Fast TV Regularization for 2D Maximum Penalized Likelihood Estimation

Total Variation-based regularization, well established for image processing applications such as denoising, was recently introduced for Maximum Penalized Likelihood Estimation (MPLE) as an effective way to estimate nonsmooth probability densities. While the estimates show promise for a variety of applications, the nonlinearity of the regularization leads to computational challenges, especially in multidimensions. In this article we present a numerical methodology, based upon the Split Bregman L1 minimization technique, that overcomes these challenges, allowing for the fast and accurate computation of 2D TV-based MPLE. We test the methodology with several examples, including V-fold cross-validation with large 2D datasets, and highlight the application of TV-based MPLE to point process crime modeling. The proposed algorithm is implemented as the Matlab function TVMPLE. The Matlab (mex) code and datasets for examples and simulations are available as online supplements.

[1]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[2]  Yosihiko Ogata,et al.  Likelihood analysis of spatial inhomogeneity for marked point patterns , 1988 .

[3]  C. J. Stone,et al.  A study of logspline density estimation , 1991 .

[4]  M. C. Jones,et al.  A reliable data-based bandwidth selection method for kernel density estimation , 1991 .

[5]  David W. Scott,et al.  Multivariate Density Estimation: Theory, Practice, and Visualization , 1992, Wiley Series in Probability and Statistics.

[6]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[7]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[8]  Y. Ogata.  Space-Time Point-Process Models for Earthquake Occurrences , 1998 .

[9]  P. Davies,et al.  Densities, spectral densities and modality , 2004, math/0410071.

[10]  Alan F Karr,et al.  Maximum Penalized Likelihood Estimation, Vol. I: Density Estimation , 2003 .

[11]  P. Larsen.  Maximum Penalized Likelihood Estimation, Volume I, Density Estimation , 2004 .

[12]  Roger D. Peng,et al.  A Space–Time Conditional Intensity Model for Evaluating a Wildfire Hazard Index , 2005 .

[13]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[14]  R. Koenker.  DENSITY ESTIMATION BY TOTAL VARIATION REGULARIZATION , 2006 .

[15]  L. He,et al.  MR Image Reconstruction from Sparse Radial Samples Using Bregman Iteration , 2006 .

[16]  Thomas J. Asaki,et al.  A Variational Approach to Reconstructing Images Corrupted by Poisson Noise , 2007, Journal of Mathematical Imaging and Vision.

[17]  Jon E. Keeley,et al.  A Critical Assessment of the Burning Index in Los Angeles County, California , 2007 .

[18]  Shane D. Johnson,et al.  Space–Time Patterns of Risk: A Cross National Assessment of Residential Burglary Victimization , 2007 .

[19]  Wotao Yin,et al.  Bregman Iterative Algorithms for (cid:2) 1 -Minimization with Applications to Compressed Sensing ∗ , 2008 .

[20]  Àlex Haro,et al.  A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi-Periodic Maps: Explorations and Mechanisms for the Breakdown of Hyperbolicity , 2008 .

[21]  Junfeng Yang,et al.  A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , 2008, SIAM J. Imaging Sci..

[22]  George E. Tita,et al.  Measuring and Modeling Repeat and Near-Repeat Burglary Effects , 2009 .

[23]  Tom Goldstein,et al.  The Split Bregman Method for L1-Regularized Problems , 2009, SIAM J. Imaging Sci..

[24]  George E. Tita,et al.  Self-Exciting Point Process Modeling of Crime , 2011 .