Image segmentation and classification still represent an active area of research since no universal solution can be identified. Established segmentation algorithms like thresholding are problem specific, treat well the easy cases and mostly relied on single parameter i.e intensity. Machine learning approaches offer alternatives where predefined features are combined into different classifiers. On the other hand, the outcome of machine learning is only as good as the underlying feature space. Differential geometry can substantially improve the outcome of machine learning since it can enrich the underlying feature space with new geometrical objects, called invariants. In this way, the geometrical features form a high-dimensional feature space for each pixel, where original objects can be resolved. Alternatives based on the geometry of the image scale-invariant interest points have been exploited successfully in the field of computer vision. Here, we integrate geometrical feature extraction based on signal processing, machine learning, and input relying on domain knowledge. The approach is exemplified on the ISBI 2012 image segmentation challenge data set. As a second application, we demonstrate powerful image classification functionality based on the same principles, which was applied to the HeLa and HEp-2 data sets. Obtained results demonstrate that feature space enrichment properly balanced with feature selection functionality can achieve performance comparable to deep learning approaches.