This book chapter is published open access.
Imaging has been playing a vital role in the development of natural sciences. Advances in sensory, information, and computer technologies have further extended the scope of influence of imaging, making digital images an essential component of our daily lives. Image reconstruction is one of the most fundamental problems in imaging. For the past three decades, we have witnessed phenomenal developments of mathematical models and algorithms in image reconstruction. In this paper, we will first review some progress of the two prevailing mathematical approaches, i.e., the wavelet frame-based and PDE-based approaches, for image reconstruction. We shall discuss the connections between the two approaches and the implications and impact of the connections. Furthermore, we will review how the studies of the links between the two approaches lead us to a mathematical understanding of deep convolutional neural networks, which has led to further developments in modeling and algorithmic design in deep learning and new applications of machine learning in scientific computing.