65 research outputs found
Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices
We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges
Multilinear Subspace Clustering
In this paper we present a new model and an algorithm for unsupervised
clustering of 2-D data such as images. We assume that the data comes from a
union of multilinear subspaces (UOMS) model, which is a specific structured
case of the much studied union of subspaces (UOS) model. For segmentation under
this model, we develop Multilinear Subspace Clustering (MSC) algorithm and
evaluate its performance on the YaleB and Olivietti image data sets. We show
that MSC is highly competitive with existing algorithms employing the UOS model
in terms of clustering performance while enjoying improvement in computational
complexity
A Tensor-Based Dictionary Learning Approach to Tomographic Image Reconstruction
We consider tomographic reconstruction using priors in the form of a
dictionary learned from training images. The reconstruction has two stages:
first we construct a tensor dictionary prior from our training data, and then
we pose the reconstruction problem in terms of recovering the expansion
coefficients in that dictionary. Our approach differs from past approaches in
that a) we use a third-order tensor representation for our images and b) we
recast the reconstruction problem using the tensor formulation. The dictionary
learning problem is presented as a non-negative tensor factorization problem
with sparsity constraints. The reconstruction problem is formulated in a convex
optimization framework by looking for a solution with a sparse representation
in the tensor dictionary. Numerical results show that our tensor formulation
leads to very sparse representations of both the training images and the
reconstructions due to the ability of representing repeated features compactly
in the dictionary.Comment: 29 page
"Plug-and-Play" Edge-Preserving Regularization
In many inverse problems it is essential to use regularization methods that
preserve edges in the reconstructions, and many reconstruction models have been
developed for this task, such as the Total Variation (TV) approach. The
associated algorithms are complex and require a good knowledge of large-scale
optimization algorithms, and they involve certain tolerances that the user must
choose. We present a simpler approach that relies only on standard
computational building blocks in matrix computations, such as orthogonal
transformations, preconditioned iterative solvers, Kronecker products, and the
discrete cosine transform -- hence the term "plug-and-play." We do not attempt
to improve on TV reconstructions, but rather provide an easy-to-use approach to
computing reconstructions with similar properties.Comment: 14 pages, 7 figures, 3 table
- …