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We present a new approach to the general problem of template-based detection, segmentation, and registration of elastically deformable objects. This joint problem is highly nonlinear and high dimensional, due to the large space of possible geometric transformations between a given template and its observed signature. Hence, any attempt to directly solve it inevitably leads to a high dimensional, nonlinear, nonconvex optimization procedure. We propose a novel parametric solution to this problem, by showing that it can be equivalently represented by a low dimensional model, which is linear in the deformation parameters, and biased by the unknown observation background. Linear Bayesian estimation is then employed to estimate the deformation parameters, and a likelihood ratio test is utilized to detect a deformed instance of the template, thus providing an explicit closed-form solution for the joint problem.