Date |
Content of the Lecture |
Assignments/Readings/Notes |
| Lecture 1: 5/1 (Mon) |
- Course overview
- Intro to compressed sensing, tomography, dictionary learning, low rank matrix recovery
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| Lecture 2: 8/1 (Thu) |
- Compressed sensing (CS): introduction and motivation
- Review of DFT and DCT, Review of JPEG, representation of a signal/image as a linear combination of basis vectors
- Sparsity of natural images in transform bases
- Candes, Romberg, Tao: puzzling experiment; Basic optimization problem for CS involving the total variation (i.e. sum total gradient magnitude of the image)
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| Lecture 3: 12/1 (Mon) |
- Whittaker-Shannon sampling theorem
- Concept of incoherence between sensing matrix and representation basis
- L0 and L1 norm optimization problems for compressed sensing; proof of uniqueness of the solution of the L0 problem
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| Lecture 4: 15/1 (Thu) |
- Theorem 1 and its relation to Shannon's sampling theorem; examples of bad/incompatible pairs of signal-support and measurement-subset
- Intuition behind incoherence
- Concept of restricted isometry property (RIP) and its relation to the nullspace property
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| Lecture 5: 19/1 (Mon) |
- Theorems 2 and 3 and comments on them
- Motivation for use of L1 norm instead of L2 norm for compressed sensing
- Interpretation of L1 norm optimization as a linear programming problem
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| Lecture 6: 22/1 (Thu) |
- Sample results for compressed sensing
- CS for piecewise constant signals: Theorem 4
- Algorithms for compressed sensing: matching pursuit (MP) and orthogonal matching pursuit (OMP)
- Iterative Shrinkage and Thresholding Algorithm (ISTA)
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| Lecture 7: 29/1 (Thu) |
- Algorithms for compressed sensing: iterated shrinkage and thresholding algorithm (ISTA)
- Rice single pixel camera, and its block-based version
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| Lecture 8: 2/2 (Mon) |
- Rice single pixel camera in video mode: separate frame-by-frame reconstruction, coupled reconstruction
- CASSI camera for compressive hyperspectral imaging
- Color filter arrays and color image demosaicing
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| Lecture 9: 7/2 (Sat) |
- Concept of tradeoff between spatial and temporal resolution in image acquisition
- Video snapshot compressive sensing: concept, hardware description, results
- Introduction to compressed sensing for pooled testing: noise model in RTPCR
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| Lecture 10: 9/2 (Mon) |
- Introduction to compressed sensing for pooled testing: Dorfman's algorithm, design of pooling matrices, use of family and contact tracing information, group sparsity
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| Lecture 11: 12/2 (Thu) |
- Proof sketch for theorem 3: use of Cauchy Schwartz Inequality, Triangle Inequality, Reverse Triangle Inequality, relationship between different norms
- Theorem 6: use of RIC of order s, instead of order 2s
- Theorem 5: problem P1 analyzed using mutual coherence
- Mutual coherence versus RIC; Gershgorin's disc theorem to derive the relationship between them
- Logan's result regarding recovery of a bandlimited signal given impulse noise
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| Lecture 12: 16/2 (Mon) |
- Sensing matrix design
- Compressive classification
- CS in MRI
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| Lecture 13: 19/2 (Thu) |
Dictionary Learning
- Concept of dictionary learning and dictionary coefficients
- Orthonormal and overcomplete dictionaries; overcompleteness and sparsity; sparse coding for orthonormal and overcomplete dictionaries
- KMeans as a special case of dictionary learning
- Dictionary learning using Olshausen's method, Method of optimal directions
- Introduction to KSVD method for dictionary learning
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| Lecture 14: 2/3 (Mon) |
- Dictionary learning using Olshausen's method, Method of optimal directions
- KSVD method for dictionary learning, with applications to lossy image compression
- Singular value decomposition: basic concept, Eckart Young theorem, reduced form of SVD
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| Lecture 15: 7/3 (Sat) |
- KSVD for denoising and inpainting; KSVD for video compressive sensing for snapshot-coded images;
- Blind compressive sensing
- Compressed sensing with overcomplete dictionaries
- Dictionary learning via unions of orthonormal bases
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| Lecture 16: 9/3 (Mon) |
- Compressed sensing with overcomplete dictionaries
- Dictionary learning via unions of orthonormal bases
- Method of optimal directions
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| Lecture 17: 12/3 (Thu) |
- PCA (principal components analysis): aim, derivation, mathematics
- PCA and face recognition
- PCA and the discrete cosine transform
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| Lecture 18: 16/3 (Mon) |
- Function handles for implementation of compressed sensing algorithms for large-sized signals
Low rank matrix recovery
- Low rank matrices in data scienc, computer vision and image processing
- Concept and key theorem + estimator for low rank matrix completion; inherent limitations of low rank matrix completion and concept of matrix coherence
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| Lecture 19: 23/3 (Mon) |
- Singular value thresholding for low rank matrix completion, with some properties and numerical results
- Robust principal components analysis: concept and use cases in video surveillance and facial images
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| Lecture 20: 26/3 (Thu) |
- Robust principal components analysis: key theorems and limitations
- RPCA algorithm
- Concept of low rank matrix recovery and key theorems for the same
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| Lecture 21: 30/3 (Mon) |
- Compressive RPCA: key theorems, SparCS algorithm based on CoSamp and applications
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| Lecture 22: 2/4 (Thu) |
Neural Networks for Compressive Sensing
- Brief overview of Multi-layer perceptrons (MLP) and convolutional neural networks (CNNs)
- Types of neural network architectures for CS
- Concept of neural network unrolling and learned ISTA
- Applications of neural network unrolling in super-resolution
- Theoretical result for LISTA
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| Lecture 23: 6/4 (Mon) |
- Theoretical result for LISTA; concept of generalized coherence
- Concept of analytic LISTA (ALISTA)
- Concept of augmented Lagrangian method (ALM) and its application for RPCA pursuit
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| Lecture 24: 9/4 (Thurs) |
- ADMM for CS; concept of ADMMNet
- Concept of untrained neural networks
- Convolutional generators
- Theoretical results for over-parameterized linear generators
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| Lecture 25: 13/4 (Mon) |
- Theoretical results for over-parameterized non-linear convolutional generators
- Convolutional untrained generators for denoising: theoretical results and concept of early stopping
- Convolutional untrained generators for compression
Bayesian compressed sensing
- Concept of MAP and MMSE in Bayesian estimation with examples
- Compressive reconstruction of signals obeying Gaussian priors: closed form expression
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| Lecture 26: 16/4 (Thu) |
- Concept of MAP and MMSE in Bayesian estimation with examples
- Compressive reconstruction of signals obeying Gaussian priors: closed form expression
- Bayesian statistics: basics, applications in image denoising and deblurring; concept of MAP, posterior probabilities with gaussian likelihood and Laplacian and Gaussian priors
- Compressive reconstruction of signals obeying Gaussian mixture model priors
- Dictionary learning technique: non-negative matrix factorization (NMF)
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