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The Hermite Transform: A Survey

Abstract

With this survey on the Hermite transformation we want to pursue the following two goals. First, we want to provide a comprehensive and up-to-date description of the Hermite transformation, its underlying philosophy, and its most important properties and their implications for applications. As so often when publications and development go hand-in-hand, new insights have led to changes in or generalizations of already published results, and not all of these changes have been considered sufficiently substantial to be published separately. As a consequence, the existing publications on the Hermite transformation do not fully reflect our most recent insights, and the current paper intends to remedy this. Second, we also want to share some new results. Two specific new results, that is, partial signal decompositions and intersection curvatures, are therefore treated in more detail than other aspects.

References

  1. 1.

    Martens J-B: The Hermite transform—applications. IEEE Transactions Acoustics, Speech, Signal Processing 1990, 38(9):1607–1618. 10.1109/29.60076

    Article  Google Scholar 

  2. 2.

    Martens J-B: The Hermite transform—theory. IEEE Transactions Acoustics, Speech, Signal Processing 1990, 38(9):1595–1606. 10.1109/29.60086

    MATH  Article  Google Scholar 

  3. 3.

    Young R: The Gaussian derivative theory of spatial vision: analysis of cortical cell receptive field line-weighting profiles. In Tech. Report GMR-4920. General Motors Research Laboratories, Detroit, Mich, USA; 1985.

    Google Scholar 

  4. 4.

    Silván-Cárdenas JL, Escalante-Ramírez B: Image coding with a directional-oriented Hermite transform on a hexagonal lattice. In Applications of Digital Image Processing XXIV, July–August 2001, San Diego, Calif, USA, Proceedings of SPIE Edited by: Tescher AG. 4472: 528–536.

    Google Scholar 

  5. 5.

    van Dijk AM, Martens J-B: Feature-based image compression with steered Hermite transforms. In Proceedings of IEEE International Conference on Image Processing (ICIP '96), September 1996, Lausanne, Switzerland. Volume 1. IEEE; 205–208.

    Google Scholar 

  6. 6.

    van Dijk AM, Martens J-B: Image representation and compression with steered Hermite transforms. Signal Processing 1997, 56(1):1–16. 10.1016/S0165-1684(96)00146-6

    MATH  Article  Google Scholar 

  7. 7.

    Escalante-Ramírez B, López-Caloca A: Image fusion with the Hermite transform. Proceedings of IEEE International Conference on Image Processing (ICIP '03), September 2003, Barcelona, Spain 145–148.

    Google Scholar 

  8. 8.

    Escalante-Ramírez B, López-Caloca A, Zambrano-Gallardo C: Multiresolution fusion of remotely sensed images with the Hermite transform. In Image and Signal Processing for Remote Sensing IX, September 2004, Barcelona, Spain, Proceedings of SPIE Edited by: Bruzzone L. 5238: 575–584.

    Google Scholar 

  9. 9.

    Escalante-Ramírez B: Optic flow estimation using the Hermite transform. In Applications of Digital Image Processing XXVII, August 2004, Denver, Colo, USA, Proceedings of SPIE Edited by: Tescher AG. 5558: 632–643.

    Google Scholar 

  10. 10.

    Silván-Cárdenas JL, Escalante-Ramírez B: Optic-flow information extraction with directional Gaussian-derivatives. Proceedings of IEEE 15th International Conference on Pattern Recognition (ICPR '00), September 2000, Barcelona, Spain 3: 3194–3197.

    Google Scholar 

  11. 11.

    Escalante-Ramírez B, Martens J-B: Noise reduction in computerized tomography images by means of polynomial transforms. Journal of Visual Communication and Image Representation 1992, 3: 272–285. 10.1016/1047-3203(92)90023-M

    Article  Google Scholar 

  12. 12.

    Escalante-Ramírez B, Martens J-B, de Ridder H: Perceptually-based digital processing techniques for noise reduction in computed tomography images. In Human Vision, Visual Processing, and Digital Display III, February 1992, San Jose, Calif, USA, Proceedings of SPIE Edited by: Rogowitz BE. 1666: 288–299.

    Google Scholar 

  13. 13.

    Martens J-B: Deblurring digital images by means of polynomial transforms. Computer Vision, Graphics and Image Processing 1990, 50(2):157–176. 10.1016/0734-189X(90)90039-X

    Article  Google Scholar 

  14. 14.

    Martens J-B: Adaptive image processing by means of polynomial transforms. In Human Vision, Visual Processing, and Digital Display III, February 1992, San Jose, Calif, USA, Proceedings of SPIE Edited by: Rogowitz BE. 1666: 276–287.

    Google Scholar 

  15. 15.

    Martens J-B: Adaptive contrast enhancement through residue-image processing. Signal Processing 1995, 44(1):1–18. 10.1016/0165-1684(95)00011-2

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Kayargadde V, Martens J-B: Estimation of edge parameters and image blur using polynomial transforms. CVGIP: Graphical Models and Image Processing 1994, 56(6):442–461. 10.1006/cgip.1994.1041

    Google Scholar 

  17. 17.

    Kayargadde V, Martens J-B: An objective measure for perceived noise. Signal Processing 1996, 49(3):187–206. 10.1016/0165-1684(96)00016-3

    MATH  Article  Google Scholar 

  18. 18.

    Kayargadde V, Martens J-B: Estimation of perceived image blur using edge features. International Journal of Imaging Systems and Technology 1996, 7(2):102–109. 10.1002/(SICI)1098-1098(199622)7:2<102::AID-IMA6>3.0.CO;2-C

    Article  Google Scholar 

  19. 19.

    Kayargadde V, Martens J-B: Perceptual characterization of images degraded by blur and noise: model. Journal of the Optical Society of America A 1996, 13(6):1178–1188. 10.1364/JOSAA.13.001178

    Article  Google Scholar 

  20. 20.

    Martens J-B: Local orientation analysis in images by means of the Hermite transformation. IEEE Transactions Image Processing 1997, 6(8):1103–1116. 10.1109/83.605408

    Article  Google Scholar 

  21. 21.

    Rivero-Moreno CJ, Bres S: Spatio-temporal primitive extraction using Hermite and Laguerre filters for early vision video indexing. Proceedings of International Conference on Image Analysis and Recognition (ICIAR '04), September–October 2004, Porto, Portugal 825–832.

    Google Scholar 

  22. 22.

    Rivero-Moreno CJ, Bres S: Texture feature extraction and indexing by Hermite filters. Proceedings of IEEE 17th International Conference on Pattern Recognition (ICPR '04), August 2004, Cambridge, UK 684–687.

    Google Scholar 

  23. 23.

    Georgeson MA, Freeman TCA: Perceived location of bars and edges in one-dimensional images: computational models and human vision. Vision Research 1997, 37(1):127–142. 10.1016/S0042-6989(96)00078-8

    Article  Google Scholar 

  24. 24.

    Martens J-B: Image Technology Design—A Perceptual Approach. Kluwer Academic, Boston, Mass, USA; 2003.

    Google Scholar 

  25. 25.

    Rivero-Moreno CJ, Bres S: Conditions of similarity between Hermite and Gabor filters as models of the human visual system. Proceedings of 10th International Conference on Computer Analysis of Images and Patterns (CAIP '03 ), August 2003, Groningen, the Netherlands 762–769.

    Google Scholar 

  26. 26.

    Brackx F, de Schepper N, Sommen F: The higher dimensional Hermite transform—a new approach. Complex Variables 2003, 48(3):189–210.

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Mallat SG: Multifrequency channel decompositions of images and wavelet models. IEEE Transactions Acoustics, Speech, Signal Processing 1989, 37(12):2091–2110. 10.1109/29.45554

    Article  Google Scholar 

  28. 28.

    Vetterli M, Le Gall D: Perfect reconstruction FIR filter banks: some properties and factorizations. IEEE Transactions Acoustics, Speech, Signal Processing 1989, 37(7):1057–1071. 10.1109/29.32283

    Article  Google Scholar 

  29. 29.

    Daubechies I: Ten Lectures on Wavelets. SIAM, Philadelphia, Pa, USA; 1992.

    Google Scholar 

  30. 30.

    Rosenfeld A: Multiresolution Image Processing and Analysis. Springer, Berlin, Germany; 1984.

    Google Scholar 

  31. 31.

    Koenderink JJ, van Doorn AJ: Generic neighbourhood operators. IEEE Transactions on Pattern Analysis and Machine Intelligence 1992, 14(6):597–605. 10.1109/34.141551

    Article  Google Scholar 

  32. 32.

    Lindeberg T: Scale-Space Theory in Computer Vision. Kluwer Academic, Boston, Mass, USA; 1994.

    Google Scholar 

  33. 33.

    Makram-Ebeid S, Mory B: Scale-space image analysis based on Hermite polynomials theory. In Proceedings of 4th International Conference on Scale Space Methods in Computer Vision (Scale-Space '03), Lecture Notes in Computer Science. Volume 2695. Edited by: Griffin LD, Lillholm M. Springer, Isle of Skye, UK; 2003:57–71.

    Google Scholar 

  34. 34.

    Martens J-B: Application of scale space to image coding. IEEE Transactions on Communications 1990, 38(9):1585–1591. 10.1109/26.61400

    Article  Google Scholar 

  35. 35.

    Florack L: Computational Imaging and Vision. Kluwer Academic, Dordrecht, the Netherlands; 1997.

    Google Scholar 

  36. 36.

    Dubois E: The Sampling and reconstruction of time-varying imagery with application in video systems. Proceedings of the IEEE 1985, 73(4):502–522.

    Article  Google Scholar 

  37. 37.

    Koenderink JJ: The structure of images. Biological Cybernetics 1984, 50(5):363–370. 10.1007/BF00336961

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    ter Haar Romeny BM: Front-End Vision and Multi-Scale Image Analysis. Kluwer Academic, Boston, Mass, USA; 2003.

    Google Scholar 

  39. 39.

    ter Haar Romeny BM (Ed): Geometry-Driven Diffusion in Computer Vision. Kluwer Academic, Dordrecht, the Netherlands; 1994.

    Google Scholar 

  40. 40.

    Weickert J, ter Haar Romeny BM, Viergever MA: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions Image Processing 1998, 7(3):398–410. 10.1109/83.661190

    Article  Google Scholar 

  41. 41.

    Lindeberg T: Edge detection and ridge detection with automatic scale selection. International Journal of Computer Vision 1998, 30(2):117–156. 10.1023/A:1008097225773

    Article  Google Scholar 

  42. 42.

    Lindeberg T: Feature detection with automatic scale selection. International Journal of Computer Vision 1998, 30(2):79–116. 10.1023/A:1008045108935

    Article  Google Scholar 

  43. 43.

    Thorpe JA: Elementary Topics in Differential Geometry. Springer, New York, NY, USA; 1979.

    Google Scholar 

  44. 44.

    Freeman WT, Adelson EH: The design and use of steerable filters. IEEE Transactions on Pattern Analysis and Machine Intelligence 1991, 13(9):891–906. 10.1109/34.93808

    Article  Google Scholar 

  45. 45.

    Meesters L, Martens J-B: A single-ended blockiness measure for JPEG-coded images. Signal Processing 2002, 82(3):369–387. 10.1016/S0165-1684(01)00177-3

    MATH  Article  Google Scholar 

  46. 46.

    van Wijk JJ: Spot noise texture synthesis for data visualization. Computer Graphics 1991, 25(4):309–318. 10.1145/127719.122751

    Article  Google Scholar 

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Correspondence to Jean-Bernard Martens.

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Martens, J. The Hermite Transform: A Survey. EURASIP J. Adv. Signal Process. 2006, 026145 (2006). https://doi.org/10.1155/ASP/2006/26145

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Keywords

  • Information Technology
  • Quantum Information
  • Current Paper
  • Partial Signal
  • Intersection Curvature