007/978-3-319-07248-7_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-07248-7_18
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07247-0
Online ISBN: 978-3-319-07248-7
eBook Packages: Computer ScienceComputer Science (R0)