||Many low-level image analysis algorithms are not very reliable in the sense that their results differ in unpredictable ways from an "ideal" outcome. Making higher-level algorithms robust against low-level errors is thus an important research direction. However, in spite of some impressive results, this alone is insufficient -- it is also necessary to improve the performance of low-level algorithms. Our group has worked on that problem in several directions: (i) We want to understand in detail the properties of image data that cause errors and the principle limitations they impose on low-level image analysis. (ii) We are contributing to a truly 2-dimensional signal theory in order to actually approach these limits. (iii) We work on image representations which allow to perform basic analysis tasks in a provably correct way. In the talk, I'll illustrate the background of our approach and describe some of our results, in particular a geometric sampling theorem, two improved tensor-based structure descriptions, and the GeoMap/GeoMapPyramid.