This paper provides linear-time algorithms for solving a class of minimization problems involving a cost function with both local and spatial terms. These problems can be viewed as a generalization of classical distance transforms of binary images, where the binary image is replaced by an arbitrary sampled function. Alternatively they can be viewed in terms of the minimum convolution of two functions, which is an important operation in grayscale morphology. A useful consequence of our techniques is a simple, fast method for computing the Euclidean distance transform of a binary image. The methods are also applicable to Viterbi decoding, belief propagation and optimal control.