Abstract: Computing locally injective mappings with low distortion in an efficient way is a fundamental task in computer graphics. By revisiting the well-known MIPS (Most-Isometric ParameterizationS) method, we introduce an advanced MIPS method that inherits the local injectivity of MIPS, achieves as low as possible distortions compared to the state-of-the-art locally injective mapping techniques, and performs one to two orders of magnitude faster in computing a mesh-based mapping. The success of our method relies on two key components. The first one is an enhanced MIPS energy function that penalizes the maximal distortion significantly and distributes the distortion evenly over the domain for both mesh-based and meshless mappings. The second is a use of the inexact block coordinate descent method in mesh-based mapping in a way that efficiently minimizes the distortion with the capability not to be trapped early by the local minimum. We demonstrate the capability and superiority of our method in various applications including mesh parameterization, mesh-based and meshless deformation, and mesh improvement.