Maximizing the predictivity of smooth deformable image warps through cross-validation

Estimating smooth image warps from landmarks is an important problem in computer vision and medical image analysis. The standard paradigm is to find the model parameters by minimizing a compound energy including a data term and a smoother, balanced by a 'smoothing parameter' that is usually fixed by trial and error. We point out that warp estimation is an instance of the general supervised machine learning problem of fitting a flexible model to data, and propose to learn the smoothing parameter while estimating the warp. The leading idea is to depart from the usual paradigm of minimizing the energy to the one of maximizing the predictivity of the warp, i.e. its ability to do well on the entire image, rather than only on the given landmarks. We use cross-validation to measure predictivity, and propose a complete framework to solve for the desired warp. We point out that the well-known non-iterative closed-form for the leave-one-out cross-validation score is actually a good approximation to the true score and show that it extends to the warp estimation problem by replacing the usual vector two-norm by the matrix Frobenius norm. Experimental results on real data show that the procedure selects sensible smoothing parameters, very close to user selected ones.