This thesis develops theoretical analysis of the approximation properties of neural networks, and algorithms to extract useful features of images in fields of deep learning, quantum energy regression and cancer image analysis. The separate applications are connected by using representation systems in harmonic analysis; we focus on deriving proper representations of data using Gabor transform in this thesis. A novel neural network with proven approximation properties dependent on its size is developed using Gabor system. In quantum energy regression, invariant representation of chemical molecules using electron densities is obtained based on the Gabor transform. Additionally, we dig into pooling functions, the feature extractor in deep neural networks, and develop a novel pooling strategy originated from the maximal function with stability property and stable performance. Anisotropic representation of data using the Shearlet transform is also explored in its ability to detect regions of interests of nuclei in cancer images.