Résumé
Dans cet article on développe quatre méthodes linéaires des moindres carrés consacrées à ľidentification des signaux non gaussiens bidimensionnels (2-D) à moyenne ajustée (ma), éventuellement à phase non minimale (pnm), ainsi qu’une relation liant ľautocorrélation et les cumulants. Ľune des méthodes est fondée uniquement sur les cumulants tandis que les autres exploitent à la fois les autocorrélations et les cumulants ďordre m (m > 2). Ces méthodes sont non surparamétrisées et non restreintes aux cas où les deux coefficients extrêmes b(qp q2) et b(0,0) du modèle ma, ďordre (q1, q2), sont non nuls. La relation présentée et trois méthodes parmi les quatre proposées dérivent de la transformation de ľéquation non linéaire de Brillinger et Rosenblatt en relations linéaires grâce à la solution explicite de Tugnait (formule modifiée de la version 2-D de ľalgorithme ‘C(q, k)’ de Giannakis). Une généralisation à ľordre m de la version 2-D de la méthode classique de Giannakis-Mendel est aussi présentée. Et par des simulations sur deux tailles du même signal ma 2-D synthétique, en ľabsence et en présence de bruit, on évalue les performances des méthodes développées et de la solution explicite de Tugnait en les comparant, puis on teste la relation proposée sous un environnement bruité, et on termine par une application à la caractérisation ďune image réelle homogène texturée par un modèle ma 2-D que nous identifierons dans le cas bruité et non bruité.
Abstract
In this paper, four batch least squares linear approaches are presented for identification of non minimum phase bidimensional non Gaussian moving average (MA) models, and a relationship between autocorrelation and cumulant sequences is given. One of the proposed methods is cumulant-based only but the others use both autocorrelations and m-th order cumulants (m. > 2). Three of them are derived from the Brillinger-Rosenblatt’s non linear relation by using the Tugnait’s closed-form solution. Also, we generalize to m-th order cumulants the 2-D version of Giannakis-Mende’s approach. By simulations, we test and compare the Tugnait’s closed-form solution and the proposed methods, and we evaluate the performance of our relationship in noisy environment. Finally, we propose to characterize textured images by a 2-D ma model witch will be identified using our approaches in noisy and free noise cases.
Bibliographie
Bakrim (M.), Aboutajdine (D.), Sur ľidentification des systèmes ma à non minimum de phase,Proc. Mediterranean Conf. on Electronics and Automatics Control (mcea), Grenoble, France, 13–15 Sept. 1995.
Bakrim (M.), Aboutajdine (D.), Identification paramétrique des signaux non Gaussiens bidimensionnels à non minimum de phase,Proc. Télécom’97, Fès, Maroc, 15–17 Oct. 1997, pp. 281–285.
Bakrim (M.), Aboutajdine (D.), Nouvelles approches ďidentification paramétrique des signaux/systèmes bidimensionnels à non minimum de phase à ľaide des statistiques ďordre supérieur,Second International Conference on Applied Mathematics and Engineering Sciences, (cimact’98), 27–29 Oct. 1998, Casablanca, Maroc, pp. 583–588, Tome 1.
Bakrim (M.), Aboutajdine (D.), Identification des signaux ma non Gaussiens à ľaide de cumulants,Revue Traitement du Signal, France,16, n° 3, pp. 175–186, 1999.
Bakrim (M.), Aboutajdine (D.), Classification et caractérisation des images textures par des modèles à moyennes ajustées (ma),International Symposium on Image/Video Communications Over Fixed and Mobile Networks, 17–20 Apr. 2000, Faculté des Sciences, Rabat, Morocco.
Bakrim (M.), Aboutajdine (D.), Identification des modèles bidimensionnels à moyenne ajustée à ľaide des statistiques ďordre supérieur,5 e Colloque sur la Recherche en Informatique, Antananarivo, Madagascar, 16–19 Octobre 2000, pp. 143–148,(Présenté aussi au Third International Conference on Applied Mathematics and Engineering Sciences, (cimaci’2000), 23–25 Octobre 2000, page 25, École Hassania des Travaux Publics, Casablanca, Maroc).
Bercher (J.-F.), Un algorithme de factorisation bispectrale.Proc. 14 e Colloque gretsi, Juan-les-pins, France, 13–16 Sept. 1993, pp. 117–120.
Brillinger (D.), Rosenblatt (L.M.), Computation and interpretation of kth order spectraSpectral Analysis of Times Signals. New York: Wiley, 1967, pp. 907–938.
Brodatz (P.), Textures, A photographic album for artists and designers, Dover, New-York, 1966.
Cadzow (J.A.), Wilkes (A.D.), Peters (R.A.), Image texture synthesis-by analysis using moving-average models,ieee Trans. Aerosp. Electrons. Syst., vol. 29, pp.1110–1122, Oct. 1993.
Chang (M.M.), Tekalp (A.M.), Erdem (A.T.), Blur identification using the bispectrum,IEEE Trans. on Signal Proc.,39, n° 10, pp. 2323–2325, Oct. 1991.
Chien (H.-M.), Yang (H.-L.), Chi (C.-Y.), Parametric cumulant based phase estimation of 1-D and 2-D nonminimum phase systems by allpass filtering,ieee Trans. on Sig. Proc.,45, n° 7, pp. 1742–1762, July 1997.
Dianat (S.A.), Raghuveer (M.R.), Fast algorithms for bispectral reconstruction of two-dimensional signals,Proc. icassp-90, nm, USA, pp. 2377–2379, April 1990.
Durbin (J.), Efficient estimation of parameters in moving averaging models.Biometrika, 46, pp. 306–316, 1959.
Erdem (A.T.), Tekalp (A.M.), Linear bispectrum of signals and identification of nonminimum phase fir system driven by colored input.ieee Trans. assp.,40, pp. 1469-1–1478, June 1992.
Fonollosa (J.A.R.), Vidal (J.), System identification using linear combination of cumulant slices.ieee Trans. Signal Proc.,41, n° 7, pp. 2405–2412, July 1993.
Francos (J.M.), Meiri (A.Z.), Porat (B.), A unified texture model based on a 2-D Wold-like decomposition.ieee Trans. Sig. Proc.,41, n° 8, pp. 2665–2677, Aug. 1993.
Francos (J. M.), Narasimhan (A.), Woods (J.W.), Maximum likelihood parameter estimation of textures using a Wold decomposition based model.ieee Trans. Image. Proc., 4, pp. 1655–1666, 1995.
Francos (J.M.), Friedlander (B.), Parameter estimation of two-dimensional moving average random fields.ieee Trans. on Sig. Proc.,46, n° 8, pp. 2157–2165, Aug. 1998.
Friedlander (B.), Porat (B.), Asymptotically optimal estimation of ma and arma parameters of non-Gaussian processes from higher-order moments.ieee Trans. Automat. Contr.,35, pp. 27–35, Jan. 1990.
Giannakis (G.B.), Cumulants: A power tools in signal processing.Proc. ieee,77, pp. 1333–1334, Sept. 1987.
Giannakis (G. B.), Mendel (J. M.), Identification of nonminimum phase systems using higher order statistics,ieee Trans. assp.,37, pp. 360–377, Mar. 1989.
Giannakis (G. B.), Swami (A.). On estimating noncausal nonminimum phase arma models of non-Gaussian processes,ieee Trans. assp.,38, pp. 478–495, Mar. 1990.
Giannakis (G. B.), Tsatsanis (M. K.), A unifying maximum-likelihood view of cumulant and polyspectral measures for non Gaussian signal classification and estimation,ieee Trans. on Information Theory,38, pp. 386–406, Mar. 1992.
Hall (T.E.), Giannakis (G. B.), Wilson (S. G.), Predictive image codig using cumulant-based causal and non causal ar/arma models.Proc. of Conf. on Info. Sciences and Systems, Johns Hopkins Univ., Baltimore, pp. 413–418, Mar. 1989.
Hall (T.E.), Wilson (S.G.), Stochastic image modeling using cumulants with application to predictive image coding,Proc. on Higher-Order Spectral Analysis, Vail, CO, June 1989, pp. 239–244.
Hall (T.E.), Giannakis (G.B.), Texture model validation using higher-order statistics,Proc. icassp., Toronto. Canada, May 1991,4, pp. 2673–2676.
Hall (T.E.), Giannakis (G. B.), Bispectral analysis and model validation of texture images,ieee. Trans. on Signal Proc.,4, n° 7, pp. 996–1009, July 1995.
Hall (T.E.), Giannakis (G.B.), Image modeling using inverse filtering criteria with application to textures,ieee Trans. on Image Processing,5, n° 6, pp. 3938–949, June 1996.
Jain (A.K.), Advances in mathematical models for image processing,ieee Trans. assp.,69, pp. 502–528, May 1981.
Krishnamurthy (R.), Woods (J.W.), Francos (J.M.), Adaptive restoration of textured images with mixed spectra using a generalized wiener filter.ieee Trans. Image, Proc.,5, pp. 648–652, 1996.
Le Caillec (J. M.), Garello (R.), Étude ďestimateurs bispec-traux pour les signaux à deux dimensions,Proc 15 e colloque cretsi, Juan-les-pins, France, pp. 101–104, Sept. 18–21, 1995.
Marzetta (T.L.), Two-dimensional linear prediction: Autocorrelation arrays, minimum-phase prediction error filters and reflection coefficients arrays,ieee Trans. assp.,28, pp. 725–733, 1980.
Mendel (J. M.), Tutorial on higher-order statistics (spectra) in signal processing and system theory: Theoretical results and some applications.Proc. ieee,79, pp. 278–305, Mar. 1991.
Nikias (C.L.), Raghuveer (R.), Bispectrum estimation: A digital signal processing framework,Proc. ieee,75, pp. 869–891, July 1987.
Nikias (C.L.),Petropulu (A.P.), Higher-Order Spectra Analysis.Prentice-Hall, 1993.
Rosenblatt (M.), Stationary Sequences and Random Fields,Birkhänser, 1985.
Sadler (B.M.), Giannakis (G. B.), Image sequence analysis and reconstruction from the bispectrum,Proc. of Conf. on Info. Sciences and Systems, Johns Hopkins Univ., Baltimore, pp. 242–246, Mar. 1989.
Stogioglou (A.G.), McLaughin (S.), ma parameter estimation and cumulant enhancement.ieee Trans. Signal Proc.,44, n° 7, pp. 1704–1717, July 1996.
Swami (A.),Giannakis (G. B.), arma modeling and phase reconstruction of multdimensional non-Gaussian processes using cumulants.Proc ieee Int. Conf. assp., April 1988, pp. 729-732.
Swami (A.), Giannakis (G. B.), Mendel (J.M.), Linear modeling of multidimensional non-Gaussian processes using cumulants.Multidimensional Syst. Signal Processing,1, pp. 11–37, Mar. 1990.
Tekalp (A.M.), Kaufman (H.), On statistical identification of a class of linear space-invariant image blurs using nonminimum-phase arma Models.ieee Trans. assp.,36, n° 8, pp. 1360–1363, Aug. 1988.
Tekalp (A.M.), Erdem (A.T.), Higher-order spectrum factorization in one and two dimensions with application in signal modeling and nonminimum phase system identification.ieee Trans. assp.,37, pp. 1537–1559, Oct. 1989.
Tsatsanis (M.K.), Giannakis (G. B.), Objet and texture classification using higher-order statistics.ieee Trans. on pami,14, n° 7, pp. 733–750, July 1992.
Tugnait (J.K.), Identification of linear stochastic systems via second and fourth-order cumulant matching,ieee Trans. on Inform. Theory,33, pp. 393–407, May 1987.
Tugnait (J.K.), Approaches for fir system identification with noisy data using higher order statistics,ieee Trans. assp,38, pp. 1307–1317, July 1990.
Tugnait (J.K.), Linear model validation and order selection using higher order Statistics,ieee Signal Processing, Workshop on Higher Order Statistic, lake Tahoe, USA, pp. 111-115, 1993.
Tugnait (J.K.), Estimation of linear parametric models of non-Gaussian discrete random fields.ieee Trans. on Image Processing,3, n° 2, pp. 109–126, March 1994.
Tugnait (J.K.). Fitting ma models to linear non-Gaussian random fields using higher order cumulants,ieee Trans. on Signal Processing,45, pp. 1045–1050, April 1997.
Xu (Y.),Crebbin (G.), Image blur identification by using higher order statistic techniques,Proc. ieee Int. Conf. assp., 1996, pp. 77-80.
Zhang (X.-D.), Zhang (Y.-S.), fir system identification using higher order statistics alone.ieee Trans. Signal Processing,42, n° 10, pp. 2854–2859, Oct. 1994.
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Bakrim, M., ABoutajdine, D. Caractérisation des images stationnaires par des modèles non gaussiens bidimensionnels à moyenne ajustée. Ann. Télécommun. 56, 523–537 (2001). https://doi.org/10.1007/BF03008830
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DOI: https://doi.org/10.1007/BF03008830
Mots clés
- Traitement image
- Signal bidimensionnel
- Signal stationnaire
- Signal non gaussien
- Signal aléatoire
- Modèle statistique
- Autocorrélation
- Cumulant
- Méthode moindre carré
- Modèle linéaire