Deconvolution When Classifying Noisy Data Involving Transformations

Abstract

In the present study, we consider the problem of classifying spatial data distorted by a linear transformation or convolution and contaminated by additive random noise. In this setting, we show that classifier performance can be improved if we carefully invert the data before the classifier is applied. However, the inverse transformation is not constructed so as to recover the original signal, and in fact, we show that taking the latter approach is generally inadvisable. We introduce a fully data-driven procedure based on cross-validation, and use several classifiers to illustrate numerical properties of our approach. Theoretical arguments are given in support of our claims. Our procedure is applied to data generated by light detection and ranging (Lidar) technology, where we improve on earlier approaches to classifying aerosols. This article has supplementary materials online.

KEY WORDS:

• Centroid classifier
• Cross-validation
• Fourier transform
• Inverse transform
• Spatial data

Acknowledgments

Carroll's research was supported by a grant from the National Cancer Institute (R37-CA057030) and in part by award number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST) and by the National Science Foundation (DMS-0914951). Delaigle's research was supported by grants and a Queen Elizabeth II Fellowship from the Australian Research Council, and Hall's research was supported by a Federation Fellowship, a Laureate Fellowship, and grants from the Australian Research Council.