Date |
Content of the Lecture |
Assignments/Readings/Notes |
| Lecture 1: 7/1 (Tue) |
- Course overview
- Intro to compressed sensing, tomography, dictionary learning, low rank matrix recovery
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| Lecture 2: 10/1 (Fri) |
- 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: 14/1 (Tue) |
- Concept of incoherence between sensing matrix and representation basis
- Theorem 1 for reconstruction guarantees
- Whittaker-Shannon sampling theorem
- Comparison between Theorem 1 and Shannon's sampling theorem.
- Further comments on theorem 1
- Intuition behind incoherence
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| Lecture 4: 17/1 (Fri) |
- Concept of restricted isometry property (RIP)
- Theorems 2 and 3 and comments on them
- Motivation for use of L1 norm instead of L2 norm for compressed sensing
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| Lecture 5: 21/1 (Tue) |
- Toy sample experiments in compressed sensing
- CS for piecewise constant signals: Theorem 4
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| Lecture 6: 24/1 (Fri) |
- Algorithms for compressed sensing: matching pursuit (MP) and orthogonal matching pursuit (OMP)
- Iterative Shrinkage and Thresholding Algorithm (ISTA)
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| Lecture 7: 28/1 (Tue) |
- 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: 31/1 (Fri) |
- 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: 4/2 (Tue) |
- Concept of tradeoff between spatial and temporal resolution in image acquisition
- Video snapshot compressive sensing: concept, hardware description, results
- CS in MRI
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| Lecture 10: 7/2 (Fri) |
- CS in MRI
- Introduction to pooled testing
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| Lecture 11: 11/2 (Tue) |
- Pooled testing: use of compressed sensing, multiplicative lognormal noise model, concept of CT value
- Dorfman's algorithm
- Use of family information for pooled testing: group sparsity
- 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: 14/2 (Fri) |
- 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
- Sensing matrix design
- Compressive classification
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| Lecture 13: 18/2 (Tue) |
- Compressive classification
Tomography
- Introduction to tomography
- Radon transforms
- Backprojection
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| Lecture 14: 21/2 (Fri) |
- Radon transforms, Fourier slice theorem
- Backprojection and filtered backprojection
- CS for tomography; concept of function handles
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| Lecture 14: 4/3 (Tue) |
- CS for tomography; concept of function handles; couple CS for reconstruction of consecutive tomographic slices
- Cryo-electron microscopy: introduction, concept of micrograph/particle
- Tomography under unknown angles (2D images, 1D projections): Helgason Ludwig consistency conditions (HLCC), inherent ambiguity in tomography under unknown angles
- Theoretical results in tomography under unknown angles (stated without proof)
- Algorithms for tomography under unknown angles: moment-based method, algorithm by Basu and Bresler based on ordering
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| Lecture 15: 7/3 (Fri) |
- Algorithms for tomography under unknown angles: moment-based method, algorithm by Basu and Bresler based on ordering
- algorithm based on dimensionality reduction using Laplacian eigenmaps
- Correction for unknown shifts in tomography under unknown angles; usage of order statistics for ordering and dimensionality reduction methods
- Tomography under unknown angles in 3D: concept of common line, algorithms for determining common lines
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| Lecture 16: 11/3 (Tue) |
- Tomography under unknown angles in 3D: concept of common line, algorithms for determining common lines
- Algorithm for finding unknown angles and underlying 3D structure from 2D projections under unknown angles - case of 3 projections only
- Algorithm or finding unknown angles (based on semi-definite programming) and underlying 3D structure from 2D projections under unknown angles - case of N projections
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| Lecture 17: 18/3 (Tue) |
- Algorithm or finding unknown angles (based on semi-definite programming) and underlying 3D structure from 2D projections under unknown angles - case of N projections
- Cryo-EM complete pipeline, concept of cryo-electron tomography and its difference from single particle cryo-EM
- Orthogonal procrustes algorithm
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| Lecture 18: 21/3 (Fri) |
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
- Experimental results for low rank matrix completion
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| Lecture 19: 25/3 (Tue) |
- Singular value thresholding for low rank matrix completion, with some properties and numerical results
- Concept of compressive low rank matrix recovery, related theorems and applications
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| Lecture 20: 28/3 (Fri) |
- Robust principal components analysis: application scenarios
- Key theorems for RPCA; some experimental results
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| Lecture 21: 1/4 (Tue) |
- Compressive RPCA: concept and key theorem
- SparCS algorithm for compressive RPCA; CoSamp algorithm for compressed sensing
- Experimental results for compressive RPCA for compressive hyperspectral image reconstruction and compressive video recovery
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
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| Lecture 22: 4/4 (Fri) |
- Dictionary learning using Olshausen's method, Method of optimal directions
- KSVD method for dictionary learning, with applications to lossy image compression
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| Lecture 23: 8/4 (Tue) |
- 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 24: 11/4 (Fri) |
- Non-negative matrix factorization (NMF), non-negative sparse coding (NNSC)
- Bayesian statistics: basics, applications in image denoising and deblurring; concept of MAP, posterior probabilities with gaussian likelihood and Laplacian and Gaussian priors
- Statistical compressed sensing using Gaussian distributions and Gaussian Mixture Models
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