Ieee Transactions on Image Processing on the Number of Rectangular Tilings

Adaptive multiscale representations via quadtree splitting and two-dimensional (2-D) wavelet packets, which amount to space and frequency decompositions, respectively, are powerful concepts that have been widely used in applications. These schemes are direct extensions of their one-dimensional counterparts, in particular, by coupling of the two dimensions and restricting to only one possible further partition of each block into four subblocks. In this paper, we consider more flexible schemes that exploit more variations of multidimensional data structure. In the meantime, we restrict to tree-based decompositions that are amenable to fast algorithms and have low indexing cost. Examples of these decomposition schemes are anisotropic wavelet packets, dyadic rectangular tilings, separate dimension decompositions, and general rectangular tilings. We compute the numbers of possible decompositions for each of these schemes. We also give bounds for some of these numbers. These results show that the new rectangular tiling schemes lead to much larger sets of 2-D space and frequency decompositions than the commonly-used quadtree-based schemes, therefore bearing the potential to obtain better representation for a given image

[1]  Martin Vetterli,et al.  Orthogonal time-varying filter banks and wavelet packets , 1994, IEEE Trans. Signal Process..

[2]  Markus H. Gross,et al.  Efficient Triangular Surface Approximations Using Wavelets and Quadtree Data Structures , 1996, IEEE Trans. Vis. Comput. Graph..

[3]  Murat Kunt,et al.  Adaptive Split-and-Merge for Image Analysis and Coding , 1986, Other Conferences.

[4]  Wee Sun Lee Tiling and adaptive image compression , 2000, IEEE Trans. Inf. Theory.

[5]  Minh N. Do,et al.  Rat e-distortion optimized tree structured compression algorithms for piecewise smooth images , 2005 .

[6]  Shih-Fu Chang,et al.  Frequency and spatially adaptive wavelet packets , 1995, 1995 International Conference on Acoustics, Speech, and Signal Processing.

[7]  Ronald R. Coifman,et al.  Entropy-based algorithms for best basis selection , 1992, IEEE Trans. Inf. Theory.

[8]  Kannan Ramchandran,et al.  Tilings of the time-frequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms , 1993, IEEE Trans. Signal Process..

[9]  D. Donoho Wedgelets: nearly minimax estimation of edges , 1999 .

[10]  Minh N. Do,et al.  Rate-distortion optimized tree-structured compression algorithms for piecewise polynomial images , 2005, IEEE Transactions on Image Processing.

[11]  Allen Gersho,et al.  Image compression with variable block size segmentation , 1992, IEEE Trans. Signal Process..

[12]  Minh N. Do,et al.  Anisotropic 2D wavelet packets and rectangular tiling: theory and algorithms , 2003, SPIE Optics + Photonics.

[13]  K Ramchandran,et al.  Best wavelet packet bases in a rate-distortion sense , 1993, IEEE Trans. Image Process..

[14]  S. Mallat A wavelet tour of signal processing , 1998 .

[15]  Gary J. Sullivan,et al.  Efficient quadtree coding of images and video , 1994, IEEE Trans. Image Process..

[16]  Charles A. Bouman,et al.  Fast search for best representations in multitree dictionaries , 2006, IEEE Transactions on Image Processing.

[17]  L. Villemoes,et al.  A Fast Algorithm for Adapted Time–Frequency Tilings , 1996 .

[18]  D. Donoho CART AND BEST-ORTHO-BASIS: A CONNECTION' , 1997 .

[19]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing, 2nd Edition , 1999 .

[20]  Minh N. Do,et al.  Anisotropic 2-D Wavelet Packets and Rectangular Tiling: Theory and Algorithms , 2003 .

[21]  Michael T. Orchard,et al.  Joint space-frequency segmentation using balanced wavelet packet trees for least-cost image representation , 1997, IEEE Trans. Image Process..

[22]  N. Sloane,et al.  Some Doubly Exponential Sequences , 1973, The Fibonacci Quarterly.

[23]  Nicholas N. Bennett Fast Algorithm for Best Anisotropic Walsh Bases and Relatives , 2000 .

[24]  M. Victor Wickerhauser,et al.  Adapted wavelet analysis from theory to software , 1994 .

[25]  M. Wickerhauser SOME PROBLEMS RELATED TO WAVELET PACKET BASES AND CONVERGENCE , 2003 .