Abstract
Principal component analysis models are widely used to model shapes in medical image analysis, computer vision, and other fields. The “Laplacian” spline approaches including thin-plate splines are also used for this purpose. These alternative approaches have complementary advantages and weaknesses: a low-rank principal component analysis model has some “knowledge” of the data being modeled, but cannot exactly fit arbitrary data, whereas spline models can fit arbitrary data but have only a generic smoothness assumption about the character of the data. In this contribution we show that the data fitting problem for these two approaches can be put into a common form, by making use of a relation between the data covariance and the Laplacian. This suggests the possibility of a unified approach that combines the advantages of each.
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Lewis, J.P., Rhee, T., Zhang, M. (2014). Principal Component Analysis and Laplacian Splines: Steps Toward a Unified Model. In: Wakayama, M., et al. The Impact of Applications on Mathematics. Mathematics for Industry, vol 1. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54907-9_24
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DOI: https://doi.org/10.1007/978-4-431-54907-9_24
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