Course content

• Image transforms and statistics of natural images
• Survey of statistical properties of image transform coefficients
• Implications of these statistics for important image processing applications such as denoising, compression, source separation, deblurring and image forensics
• Non-local self-similarity in images
• Dictionary learning and sparse representations in image processing
• Overview of Principal Components Analysis (PCA), Singular Value Decomposition (SVD) and Independent Components Analysis (ICA); PCA, SVD and ICA in the context of image processing
• Sparse PCA
• Concept of overcomplete dictionaries
• Greedy pursuit algorithms: matching pursuit (MP), orthogonal matching pursuit (OMP) and basis pursuit (BP)
• Popular dictionary learning techniques: Method of Optimal Directions (MOD), Unions of Orthonormal Bases, K-SVD, Non-negative sparse coding – along with applications in image compression, denoising, inpainting and deblurring
• Sparsity-seeking algorithms: iterative shrinkage and thresholding (ISTA) (3) Compressed Sensing (CS)
• Concept and need for CS
• Theoretical treatment: concept of coherence, null-space property and restricted isometry property, proof of a key theorem in CS
• Algorithms for CS (covered in part 2) and some key properties of these algorithms
• Applications of CS: Rice Single Pixel Camera and its variants, Video compressed sensing, Color and Hyperspectral CS, Applications in Magnetic Resonance Imaging (MRI), Implications for Computed Tomography
• CS under Forward Model Perturbations: a few key results and their proofs as well as applications
• Designing Forward Models for CS
• Low-rank matrix estimation and Robust Principal Components Analysis: concept and application scenarios in image processing, statement of some key theorems, and proof of one important theorem

References

• We will extensively refer to the following textbooks, besides a number of research papers from journals such as IEEE Transactions on Image Processing, IEEE Transactions onSignal Processing, and IEEE Transactions on Pattern Analysis and Machine Intelligence:
• "Natural Image Statistics" by Aapo Hyvarinen, Jarmo Hurri and Patrick Hoyer,Springer Verlag 2009 (http://www.naturalimagestatistics.net/ - freely downloadable online)
• "A Mathematical Introduction to Compressive Sensing" by Simon Foucart andHolger Rauhut, Birkhauser,2013 (http://www.springer.com/us/book/9780817649470)
• Fung, Y. C.: Biomechanics: Mechanical Properties of Living Tissues. 2nd Ed., Springer.
• R. Kamm and M. K. Mofrad. Cytoskeletal Mechanics: Models and Measurements. Cambridge University Press.