Explore a comprehensive lecture on splines and imaging, covering the journey from compressed sensing to deep neural networks. Delve into the optimality of splines for solving inverse problems in imaging and designing deep neural networks. Examine a representer theorem that demonstrates how extremal points of linear inverse problems with generalized total-variation regularization are adaptive splines. Discover the connection between continuous-domain solutions and compressed sensing algorithms. Investigate the application of the theorem to optimize activation shapes in deep neural networks, leading to the concept of "optimal" deep-spline networks with piecewise-linear spline activations. Gain insights into the variational justification of ReLU architecture and explore new computational challenges in determining optimal activations. Learn about the structure of iterative reconstruction algorithms, the algebra of CPWL functions, and the implications for deep ReLU neural networks.