Generalizing the Prediction Sum of Squares Statistic and Formula, Application to Linear Fractional Image Warp and Surface Fitting

The prediction sum of squares statistic uses the principle of leave-one-out cross-validation in linear least squares regression. It is computationally attractive, as it can be computed non-iteratively. However, it has limitations: it does not handle coupled measurements, which should be held out simultaneously, and is specific to the principle of leave-one-out, which is known to overfit when used for selecting a model’s complexity. We propose multiple-exclusion PRESS (MEXPRESS), which generalizes PRESS to coupled measurements and other types of cross-validation, while retaining computational efficiency with the non-iterative MEXPRESS formula. Using MEXPRESS, various strategies to resolve overfitting can be efficiently implemented. The core principle is to exclude training data too ‘close’ or too ‘similar’ to the validation data. We show that this allows one to select the number of control points automatically in three cases: (i) the estimation of linear fractional warps for dense image registration from point correspondences, (ii) surface reconstruction from a dense depth-map obtained by a depth sensor and (iii) surface reconstruction from a sparse point cloud obtained by shape-from-template.