Dimension Reduction and Remote Sensing Using Modern Harmonic Analysis

  • John J. Benedetto
  • Wojciech Czaja
Living reference work entry


Harmonic analysis has interleaved creatively and productively with remote sensing to address effectively some of the most difficult dimension reduction problems of modern times. These problems are part and parcel of fundamental ideas in machine learning and data mining, dealing with a host of data collection and data fusion technologies. Linear dimension reduction methods are the starting point herein, which themselves lead to the formulation of non-linear dimension reduction algorithms necessary to resolve information preserving dimension reduction associated with the likes of hyperspectral imagery and LIDAR data. Harmonic analysis arises in the form of data dependent non-linear kernel eigenmap methods, and it is fundamental to design and optimize techniques such as Laplacian and Schroedinger eigenmaps. These are exposited. Further, the fundamental roles in remote sensing of the theories of frames, compressed sensing, sparse representations, and diffusion-based image processing are explained. Significant examples and major applications are described.


Compress Sense Tight Frame Dual Frame Locally Linear Embedding Restricted Isometry Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The first named author gratefully acknowledges the support of MURI-ARO Grant W911NF-090-1-0383, NGA Grant HM - 1582-08-1-0009, and DTRA Grant HDTRA 1-13-1-0015. The Second named author gratfully acknowledges the support of NSF through grant CBET 0854233, NGA through grant HM - 1582-08-1-0009 and DTRA though grant HDTRA 1-13-1-0015.


  1. Aldroubi A, Cabrelli C, Molter U (2004) Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for \(L^{2}(\mathbb{R}^{d})\). Appl Comput Harmon Anal 17(2):11–140MathSciNetCrossRefGoogle Scholar
  2. Bachmann CM, Ainsworth TL, Fusina RA (2005) Exploiting manifold geometry in hyperspectral imagery. IEEE Trans Geosci Remote Sens 43(3):441–454CrossRefGoogle Scholar
  3. Banerjee A, Burlina P, Broadwater J (2007) A machine learning approach for finding hyperspectral endmembers. In: IEEE international geoscience and remote sensing symposium, Barcelona, 2007, pp 3817–3820Google Scholar
  4. Baraniuk RG, Wakin MB (2009) Random projections of smooth manifolds. Found Comput Math 9(1):51–77MathSciNetCrossRefzbMATHGoogle Scholar
  5. Baraniuk R, Davenport M, DeVore R, Wakin M (2008) A simple proof of the restricted isometry property for random matrices. Constr Approx 28(3):253–263MathSciNetCrossRefzbMATHGoogle Scholar
  6. Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15(6):1373–1396CrossRefzbMATHGoogle Scholar
  7. Belkin M, Niyogi P (2008) Towards a theoretical foundation for Laplacian-based manifold methods. J Comput Syst Sci 74(8):1289–1308MathSciNetCrossRefzbMATHGoogle Scholar
  8. Benedetto JJ (1994) Frame decompositions, sampling and uncertainty principle inequalities. In: Benedetto J, Frazier M (eds) Wavelets: mathematics and applications. CRC, Boca Raton, pp 247–304Google Scholar
  9. Benedetto JJ, Czaja W, Dobrosotskaya J, Doster T, Duke K, Gillis D (2012a) Semi-supervised learning of heterogeneous data in remote sensing imagery. In: Independent component analyses, compressive sampling, wavelets, neural net, biosystems, and nanoengineering X, Baltimore. Proceedings of SPIE, vol 8401, 8401-03Google Scholar
  10. Benedetto JJ, Czaja W, Dobrosotskaya J, Doster T, Duke K, Gillis D (2012b) Integration of heterogeneous data for classification in hyperspectral satellite imagery. In: Algorithms and technologies for multispectral, hyperspectral, and ultraspectral imagery XVIII, Baltimore. Proceedings of SPIE, vol 8390, 8390-78Google Scholar
  11. Benedetto JJ, Czaja W, Ehler M, Flake C, Hirn M (2010) Wavelet packets for multi- and hyper-spectral imagery. In Wavelet applications in industrial processing VII, San Jose. Proceedings of SPIE, vol 7535, 7535-08Google Scholar
  12. Benedetto JJ, Czaja W, Flake JC, Hirn M (2009) Frame based kernel methods for automatic classification in hyperspectral data. In: IEEE IGARSS, Cape TownGoogle Scholar
  13. Benedetto JJ, Fickus M (2003) Finite normalized tight frames. Adv Comput Math 18:357–385MathSciNetCrossRefzbMATHGoogle Scholar
  14. Benedetto JJ, Walnut D (1994) Gabor frames for L 2 and related spaces. In: Benedetto J, Frazier M (eds) Wavelets: mathematics and applications. CRC, Boca Raton, pp 97–162Google Scholar
  15. Benedetto JJ, Dellomo M (2015, preprint) Reactive sensing and multiplicative framesGoogle Scholar
  16. Bertozzi A, Esedoglu S, Gillette A (2007) Analysis of a two-scale Cahn-Hilliard model for image inpainting. Multiscale Model Simul 6(3):913–936MathSciNetCrossRefzbMATHGoogle Scholar
  17. Boardman J, Kruse F, Green R (1995) Mapping target signatures via partial unmixing of aviris data. In: Fifth JPL Airborne Earth Science Workshop, Pasadena. Volume 1 of JPL Publication 95-1, pp 23–26Google Scholar
  18. Bowles J, Palmadesso P, Antoniades J, Baumbeck M, Rickard L (1995) Use of filter vectors in hyperspectral data analysis. Proc SPIE 2553:148–157CrossRefGoogle Scholar
  19. Bosch EH, Castrodad A, Cooper JS, Czaja W, Dobrosotskaya J (2013) Multiscale and multidirectional tight frames for image analysis. Proc SPIE 8750Google Scholar
  20. Bosch EH, González A, Vivas J, Easley G (2009) Directional wavelets and a wavelet variogram for two-dimensional data. Math Geosci 41(6):611–641MathSciNetCrossRefzbMATHGoogle Scholar
  21. Candès EJ (2008) The restricted isometry property and its implications for compressed sensing. Compte Rendus de l’Academie des Sci 346:589–592CrossRefzbMATHGoogle Scholar
  22. Candès EJ, Donoho DL (2002) New tight frames of curvelets and optimal representations of objects with piecewise-C 2 singularities. Commun Pure Appl Math 57:219–266CrossRefGoogle Scholar
  23. Candès EJ, Romberg J, Tao T (2006a) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52:489–509CrossRefzbMATHGoogle Scholar
  24. Candès EJ, Romberg J, Tao T (2006b) Stable signal recovery from incomplete and inaccurate measurements. Commun Pure Appl Math 59:1207–1223CrossRefzbMATHGoogle Scholar
  25. Candès EJ, Tao T (2005) Decoding by linear programming. IEEE Trans Inf Theory 51:4203–4215CrossRefzbMATHGoogle Scholar
  26. Candès EJ, Tao T (2006) Near-optimal signal recovery from random projections: universal encoding strategies. IEEE Trans Inf Theory 52:5406–5425CrossRefGoogle Scholar
  27. Casazza P (1999) The art of frame theory. arXiv preprint math/9910168Google Scholar
  28. Casazza P, Kutyniok G (2003) Frames of subspaces. In: Wavelets, frames and operator theory. Contemporary mathematics, vol 345. American Mathematical Society, Providence, pp 87–113Google Scholar
  29. Castrodad A (2009) Graph-based denoising and classification of hyperspectral imagery using nonlocal operators. In: Algorithms and technologies for multispectral, hyperspectral, and ultraspectral imagery XV, Orlando. Proceedings of SPIE, vol 7334, 7334-0EGoogle Scholar
  30. Chambolle A, Lions P-L (1997) Image recovery via total variation minimization and related problems. Numer Math 76:167–188MathSciNetCrossRefzbMATHGoogle Scholar
  31. Chambolle A, DeVore RA, Lee N, Lucier BJ (1998) Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans Image Process 7(3):319–333MathSciNetCrossRefzbMATHGoogle Scholar
  32. Chan TF, Shen J (2005) Image processing and analysis: variational, PDE, wavelet, and stochastic methods. SIAM, PhiladelphiaCrossRefGoogle Scholar
  33. Chan TF, Shen J, Zhou H-M (2006) Total variation wavelet inpainting. J Math Imaging Vis 25: 107–125MathSciNetCrossRefGoogle Scholar
  34. Charles AS, Olshausen BA, Rozell CJ (2011) Learning sparse codes for hyperspectral imagery. Sel Top Signal Process 5(5):963–978CrossRefGoogle Scholar
  35. Christensen O (2003) An introduction to frames and Riesz bases. Birkhauser, BostonCrossRefzbMATHGoogle Scholar
  36. Christensen O, Eldar Y (2004) Oblique dual frames and shift-invariant spaces. Appl Comput Harmon Anal 17(1):48–68MathSciNetCrossRefzbMATHGoogle Scholar
  37. Chui CK, Wang J (2010) Randomized anisotropic transform for nonlinear dimensionality reduction. Int J Geomath 1(1):23–50MathSciNetCrossRefzbMATHGoogle Scholar
  38. Chui CK, Wang J (2010) Dimensionality reduction of hyper-spectral imagery data for feature classification. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, vol 1. Springer, Berlin/Heidelberg, pp 1005–1048CrossRefGoogle Scholar
  39. Chung FRK (1997) Spectral graph theory. CBMS regional conference series in mathematics, vol 92. American Mathematical Society, ProvidenceGoogle Scholar
  40. Coifman RR, Lafon S (2006) Geometric harmonics: a novel tool for multiscale out-of-sample extension of empirical functions. Appl. Comput. Harmon. Anal 21(1):31–52MathSciNetCrossRefzbMATHGoogle Scholar
  41. Coifman RR, Lafon S, Lee AB, Maggioni M, Nadler B, Warner FJ, Zucker SW (2005) Geometric diffusions as a tool for harmonic analysis and structure definition of data. Part i: diffusion maps Proc Natl Acad Sci 102:7426–7431Google Scholar
  42. Coifman RR, Maggioni M (2006) Diffusion wavelets. Appl Comput Harmon Anal 21(1):53–94MathSciNetCrossRefzbMATHGoogle Scholar
  43. Czaja W, Ehler M (2013) Schroedinger eigenmaps for the analysis of biomedical data. IEEE Trans Pattern Anal Mach Intell 35(5):1274–1280CrossRefGoogle Scholar
  44. Czaja W, Halevy A (2011, preprint) On convergence of Schroedinger eigenmapsGoogle Scholar
  45. Czaja W, Dobrosotskaya J, Manning B (2013) Composite wavelet representations for reconstruction of missing data. Proc SPIE 8750Google Scholar
  46. Dasgupta S, Gupta A (1999) An elementary proof of the Johnson-Lindenstrauss lemma. Technical report 99-006, UC BerkeleyGoogle Scholar
  47. Deloye CJ, Flake JC, Kittle D, Bosch EH, Rand RS, Brady DJ (2013) Exploitation performance and characterization of a prototype compressive sensing imaging spectrometer. In: Excursions in harmonic analysis, vol. 1. Applied and numerical harmonic analysis. Birkhäuser, Boston, pp 151–171Google Scholar
  48. Do MN, Vetterli M (2002) Contourlets: a directional multiresolution image representation. In: Proceedings of IEEE international conference on image processing (ICIP), RochesterGoogle Scholar
  49. Dobrosotskaya J, Bertozzi A (2008) A wavelet-laplace Variational technique for image deconvolution and inpainting. IEEE Trans Image Process 17(5):657–663MathSciNetCrossRefGoogle Scholar
  50. Dobrosotskaya J, Bertozzi A (2010) Wavelet analogue of the Ginzburg-Landau energy and its gamma-convergence. Interfaces Free Bound 12(2):497–525MathSciNetCrossRefzbMATHGoogle Scholar
  51. Dobrosotskaya J, Bertozzi A (2013) Analysis of the wavelet Ginzburg-Landau energy in image applications with edges. SIAM J Imaging Sci 6(1):698–729MathSciNetCrossRefzbMATHGoogle Scholar
  52. Dobrosotskaya J, Czaja W (2013, preprint) Shearlet Ginzburg-Landau energy, its gamma convergence and applicationsGoogle Scholar
  53. Donoho DL (1999) Wedgelets: nearly minimax estimation of edges. Ann Stat 27(3):859–897MathSciNetCrossRefzbMATHGoogle Scholar
  54. Donoho DL (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306MathSciNetCrossRefzbMATHGoogle Scholar
  55. Donoho DL, Grimes C (2003) Hessian eigenmaps: new locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci 100:5591–5596MathSciNetCrossRefzbMATHGoogle Scholar
  56. Donoho D, Tanner J (2005) Sparse nonnegative solutions of underdetermined linear equations by linear programming. Proc Natl Acad Sci 102(27):9446–9451MathSciNetCrossRefGoogle Scholar
  57. Duffin RJ, Schaeffer AC (1952) A class of nonharmonic Fourier series. Trans Am Math Soc 72:341–366MathSciNetCrossRefzbMATHGoogle Scholar
  58. Duke K (2012) A study of the relationship between spectrum and geometry through Fourier frames and Laplacian eigenmaps. Ph.D. thesis, University of Maryland, College ParkGoogle Scholar
  59. Easley GR, Labate D, Colonna F (2009) Shearlet based total variation for denoising. IEEE Trans Image Process 18(2):260–268MathSciNetCrossRefGoogle Scholar
  60. Elad M, Starck JL, Querre P, Donoho DL (2005) Simultaneous cartoon texture image inpaitning using morphological component analysis. Appl Comput Harmon Anal 19:340–358MathSciNetCrossRefzbMATHGoogle Scholar
  61. Emmerich H (2003) Diffuse interface approach in materials science thermodynamic concepts and applications of phase-field models. Springer, Berlin/New YorkzbMATHGoogle Scholar
  62. Flake JC (2010) The multiplicative Zak transform, dimension reduction, and wavelet analysis of LIDAR data. Ph.D. thesis, University of Maryland, College ParkGoogle Scholar
  63. Gillis D, Bowles J (2013) An introduction to hyperspectral image data modeling. In: Excursions in harmonic analysis, vol 1. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, pp 173–194Google Scholar
  64. Ginzburg VL, Landau LD (1950) Zh Eksp Teor Fiz 20:1064Google Scholar
  65. Goldberg Y, Zakai A, Kushnir D, Ritov Y (2008) Manifold learning: the price of normalization. J Mach Learn Res 9:1909–1939MathSciNetzbMATHGoogle Scholar
  66. Greer JB (2013) Hyperspectral demixing: sparse recovery of highly correlated endmembers. In: Excursions in harmonic analysis, vol 1. Applied and numerical harmonic analysis. Birkhauser, Boston, pp 195–210Google Scholar
  67. Guo K, Labate D, Lim W-Q, Weiss G, Wilson E (2006) The theory of wavelets with composite dilations. In: Heil C (ed) Harmonic analysis and applications. Applied and numerical harmonic analysis. Birkhauser, Boston, pp 231–250CrossRefGoogle Scholar
  68. Halevy A (2011) Extensions of Laplacian eigenmaps for manifold learning. Ph.D. thesis, University of Maryland, College ParkGoogle Scholar
  69. Hirn M (2009) Enumeration of harmonic frames and frame based dimension reduction. Ph.D. thesis, University of Maryland, College ParkGoogle Scholar
  70. Johnson WB, Lindenstrauss J (1984) Extensions of Lipschitz mappings into a Hilbert space. Contemp Math 26:189–206MathSciNetCrossRefzbMATHGoogle Scholar
  71. Kovačević J, Chebira A (2007) Life beyond bases: the advent of frames (parts I and II). IEEE Signal Process Mag 24(4):86–104 and 24(5):115–125CrossRefGoogle Scholar
  72. Kovačević J, Chebira A (2008) Introduction to frames. Foundations and trends in signal processing, vol 2(1). Now Publishers, BostonGoogle Scholar
  73. Labate D, Lim W, Kutyniok G, Weiss G (2005) Sparse multidimensional representation using shearlets. In: Wavelets XI, San Diego. SPIE proceedings, vol 5914, pp 254–262Google Scholar
  74. Lammers M, Powell A, Yilmaz Ö (2009) Alternative dual frames for digital-to-analog conversion in sigma-delta quantization. Adv Comput Math 32(1):73–102MathSciNetCrossRefGoogle Scholar
  75. Lee JA, Verleysen M (2007) Nonlinear dimensionality reduction, Springer, New York/LondonCrossRefzbMATHGoogle Scholar
  76. Li S, Ogawa H (2004) Pseudoframes for subspaces with applications. J Fourier Anal Appl 10(4):409–431MathSciNetCrossRefzbMATHGoogle Scholar
  77. Mallat S (1999) Wavelet tour of signal processing. Academic, San DiegozbMATHGoogle Scholar
  78. Meyer F, Coifman R (1997) Brushlets: a tool for directional image analysis and image compression. Appl Comput Harmon Anal 4:147–187MathSciNetCrossRefzbMATHGoogle Scholar
  79. Mohan A, Sapiro G, Bosch E (2007) Spatially coherent nonlinear dimensionality reduction and segmentation of hyperspectral images. IEEE Geosci Remote Sens Lett 4(2):206–210CrossRefGoogle Scholar
  80. Patel VM, Easley GR, Healy DM Jr, Chellappa R (2010) Compressed synthetic aperture radar. IEEE J Sel Top Signal Process 4(2):244–254CrossRefGoogle Scholar
  81. Pearson K (1901) On lines and planes of closest fit to systems of points in space. Philos Mag 2(6):559–572CrossRefGoogle Scholar
  82. Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500):2323–2326CrossRefGoogle Scholar
  83. Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D 60:259–268CrossRefzbMATHGoogle Scholar
  84. Schölkopf B, Smola A, Müller K (1998) Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput 10(5):1299–1319CrossRefGoogle Scholar
  85. Strawn N (2011) Geometric structures and optimization on spaces of finite frames. Ph.D. thesis, University of Maryland, College ParkGoogle Scholar
  86. Sun W (2006) G-frames and g-Riesz bases. J Math Anal Appl 322(1):437–452MathSciNetCrossRefzbMATHGoogle Scholar
  87. Tenenbaum V, Silva J, Langford J (2000) A global geometric framework for nonlinear dimensionality reduction. Science 209:2319–2323CrossRefGoogle Scholar
  88. Wang R (2013) Global geometric conditions on dictionaries for the convergence of L 1 minimization algorithms. Ph.D. thesis, University of Maryland, College ParkGoogle Scholar
  89. Widemann D (2008) Dimensionality reduction for hyperspectral data. Ph.D. thesis, University of Maryland, College ParkGoogle Scholar
  90. Winter M (1999) N-FINDR: an algorithm for fast autonomous spectral endmember determination in hyperspectral data. Proc SPIE 3753:266–275CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Norbert Wiener Center, Department of MathematicsUniversity of MarylandCollege Park, MDUSA

Personalised recommendations