## Discrimination of isotrigon textures using the Rényi entropy of Allan variances

JOSA A, Vol. 25, Issue 9, pp. 2309-2319 (2008)

http://dx.doi.org/10.1364/JOSAA.25.002309

Enhanced HTML Acrobat PDF (2186 KB)

### Abstract

We present a computational algorithm for isotrigon texture discrimination. The aim of this method consists in discriminating isotrigon textures against a binary random background. The extension of the method to the problem of multitexture discrimination is considered as well. The method relies on the fact that the information content of time or space–frequency representations of signals, including images, can be readily analyzed by means of generalized entropy measures. In such a scenario, the Rényi entropy appears as an effective tool, given that Rényi measures can be used to provide information about a local neighborhood within an image. Localization is essential for comparing images on a pixel-by-pixel basis. Discrimination is performed through a local Rényi entropy measurement applied on a spatially oriented 1-D pseudo-Wigner distribution (PWD) of the test image. The PWD is normalized so that it may be interpreted as a probability distribution. Prior to the calculation of the texture’s PWD, a preprocessing filtering step replaces the original texture with its localized spatially oriented Allan variances. The anisotropic structure of the textures, as revealed by the Allan variances, turns out to be crucial later to attain a high discrimination by the extraction of Rényi entropy measures. The method has been empirically evaluated with a family of isotrigon textures embedded in a binary random background. The extension to the case of multiple isotrigon mosaics has also been considered. Discrimination results are compared with other existing methods.

© 2008 Optical Society of America

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(150.1135) Machine vision : Algorithms

(100.4992) Image processing : Pattern, nonlinear correlators

**ToC Category:**

Machine Vision

**History**

Original Manuscript: January 22, 2008

Revised Manuscript: April 25, 2008

Manuscript Accepted: May 27, 2008

Published: August 20, 2008

**Citation**

Salvador Gabarda and Gabriel Cristóbal, "Discrimination of isotrigon textures using the Rényi entropy of Allan variances," J. Opt. Soc. Am. A **25**, 2309-2319 (2008)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-25-9-2309

Sort: Year | Journal | Reset

### References

- B. Julesz, E. N. Gilbert, and J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137-140 (1978). [CrossRef] [PubMed]
- T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vision Res. 44, 1093-1113 (2004). [CrossRef] [PubMed]
- K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Natl. Acad. Sci. U.S.A. 91, 8482-8486 (1994). [CrossRef] [PubMed]
- T. Maddess and Y. Nagai, “Discriminating isotrigon textures,” Vision Res. 41, 3837-3860 (2001). [CrossRef] [PubMed]
- D. W. Allan, “Statistics of atomic frequency standard,” Proc. IEEE 54, 223-231 (1966). [CrossRef]
- R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis (Prentice Hall, 1992).
- J. L. Starck, E. Candes, and D. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670-684 (2002). [CrossRef]
- M. N. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14, 2091-2106 (2005). [CrossRef] [PubMed]
- D. Donoho, “Wedgelets: nearly minimax estimation of edges,” Ann. Stat. 27, 859-867 (1999). [CrossRef]
- V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L. Dragotti, “Directionlets: anisotropic multidirectional representation with separable filtering,” IEEE Trans. Image Process. 15, 1916-1933 (2006). [CrossRef] [PubMed]
- A. Cumani, “Edge detection in multispectral images,” Comput. Vis. Graph. Image Process. 53, 40-51 (1991).
- L. D. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representation,” Signal Process. 14, 37-68 (1988). [CrossRef]
- E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932). [CrossRef]
- L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781-786 (1966). [CrossRef]
- T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 237-250 (1980).
- T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part II. Discrete-time signals,” Philips J. Res. 35, 276-300 (1980).
- T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372-389 (1980).
- K. H. Brenner, “A discrete version of the Wigner distribution function,” EURASIP J. Appl. Signal Process. 2005, 307-309.
- L. Stankovic, “A measure of some time-frequency distributions concentration,” Signal Process. 81, 623-631 (2001). [CrossRef]
- C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, 1949).
- N. Wiener, Cybernetics (Wiley, 1948).
- A. Rényi, “Some fundamental questions of information theory,” in Selected Papers of Alfréd Rényi, P.Turán, ed. (Akadémiai Kiadó, 1976), pp. 526-552 (1976). [Originally published in Magy. Tud. Akad. Mat. Fiz Tud. Oszt. Kozl., 10, 251-282 (1960)].
- W. J. Williams, M. L. Brown, and A. O. Hero, “Uncertainty, information and time-frequency distributions,” Proc. SPIE 1566, 144-156 (1991). [CrossRef]
- T. H. Sang and W. J. Williams, “Rényi information and signal dependent optimal kernel design.” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1995), Vol. 2, pp. 997-1000.
- P. Flandrin, R. G. Baraniuk, and O. Michel, “Time-frequency complexity and information,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1994), Vol. 3, pp. 329-332.
- R. Eisberg and R. Resnick, Quantum Physics (Wiley, 1974).
- G. Süßmann, “Uncertainty relation: from inequality to equality,” Z. Naturforsch. 52, 49-52 (1997).
- J. J. Wlodarz, “Entropy and Wigner distribution functions revisited,” Int. J. Theor. Phys. 42, 1075-1084 (2003). [CrossRef]
- J. Ville, “Théorie and applications de la Notion de Signal Analytique,” Cables Transm. 2A, 61-74 (1948).
- D. Dragoman, “Applications of the Wigner distribution function in signal processing,” EURASIP J. Appl. Signal Process. 10, 1520-1534 (2005).
- B. Yegnanarayana, P. Pavan Kumar, and S. Das, “One-dimensional Gabor filtering for texture edge detection,” Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (IEEE, 1998), pp. 231-237.
- MATLAB code for segmentation and classification of multitexture images is available from http://www.cse.iitk.ac.in/~amit/courses/768/00/rajrup/.
- P. de Rivaz and N. Kingsbury, “Complex wavelet features for fast texture image retrieval,” Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 25-28.
- M. E. Barilla, M. G. Forero, and M. Spann, “Color-based texture segmentation,” in Information Optics, 5th International Workshop, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds. (AIP, 2006), pp. 401-409.
- G. Smith and I. Burns, “MeasTex image texture database and test suite” (1997). Available online at http://www.texturesynthesis.com/meastex/meastex.html.
- C. W. Tyler, “Stereoscopic tilt and size effects,” Perception 4, 287-192 (1975). [CrossRef]
- R. Schumer and L. Ganz, “Independent stereoscopic channels for different extents of spatial pooling,” Vision Res. 19, 1303-1314 (1979). [CrossRef] [PubMed]
- T. Maddess and Y. Nagai, “Lessons from biological processing of image texture,” in International Congress Series (Elsevier, 2004), Vol. 1269, pp. 26-29. [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.