Electronic Journal of Statistics

Statistical multiresolution Dantzig estimation in imaging: Fundamental concepts and algorithmic framework

Klaus Frick, Philipp Marnitz, and Axel Munk

Full-text: Open access


In this paper we are concerned with fully automatic and locally adaptive estimation of functions in a “signal + noise”-model where the regression function may additionally be blurred by a linear operator, e.g. by a convolution. To this end, we introduce a general class of statistical multiresolution estimators and develop an algorithmic framework for computing those. By this we mean estimators that are defined as solutions of convex optimization problems with -type constraints. We employ a combination of the alternating direction method of multipliers with Dykstra’s algorithm for computing orthogonal projections onto intersections of convex sets and prove numerical convergence. The capability of the proposed method is illustrated by various examples from imaging and signal detection.

Article information

Electron. J. Statist. Volume 6 (2012), 231-268.

First available in Project Euclid: 29 February 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 90C06: Large-scale problems
Secondary: 68U10: Image processing

alternating direction method of multipliers (ADMM) biophotonics Dantzig selector Dykstra’s projection algorithm local adaption signal detection statistical imaging statistical multiscale analysis statistical regularization


Frick, Klaus; Marnitz, Philipp; Munk, Axel. Statistical multiresolution Dantzig estimation in imaging: Fundamental concepts and algorithmic framework. Electron. J. Statist. 6 (2012), 231--268. doi:10.1214/12-EJS671. https://projecteuclid.org/euclid.ejs/1330524559

Export citation


  • [1] Aujol, J.-F., Aubert, G., Blanc-Féraud, L. and Chambolle, A. (2005). Image decomposition into a bounded variation component and an oscillating component., J. Math. Imaging Vision 22 71–88.
  • [2] Becker, S., Candès, E. and Grant, M. (2011). Templates for convex cone problems with applications to sparse signal recovery., Math. Program. Comput. 3 165-218.
  • [3] Bertalmio, M., Caselles, V., Rougé, B. and Solé, A. (2003). TV based image restoration with local constraints., J. Sci. Comput. 19 95–122. Special issue in honor of the sixtieth birthday of Stanley Osher.
  • [4] Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector., Ann. Statist. 37 1705–1732.
  • [5] Boyle, J. P. and Dykstra, R. L. (1986). A method for finding projections onto the intersection of convex sets in Hilbert spaces. In, Advances in order restricted statistical inference (Iowa City, Iowa, 1985). Lecture Notes in Statist. 37 28–47. Springer, Berlin.
  • [6] Boysen, L., Kempe, A., Liebscher, V., Munk, A. and Wittich, O. (2009). Consistencies and rates of convergence of jump-penalized least squares estimators., Ann. Statist. 37 157–183.
  • [7] Candès, E. and Tao, T. (2007). The Dantzig selector: statistical estimation when, p is much larger than n. Ann. Statist. 35 2313–2351.
  • [8] Chen, S. S., Donoho, D. L. and Saunders, M. A. (2001). Atomic Decomposition by Basis Pursuit., SIAM Review 43 129-159.
  • [9] Csiszár, I. (1991). Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems., Ann. Statist. 19 2032–2066.
  • [10] Davies, P. L. and Kovac, A. (2001). Local extremes, runs, strings and multiresolution., Ann. Statist. 29 1–65. With discussion and rejoinder by the authors.
  • [11] Davies, P. L. and Kovac, A. (2004). Densities, spectral densities and modality., Ann. Statist. 32 1093–1136.
  • [12] Davies, P. L., Kovac, A. and Meise, M. (2009). Nonparametric regression, confidence regions and regularization., Ann. Statist. 37 2597–2625.
  • [13] Deutsch, F. and Hundal, H. (1994). The rate of convergence of Dykstra’s cyclic projections algorithm: the polyhedral case., Numer. Funct. Anal. Optim. 15 537–565.
  • [14] Dobson, D. C. and Vogel, C. R. (1997). Convergence of an iterative method for total variation denoising., SIAM J. Numer. Anal. 34 1779–1791.
  • [15] Dong, Y., Hintermüller, M. and Rincon-Camacho, M. (May 2011). Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration., J. Math. Imaging Vision 40 82-104(23).
  • [16] Dümbgen, L. and Johns, R. B. (2004). Confidence Bands for Isotonic Median Curves Using Sign Tests., J. Comput. Graph. Statist 13 519-533.
  • [17] Dümbgen, L. and Spokoiny, V. G. (2001). Multiscale testing of qualitative hypotheses., Ann. Statist. 29 124–152.
  • [18] Dümbgen, L. and Walther, G. (2008). Multiscale inference about a density., Ann. Statist. 36 1758–1785.
  • [19] Ekeland, I. and Temam, R. (1976)., Convex analysis and variational problems. Studies in Mathematics and its Applications 1. North-Holland Publishing Co., Amsterdam-Oxford.
  • [20] Fortin, M. and Glowinski, R. (1983)., Augmented Lagrangian methods. Studies in Mathematics and its Applications 15. North-Holland Publishing Co., Amsterdam. Applications to the numerical solution of boundary value problems, Translated from the French by B. Hunt and D. C. Spicer.
  • [21] Frick, K., Marnitz, P. and Munk, A. (2010). Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis Technical Report. Available at, http://arxiv.org/abs/1003.3323.
  • [22] Frick, K. and Scherzer, O. (2010). Regularization of ill-posed linear equations by the non-stationary Augmented Lagrangian Method., J. Integral Equations Appl. 22 217-257.
  • [23] Gaffke, N. and Mathar, R. (1989). A Cyclic Projection Algorithm Via Duality., Metrika 36 29–54.
  • [24] Grasmair, M. (2007). The equivalence of the taut string algorithm and BV-regularization., J. Math. Imaging Vision 27 59–66.
  • [25] Hawkins, D. M. and Wixley, R. A. J. (1986). A Note on the Transformation of Chi-Squared Variables to Normality., Amer.Statist. 40 296–298.
  • [26] Hell, S. W. (2007). Far-Field Optical Nanoscopy., Science 316 1153-1158.
  • [27] Hell, S. W. and Wichmann, J. (1994). Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy., Opt. Lett. 19 780–782.
  • [28] Hotz, T., Marnitz, P., Stichtenoth, R., Davies, L., Kabluchko, Z. and Munk, A. (2012). Locally adaptive image denoising by a statistical multiresolution criterion., Comput. Stat. Data An. 56 543 - 558.
  • [29] James, G. M., Radchenko, P. and Lv, J. (2009). DASSO: connections between the Dantzig selector and lasso., J. R. Stat. Soc. Ser. B Stat. Methodol. 71 127–142.
  • [30] Kabluchko, Z. (2011). Extremes of the standardized Gaussian noise., Stochastic Process. Appl. 121 515–533.
  • [31] Kabluchko, Z. and Munk, A. (2009). Shao’s theorem on the maximum of standardized random walk increments for multidimensional arrays., ESAIM Probab. Stat. 13 409–416.
  • [32] Kaipio, J. and Somersalo, E. (2005)., Statistical and computational inverse problems. Applied Mathematical Sciences 160. Springer-Verlag, New York.
  • [33] Lu, Z., Pong, T. K. and Zhang, Y. (2010). An Alternating Direction Method for Finding Dantzig Selectors Technical Report. Available at, http://arxiv.org/abs/1011.4604v1.
  • [34] Mammen, E. and van de Geer, S. (1997). Locally adaptive regression splines., Ann. Statist. 25 387–413.
  • [35] Meyer, Y. (2001)., Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series 22. American Mathematical Society, Providence, RI. The fifteenth Dean Jacqueline B. Lewis memorial lectures.
  • [36] Mildenberger, T. (2008). A geometric interpretation of the multiresolution criterion in nonparametric regression., J. Nonparametr. Stat. 20 1048-5252.
  • [37] Mohler, G. O., Bertozzi, A. L., Goldstein, T. A. and Osher, S. J. (2011). Fast TV Regularization for 2D Maximum Penalized Likelihood Estimation., J. Comput. Graph. Statist 20 479-491.
  • [38] Munk, A., Bissantz, N., Wagner, T. and Freitag, G. (2005). On difference-based variance estimation in nonparametric regression when the covariate is high dimensional., J. R. Stat. Soc. Ser. B Stat. Methodol. 67 19-41.
  • [39] Pawley, J. B. (2006)., Handbook of Biological Confocal Microscopy. Springer.
  • [40] Polzehl, J. and Spokoiny, V. G. (2000). Adaptive weights smoothing with applications to image restoration., J. R. Stat. Soc. Ser. B Stat. Methodol. 62 335–354.
  • [41] Romberg, J. K. (2008). The Dantzig selector and generalized thresholding. In, CISS 22-25. IEEE.
  • [42] Siegmund, D. and Yakir, B. (2000). Tail probabilities for the null distribution of scanning statistics., Bernoulli 6 191–213.
  • [43] Tibshirani, R. (1994). Regression Shrinkage and Selection Via the Lasso., J. R. Stat. Soc. Ser. B Stat. Methodol. 58 267–288.
  • [44] Vogel, C. R. (2002)., Computational methods for inverse problems. Frontiers in Applied Mathematics 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. With a foreword by H. T. Banks.
  • [45] Wang, Z., Bovik, A. C., Sheikh, H. R. and Simoncelli, E. P. (2004). Image Quality Assessment: From Error Visibility to Structural Similarity., IEEE Trans. Image Process. 13 600–612.
  • [46] Xu, S. (2000). Estimation of the convergence rate of Dykstra’s cyclic projections algorithm in polyhedral case., Acta Math. Appl. Sinica (English Ser.) 16 217–220.