Deep neural network has become a very popular machine learning method for the last decade or so. It has been successfully applied in various fields, such as image processing, computer vision and computer graphics. The obtained results are pleasing and outperformed other existing methods. However, the performance of the method depends largely on the architectural of the network. It also takes tremendous computational time to train the network and the convergence of the computational method is usually unknown. It is therefore necessary to have a better mathematical understanding of the deep neural network, in order to develop more effective deep network structures for various applications. As such, we plan to mathematically formulate the approximation accuracy of the different deep neural network architectures. The goal is to explore the possibility to build a deep neural network structure that gives high approximation accuracy, while requiring the least data input. With the theoretical understanding the network structure, we plan to develop efficient numerical algorithms to train the network, whose convergence and accuracy can be theoretically guaranteed. As such, the developed deep learning based method for problem solving can be controllable and can also be adjusted according to the specific applications.

Image processing has been a hot topic in the fields of applied mathematics and engineering. Applications can be found in computer graphics, medicine, chemistry, life sciences and so on. Various mathematical models have been developed recently, which can give quite satisfactory results. However, with the availability of big data, it is believed the combination of data information with the mathematical models can significantly enhance the performance. As such, we plan to develop data-driven based image processing models to tackle various imaging problems, such as image denoising, debluring, segmentation, registration and so on. We will carefully study how the data information can be incorporated into mathematical models. It involves careful mathematical formulation of the problem as well as rigorous analysis of the developed models.

Techniques for data analysis is especially important for medical imaging problems. On one hand, data information can guide the processing of medical images, especially for corrupted or degraded images. On the other hand, data information can give meaningful feature vectors to analyse medical images for disease analysis. In collaboration with CUHK medical school, we plan to develop data analytics methods to extract meaningful feature vectors for various medical image processing and analysis tasks. In particular, we plan to develop deep learning based method to extract important and meaningful feature vectors. Due to the lack of medical data in some scenario, data-augmentation techniques will also be considered.

Deep learning methods have become an omnipresent and highly successful part of recent approaches in imaging and vision. However, in most cases they are used on a purely empirical basis without real understanding of their behavior. From a scientific viewpoint, this is unsatisfying. Many mathematically inclined researchers have a strong desire to understand the theoretical reasons for the success of these approaches and to find relations between deep learning and mathematically well-established techniques in imaging science. The goal of this topic is to showcase their latest research results and to promote future research in this direction. Indeed, we are interested in:

A. gaining mathematical introspection into the behavior of deep learning methods, e.g. through: theoretical insights into their expressive power, quality, stability, and efficiency analysis of their ability to handle the curse of dimensionality; investigation of their generalization properties; theoretical bounds on their necessary complexity; theories for architectural design characterization of their loss surface; analysis of optimization algorithms; mathematical theories for generative adversarial networks;

B. establishing connections between deep learning and successful mathematical concepts in image analysis such as: radial basis functions, splines, and harmonic analysis sparsity; compressed sensing, and dictionary learning subspace methods; inverse problems, regularization theory, and operator learning variational methods; optimization, and optimal control ordinary and partial differential equations information theory, information geometry, and the physics of information; statistical learning theory.