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Inverse problems, such as the reconstruction problems that arise in early vision, tend to be mathematically ill-posed. Through regularization, they may be reformulated as well-posed variational principles whose solutions are computable. Standard regularization theory employs quadratic stabilizing functionals that impose global smoothness constraints on possible solutions. Discontinuities present serious difficulties to standard regularization, however, since their reconstruction requires a precise spatial control over the smoothing properties of stabilizers. This paper proposes a general class of controlled-continuity stabilizers which provide the necessary control over smoothness. These nonquadratic stabilizing functionals comprise multiple generalized spline kernels combined with (noncontinuous) continuity control functions. In the context of computational vision, they may be thought of as controlled-continuity constraints. These generic constraints are applicable to visual reconstruction problems that involve both continuous regions and discontinuities, for which global smoothness constraints fail.