Date |
Content of the Lecture |
Assignments/Readings/Notes |
4th Jan |
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Slides (check moodle) |
8th Jan |
Compressed Sensing
- Compressed sensing (CS): introduction and motivation
- Review of DFT and DCT, Review of JPEG, Review of the Sampling theorem and its limitations
- Sparsity of natural images in transform bases
- Candes, Romberg, Tao: puzzling experiment
- Basic optimization problem for CS
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- Slides (check moodle)
- Research papers (check moodle)
- Candes and wakin, "Introduction to Compressive sampling", IEEE Signal Processing Magazine, 2008
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11th Jan |
- Three key theorems for compressed sensing and comments on the theorems
- Concept of incoherence between sensing matrix and representation basis
- Basic CS optimization problem: comments on the uniqueness of its solution
- Concept of restricted isometry property
|
- Slides (check moodle)
- Research papers (check moodle)
- Candes and wakin, "Introduction to Compressive sampling", IEEE Signal Processing Magazine, 2008
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15th Jan |
- 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
- Rice single pixel camera
|
- Slides (check moodle)
- Research papers (check moodle)
- Candes and wakin, "Introduction to Compressive Sampling", IEEE Signal Processing Magazine, 2008
- Romberg,"Imaging via Compressive Sampling", IEEE Signal processing Magazine, 2008
|
18th Jan |
- Rice single pixel camera
- Block-wise single pixel camera by El-Gamal
- Video version of the Rice SPC
- The CASSI architecture for compressed sensing of hyperspectral images
|
- Slides (check moodle)
- Research papers (check moodle)
- Duarte et al, "Single-pixel imaging via Compressive sampling", IEEE Signal processing magazine
- Kittle et al, "Multiframe image estimation for coded aperture snapshot spectral imagers", Applied Optics
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22nd Jan |
- Discussion on compressive architectures: video version of Rice SPC, CASSI
- Color image acquisition via color filter arrays and its relationship with CASSI acqusition
- Video compressive sensing using coded snapshots
|
- Slides (check moodle)
- Research papers (check moodle)
- Hitomi et al, "Video from a Single Coded Exposure Photograph using a Learned Over-Complete Dictionary", ICCV 2011
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25th Jan |
- Video compressive sensing using coded snapshots
- CS in Magnetic Resonance Imaging (MRI)
- CS Theory: Reconstruction of piecewise flat signals
- Concept of mutual coherence and its comparison to RIP/RIC: Theorem 5
|
- Slides (check moodle)
- Research papers (check moodle)
- Lustig et al, "Compressed Sensing MRI", IEEE Signal Processing Magazine, 2008
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29th Jan |
- CS Algorithms: Matching pursuit and Orthogonal Matching pursuit
- Iterative Shrinkage and Thresholding Algorithm (ISTA)
|
- Slides (check moodle)
- Research papers (check moodle)
|
1st Feb |
- Compressed Sensing Theorem 3: sketch of the proof - tube constraint, cone constraint, various vector inequalities
- Concept of overcomplete dictionaries
- Concept of design of CS matrices my minimizing mutual coherence
- Compressive classification
|
- Slides (check moodle)
- Research papers (check moodle)
- Candes, "The restricted isometry property and its implications for compressed sensing", Comptes Rendus de Mathematiques, 2008
|
5th Feb |
- Compressive classification: maximum likelihood classifier, matched filter, smashed filter, relation to RIP
- Concept of design of CS matrices my minimizing mutual coherence: method of Duarte-Carvajalino and Sapiro
Tomography
- Concept of tomography and tompgraphic projection/Radon transform
- Concept of backprojection and its limitations
- Generations of CT (computed tomography) machines
- Fourier slice theorem in tomography
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- Slides (check moodle)
- Research papers (check moodle)
- Davenport et al, "The smashed filter for compressive classification and target recognition", SPIE, 2007
|
8th Feb |
- Fourier slice theorem in tomography
- Concept of filtered backprojection
- Tomography as a compressed sensing problem
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12th Feb |
- Tomography as a compressed sensing problem
- Tomography under unknown angles: motivation, algorithms for 2D images
- Moment based approach - Helgason Ludwig consistency conditions (2D images)
- Moment-based algorithm (2D images)
|
|
15th Feb |
- Ordering-based algorithm: nearest neighbor algorithm (2D images)
- Approaches based on dimensionality reduction: Graph Laplacian algorithm for dimensionality reduction (2D images)
- Issue of unknown shifts
|
|
19th Feb |
- PCA-based algorithm for denoising tomographic projections
- Tomography under unknown angles for 3D images - common lines constraint, detailed algorithm
- Use of the SVD in the aforementioned algorithm, orthogonal Procrustes problem
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19th Feb |
- Tompgraphy under unknown angles for 3D images - common lines constraint, detailed algorithm
- Use of the SVD in the aforementioned algorithm, orthogonal Procrustes problem
- Discussion session for midsem
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2nd March |
- Distribution of midsem papers and discussion of questions
Dictionary Learning
- Introduction to basics of dictionary learning
- Principal components analysis (PCA)
- Eigenfaces algorithms - efficient variant, eigenvalue decay for face images
|
|
5th March |
- Eigenfaces algorithms - efficient variant, eigenvalue decay for face images
- Derivation for PCA - as an algorithm to infer an orthonormal matrix that minimizes reconstruction error, and hence maximizes variance
- Extension of this derivation to handle multiple directions
- Robust versions of PCA (eg: L1 PCA)
- Applications of PCA in image compression
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|
7th March (extra lecture) |
- Eigenfaces algorithms - efficient variant, eigenvalue decay for face images
- Derivation for PCA - as an algorithm to infer an orthonormal matrix that minimizes reconstruction error, and hence maximizes variance
- Extension of this derivation to handle multiple directions
- Robust versions of PCA (eg: L1 PCA)
- Applications of PCA in image compression
|
|
8th March |
- Relationship between PCA of tiny image patches and the DCT
- 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
- A basic algorithm for simultaneously obtaining dictionary and sparse code - using projected gradient descent with adaptive step-size
|
|
10th March (extra lecture) |
- 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
|
|
12th March |
- KSVD method for compression, denoising, inpainting
|
|
14th March (extra lecture) |
- Concept of blind compressed sensing: the compressive KSVD method
- Requirements for blind compressed sensing
- Non-negative matrix factorization (NMF)
- Non-negative sparse coding (NNSC)
|
|
2nd April |
- NNSC for Poisson noise; The Poisson noise model in imaging
- 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
|
|
5th April |
- 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
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
|
|
7th April (extra lecture) |
- 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
|
|
9th April |
- 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
- Revision of basic Bayesian statistics, concept of MAP and MMSE, posterior probabilities with gaussian likelihood and Lapalcian and Gaussian priors
- Compressed sensing based on statistical priors
- Statistics of wavelet coefficients of natural images, use of dependencies between wavelet coefficients as priors for image denoising and in compression
|
|
12th April |
- Statistics of wavelet coefficients of natural images, use of dependencies between wavelet coefficients as priors for image denoising and in compression: linear model for variance of a wavelet coefficient given a set of neighbors
Low rank matrix recovery
- 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
|
|
14th April (extra lecture) |
- 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
- Empirical results for low rank matrix completion
- Concept of low rank matrix recovery
- Introduction to Robust Principal Components Analysis (PCA)
- Motivating examples for RPCA: background subtraction in videos, specularity and shadow removal from face images
|
|
16th April |
- 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
- 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
- Basic theorem for RPCA with incompletely observed entries
|
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19th April |
- Singular Value Thresholding Algorithm (SVT) for matrix completion
- Augmented Lagrangian Method (ALM): ALM for RPCA
- Compressive RPCA: Greedy Algorithm by Waters et al, applications in video and hyperspectral CS, and RPCA with missing entries
- Basic Theorem for Compressive RPCA
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