Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures

2015 ◽  
Vol 23 (2) ◽  
Author(s):  
Michael V. Klibanov ◽  
Vladimir G. Romanov
Keyword(s):  
Born Approximation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattered Wave ◽  
Scattering Data ◽  
Unknown Coefficient ◽  
X Rays ◽  
Reconstruction Formula ◽  
Nano Structures ◽  
Complex Valued

AbstractImaging of nano structures is necessary for the quality control in their manufacturing. In the case when X-rays probe the medium, only the modulus of the complex valued scattered wave field can be measured. The phase cannot be measured. In the case of the Born approximation, we obtain an explicit reconstruction formula for the unknown coefficient from the phaseless scattering data.

scholarly journals On the Uniqueness and Reconstruction of Rough and Complex Obstacles from Acoustic Scattering Data

2011 ◽  
Vol 11 (1) ◽  
pp. 83-104 ◽  
Author(s):  
Mourad Sini
Keyword(s):  
Boundary Conditions ◽  
Surface Impedance ◽  
Scattering Problem ◽  
Acoustic Scattering ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
Singular Solutions ◽  
Regularity Conditions ◽  
Unique Continuation Property ◽  
Complex Valued

Abstract We deal with the inverse scattering problem by an obstacle at a fixed frequency. The obstacle is characterized by its shape, the type of boundary conditions on its surface and the eventual coefficients distributed on this surface. In this paper, we assume that the surface ∂D of the obstacle D is Lipschitz and the surface impedance, λ, is given by a complex valued, measurable and bounded function. We prove uniqueness of (∂D,λ) from the far field map under these regularity conditions. The usual proof of uniqueness for obstacles, based on the use of singular solutions, is divided into two steps. The first one consists of the use of Rellich type lemma to go from the far fields to the near fields and then use the singularities of the singular solutions, via orthogonality relations, to show uniqueness of ∂D. The second step is to use the boundary conditions to prove uniqueness of λ on ∂D via the unique continuation property. This last step requires the surface impedance to be continuous. We propose an approach using layer potentials to transform the inverse problem to the invertibility of integral equations of second kind involving the unknowns ∂D and λ. This enables us to weaken the required regularity conditions by assuming ∂D to be Lipschitz and λ to be only bounded. The procedure of the proof is reconstructive and provides a method to compute the complex valued and bounded surface impedance λ on ∂D by inverting an invertible integral equation. In addition, assuming ∂D to be C^2 regular and λ to be of class C^{0,α}, with α>0, we give a direct and stable formula as another method to reconstruct the surface impedance on ∂D.

TWO VERSIONS OF QUASI-NEWTON METHOD FOR MULTIDIMENSIONAL INVERSE SCATTERING PROBLEM

1993 ◽  
Vol 01 (02) ◽  
pp. 197-228 ◽  
Author(s):  
SEMION GUTMAN ◽  
MICHAEL KLIBANOV
Keyword(s):  
Newton Method ◽  
Inverse Scattering ◽  
Born Approximation ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Circular Cylinders ◽  
Iterative Sequence ◽  
Scattering Field ◽  
Quasi Newton

Suppose that a medium with slowly changing spatial properties is enclosed in a bounded 3-dimensional domain and is subjected to a scattering by plane waves of a fixed frequency. Let measurements of the wave scattering field induced by this medium be available in the region outside of this domain. We study how to extract the properties of the medium from the information contained in the measurements. We are concerned with the weak scattering case of the above inverse scattering problem (ISP), that is, the unknown. spatial variations of the medium are assumed to be close to a constant. Examples can be found in the studies of the wave propagation in oceans, in the atmosphere, and in some biological media. Since the problems are nonlinear, the methods for their linearization (the Born approximation) have been developed. However, such an approach often does not produce good results. In our method, the Born approximation is just the first iteration and further iterations improve the identification by an order of magnitude. The iterative sequence is defined in the framework of a Quasi-Newton method. Using the measurements of the scattering field from a carefully chosen set of directions we are able to recover (finitely many) Fourier coefficients of the sought parameters of the model. Numerical experiments for the scattering from coaxial circular cylinders as well as for simulated data are presented.

Approximate solution of nonlinear double-scattering inversion for true amplitude imaging

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. R43-R55 ◽  
Author(s):  
Wei Ouyang ◽  
Weijian Mao ◽  
Xuelei Li
Keyword(s):  
Scattered Field ◽  
Born Approximation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Single Scattering ◽  
Complex Model ◽  
Earth Model ◽  
Model Parameters ◽  
Double Scattering ◽  
Integral Equation Formulation

Linearized inversion algorithms are the main techniques in seismic imaging that apply the single-scattering (Born) approximation to the scattered field, and therefore, have difficulty handling the strong perturbation of model parameters and nonlinear multiple-scattering effects. To theoretically overcome these drawbacks in the linearization of the inverse scattering problem, we have developed an approach to deal with nonlinear double-scattering inversion. We first used an integral equation formulation associated with the scattered field consisting of single and double scattering in an acoustic earth model based on the second-order Born approximation, and we found that the approximation of the scattered field can be naturally related to the generalized Radon transform (GRT). We then adopted the inverse GRT to obtain the corresponding quadratic approximate inversion solution. Our inversion scheme can appropriately handle the first-order transmission effects from double scattering in a local area, which gives a significant amplitude correction for the inversion results and ultimately results in a more accurate image with true amplitude. We conducted numerical experiments that showed the conventional single-scattering inversion was good in amplitude only for perturbation up to 10% of the background medium, but our approach can work for up to 40% or more. Test results indicated that our inversion scheme can effectively relax the requirement of the weak perturbation of the model parameter in the Born approximation and can deal with the complex model directly. The computational complexity of our new scheme is almost the same as conventional linear scattering inversion processing. Therefore, the cost of our approach is at a similar level to that of linear scattering inversion.

Inverse scattering for three-dimensional quasi-linear biharmonic operator

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Markus Harju ◽  
Jaakko Kultima ◽  
Valery Serov
Keyword(s):  
Inverse Scattering ◽  
Space Variable ◽  
Scattering Amplitude ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Regularity Conditions ◽  
Biharmonic Operator ◽  
Essential Information ◽  
Complex Valued

Abstract We consider an inverse scattering problem of recovering the unknown coefficients of a quasi-linearly perturbed biharmonic operator in the three-dimensional case. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove Saito’s formula and uniqueness theorem of recovering some essential information about the unknown coefficients from the knowledge of the high frequency scattering amplitude.

AN INVERSE SCATTERING PROBLEM FOR NONLOCAL POTENTIALS I: THE FAMILY OF PHASE EQUIVALENT WAVEFUNCTIONS

1986 ◽  
Vol 01 (07) ◽  
pp. 449-454 ◽  
Author(s):  
V.M. MUZAFAROV
Keyword(s):  
Integral Equation ◽  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
The Family ◽  
Consistent Approach ◽  
Nonlocal Potentials

We develop a consistent approach to an inverse scattering problem for the Schrodinger equation with nonlocal potentials. The main result presented in this paper is that for the two-body scattering data, given the problem of reconstructing both the family of phase equivalent two-body wavefunctions and the corresponding family of phase equivalent half-off-shell t-matrices, is reduced to solving a regular integral equation. This equation may be regarded as a generalization of the Gel’fand-Levitan equation.

scholarly journals A robust expectation-maximization method for the interpretation of small-angle scattering data from dense nanoparticle samples

2019 ◽  
Vol 52 (5) ◽  
pp. 926-936
Author(s):  
M. Bakry ◽  
H. Haddar ◽  
O. Bunău
Keyword(s):  
Small Angle ◽  
Expectation Maximization ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Small Angle Scattering ◽  
Fine Tuning ◽  
Scattering Data ◽  
Model Parameters ◽  
Dry Powders ◽  
Angle Scattering

The local monodisperse approximation (LMA) is a two-parameter model commonly employed for the retrieval of size distributions from the small-angle scattering (SAS) patterns obtained from dense nanoparticle samples (e.g. dry powders and concentrated solutions). This work features a novel implementation of the LMA model resolution for the inverse scattering problem. The method is based on the expectation-maximization iterative algorithm and is free of any fine-tuning of model parameters. The application of this method to SAS data acquired under laboratory conditions from dense nanoparticle samples is shown to provide good results.

scholarly journals Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 65 ◽  
Author(s):  
Eugen Mircea Anitas
Keyword(s):  
Small Angle ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Theoretical Models ◽  
Small Angle Scattering ◽  
X Rays ◽  
Fractal Structures ◽  
Practical Applications ◽  
Additional Information ◽  
Angle Scattering

Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific theoretical models which exploit the fractal symmetry have been developed to extract additional information from SAS data. Although this problem can be solved exactly for many particular fractal structures, due to the intrinsic limitations of the SAS method, the inverse scattering problem, i.e., determination of the fractal structure from the intensity curve, is ill-posed. However, fractals can be divided into various classes, not necessarily disjointed, with the most common being random, deterministic, mass, surface, pore, fat and multifractals. Each class has its own imprint on the scattering intensity, and although one cannot uniquely identify the structure of a fractal based solely on SAS data, one can differentiate between various classes to which they belong. This has important practical applications in correlating their structural properties with physical ones. The article reviews SAS from several fractal models with an emphasis on describing which information can be extracted from each class, and how this can be performed experimentally. To illustrate this procedure and to validate the theoretical models, numerical simulations based on Monte Carlo methods are performed.

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