Date |
Content of the Lecture |
Assignments/Readings/Notes |
3rd Jan (Thu) |
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7th Jan (Mon) |
Compressed Sensing
- Compressed sensing (CS): introduction and motivation
- Review of DFT and DCT, Review of JPEG
- Sparsity of natural images in transform bases
- Candes, Romberg, Tao: puzzling experiment; Basic optimization problem for CS involving the total variation
- Concept of incoherence between sensing matrix and representation basis
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- Slides (check moodle)
- E. Candes and M. Wakin, "Introduction to Compressive sampling", IEEE Signal Processing Magazine, 2008
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10th Jan (Thu) |
- Shannon's sampling theorem and the Whittaker-Shannon Interpolation Formula
- Basic CS optimization problem: comments on the uniqueness of its solution
- Basic theorem of compressed sensing involving coherence (due to Candes, Romberg, Tao)
- Intuition behind role of incoherence in CS
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- Slides (check moodle)
- E. Candes and M. Wakin, "Introduction to Compressive sampling", IEEE Signal Processing Magazine, 2008
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14th Jan (Mon) |
- Restricted isometry property
- Basic theorems of CS: reconstruction of compressible signals
- Compressed sensing under noise (theorem 3): choice of regularization parameter under different noise models
- The role of randomness
- Basis pursuit and its efficiency - Basis pursuit as a linear programming problem
- Intuitive explanation: benefits of L1 or L2 penalty for signal sparsity
- Toy examples of CS results
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- Slides (check moodle)
- E. Candes and M. Wakin, "Introduction to Compressive sampling", IEEE Signal Processing Magazine, 2008
- J. Romberg, "Imaging via Compressive sampling", IEEE Signal Processing Magazine, 2008
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17th Jan (Thu) |
- Sketch of a conventional camera
- Rice single pixel camera and its patchwise variant from Stanford
- Video version of the Rice SPC
- Conventional hyperspectral camera
- Architecture of CASSI: Coded aperture snapshot spectral imager
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- Slides (check moodle)
- Duarte et al, "Single-pixel imaging via Compressive sampling", IEEE Signal processing magazine, 2008
- Kittle et al, "Multiframe image estimation for coded aperture snapshot spectral imagers", Applied Optics, 2010
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21st Jan (Mon) |
- Architecture of CASSI: Coded aperture snapshot spectral imager
- Concept for color filter arrays (CFA), Comparison of CASSI with color image demosaicing
- Video compressive sensing using coded snapshots
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- Slides (check moodle)
- Kittle et al, "Multiframe image estimation for coded aperture snapshot spectral imagers", Applied Optics, 2010
- Hitomi et al, "Video from single exposure coded snapshots", ICCV 2011
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24th Jan (Thu) |
- Video compressive sensing using coded snapshots
- Compressed sensing in MRI: concept of different MRI trajectories
- CS algorithms: matching pursuit (MP) and orthogonal matching pursuit (OMP)
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28th Jan (Mon) |
- CS algorithms: ISTA (iterative shrinkage/thresholding algorithm)
- Concept of mutual coherence and its relationship with the restricted isometry property/constant via Gershgorin's disc theorem
- Comparison between mutual coherence and RIC
- Theorem for performance bounds using mutual coherence
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31st Jan (Thu) |
- RIC interpreted in terms of singular values, relationship between mutual coherence and RIC
- CS bounds based on mutual coherence (Theorem 5)
- CS bounds based on RIC of order s (as opposed to 2s): Theorem 6
- Compressive classification: maximum likelihood classifier, matched filter, smashed filter, relation to RIP
- Sketch of proof of Theorem 3
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- Slides (check moodle)
- Candes, "The restricted isometry property and its implications for compressed sensing", CRM 2008
- Davenport et al, "The smashed filter for compressive classification and target recognition", SPIE 2007
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4th Feb (Mon) |
- Theorem 4: CS for piecewise constant signals
- CS with tight frames
- Clarification regarding RIP of Random Discrete Fourier Measurement Matrices
Tomography
- Concept of tomography and tompgraphic projection/Radon transform
- Concept of backprojection and its limitations
- Generations of CT (computed tomography) machines
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7th Feb (Thur) |
- Fourier slice theorem in tomography
- Concept of filtered backprojection and its relationship to (unfiltered) backprojection
- Tomography as a compressed sensing problem
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9th Feb (Sat, extra lecture) |
- Tomography as a compressed sensing problem
- Tomography under unknown angles: motivation in cryo-EM, algorithms for 2D images
- Moment based approach - Helgason Ludwig consistency conditions (2D images)
- Concept of micrograph and particle picking in cryo-EM
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11th Feb (Mon) |
- Moment-based algorithm (2D images)
- Ordering-based algorithm: nearest neighbor algorithm (2D images)
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15th Feb (Thu) |
- Graph-laplacian algorithm for dimensionality reduction (due to Belkin and Niyogi); its application to tomography under unknown angles
- PCA-based algorithm for denoising of tomographic projections
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21st Feb (Thu) |
- PCA-based algorithm for denoising of tomographic projections; derivation of a Wiener filter in the transform domain
- Tomography under unknown angles in 3D (i.e. 2D projection images): the concept of common lines, algorithm for finding common lines in spatial or Fourier domain
- Algorithm for finding unknown angles and underlying 3D structure from 2D projections under unknown angles - case of 3 projections only
- Orthogonal procrustes algorithm
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4th March (Mon) |
- Distribution of midsem papers and discussion of solutions
- Algorithm for finding unknown angles and underlying 3D structure from 2D projections under unknown angles - case of 3 projections only
- Orthogonal procrustes algorithm
- Algorithm for 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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7th March (Thurs) |
Dictionary Learning
- Introduction to overcomplete dictionaries
- Concept of overcomplete dictionaries: sinusoids and spikes examples
- Issues of uniqueness and sparsity of representation in overcomplete dictionaries, dictionary learning as a generalization of K-means
- Principal Components Analysis: Eigenfaces; relationship between DCT and PCA
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11th March (Mon) |
- A basic algorithm for simultaneously obtaining dictionary and sparse code - using projected gradient descent with adaptive step-size
- Method of Optimal Directions (MOD)
- Union of Orthonormal Bases
- KSVD method
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14th March (Thurs) |
- KSVD method
- KSVD method for compression, denoising, inpainting
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18th March (Mon) |
- Blind compressed sensing
- Compressive KSVD algorithm
- Requirement for blind compressed sensing: diversity of measurement matrices
- Fisher's linear discriminant: case of 2 classes
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25th March (Mon) |
- Fisher's linear discriminant: case of 2 classes
- Fisher's linear discriminant: case of more than 2 classes
- Limitations of FLD, introduction to mutual information for classification, concept of mutual information and its properties
- Mutual information for optimal transform learning for classification
- Kernel density estimation
- Optimization of quadratic mutual information
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28th March (Thu) |
Low Rank Matrix Recovery and Beyond
- Ubiquitousness of low rank matrices in image processing and machine learning: recommender systems, image patches assembled to form a matrix, distance matrices, applications in structrue from motion, applications in face recognition
- The problem of low rank matrix completion
- Informal statement of the basic theorem, concept of coherence of subspaces, sufficient conditions for successful recovery and pathological cases
- Formal statement of theorem of low rank matrix completion; extension to the case of noisy matrix completion
- Empirical results for low rank matrix completion
- Concept of low rank matrix recovery
- Basic theorems for low rank matrix recovery, concept of RIP of linear maps/operators, relation between low rank matrix recovery and low rank matrix completion
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1st April (Mon) |
- Introduction to Robust Principal Components Analysis (PCA)
- Motivating examples for RPCA: background subtraction in videos, specularity and shadow removal from face images
- Basic theorem for RPCA; extension to the case when there is noise in observed matrix
- Basic theorem for RPCA with incompletely observed entries
- Singular Value Thresholding Algorithm (SVT) for matrix completion
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4th April (Thurs) |
- Introduction to Robust Principal Components Analysis (PCA)
- Motivating examples for RPCA: background subtraction in videos, specularity and shadow removal from face images
- Basic theorem for RPCA; extension to the case when there is noise in observed matrix
- Basic theorem for RPCA with incompletely observed entries
- Singular Value Thresholding Algorithm (SVT) for matrix completion
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8th April (Mon) |
Inverse Problems using Deep Neural Networks
- Inverse problems from a Bayesian perspective; limitations of analytical models
- Deep neural networks for inverse problems: introduction
- Multi-layer perceptrons; Image denoising using MLPs
- Motivation for convolutional neural networks; their use for inverse problems
- Encoder-decoder CNNs
- Auto-encoders for super-resolution
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11th April (Thurs) |
Low Rank Matrix recovery and Beyond
- Compressive RPCA: Greedy Algorithm by Waters et al, applications in video and hyperspectral CS, and RPCA with missing entries
- Basic Theorem for Compressive RPCA
Statistics of Natural Images
- Power law
- Correlation of pixel values in natural images
- Statistics of DCT coefficients of natural images: laplacian models and justification for the same
- Brief introduction to the Haar wavelet transform
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15th April (Mon) |
- Brief introduction to the Haar wavelet transform
- Revision of basic Bayesian statistics, concept of MAP and MMSE, posterior probabilities with gaussian likelihood and Lapalcian and Gaussian priors
- Statistics of wavelet coefficients of natural images, use of dependencies between wavelet coefficients as priors for image denoising and in compression
- Compressed sensing based on statistical priors
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18th April (Thu) |
- Statistics of natural image categories: natural, man-made,etc. Image and scene scale
- A semi-automated method for reflection removal using statistics of image gradients: Iteratively reweighted least squares algorithm
- Results of the IRLS algorithm for reflection removal, mixture of Laplacian prior for image gradients
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