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Inverse Problems and Imaging

May 2016 , Volume 10 , Issue 2

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Restoration of manifold-valued images by half-quadratic minimization
Ronny Bergmann, Raymond H. Chan, Ralf Hielscher, Johannes Persch and Gabriele Steidl
2016, 10(2): 281-304 doi: 10.3934/ipi.2016001 +[Abstract](4961) +[PDF](8903.1KB)
The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the $c$-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group $SO(3)$, and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with $SO(3)$-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.
Forward and backward filtering based on backward stochastic differential equations
Dariusz Borkowski
2016, 10(2): 305-325 doi: 10.3934/ipi.2016002 +[Abstract](3667) +[PDF](1854.3KB)
In this paper we explore the problem of reconstruction of blurred and noisy images. The idea presented here provides a new methodology based on advanced tools of stochastic analysis which can be successfully used to solve the inverse problem. In order to solve this problem we use backward stochastic differential equations. The reconstructed image is characterized by smoothing noisy pixels and at the same time enhancing and sharpening edges. Our experiments show that the new approach gives very good results and compares favourably with deterministic partial differential equation methods.
On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives
Fabien Caubet, Carlos Conca and Matías Godoy
2016, 10(2): 327-367 doi: 10.3934/ipi.2016003 +[Abstract](3591) +[PDF](3300.7KB)
We consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain $\Omega$ from partial boundary measurements in the two dimensional case. The fluid flow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape.
On a transmission eigenvalue problem for a spherically stratified coated dielectric
David Colton and Yuk-J. Leung
2016, 10(2): 369-378 doi: 10.3934/ipi.2016004 +[Abstract](2477) +[PDF](355.9KB)
Suppose that the boundary of the unit ball in $R^3$ is coated with a very thin layer of a highly conductive material and the refractive index $n(x)$ inside the ball is spherically stratified. We show that in this case the set of transmission eigenvalues behave quite differently than in the previous studied case of an uncoated ball. In particular, if the index of refraction varies smoothly across the boundary of the unit ball we show that complex eigenvalues always exist and accumulate on the real axis and that the real and complex eigenvalues uniquely determine the index of refraction without any restriction on its magnitude.
Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems
Jérémi Dardé
2016, 10(2): 379-407 doi: 10.3934/ipi.2016005 +[Abstract](3113) +[PDF](1265.6KB)
We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method proves to be efficient even with highly corrupted data.
Ghost imaging in the random paraxial regime
Josselin Garnier
2016, 10(2): 409-432 doi: 10.3934/ipi.2016006 +[Abstract](2445) +[PDF](437.5KB)
In this paper we analyze a wave-based imaging modality called ghost imaging that can produce an image of an object illuminated by a partially coherent source. The image of the object is obtained by correlating the intensities measured by two detectors, one that does not view the object and another one that does view the object. More exactly, a high-resolution detector measures the intensity of a wave field emitted by a partially coherent source which has not interacted with the object to be imaged. A bucket (or single-pixel) detector collects the total (spatially-integrated) intensity of the wave field emitted by the same source that has interacted with the object. The correlation of the intensity measured at the high-resolution detector with the intensity measured by the bucket detector gives an image of the object. In this paper we analyze this imaging modality when the medium through which the waves propagate is random. We discuss the relation with time reversal focusing and with correlation-based imaging using ambient noise sources. We clarify the role of the partial coherence of the source and we study how scattering affects the resolution properties of the ghost imaging function in the paraxial regime: the image resolution is all the better as the source is less coherent, and all the worse as the medium is more scattering.
Efficient tensor tomography in fan-beam coordinates
François Monard
2016, 10(2): 433-459 doi: 10.3934/ipi.2016007 +[Abstract](2802) +[PDF](1394.4KB)
We propose a thorough analysis of the tensor tomography problem on the Euclidean unit disk parameterized in fan-beam coordinates. This includes, for the inversion of the Radon transform over functions, using another range characterization first appearing in [32] to enforce in a fast way classical moment conditions at all orders. When considering direction-dependent integrands (e.g., tensors), a problem where injectivity no longer holds, we propose a suitable representative (other than the traditionally sought-after solenoidal candidate) to be reconstructed, as well as an efficient procedure to do so. Numerical examples illustrating the method are provided at the end.
Color image processing by vectorial total variation with gradient channels coupling
Juan C. Moreno, V. B. Surya Prasath and João C. Neves
2016, 10(2): 461-497 doi: 10.3934/ipi.2016008 +[Abstract](4758) +[PDF](13407.7KB)
We study a regularization method for color images based on the vectorial total variation approach along with channel coupling for color image processing, which facilitates the modeling of inter channel relations in multidimensional image data. We focus on penalizing channel gradient magnitude similarities by using $L^{2}$ differences, which allow us to explicitly couple all the channels along with a vectorial total variation regularization for edge preserving smoothing of multichannel images. By using matched gradients to align edges from different channels we obtain multichannel edge preserving smoothing and decomposition. A detailed mathematical analysis of the vectorial total variation with penalized gradient channels coupling is provided. We characterize some important properties of the minimizers of the model as well as provide geometrical results regarding the regularization parameter. We are interested in applying our model to color image processing and in particular to denoising and decomposition. A fast global minimization based on the dual formulation of the total variation is used and convergence of the iterative scheme is provided. Extensive experiments are given to show that our approach obtains good decomposition and denoising results in natural images. Comparison with previous color image decomposition and denoising methods demonstrate the advantages of our approach.
A variational approach to edge detection
Monika Muszkieta
2016, 10(2): 499-517 doi: 10.3934/ipi.2016009 +[Abstract](3237) +[PDF](1052.8KB)
In this paper, using the variational framework and elements of topological asymptotic analysis, we derive an algorithm for edge detection in a digital image in which an optimal value for the threshold is computed automatically. In order to examine this algorithm, we perform a simple experiment on synthetic images composed of two objects with different values of intensity and size. In this case, we are be able to find an exact condition which has to be satisfied so that an edge of object with lower contrast would be detected. At the end, we compare results of numerical experiments obtained by application of our algorithm and the algorithm proposed by Desolneux et al. [7,8]. We indicate some similarities between these two approaches to edge detection and discuss their differences.
The factorization method for the Drude-Born-Fedorov model for periodic chiral structures
Dinh-Liem Nguyen
2016, 10(2): 519-547 doi: 10.3934/ipi.2016010 +[Abstract](2886) +[PDF](1502.1KB)
We consider the electromagnetic inverse scattering problem for the Drude-Born-Fedorov model for periodic chiral structures known as chiral gratings both in $\mathbb{R}^2$ and $\mathbb{R}^3$. The Factorization method is studied as an analytical as well as a numerical tool for solving this inverse problem. The method constructs a simple criterion for characterizing shape of the periodic scatterer which leads to a fast imaging algorithm. This criterion is necessary and sufficient which gives a uniqueness result in shape reconstruction of the scatterer. The required data consists of certain components of Rayleigh sequences of (measured) scattered fields caused by plane incident electromagnetic waves. We propose in this electromagnetic plane wave setting a rigorous analysis for the Factorization method. Numerical examples in two and three dimensions are also presented for showing the efficiency of the method.
The relationship between backprojection and best linear unbiased estimation in synthetic-aperture radar imaging
Kaitlyn Muller
2016, 10(2): 549-561 doi: 10.3934/ipi.2016011 +[Abstract](2234) +[PDF](349.1KB)
In this paper we investigate the relationship between two different techniques typically used in imaging and estimation problems. We focus on synthetic-aperture radar imaging and compare the methods of backprojection (standard for imaging) and best linear unbiased estimation (BLUE). We aim to reconstruct or estimate the reflectivity function of an object present in a scene of interest. We find that the estimate of the reflectivity (calculated using BLUE) and the reconstructed image (calculated using filtered backprojection) are the same when we utilize a criterion from microlocal analysis to define the optimal backprojection filter and assume the measured data is corrupted by zero-mean independently identically distributed (white) noise. In particular we show that the microlocal criterion for the optimal backprojection filter is equivalent to the unbiased constraint present in the BLUE technique.
A fast patch-dictionary method for whole image recovery
Yangyang Xu and Wotao Yin
2016, 10(2): 563-583 doi: 10.3934/ipi.2016012 +[Abstract](3289) +[PDF](1580.2KB)
Many dictionary based methods in image processing use dictionary to represent all the patches of an image. We address the open issue of modeling an image by its overlapping patches: due to overlapping, there are a large number of patches, and to recover these patches, one must determine an excessive number of their dictionary coefficients. With very few exceptions, this issue has limited the applications of image-patch methods to the ``local'' tasks such as denoising, inpainting, cartoon-texture decomposition, super-resolution, and image deblurring, where one can process a few patches at a time. Our focus is the global imaging tasks such as compressive sensing and medical image recovery, where the whole image is encoded together in each measurement, making it either impossible or very ineffective to update a few patches at a time.
    Our strategy is to divide the sparse recovery into multiple subproblems, each of which handles a subset of non-overlapping patches, and then the results of the subproblems are averaged to yield the final recovery. This simple strategy is surprisingly effective in terms of both quality and speed.
    In addition, we accelerate computation of the learned dictionary by applying a recent block proximal-gradient method, which not only has a lower per-iteration complexity but also takes fewer iterations to converge, compared to the current state-of-the-art. We also establish that our algorithm globally converges to a stationary point. Numerical results on synthetic data demonstrate that our algorithm can recover a more faithful dictionary than two state-of-the-art methods.
    Combining our image-recovery and dictionary-learning methods, we numerically simulate image inpainting, compressive sensing recovery, and deblurring. Our recovery is more faithful than those by a total variation method and a method based on overlapping patches. Our Matlab code is competitive in terms of both speed and quality.

2020 Impact Factor: 1.639
5 Year Impact Factor: 1.720
2020 CiteScore: 2.6




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