Date |
Content of the Lecture |
Assignments/Readings/Notes |
2nd Jan |
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Slides (check moodle) |
5th Jan |
Statistics of natural images
- Power law
- Correlation between a pixel and its neighbors
- Sparsity of DCT coefficients - Laplacian model
- Sparsity of wavelet coefficients, dependencies between
wavelet coefficients in different sub-bands
- Bayesian models: likelihood and prior probability or probability density with examples
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Slides (check moodle) |
9th Jan |
- Bayesian models: likelihood and prior probability or probability density with examples
- Denoising or deblurring using a Laplacian signal prior; derivation of the ISTA algorithm in detail
- Denoising or deblurring using a Gaussian signal prior - leading to the Wiener filter
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Slides (check moodle) |
12th Jan |
- Genesis of the Laplacian model for DCT coefficients of natural images
- Lindeberg's central limit theorem, exponential distribution for patch variances in natural images
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16th Jan |
- Denoising using dependencies between wavelet coefficients, and a modified Wiener filter update
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19th Jan |
- Semi-automated method for reflection removal using statistical properties of natural images
- Iterative Reweighted Least squares Algorithm (IRLS)
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23rd Jan |
Compressed Sensing
- Conventional sensing versus compressed sensing
- Application areas of compressed sensing: MRI, video, CT, hyperspectral images
- Shannon's sampling theorem and its limitations
- Candes' puzzling experiment
- The role of sparsity
- Concept of sensing matrix, representation matrix and incoherence between the two
- The key optimization problem in compressed sensing using L0 norm and its softening to L1 norm
- Number of independent columns of a sensing matrix
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30th Jan |
- Softening to L1 norm: linear programming
- Theorem by Candes, Romberg, Tao involving incoherence and sparsity
- Corollary to the theorem involving Fourier sensing matrix and signals sparse in canonical basis: comparison to Shannon's sampling theorem
- Intuition behind incoherence
- Concept of the restricted isometry property
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2nd Feb |
- Concept of the restricted isometry property
- Sufficient condition for compressive reconstruction of compressible signals with and without noise (theorem 3 and theorem 2 in the slides)
- Comparison between Theorems 1, 2, 3
- Random sensing matrices and the restricted isometry property
- L1 versus L2 norm in compressed sensing
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6th Feb |
- L1 versus L2 norm in compressed sensing
- Candes' experiments and its results in terms of theorem 1 (leading to theorem 4): reconstruction of piecewise constant signals/images
- Concept of mutual coherence
- Theorem 5: sufficient conditions for compressive recovery using mutual coherence
- Gershgorin's disc theorem: relation between restricted isometry constant (RIC) and mutual coherence; comparison between the two
- Greedy algorithms for compressive reconstruction: matching pursuit (MP) and orthogonal matching pursuit (OMP)
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9th Feb |
- Rice single pixel camera (SPC)
- Rice (SPC) in video mode
- Architecture of compressive camera by El Gamal
- Video compressive sensing based on coded snapshots
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13th Feb |
- Video compressive sensing based on coded snapshots
- CASSI camera for hyperspectral imaging
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16th Feb |
- Applications of CS in Magnetic Resonance Imaging
- Discussion of project topics
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27th Feb |
- Midterm paper distribution
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2nd March |
- Sketch of proof of key theorem on CS (theorem 3 in the slides)
- Associated lemmas on the RIP and other simple vector properties for the aforementioned proof
- Statement of improved version of theorem 3 - theorem 6, with RIP of order s instead of 2s (for s-sparse signals)
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13th March |
- Designing of compressed sensing matrices by minimization of mutual coherence: applications to the Hitomi video camera and demosaicing
- Method of Duarte-Carvajalino et al
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16th March |
Dictionary Learning
- Dictionary learning: problem definition, sparse coding: problem definition
- Principal Components Analysis (PCA): derivation
- Application of PCA to face recognition (eigenfaces), image compression; PCA on natural image patches and its relation to DCT
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20th March |
- Motivation for overcomplete dictionaries
- Method of Olshausen and Field: sparsity constraints on sparse codes and gradient descent based dictionary updates
- Method of Optimal Directions (MOD) for Overcomplete dictionary learning
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23rd March |
- KSVD algorithm: sparse coding through OMP, dictionary update using Eckhart Young theorem for rank 1 approximations
- Applications of KSVD: compression, denoising, inpainting
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27th March |
- Blind compressed sensing: inferring KSVD dictionaries directly from compressive measurements
- Non-negative matrix factorization (NMF) and Non-negative sparse coding (NNSC)
- Poisson noise in images, Applications of NNSC in removal of Poisson noise
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30th March |
- Method of union of orthonormal basis
- Orthogonal procrustes for inferring orthonormal bases (applications of SVD)
Tomographic Rconstruction
- Problem statement and definition
- Concept of radon transform and its relationship to tomographic projections
- Back-projection for tomography and its limitations
- Applications of tomography, Beer's law, 1st to 4th generation CT
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3rd April |
- Filtered backprojection: detailed derivation and use of Ram-Lak filter; relation between backprojection and the "true" Radon inverse
- Comparison between filtered backprojection and backprojection
- Tomography as a compressed sensing problem: empirical comparison to FBP
- Limitations in theory: Radon matrix does not obey RIP, incoherence properties
- Coupled tomographic reconstruction of similar slices
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6th April |
- Tomography under unknown angles: application scenarios
- Concept of image and projection moments, relation between image and projection moments and the angles of projection (in parallel beam tomography)
- Fundamental rotational ambiguity tomography under unknown angles
- Moment-based method for estimating projection angles and image moments from tomographic projections under unknown angles
- Ordering based method for tomography under unknown angles, assuming known distribution of the unknown angles - nearest neighbor algorithm (due to Basu and Bresler)
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10th April |
- Ordering based method for tomography under unknown angles, assuming known distribution of the unknown angles - nearest neighbor algorithm (due to Basu and Bresler)
- Comparison between ordering-based and moment based methods
- Laplacian eigenmaps for dimensionality reduction, with toy examples
- Application of Laplacian Eigenmaps for tomography under unknown angles
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13th April |
- PCA-based denoising for tomography from noisy measurements under unknown angles
Compressive classification
- Classification from compressive measurements - maximum likelihood classifier, generalized maximum likelihood classifier, matched filter, smashed filter
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- Slides (check moodle)
- Slides for compressive classification (check moodle)
- HW5 out
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