This paper generalizes the geodesic transforms used in mathematical morphology for binary image processing by introducing the concept of morphological operations constrained by an initial set and a transformable set. These new operations allow the efficient extension of geodesic constraints to grayscale image processing as they can be implemented with simple algorithms. They also make possible the application of new constraints, leading to new transforms with new properties. Two particular cases of these new transforms are then studied thoroughly: erosion and dilation operations constrained by an initial set and a transformable set that are iterated until idempotence is reached, or idempotent geodesic transforms (IGT). A sequential local transform (SLT) algorithm is presented that greatly speeds up the computation of IGT; the conditions for convergence for the algorithm and the relation between speed of convergence and the complexity of the initial set and the transformable set are investigated.