Cramér-Rao Bound for Line Constrained Trajectory Tracking

In this paper, target tracking constrained to short-term linear trajectories is explored. The problem is viewed as an extension of the matrix decomposition problem into low-rank and sparse components by incorporating an additional line constraint. The Cramér-Rao Bound (CRB) for the trajectory estimation is derived; numerical results show that an alternating algorithm which estimates the various components of the trajectory image is near optimal due to proximity to the computed CRB. In addition to the theoretical contribution of incorporating an additional constraint in the estimation problem, the alternating algorithm is applied to real video data and shown to be effective in estimating the trajectory despite it not being exactly linear.

[1]  Xin Li,et al.  Simultaneous Video Stabilization and Moving Object Detection in Turbulence , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  Urbashi Mitra,et al.  Nested Sparse Approximation: Structured Estimation of V2V Channels Using Geometry-Based Stochastic Channel Model , 2014, IEEE Transactions on Signal Processing.

[3]  Song Wang,et al.  Object tracking via appearance modeling and sparse representation , 2011, Image Vis. Comput..

[4]  Ebroul Izquierdo,et al.  Efficient background subtraction with low-rank and sparse matrix decomposition , 2015, 2015 IEEE International Conference on Image Processing (ICIP).

[5]  Gaurav S. Sukhatme,et al.  A topological approach to using cables to separate and manipulate sets of objects , 2013, Int. J. Robotics Res..

[6]  Larry S. Davis,et al.  Learning Structured Low-Rank Representations for Image Classification , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

[7]  Urbashi Mitra,et al.  Multi-Scale Multi-Lag Channel Estimation Using Low Rank Approximation for OFDM , 2015, IEEE Transactions on Signal Processing.

[8]  Zhixun Su,et al.  Linearized Alternating Direction Method with Adaptive Penalty for Low-Rank Representation , 2011, NIPS.

[9]  Gongguo Tang,et al.  Lower Bounds on the Mean-Squared Error of Low-Rank Matrix Reconstruction , 2011, IEEE Transactions on Signal Processing.

[10]  Xin Liu,et al.  Background subtraction based on low-rank and structured sparse decomposition. , 2015, IEEE transactions on image processing : a publication of the IEEE Signal Processing Society.

[11]  X. R. Li,et al.  Survey of maneuvering target tracking. Part I. Dynamic models , 2003 .

[12]  Namrata Vaswani,et al.  Online matrix completion and online robust PCA , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[13]  Yonina C. Eldar,et al.  Simultaneously Structured Models With Application to Sparse and Low-Rank Matrices , 2012, IEEE Transactions on Information Theory.

[14]  Richard Egli,et al.  Old and new straight-line detectors: Description and comparison , 2008, Pattern Recognit..

[15]  Zhixun Su,et al.  Linearized alternating direction method with parallel splitting and adaptive penalty for separable convex programs in machine learning , 2013, Machine Learning.

[16]  Angshul Majumdar,et al.  Multi-spectral demosaicing: A joint-sparse elastic-net formulation , 2015, 2015 Eighth International Conference on Advances in Pattern Recognition (ICAPR).

[17]  Gongguo Tang,et al.  Constrained Cramér–Rao Bound on Robust Principal Component Analysis , 2011, IEEE Transactions on Signal Processing.

[18]  Lawrence Carin,et al.  Bayesian Robust Principal Component Analysis , 2011, IEEE Transactions on Image Processing.

[19]  B. C. Ng,et al.  On the Cramer-Rao bound under parametric constraints , 1998, IEEE Signal Processing Letters.

[20]  Xiaoming Yuan,et al.  Recovering Low-Rank and Sparse Components of Matrices from Incomplete and Noisy Observations , 2011, SIAM J. Optim..

[21]  Yi Ma,et al.  The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices , 2010, Journal of structural biology.

[22]  Alfred O. Hero,et al.  Lower bounds for parametric estimation with constraints , 1990, IEEE Trans. Inf. Theory.

[23]  Urbashi Mitra,et al.  Low-rank, sparse and line constrained estimation: Applications to target tracking and convergence , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).