×

Image denoising: Pointwise adaptive approach. (English) Zbl 1018.62047

Summary: A new method of pointwise adaptation has been proposed and studied by V. G. Spokoiny [Ann. Stat. 26, No. 4, 1356-1378 (1998; Zbl 0934.62037)] in the context of estimation of piecewise smooth univariate functions. The present paper extends that method to estimation of bivariate grey-scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design. The proposed estimator \(\widehat f(x)\) at a point \(x\) is simply the average of observations over a window \(\widehat U(x)\) selected in a data-driven way. The theoretical properties of the procedure are studied for the case of piecewise constant images. We present a nonasymptotic bound for the accuracy of estimation at a specific grid point \(x\) as a function of the number of pixels \(n\), of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary.
It is also shown that the proposed method provides a near-optimal rate of estimation near edges and inside homogeneous regions. We briefly discuss algorithmic aspects and the complexity of the procedure. The numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues. An example from satellite (SAR) imaging illustrates the applicability of the method.

MSC:

62H35 Image analysis in multivariate analysis
62G08 Nonparametric regression and quantile regression
65C60 Computational problems in statistics (MSC2010)
62G07 Density estimation

Citations:

Zbl 0934.62037

Software:

wavethresh; AWS; spatial
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] BARRON, A., BIRGÉ, L. and MASSART, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301-413. · Zbl 0946.62036
[2] BLAKE, A. and ZISSERMAN, A. (1987). Visual Reconstruction. MIT Press.
[3] CHU, C. K., GLAD, I. K., GODTLIEBSEN, F. and MARRON, J. S. (1998). Edge-preserving smoothers for image processing (with discussion). J. Amer. Statist. Assoc. 93 526-556. JSTOR: · Zbl 0954.62115
[4] DONOHO, D. L. (1999). Wedgelets: Nearly minimax estimation of edges. Ann. Statist. 27 859-897. · Zbl 0957.62029
[5] ENGEL, J. (1994). A simple wavelet approach to nonparametric regression from recursive partitioning schemes. J. Multivariate Anal. 49 242-254. · Zbl 0795.62034
[6] GEMAN, S. and GEMAN, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Analy sis Machine Intelligence 6 721-741. · Zbl 0573.62030
[7] GIRARD, D. (1990). From template matching to optimal approximation by piecewise smooth curves. In Curves and Surfaces in Computer Vision and Graphics (L. A. Ferrari and R. S. de Figueiredo, eds.) 174-182. SPIE, Bellingham, WA.
[8] GLASBEY, C. A. and HORGAN, G. W. (1995). Image Analy sis for the Biological Sciences. Wiley, New York. · Zbl 0876.92001
[9] GODTLIEBSEN, F., SPJØTVOLL, E. and MARRON, J. S. (1997). A nonlinear Gaussian filter applied to images with discontinuities. J. Nonparametr. Statist. 8 21-43. · Zbl 0885.62111
[10] GRENANDER, U. (1976, 1978, 1981). Lectures in Pattern Theory 1, 2, 3. Springer, New York. · Zbl 0334.68009
[11] HALL, P. and RAIMONDO, M. (1998). On global performance of approximations to smooth curves using gridded data. Ann. Statist. 26 2206-2217. · Zbl 0933.62026
[12] HARALICK, R. M. (1980). Edge and region analysis for digital image data. Comput. Graphics Image Processing 12 60-73.
[13] HUANG, J. S. and TSENG, D. H. (1988). Statistical theory of edge detection. Comput. Vision Graphics Image Processing 43 337-346.
[14] KHINTCHINE, A. I. (1949). Continued Fractions. ITTL, Moscow.
[15] KOROSTELEV, A. and TSy BAKOV, A. (1993). Minimax Theory of Image Reconstruction. Springer, New York. · Zbl 0833.62039
[16] LEE, J. S. (1983). Digital image smoothing and the sigma filter. Comput. Vision Graphics Image Processing 24 255-269.
[17] LEPSKI, O. V. (1990). A problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454-466. · Zbl 0745.62083
[18] LEPSKI, O. V. (1992). Asy mptotically minimax adaptive estimation. II. Statistical model without optimal adaptation. Adaptive estimators. Theory Probab. Appl. 37 433-448. · Zbl 0787.62087
[19] LEPSKI, O., MAMMEN, E. and SPOKOINY, V. (1997). Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 929-947. · Zbl 0885.62044
[20] LEPSKI, O. and SPOKOINY, V. (1997). Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 2512-2546. · Zbl 0894.62041
[21] MAMMEN, E. and TSy BAKOV, A. (1995). Asy mptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 502-524. · Zbl 0834.62038
[22] MARR, D. (1982). Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. Freeman, San Francisco.
[23] MÜLLER, H.-G. and SONG, K. S. (1994). Maximin estimation of multidimensional boundaries. J. Multivariate Anal. 50 265-281. · Zbl 0798.62053
[24] MUMFORD, D. and SHAH, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 577-685. · Zbl 0691.49036
[25] NASON, G. P. and SILVERMAN, B. W. (1994). The discrete wavelet transform in S. J. Comput. Graph. Statist. 3 163-191.
[26] POLZEHL, J. and SPOKOINY, V. (2000). Adaptive weights smoothing with applications to image restoration. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 335-354. JSTOR: · Zbl 04558575
[27] PRATT, W. K. (1978). Digital Image Processing. Wiley, New York.
[28] QIU, P. (1998). Discontinuous regression surfaces fitting. Ann. Statist. 26 2218-2245. · Zbl 0927.62041
[29] RIPLEY, B. D. (1988). Statistical Inference for Spatial Processes. Cambridge Univ. Press. · Zbl 0716.62100
[30] ROSENFELD, A. and KAK, A. C. (1982). Digital Picture Processing. Academic Press, London. · Zbl 0564.94002
[31] SCOTT, D. W. (1992). Multivariate Density Estimation. Wiley, New York. · Zbl 0850.62006
[32] SPOKOINY, V. (1998). Estimation of a function with discontinuities via local poly nomial fit with an adaptive window choice. Ann. Statist. 26 1356-1378. · Zbl 0934.62037
[33] TITTERINGTON, D. M. (1985). Common structure of smoothing techniques in statistics. Internat. Statist. Rev. 53 141-170. JSTOR: · Zbl 0569.62026
[34] TSy BAKOV, A. (1989). Optimal orders of accuracy of the estimation of nonsmooth images. Problems Inform. Transmission 25 180-191.
[35] WINKLER, G. (1995). Image Analy sis, Random Fields and Dy namic Monte Carlo Methods. Springer, Berlin. · Zbl 0821.68125
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.