159 research outputs found
Randomized Extended Kaczmarz for Solving Least-Squares
We present a randomized iterative algorithm that exponentially converges in
expectation to the minimum Euclidean norm least squares solution of a given
linear system of equations. The expected number of arithmetic operations
required to obtain an estimate of given accuracy is proportional to the square
condition number of the system multiplied by the number of non-zeros entries of
the input matrix. The proposed algorithm is an extension of the randomized
Kaczmarz method that was analyzed by Strohmer and Vershynin.Comment: 19 Pages, 5 figures; code is available at
https://github.com/zouzias/RE
Sparse Randomized Kaczmarz for Support Recovery of Jointly Sparse Corrupted Multiple Measurement Vectors
While single measurement vector (SMV) models have been widely studied in
signal processing, there is a surging interest in addressing the multiple
measurement vectors (MMV) problem. In the MMV setting, more than one
measurement vector is available and the multiple signals to be recovered share
some commonalities such as a common support. Applications in which MMV is a
naturally occurring phenomenon include online streaming, medical imaging, and
video recovery. This work presents a stochastic iterative algorithm for the
support recovery of jointly sparse corrupted MMV. We present a variant of the
Sparse Randomized Kaczmarz algorithm for corrupted MMV and compare our proposed
method with an existing Kaczmarz type algorithm for MMV problems. We also
showcase the usefulness of our approach in the online (streaming) setting and
provide empirical evidence that suggests the robustness of the proposed method
to the distribution of the corruption and the number of corruptions occurring.Comment: 13 pages, 6 figure
Preasymptotic Convergence of Randomized Kaczmarz Method
Kaczmarz method is one popular iterative method for solving inverse problems,
especially in computed tomography. Recently, it was established that a
randomized version of the method enjoys an exponential convergence for
well-posed problems, and the convergence rate is determined by a variant of the
condition number. In this work, we analyze the preasymptotic convergence
behavior of the randomized Kaczmarz method, and show that the low-frequency
error (with respect to the right singular vectors) decays faster during first
iterations than the high-frequency error. Under the assumption that the inverse
solution is smooth (e.g., sourcewise representation), the result explains the
fast empirical convergence behavior, thereby shedding new insights into the
excellent performance of the randomized Kaczmarz method in practice. Further,
we propose a simple strategy to stabilize the asymptotic convergence of the
iteration by means of variance reduction. We provide extensive numerical
experiments to confirm the analysis and to elucidate the behavior of the
algorithms.Comment: 20 page
Unraveling Nanostructured Spin Textures in Bulk Magnets
One of the key challenges in magnetism remains the determination of the
nanoscopic magnetization profile within the volume of thick samples, such as
permanent ferromagnets. Thanks to the large penetration depth of neutrons,
magnetic small-angle neutron scattering (SANS) is a powerful technique to
characterize bulk samples. The major challenge regarding magnetic SANS is
accessing the real-space magnetization vector field from the reciprocal
scattering data. In this letter, a fast iterative algorithm is introduced that
allows one to extract the underlying two-dimensional magnetic correlation
functions from the scattering patterns. This approach is used here to analyze
the magnetic microstructure of Nanoperm, a nanocrystalline alloy which is
widely used in power electronics due to its extraordinary soft magnetic
properties. It can be shown that the computed correlation functions clearly
reflect the projection of the three-dimensional magnetization vector field onto
the detector plane, which demonstrates that the used methodology can be applied
to probe directly spin-textures within bulk samples with nanometer-resolution.Comment: 9 pages, 3 figure
Real-time Image Generation for Compressive Light Field Displays
With the invention of integral imaging and parallax barriers in the beginning of the 20th century, glasses-free 3D displays have become feasible. Only today—more than a century later—glasses-free 3D displays are finally emerging in the consumer market. The technologies being employed in current-generation devices, however, are fundamentally the same as what was invented 100 years ago. With rapid advances in optical fabrication, digital processing power, and computational perception, a new generation of display technology is emerging: compressive displays exploring the co-design of optical elements and computational processing while taking particular characteristics of the human visual system into account. In this paper, we discuss real-time implementation strategies for emerging compressive light field displays. We consider displays composed of multiple stacked layers of light-attenuating or polarization-rotating layers, such as LCDs. The involved image generation requires iterative tomographic image synthesis. We demonstrate that, for the case of light field display, computed tomographic light field synthesis maps well to operations included in the standard graphics pipeline, facilitating efficient GPU-based implementations with real-time framerates.United States. Defense Advanced Research Projects Agency. Soldier Centric Imaging via Computational CamerasNational Science Foundation (U.S.) (Grant IIS-1116452)United States. Defense Advanced Research Projects Agency. Maximally scalable Optical Sensor Array Imaging with Computation ProgramAlfred P. Sloan Foundation (Research Fellowship)United States. Defense Advanced Research Projects Agency (Young Faculty Award
Randomized Kaczmarz solver for noisy linear systems
The Kaczmarz method is an iterative algorithm for solving systems of linear
equations Ax=b. Theoretical convergence rates for this algorithm were largely
unknown until recently when work was done on a randomized version of the
algorithm. It was proved that for overdetermined systems, the randomized
Kaczmarz method converges with expected exponential rate, independent of the
number of equations in the system. Here we analyze the case where the system
Ax=b is corrupted by noise, so we consider the system where Ax is approximately
b + r where r is an arbitrary error vector. We prove that in this noisy
version, the randomized method reaches an error threshold dependent on the
matrix A with the same rate as in the error-free case. We provide examples
showing our results are sharp in the general context
Binary Tomography Reconstruction by Particle Aggregation
This paper presents a novel reconstruction algorithm for bi- nary tomography based on the movement of particles. Particle Aggregate Reconstruction Technique (PART) supposes that pixel values are particles, and that the particles can diffuse through the image, sticking together in regions of uniform pixel value known as aggregates. The algorithm is tested on four phantoms of varying sizes and numbers of forward projections and compared to a random search algorithm and to SART, a standard algebraic reconstruction method. PART, in this small study, is shown to be capable of zero error reconstruction and compares favourably with SART and random search
Reduced projection angles for binary tomography with particle aggregation
This paper extends particle aggregate reconstruction technique (PART), a reconstruction algorithm for binary tomography based on the movement of particles. PART supposes that pixel values are particles, and that particles diffuse through the image, staying together in regions of uniform pixel value known as aggregates. In this work, a variation of this algorithm is proposed and a focus is placed on reducing the number of projections and whether this impacts the reconstruction of images. The algorithm is tested on three phantoms of varying sizes and numbers of forward projections and compared to filtered back projection, a random search algorithm and to SART, a standard algebraic reconstruction method. It is shown that the proposed algorithm outperforms the aforementioned algorithms on small numbers of projections. This potentially makes the algorithm attractive in scenarios where collecting less projection data are inevitable
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