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The geometry and the analytic properties of isotropic multiresolution analysis

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Abstract

In this paper we investigate Isotropic Multiresolution Analysis (IMRA), isotropic refinable functions, and wavelets. The main results are the characterization of IMRAs in terms of the Lax–Wiener Theorem, and the characterization of isotropic refinable functions in terms of the support of their Fourier transform. As an immediate consequence of these results, there are no compactly supported (in the space domain) isotropic refinable functions in many dimensions. Next we study the approximation properties of IMRAs. Finally, we discuss the application of IMRA wavelets to 2D and 3D-texture segmentation in natural and biomedical images.

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References

  1. Adelson, E.H., Simoncelli, E., Hingoranp, R.: Orthogonal pyramid transforms for image coding. In: Visual Communications and Image Processing II, Proceedings SPIE, vol. 845, pp. 50–58 (1987)

  2. Aldroubi, A., Cabrelli, C., Molter, U.: Wavelets on irregular grids with arbitrary dilation matrices and frame atoms on L 2(R d). Appl. Comput. Harmon. Anal. 17(2), 119–140. Special Issue: Frames in Harmonic Analysis, Part II (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alexander, S.K., Azencott, R., Papadakis, M.: Isotropic multiresolution analysis for 3D-textures and applications in cardiovascular imaging. In: Van Der Ville, D., Goyal, V., Papadakis, M. (eds.) Wavelets XII, SPIE, vol. 6701, pp. 67011S-1–67011S-12 (2007)

  4. Antonini, M., Barlaud, M., Mathieu, P.: Image coding using lattice vector quantization of wavelet coefficients. In: IEEE Internat. Conf. Acoust. Signal Speech Process., pp. 2273–2276 (1991)

  5. Ayache, A.: Construction de bases d’ondelettes orthonormées de L 2(R 2) non séparables, à support compact et de régularité arbitrairement grande. Comptes Rendus Académie des Sciences de Paris 325, 17–20 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ayache, A.: Construction of non separable dyadic compactly supported wavelet bases for L 2(R 2) of arbitrarily high regularity. Rev. Mat. Iberoam 15 15(1), 37–58 (1999)

    MATH  MathSciNet  Google Scholar 

  7. Ayache, A.: Some methods for constructing non separable, orthonormal, compactly supported wavelet bases. Appl. Comput. Harmon. Anal. 10, 99–111 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Baggett, L.W., Jorgensen, P.E.T., Merrill, K.D., Packer, J.A.: Construction of Parseval wavelets from redundant filter systems. J. Math. Phys. 46, 083502–1–083502–28 (2005)

    Article  MathSciNet  Google Scholar 

  9. Belogay, E., Wang, Y.: Arbitrarily smooth orthogonal nonseparable wavelets in R 2. SIAM J. Math. Anal. 30, 678–697 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Benedetto, J.J., Romero, J.R.: The construction of d-dimensional multiresolution analysis frames. J. Appl. Func. Anal. 2(4), 403–426 (2007)

    MATH  MathSciNet  Google Scholar 

  11. Benedetto, J.J., Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5, 389–427 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Benedetto, J.J., Treiber, O.M.: Wavelet frames: multiresolution analysis and the unitary extension principle. In: Debnath, L. (ed.) Wavelet Transforms and Time-Frequency Signal Analysis, pp. 3–36. Birkhauser, Boston MA (2001)

    Google Scholar 

  13. Bodmann, B.G., Papadakis, M., et al.: Frame isotropic multiresolution analysis for micro CT scans of coronary arteries. In: Papadakis, M., Laine, A., Unser, M. (eds.) Wavelets XI, SPIE, vol. 5914, pp. 59141O/1–12 (2005)

  14. Bownik, M.: The structure of shift invariant subspaces of L 2(ℝn). J. Funct. Anal. 177, 282–309 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bownik, M.: A characterization of affine dual frames in L 2(ℝn). Appl. Comput. Harmon. Anal. 8, 203–221 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cabrelli, C., Heil, C., Molter, U.: Self-similarity and multiwavelets in higher dimensions. In: Memoirs, Amer. Math. Soc. AMS, vol. 170 (2004)

  17. Candes, E.J., Donoho, D.L.: Ridgelets: a key to higher dimensional intermittency? Phil. Trans. R. Soc. Lond. A, 2495–2509 (1999)

    MathSciNet  Google Scholar 

  18. Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 36, 961–1005 (1986)

    Google Scholar 

  19. Christensen, O.: Frames and Riesz Bases. Birkhäuser, Boston (2005)

    Google Scholar 

  20. Coggins, J.M., Jain, A.K.: A spatial filtering approach to texture analysis. Pattern recogn. Lett. 3(3), 195–203 (1985)

    Article  Google Scholar 

  21. Cohen, A., Daubechies, I.: Nonseparable bidimensional wavelet bases. Rev. Mat. Iberoam 9, 51–137 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Condat, L., Forster-Heinlein, B., Van Der Ville, D.: A new family of rotation-covariant wavelets on the hexagonal lattice. In: Van Der Ville, D., Goyal, V., Papadakis, M. (eds.) Wavelets XII, SPIE, vol. 6701, pp. 67010B-1–67010B-9 (2007)

  23. Conway, J.B.: A Course in Operator Theory. Graduate Studies in Mathematics, vol. 21. American Mathematical Society, Providence, RI (2000)

    Google Scholar 

  24. Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput Harmon. Anal. 14(1), 1–46 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. de Boor, C., DeVore, R., Ron, A.: Approximation from shift-invariant subspaces of L 2(R d). Trans. Am. Math. Soc. 341, 787–806 (1994)

    Article  MATH  Google Scholar 

  26. Derado, J.: Nonseprable, compactly supported interpolating refinable functions with arbitrary smoothness. Appl. Comput. Harmon. Anal. 10(2), 113–138 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Dettori, L., Zayed, A.I.: Texture identification of tissues using directional wavelet, ridgelet and curvelet transforms. In: Larson, D.R., Massopust, P.R., Nashed, Z., Nguyen, M.-C., Papadakis, M., Zayed, A.I. (eds.) Frames and Operator Theory in Analysis and Signal Processing, Contemp. Mathem., Amer. Math. Soc., vol. 451, pp. 89–118 (2008)

  28. Donoho, D.L., Huo, X.: Beamlets and multiscale image analysis. In: Barth, T.J., Chan, T., Haimes, R. (eds.) Springer Lecture Notes in Computational Science and Engineering, vol. 20, pp. 149–196 (2002)

  29. Epperson, J., Frazier, M.: An almost orthogonal radial wavelet expansion for radial distributions. J. Fourier Anal. Appl. 1(3), 311–353 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  30. Epperson, J., Frazier, M.: Polar wavelets and associated Littlewood–Paley theory. Diss. Math. 348, 1–51 (1996)

    MathSciNet  Google Scholar 

  31. Fickus, M., Mixon, D.G.: Isotropic moments over integer lattices. Appl. Comp. Harmon. Anal. 26(1), 77–96 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. Fickus, M., Seetharaman, G.S., Oxley, M.E.: Multiscale moment transforms over the integer lattice. In: Van Der Ville, D., Goyal, V., Papadakis, M. (eds.) Wavelets XII, SPIE, vol. 6701, pp. 67011N-1–67011N-8 (2007)

  33. Forster, B., Blu, T., Van Der Ville, D., Unser, M.: Shift-invariant spaces from rotation-covariant functions. Appl. Comp. Harmon. Anal. 25(2), 240–265 (2008)

    Article  MATH  Google Scholar 

  34. Freeman, W.T., Adelson, E.H.: The design and use of steerable filters. IEEE Trans. Pattern Anal. Mach. Intell. 13(9), 891–906 (1991)

    Article  Google Scholar 

  35. Gertz, S.D., Cherukuri, P., Bodmann, B.G., Gladdish, G., Wilner, W.T., Conyers, J.L., Aboshady, I., Madjid, M., Vela, D., Lukovenkov, S., Papadakis, M., Kouri, D.J., Mohammadi Mazraeshahi, R., Frazier, L., Elrod, D., Willerson, J.T., Casscells, S.W.: Usefulness of multi-detector computed tomography for non-envasive evaluation of coronary arteries in asymptomatic patients. Am. J. Cardiol. 97, 287–293 (2006)

    Article  Google Scholar 

  36. Gertz, S.D., Bodmann, B.G., Vela, D., Papadakis, M., Aboshady, I., Cherukuri, P., Alexander, S.K., Kouri, D.J., Baid, S., Gittens, A.A., Gladish, G.W., Conyers, J.L., Cody, D.D., Gavish, L., Mazraeshahi, R.M., Wilner, W.T., Frazier, L., Madjid, M., Zarrabi, A., Lukovenkov, S., Ahmed, A., Willerson, J.T., Casscells, S.W.: Three-dimensional isotropic wavelets for post-acquisitional extraction of latent images of atherosclerotic plaque components from micro-computed tomography of human coronary arteries. Acad. Radiol. 17, 1509–1519 (2007)

    Article  Google Scholar 

  37. Grochenig, K., Madych, W.: Multiresolution analysis, Haar bases and self-similar tilings. IEEE Trans. Inf. Theory 38, 558–568 (1992)

    Article  MathSciNet  Google Scholar 

  38. Guo, K., Labate, D., Lim, W., Weiss, G., Wilson, E.: Wavelets with composite dilations and their MRA properties. Appl. Comput. Harmon. Anal. 20 (2006)

  39. Han, B.: On dual tight wavelet frames. Appl. Comput. Harmon. Anal. 4(4), 380–413 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  40. He, W., Lai, M.J.: Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets. In: Unser, M., Aldroubi, A., Laine, A. (eds.) Wavelet Applications in Signal and Image Processing IV, Proceedings SPIE, vol. 3169, pp. 303–314 (1997)

  41. Heinlein, P., Drexl, J., Schneider, W.: Integrated wavelets for enhancement of microcalcifications in digital mamography. IEEE Trans. Med. Imag. 22(3), 402–413 (2003)

    Article  Google Scholar 

  42. Hernández, E., Weiss, G.: A First Course on Wavelets. CRC, Boca Raton, FL (1996)

    MATH  Google Scholar 

  43. Jain, A.K., Farrokhnia, F.: Unsupervised texture segmentation using Gabor filters. IEEE Trans. Pattern Anal. Mach. Intell. 18(2), 1167–1186 (1991)

    Google Scholar 

  44. Jain, S., Papadakis, M., Dussaud, E.: Explicit schemes in seismic migration and isotropic multiscale representations. In: Grinberg, E., Larson, D.R., Jorgensen, P.E.T., Massopust, P., Olafsson, G., Quinto, E.T., Rubin, B. (eds.) Radon Transforms, Geometry, and Wavelets, Contemp. Mathem., Amer. Math. Soc., vol. 464, pp. 177–200 (2008)

  45. Karasaridis, A., Simoncelli, E.: A filter design technique for steerable pyramid image transforms. In: Acoustics Speech and Signal Processing, ICASP, Atlanta, GA (1996)

  46. Kingsbury, N.: Image processing with complex wavelets. Phil. Trans. R. Soc. London A 357, 2543–2560 (1999)

    Article  MATH  Google Scholar 

  47. Kovačević, J., Vetterli, M.: Nonseparable multidimensional perfect reconstruction filter-banks. IEEE Trans. Inf. Theory 38, 533–555 (1992)

    Article  Google Scholar 

  48. Labate, D., Lim, W., Kutyniok, G., Weiss, G.: Sparse multidimensional representation using shearlets. In: Papadakis, M., Laine, A., Unser, M. (eds.) Wavelets XI, SPIE Proceedings, vol. 5914, pp. 247–255, January 2005

  49. Lawton, W., Resnikoff, H.L.: Multidimensional wavelet bases. Technical Report, Aware, Inc., Bedford, MA, February 1991

  50. Lax, P.D.: Translation invariant spaces. Acta Math. 101, 163–178 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  51. Lindemann, M.: Approximation Properties of Non-separable Wavelet Bases with Isotropic Scaling Matrices and their Relation to Besov Spaces. Ph.D. Thesis, Universitaet Bremen (2005)

  52. Marr, D.: Vision, a Computational Investigation into the Human Representation and Processing of Visual Information. Freeman, New York, NY (1982)

    Google Scholar 

  53. Olafsson, G., Speegle, D.: Wavelets, wavelet sets and linear actions on ℝn. In: Heil, C., Jorgensen, P., Larson, D. (eds.) Wavelets, Frames and Operator Theory, Contemporary Mathematics, vol. 345, pp. 253–281 (2004)

  54. Papadakis, M.: Frames of translates and the generalized frame multiresolution analysis. In: Kopotun, K., Lyche, T., Neamtu, M. (eds.) Trends in Approximation Theory, Innovations in Applied Mathematics, pp. 353–362. Vanderbilt University Press, Nashville, TN (2001)

  55. Papadakis, M., Bodmann, B.G., Alexander, S.K., Vela MD, D., Baid, S., Gittens, A.A., Kouri, D.J, Gertz MD, S.D., Jain, S., Romero, J.R, Li, X., Cherukuri, P., Cody, D.D., Gladish MD, G.W., Aboshady MD, I., Conyers, J.L., Casscells, S.W.: Texture-based tissue characterization for high-resolution CT-scans of coronary arteries. Commun. Numer. Methods Eng. (2009, in press)

  56. Papadakis, M., Gogoshin, G., Kakadiaris, I.A., Kouri, D.J., Hoffman, D.K.: Non-separable radial frame multiresolution analysis in multidimensions. Numer. Func. Anal. Optim. 24, 907–928 (2003)

    Article  MATH  Google Scholar 

  57. Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P.: Image denoising using Gaussian scale mixtures in the wavelet domain. In: IEEE Trans. Image Processing, vol.12(11), pp. 1338–1351. Computer Science Techn. Rep. nr. TR2002-831, Courant Institute of Mathematical Sciences (2003)

  58. Rajpoot, N., Wilson, R., Yao, Z.: Planelets: a new analysis tool for planar feature extraction. In: Proceedings 5th International Workshop on Image Analysis for Multimedia Interactive Services (WIAMIS’04), Lisbon, Portugal, April 2004

  59. Ron, A., Shen, Z.: Affine system in L 2(R d): the analysis of the Analysis operator. J. Func. Anal. 148, 408–447 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  60. Selesnick, I.W.: Smooth wavelet tight frames with zero moments. Appl. Comput. Harmon. Anal. 10(2), 163–181 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  61. Selesnick, I.W., Sendur, L.: Iterated oversampled filter banks and wavelet frames. In: Unser, M., Aldroubi, A., Laine, A. (eds.) Wavelet Applications in Signal and Image Processing VIII. Proceedings of SPIE, vol. 4119 (2000)

  62. Simoncelli, E.P., Freeman, W.T., Adelson, E.H., Heeger, D.J.: Shiftable multi-scale transforms. IEEE Trans. Inf. Theory 38(2), 587–607 (1992)

    Article  MathSciNet  Google Scholar 

  63. Starck, J.-L., Moudden, Y., Bobin, J., Elad, M., Donoho, D.L.: Morphological component analysis. In: Papadakis, M., Laine, A.F., Unser, M. (eds.) Wavelets XI, SPIE, vol. 5914, pp. 59140Q-1–59140Q-15 (2005)

  64. Tzagkarakis, G., Beferull-Lozano, B., Tsakalides, P.: Rotation-invariant texture retrieval with Gaussianized steerable pyramids. IEEE Trans. Image Process. 15(9), 2702–2718 (2006)

    Article  Google Scholar 

  65. Vetterli, M., Kovačević, J.: Wavelets and Subband Coding. Prentice Hall PTR, Englewood Cliffs, NJ (1995)

    MATH  Google Scholar 

  66. Wan, Y., Nowak, R.D.: Quasi-circular rotation invariance in image denoising. In: ICIP, vol. 1, pp. 605–609 (1999)

  67. Wheeden, R.L., Zygmund, A.: Measure and Integral. An Introduction to Real Analysis. Pure and Applied Mathematics. Marcel-Dekker, New York (1977)

    Google Scholar 

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  1. Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Mayagüez, Puerto Rico

    Juan R. Romero

  2. Department of Mathematics, University of Houston, Houston, TX, USA

    Simon K. Alexander, Shikha Baid, Saurabh Jain & Manos Papadakis

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  1. Juan R. Romero
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Correspondence to Manos Papadakis.

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Communicated by Lixin Shen and Yuesheng Xu.

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Romero, J.R., Alexander, S.K., Baid, S. et al. The geometry and the analytic properties of isotropic multiresolution analysis. Adv Comput Math 31, 283–328 (2009). https://doi.org/10.1007/s10444-008-9111-6

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