- Research Article
- Open Access
The Hermite Transform: A Survey
EURASIP Journal on Advances in Signal Processing volume 2006, Article number: 026145 (2006)
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.
Martens J-B: The Hermite transform—applications. IEEE Transactions Acoustics, Speech, Signal Processing 1990, 38(9):1607–1618. 10.1109/29.60076
Martens J-B: The Hermite transform—theory. IEEE Transactions Acoustics, Speech, Signal Processing 1990, 38(9):1595–1606. 10.1109/29.60086
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.
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.
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.
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
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.
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.
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.
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.
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
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.
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
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.
Martens J-B: Adaptive contrast enhancement through residue-image processing. Signal Processing 1995, 44(1):1–18. 10.1016/0165-1684(95)00011-2
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
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
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
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
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
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.
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.
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
Martens J-B: Image Technology Design—A Perceptual Approach. Kluwer Academic, Boston, Mass, USA; 2003.
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.
Brackx F, de Schepper N, Sommen F: The higher dimensional Hermite transform—a new approach. Complex Variables 2003, 48(3):189–210.
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
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
Daubechies I: Ten Lectures on Wavelets. SIAM, Philadelphia, Pa, USA; 1992.
Rosenfeld A: Multiresolution Image Processing and Analysis. Springer, Berlin, Germany; 1984.
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
Lindeberg T: Scale-Space Theory in Computer Vision. Kluwer Academic, Boston, Mass, USA; 1994.
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.
Martens J-B: Application of scale space to image coding. IEEE Transactions on Communications 1990, 38(9):1585–1591. 10.1109/26.61400
Florack L: Computational Imaging and Vision. Kluwer Academic, Dordrecht, the Netherlands; 1997.
Dubois E: The Sampling and reconstruction of time-varying imagery with application in video systems. Proceedings of the IEEE 1985, 73(4):502–522.
Koenderink JJ: The structure of images. Biological Cybernetics 1984, 50(5):363–370. 10.1007/BF00336961
ter Haar Romeny BM: Front-End Vision and Multi-Scale Image Analysis. Kluwer Academic, Boston, Mass, USA; 2003.
ter Haar Romeny BM (Ed): Geometry-Driven Diffusion in Computer Vision. Kluwer Academic, Dordrecht, the Netherlands; 1994.
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
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
Lindeberg T: Feature detection with automatic scale selection. International Journal of Computer Vision 1998, 30(2):79–116. 10.1023/A:1008045108935
Thorpe JA: Elementary Topics in Differential Geometry. Springer, New York, NY, USA; 1979.
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
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
van Wijk JJ: Spot noise texture synthesis for data visualization. Computer Graphics 1991, 25(4):309–318. 10.1145/127719.122751
About this article
Cite this article
Martens, JB. The Hermite Transform: A Survey. EURASIP J. Adv. Signal Process. 2006, 026145 (2006). https://doi.org/10.1155/ASP/2006/26145
- Information Technology
- Quantum Information
- Current Paper
- Partial Signal
- Intersection Curvature