Open Access

Quadratic Interpolation and Linear Lifting Design

EURASIP Journal on Image and Video Processing20072007:037843

DOI: 10.1155/2007/37843

Received: 11 August 2006

Accepted: 28 December 2006

Published: 7 March 2007


A quadratic image interpolation method is stated. The formulation is connected to the optimization of lifting steps. This relation triggers the exploration of several interpolation possibilities within the same context, which uses the theory of convex optimization to minimize quadratic functions with linear constraints. The methods consider possible knowledge available from a given application. A set of linear equality constraints that relate wavelet bases and coefficients with the underlying signal is introduced in the formulation. As a consequence, the formulation turns out to be adequate for the design of lifting steps. The resulting steps are related to the prediction minimizing the detail signal energy and to the update minimizing the l2-norm of the approximation signal gradient. Results are reported for the interpolation methods in terms of PSNR and also, coding results are given for the new update lifting steps.


Authors’ Affiliations

Department of Signal Theory and Communications, Technical University of Catalonia (UPC)


  1. Sweldens W: The lifting scheme: a custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis 1996,3(2):186-200. 10.1006/acha.1996.0015MATHMathSciNetView ArticleGoogle Scholar
  2. Gouze A, Antonini M, Barlaud M, Macq B: Design of signal-adapted multidimensional lifting scheme for lossy coding. IEEE Transactions on Image Processing 2004,13(12):1589-1603. 10.1109/TIP.2004.837556View ArticleGoogle Scholar
  3. Li H, Liu G, Zhang Z: Optimization of integer wavelet transforms based on difference correlation structures. IEEE Transactions on Image Processing 2005,14(11):1831-1847.MathSciNetView ArticleGoogle Scholar
  4. Hattay J, Benazza-Benyahia A, Pesquet J-C: Adaptive lifting for multicomponent image coding through quadtree partitioning. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '05), March 2005, Philadelphia, Pa, USA 2: 213-216.Google Scholar
  5. Solé J, Salembier P: A common formulation for interpolation, prediction, and update lifting design. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '06), May 2006, Toulouse, France 2: 13-16.Google Scholar
  6. Solé J, Salembier P: Adaptive quadratic interpolation methods for lifting steps construction. Proceedings of the 6th IEEE International Symposium on Signal Processing and Information Technology (ISSPIT '06), August 2006, Vancouver, Canada 691-696.Google Scholar
  7. Muresan DD, Parks TW: Adaptively Quadratic (AQua) image interpolation. IEEE Transactions on Image Processing 2004,13(5):690-698. 10.1109/TIP.2004.826097View ArticleGoogle Scholar
  8. Boyd S, Vandenberghe L: Convex Optimization. Cambridge University Press, Cambridge, UK; 2004.MATHView ArticleGoogle Scholar
  9. ISO/IEC 15444-1: JPEG 2000 image coding system ISO/IEC, 2000
  10. Deever AT, Hemami SS: Lossless image compression with projection-based and adaptive reversible integer wavelet transforms. IEEE Transactions on Image Processing 2003,12(5):489-499. 10.1109/TIP.2003.812374MathSciNetView ArticleGoogle Scholar


© J. Solé and P. Salembier 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.