Finding Curvilinear Features in Spatial Point Patterns: Principal Curve Clustering with Noise

Clustering about principal curves combines parametric modeling of noise with nonparametric modeling of feature shape. This is useful for detecting curvilinear features in spatial point patterns, with or without background noise. Applications include the detection of curvilinear minefields from reconnaissance images, some of the points in which represent false detections, and the detection of seismic faults from earthquake catalogs. Our algorithm for principal curve clustering is in two steps: The first is hierarchical and agglomerative (HPCC) and the second consists of iterative relocation based on the classification EM algorithm (CEM-PCC). HPCC is used to combine potential feature clusters, while CEM-PCC refines the results and deals with background noise. It is important to have a good starting point for the algorithm: This can be found manually or automatically using, for example, nearest neighbor clutter removal or model-based clustering. We choose the number of features and the amount of smoothing simultaneously, using approximate Bayes factors.

[1]  P. Green.  On Use of the EM Algorithm for Penalized Likelihood Estimation , 1990 .

[2]  G. Celeux,et al.  A Classification EM algorithm for clustering and two stochastic versions , 1992 .

[3]  P. Bloomfield,et al.  Spline Functions in Data Analysis. , 1974 .

[4]  L. Wasserman,et al.  Practical Bayesian Density Estimation Using Mixtures of Normals , 1997 .

[5]  Scott MacKinnon,et al.  Statistical methods for visual defect metrology , 1998 .

[6]  R. Tibshirani,et al.  Local Likelihood Estimation , 1987 .

[7]  J. D. Wilson,et al.  A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography , 1990 .

[8]  S. Wold.  Spline Functions in Data Analysis , 1974 .

[9]  R. Tibshirani.  Principal curves revisited , 1992 .

[10]  C. Fraley,et al.  Nonparametric Maximum Likelihood Estimation of Features in Spatial Point Processes Using Voronoï Tessellation , 1997 .

[11]  Robert S. Anderssen,et al.  Assessing quantification for the EMS algorithm , 1994 .

[12]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[13]  Fionn Murtagh,et al.  Interpreting the Kohonen self-organizing feature map using contiguity-constrained clustering , 1995, Pattern Recognit. Lett..

[14]  A. Raftery,et al.  Detecting features in spatial point processes with clutter via model-based clustering , 1998 .

[15]  G. Govaert,et al.  Constrained clustering and Kohonen Self-Organizing Maps , 1996 .

[16]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[17]  Carsten Steger,et al.  An Unbiased Detector of Curvilinear Structures , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  Adam Krzyzak,et al.  A Polygonal Line Algorithm for Constructing Principal Curves , 1998, NIPS.

[19]  A. Raftery,et al.  Ice Floe Identification in Satellite Images Using Mathematical Morphology and Clustering about Principal Curves , 1992 .

[20]  Scott MacKinnon,et al.  Statistical methods for visual defect metrology , 1998 .

[21]  R. Prim.  Shortest connection networks and some generalizations , 1957 .

[22]  Adrian E. Raftery,et al.  How Many Clusters? Which Clustering Method? Answers Via Model-Based Cluster Analysis , 1998, Comput. J..

[23]  Charles T. Zahn,et al.  Graph-Theoretical Methods for Detecting and Describing Gestalt Clusters , 1971, IEEE Transactions on Computers.

[24]  L. Wasserman,et al.  A Reference Bayesian Test for Nested Hypotheses and its Relationship to the Schwarz Criterion , 1995 .

[25]  A. Raftery,et al.  Model-based Gaussian and non-Gaussian clustering , 1993 .

[26]  D. Nychka.  Some properties of adding a smoothing step to the EM algorithm , 1990 .

[27]  A. Raftery,et al.  Nearest-Neighbor Clutter Removal for Estimating Features in Spatial Point Processes , 1998 .

[28]  Peter Aundal Toft,et al.  Fast curve estimation using preconditioned generalized Radon transform , 1996, IEEE Trans. Image Process..

[29]  R. Tibshirani,et al.  Adaptive Principal Surfaces , 1994 .

[30]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[31]  Josef Kittler,et al.  A survey of the hough transform , 1988, Comput. Vis. Graph. Image Process..

[32]  A. Krzyzak,et al.  Principal curves: learning and convergence , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[33]  P. Delicado.  Another Look at Principal Curves and Surfaces , 2001 .

[34]  Geoff A. Latham,et al.  Existence of EMS Solutions and a Priori Estimates , 1995, SIAM J. Matrix Anal. Appl..