Plug-and-Play Based Optimization Algorithm for New Crime Density Estimation
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Different from a general density estimation, the crime density estimation usually has one important factor: the geographical constraint. In this paper, a new crime density estimation model is formulated, in which the regions where crime is impossible to happen, such as mountains and lakes, are excluded. To further optimize the estimation method, a learning-based algorithm, named Plug-and-Play, is implanted into the augmented Lagrangian scheme, which involves an off-the-shelf filtering operator. Different selections of the filtering operator make the algorithm correspond to several classical estimation models. Therefore, the proposed Plug-and-Play optimization based estimation algorithm can be regarded as the extended version and general form of several classical methods. In the experiment part, synthetic examples with different invalid regions and samples of various distributions are first tested. Then under complex geographic constraints, we apply the proposed method with a real crime dataset to recover the density estimation. The state-of-the-art results show the feasibility of the proposed model.
Keywordscrime density estimation augmented Lagrangian strategy Plug-and-Play filtering operator
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We would like to thank the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially, and thank Dr. Shi-Jun Wang of Research Center of Beijing Visystem Co. Ltd. for the fruitful discussion.
- Silverman B W. Density Estimation for Statistics and Data Analysis. Chapman and Hall, 1986.Google Scholar
- Eqstein C L. Introduction to the Mathematics of Medical Imaging (2nd edition). Society for Industrial and Applied Mathematics, 2007.Google Scholar
- Eggermont P P B, LaRiccia V N, LaRiccia V N. Maximum Penalized Likelihood Estimation: Volume I: Density Estimation. Springer, 2001.Google Scholar
- Koenker R, Mizera I. Density estimation by total variation regularization. In Advances in Statistical Modeling and Inference, Essays in Honor of Kjell A Doksum, Nair V (ed.), World Scientific Publishing Company, 2007, pp.613-633.Google Scholar
- Smith L M, Keegan M S, Wittman T, Mohler G O, Bertozzi A L. Improving density estimation by incorporating spatial information. EURASIP Journal on Advances in Signal Processing, 2010, 2010: Article No. 265631.Google Scholar
- Woodworth J T, Mohler G O, Bertozzi A L et al. Nonlocal crime density estimation incorporating housing information. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2014, 372(2028): Article No. 20130403.Google Scholar
- Tikhonov A N, Arsenin V I A. Solutions of Ill-Posed Problems. Washington, DC: Winston, 1977.Google Scholar
- Venkatakrishnan S V, Bouman C A, Wohlberg B. Plug-and-Play priors for model based reconstruction. In Proc. the 2013 IEEE Global Conference on Signal and Information Processing, Dec. 2013, pp.945-948.Google Scholar
- Romano Y, Elad M, Milanfar P. The little engine that could: Regularization by denoising. arXiv:1611.02862, 2016. https://arxiv.org/pdf/1611.02862.pdf, November 2018.
- Gastal E S L, Oliveira M M. Domain transform for edge-aware image and video processing. ACM Transactions on Graphics, 2011, 30(4): Article No. 69.Google Scholar
- Zhao C, Feng X, Wang W et al. A cartoon-texture decomposition based multiplicative noise removal method. Mathematical Problems in Engineering, 2016: Article No. 5130346.Google Scholar
- Xu J, Feng X, Hao Yet al. Image decomposition and staircase effect reduction based on total generalized variation. Journal of Systems Engineering and Electronics, 2014, 25(1): 168-174.Google Scholar