2,119 research outputs found
Fourier-Domain Optimization for Image Processing
Image optimization problems encompass many applications such as spectral
fusion, deblurring, deconvolution, dehazing, matting, reflection removal and
image interpolation, among others. With current image sizes in the order of
megabytes, it is extremely expensive to run conventional algorithms such as
gradient descent, making them unfavorable especially when closed-form solutions
can be derived and computed efficiently. This paper explains in detail the
framework for solving convex image optimization and deconvolution in the
Fourier domain. We begin by explaining the mathematical background and
motivating why the presented setups can be transformed and solved very
efficiently in the Fourier domain. We also show how to practically use these
solutions, by providing the corresponding implementations. The explanations are
aimed at a broad audience with minimal knowledge of convolution and image
optimization. The eager reader can jump to Section 3 for a footprint of how to
solve and implement a sample optimization function, and Section 5 for the more
complex cases
On learning with shift-invariant structures
We describe new results and algorithms for two different, but related,
problems which deal with circulant matrices: learning shift-invariant
components from training data and calculating the shift (or alignment) between
two given signals. In the first instance, we deal with the shift-invariant
dictionary learning problem while the latter bears the name of (compressive)
shift retrieval. We formulate these problems using circulant and convolutional
matrices (including unions of such matrices), define optimization problems that
describe our goals and propose efficient ways to solve them. Based on these
findings, we also show how to learn a wavelet-like dictionary from training
data. We connect our work with various previous results from the literature and
we show the effectiveness of our proposed algorithms using synthetic, ECG
signals and images
Algorithmic Design of Majorizers for Large-Scale Inverse Problems
Iterative majorize-minimize (MM) (also called optimization transfer)
algorithms solve challenging numerical optimization problems by solving a
series of "easier" optimization problems that are constructed to guarantee
monotonic descent of the cost function. Many MM algorithms replace a
computationally expensive Hessian matrix with another more computationally
convenient majorizing matrix. These majorizing matrices are often generated
using various matrix inequalities, and consequently the set of available
majorizers is limited to structures for which these matrix inequalities can be
efficiently applied. In this paper, we present a technique to algorithmically
design matrix majorizers with wide varieties of structures. We use a novel
duality-based approach to avoid the high computational and memory costs of
standard semidefinite programming techniques. We present some preliminary
results for 2D X-ray CT reconstruction that indicate these more exotic
regularizers may significantly accelerate MM algorithms.Comment: 10 pages, 4 figure
ADMM for Block Circulant Model Predictive Control
This paper deals with model predictive control problems for large scale
dynamical systems with cyclic symmetry. Based on the properties of block
circulant matrices, we introduce a complex-valued coordinate transformation
that block diagonalizes and truncates the original finite-horizon optimal
control problem. Using this coordinate transformation, we develop a modified
alternating direction method of multipliers (ADMM) algorithm for general
constrained quadratic programs with block circulant blocks. We test our
modified algorithm in two different simulated examples and show that our
coordinate transformation significantly increases the computation speed
A new sufficient condition for sum-rate tightness in quadratic Gaussian multiterminal source coding
This work considers the quadratic Gaussian multiterminal (MT) source coding
problem and provides a new sufficient condition for the Berger-Tung sum-rate
bound to be tight. The converse proof utilizes a set of virtual remote sources
given which the MT sources are block independent with a maximum block size of
two. The given MT source coding problem is then related to a set of
two-terminal problems with matrix-distortion constraints, for which a new lower
bound on the sum-rate is given. Finally, a convex optimization problem is
formulated and a sufficient condition derived for the optimal BT scheme to
satisfy the subgradient based Karush-Kuhn-Tucker condition. The set of sum-rate
tightness problems defined by our new sufficient condition subsumes all
previously known tight cases, and opens new direction for a more general
partial solution
Joint Inverse Covariances Estimation with Mutual Linear Structure
We consider the problem of joint estimation of structured inverse covariance
matrices. We perform the estimation using groups of measurements with different
covariances of the same unknown structure. Assuming the inverse covariances to
span a low dimensional linear subspace in the space of symmetric matrices, our
aim is to determine this structure. It is then utilized to improve the
estimation of the inverse covariances. We propose a novel optimization
algorithm discovering and exploiting the underlying structure and provide its
efficient implementation. Numerical simulations are presented to illustrate the
performance benefits of the proposed algorithm
Deconvolving Images with Unknown Boundaries Using the Alternating Direction Method of Multipliers
The alternating direction method of multipliers (ADMM) has recently sparked
interest as a flexible and efficient optimization tool for imaging inverse
problems, namely deconvolution and reconstruction under non-smooth convex
regularization. ADMM achieves state-of-the-art speed by adopting a divide and
conquer strategy, wherein a hard problem is split into simpler, efficiently
solvable sub-problems (e.g., using fast Fourier or wavelet transforms, or
simple proximity operators). In deconvolution, one of these sub-problems
involves a matrix inversion (i.e., solving a linear system), which can be done
efficiently (in the discrete Fourier domain) if the observation operator is
circulant, i.e., under periodic boundary conditions. This paper extends
ADMM-based image deconvolution to the more realistic scenario of unknown
boundary, where the observation operator is modeled as the composition of a
convolution (with arbitrary boundary conditions) with a spatial mask that keeps
only pixels that do not depend on the unknown boundary. The proposed approach
also handles, at no extra cost, problems that combine the recovery of missing
pixels (i.e., inpainting) with deconvolution. We show that the resulting
algorithms inherit the convergence guarantees of ADMM and illustrate its
performance on non-periodic deblurring (with and without inpainting of interior
pixels) under total-variation and frame-based regularization.Comment: Submitted to the IEEE Transactions on Image Processing in August 201
TripleSpin - a generic compact paradigm for fast machine learning computations
We present a generic compact computational framework relying on structured
random matrices that can be applied to speed up several machine learning
algorithms with almost no loss of accuracy. The applications include new fast
LSH-based algorithms, efficient kernel computations via random feature maps,
convex optimization algorithms, quantization techniques and many more. Certain
models of the presented paradigm are even more compressible since they apply
only bit matrices. This makes them suitable for deploying on mobile devices.
All our findings come with strong theoretical guarantees. In particular, as a
byproduct of the presented techniques and by using relatively new
Berry-Esseen-type CLT for random vectors, we give the first theoretical
guarantees for one of the most efficient existing LSH algorithms based on the
structured matrix ("Practical
and Optimal LSH for Angular Distance"). These guarantees as well as theoretical
results for other aforementioned applications follow from the same general
theoretical principle that we present in the paper. Our structured family
contains as special cases all previously considered structured schemes,
including the recently introduced -model. Experimental evaluation confirms
the accuracy and efficiency of TripleSpin matrices
Filter-And-Forward Relay Design for MIMO-OFDM Systems
In this paper, the filter-and-forward (FF) relay design for multiple-input
multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM)
systems is considered. Due to the considered MIMO structure, the problem of
joint design of the linear MIMO transceiver at the source and the destination
and the FF relay at the relay is considered. As the design criterion, the
minimization of weighted sum mean-square-error (MSE) is considered first, and
the joint design in this case is approached based on alternating optimization
that iterates between optimal design of the FF relay for a given set of MIMO
precoder and decoder and optimal design of the MIMO precoder and decoder for a
given FF relay filter. Next, the joint design problem for rate maximization is
considered based on the obtained result regarding weighted sum MSE and the
existing result regarding the relationship between weighted MSE minimization
and rate maximization. Numerical results show the effectiveness of the proposed
FF relay design and significant performance improvement by FF relays over
widely-considered simple AF relays for MIMO-ODFM systems.Comment: 29 pages, 6 figure
On Input Design for Regularized LTI System Identification: Power-constrained Input
Input design is an important issue for classical system identification
methods but has not been investigated for the kernel-based regularization
method (KRM) until very recently. In this paper, we consider in the time domain
the input design problem of KRMs for LTI system identification. Different from
the recent result, we adopt a Bayesian perspective and in particular make use
of scalar measures (e.g., the -optimality, -optimality, and
-optimality) of the Bayesian mean square error matrix as the design criteria
subject to power-constraint on the input. Instead to solve the optimization
problem directly, we propose a two-step procedure. In the first step, by making
suitable assumptions on the unknown input, we construct a quadratic map
(transformation) of the input such that the transformed input design problems
are convex, the number of optimization variables is independent of the number
of input data, and their global minima can be found efficiently by applying
well-developed convex optimization software packages. In the second step, we
derive the expression of the optimal input based on the global minima found in
the first step by solving the inverse image of the quadratic map. In addition,
we derive analytic results for some special types of fixed kernels, which
provide insights on the input design and also its dependence on the kernel
structure
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