In this paper, we propose an image super-resolution approach using a novel generic image prior - gradient profile prior, which is a parametric prior describing the shape and the sharpness of the image gradients. Using the gradient profile prior learned from a large number of natural images, we can provide a constraint on image gradients when we estimate a hi-resolution image from a low-resolution image. With this simple but very effective prior, we are able to produce state-of-the-art results. The reconstructed hi-resolution image is sharp while has rare ringing or jaggy artifacts.

Image super-resolution using gradient profile prior

Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

Systems and methods for implementing array cameras configured to perform super- resolution processing to generate higher resolution super-resolved images using a plurality of captured images and lens stack arrays that can be utilized in array cameras are disclosed. Lens stack arrays in accordance with many embodiments of the invention include lens elements formed on substrates separated by spacers, where the lens elements, substrates and spacers are configured to form a plurality of optical channels, at least one aperture located within each optical channel, at least one spectral filter located within each optical channel, where each spectral filter is configured to pass a specific spectral band of light, and light blocking materials located within the lens stack array to optically isolate the optical channels.

Capturing and processing of images using monolithic camera array with heterogeneous imagers

A critical problem in inversion of geophysical data is developing a stable inverse problem solution that can simultaneously resolve complicated geological structures. The traditional way to obtain a stable solution is based on maximum smoothness criteria. This approach, however, provides smoothed unfocused images of real geoelectrical structures. Recently, a new approach to reconstruction of images has been developed based on a total variational stabilizing functional. However, in geophysical applications it still produces distorted images. In this paper we develop a new technique to solve this problem which we call focusing inversion images. It is based on specially selected stabilizing functionals, called minimum gradient support (MGS) functionals, which minimize the area where strong model parameter variations and discontinuity occur. We demonstrate that the MGS functional, in combination with the penalization function, helps to generate clearer and more focused images for geological structures than conventional maximum smoothness or total variation functionals. The method has been successfully tested on synthetic models and applied to real gravity data.

Focusing geophysical inversion images

SUMMARY We present a novel technique for the determination of resistivity structures associated with arbitrary surface topography. The approach represents a triple-grid inversion technique that is based on unstructured tetrahedral meshes and finite-element forward calculation. The three grids are characterized as follows: A relatively coarse parameter grid defines the elements whose resistivities are to be determined. On the secondary field grid the forward calculations in each inversion step are carried out using a secondary potential (SP) approach. The primary fields are provided by a one-time simulation on the highly refined primary field grid at the beginning of the inversion process. We use a Gauss‐Newton method with inexact line search to fit the data within error bounds. A global regularization scheme using special smoothness constraints is applied. The regularization parameter compromising data misfit and model roughness is determined by an L-curve method and finally evaluated by the discrepancy principle. To solve the inverse subproblem efficiently, a least-squares solver is presented. We apply our technique to synthetic data from a burial mound to demonstrate its effectiveness. A resolution-dependent parametrization helps to keep the inverse problem small to cope with memory limitations of today’s standard PCs. Furthermore, the SP calculation reduces the computation time significantly. This is a crucial issue since the forward calculation is generally very time consuming. Thus, the approach can be applied to large-scale 3-D problems as encountered in practice, which is finally proved on field data. As a by-product of the primary potential calculation we obtain a quantification of the topography effect and the corresponding geometric factors. The latter are used for calculation of apparent resistivities to prevent the reconstruction process from topography induced artefacts.

Three‐dimensional modelling and inversion of dc resistivity data incorporating topography – II. Inversion

Preface. I. Introduction to Inversion Theory. 1. Forward and inverse problems in geophysics. 1.1 Formulation of forward and inverse problems for different geophysical fields. 1.2 Existence and uniqueness of the inverse problem solutions. 1.3 Instability of the inverse problem solution. 2. Ill-posed problems and the methods of their solution. 2.1 Sensitivity and resolution of geophysical methods. 2.2 Formulation of well-posed and ill-posed problems. 2.3 Foundations of regularization methods of inverse problem solution. 2.4 Family of stabilizing functionals. 2.5 Definition of the regularization parameter. II. Methods of the Solution of Inverse Problems. 3. Linear discrete inverse problems. 3.1 Linear least-squares inversion. 3.2 Solution of the purely under determined problem. 3.3 Weighted least-squares method. 3.4 Applying the principles of probability theory to a linear inverse problem. 3.5 Regularization methods. 3.6 The Backus-Gilbert method. 4. Iterative solutions of the linear inverse problem. 4.1 Linear operator equations and their solution by iterative methods. 4.2 A generalized minimal residual method. 4.3 The regularization method in a linear inverse problem solution. 5. Nonlinear inversion technique. 5.1 Gradient-type methods. 5.2 Regularized gradient-type methods in the solution of nonlinear inverse problems. 5.3 Regularized solution of a nonlinear discrete inverse problem. 5.4 Conjugate gradient re-weighted optimization. III. Geopotential Field Inversion. 6. Integral representations in forward modeling of gravity and magnetic fields. 6.1 Basic equations for gravity and magnetic fields. 6.2 Integral representations of potential fields based on the theory of functions of a complex variable. 7. Integral representations in inversion of gravity and magnetic data. 7.1 Gradient methods of gravity inversion. 7.2 Gravity field migration. 7.3 Gradient methods of magnetic anomaly inversion. 7.4 Numerical methods in forward and inverse modeling. IV. Electromagnetic Inversion. 8. Foundations of electromagnetic theory. 8.1 Electromagnetic field equations. 8.2 Electromagnetic energy flow. 8.3 Uniqueness of the solution of electromagnetic field equations. 8.4 Electromagnetic Green's tensors. 9. Integral representations in electromagnetic forward modeling. 9.1 Integral equation method. 9.2 Family of linear and nonlinear integral approximations of the electromagnetic field. 9.3 Linear and non-linear approximations of higher orders. 9.4 Integral representations in numerical dressing. 10. Integral representations in electromagnetic inversion. 10.1 Linear inversion methods. 10.2 Nonlinear inversion. 10.3 Quasi-linear inversion. 10.4 Quasi-analytical inversion. 10.5 Magnetotelluric (MT) data inversion. 11. Electromagnetic migration imaging. 11.1 Electromagnetic migration in the frequency domain. 11.2 Electromagnetic migration in the time domain. 12. Differential methods in electromagnetic modeling and inversion. 12.1 Electromagnetic modeling as a boundary-value problem. 12.2 Finite difference approximation of the boundary-value problem. 12.3 Finite element solution of boundary-value problems. 12.4 Inversion based on differential methods. V. Seismic Inversion. 13. Wavefield equations. 13.1 Basic equations of elastic waves. 13.2 Green's functions for wavefield equations. 13.3 Kirchhoff integral formula and its analogs. 13.4 Uniqueness of the solution of the wavefield equations. 14. Integral representations in wavefield theory. 14.1 Integral equation method in acoustic wavefield analysis. 14.2 Integral approximations of the acoustic wavefield. 14.3 Method of integral equations in vector wavefield analysis. 14.4 Integral approximations of the vector wavefield. 15. Integral representations in wavefield inversion. 15.1 Linear inversion methods. 15.2 Quasi-linear inversion. 15.3 Nonlinear inversion. 15.4 Principles of wavefield migration. 15.5 Elastic field inversion. A. Functional spaces of geophysical models and data. A.1 Euclidean space. A.2 Metric space. A.3 Linear vector spaces. A.4 Hilbert spaces. A.5 Complex Euclidean and Hilbert spaces. A.6 Examples of linear vector spaces. B. Operators in the spaces of models and data. B.1 Operators in functional spaces. B.2 Linear operators. B.3 Inverse operators. B.4 Some approximation problems in the Hilbert spaces of geophysical data. B.5 Gram - Schmidt orthogonalization process. C. Functionals in the spaces of geophysical models. C.1 Functionals and their norms. C.2 Riesz representation theorem. C.3 Functional representation of geophysical data and an inverse problem. D. Linear operators and functionals revisited. D.1 Adjoint operators. D.2 Differentiation of operators and functionals. D.3 Concepts for variational calculus. E. Some formulae and rules from matrix algebra. E.1 Some formulae and rules of operation on matrices. E.2 Eigenvalues and eigenvectors. E.3 Spectral decomposition of a symmetric matrix. E.4 Singular value decomposition (SVD). E.5 The spectral Lanczos decomposition method. F. Some formulae and rules from tensor calculus. F.1 Some formulae and rules of operation on tensor functions. F.2 Tensor statements of the Gauss and Green's formulae. F.3 Green's tensor and vector formulae for Lame and Laplace operators. Bibliography. Index.

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Geophysical Inverse Theory and Regularization Problems

We develop a method of 3‐D magnetic anomaly inversion based on traditional Tikhonov regularization theory. We use a minimum support stabilizing functional to generate a sharp, focused inverse image. An iterative inversion process is constructed in the space of weighted model parameters that accelerates the convergence and robustness of the method. The weighting functions are selected based on sensitivity analysis. To speed up the computations and to decrease the size of memory required, we use a compression technique based on cubic interpolation.Our method is designed for inversion of total magnetic anomalies, assuming the anomalous field is caused by induced magnetization only. The method is applied to synthetic data for typical models of magnetic anomalies and is tested on real airborne data provided by ExxonMobil Upstream Research Company.

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3‐D magnetic inversion with data compression and image focusing

We develop a new method for interpretation of tensor gravity field component data, based on regularized focusing inversion. The focusing inversion makes its possible to reconstruct a sharper image of the geological target than conventional maximum smoothness inversion. This new technique can be efficiently applied for the interpretation of gravity gradiometer data, which are sensitive to local density anomalies. The numerical modeling and inversion results show that the resolution of the gravity method can be improved significantly if we use tensor gravity data for interpretation. We also apply our method for inversion of the gradient gravity data collected by BHP Billiton over the Cannington Ag-Pb-Zn orebody in Queensland, Australia. The comparison with the drilling results demonstrates a remarkable correlation between the density anomaly reconstructed by the gravity gradient data and the true structure of the orebody. This result indicates that the emerging new geophysical technology of the airborne gravity gradient observations can improve significantly the practical effectiveness of the gravity method in mineral exploration.

Three-dimensional regularized focusing inversion of gravity gradient tensor component data

It is often argued that 3D inversion of entire airborne electromagnetic (AEM) surveys is impractical, and that 1D methods provide the only viable option for quantitative interpretation. However, real geological formations are 3D by nature and 3D inversion is required to produce accurate images of the subsurface. To that end, we show that it is practical to invert entire AEM surveys to 3D conductivity models with hundreds of thousands if not millions of elements. The key to solving a 3D AEM inversion problem is the application of a moving footprint approach. We have exploited the fact that the area of the footprint of an AEM system is significantly smaller than the area of an AEM survey, and developed a robust 3D inversion method that uses a moving footprint. Our implementation is based on the 3D integral equation method for computing data and sensitivities, and uses the re-weighted regularised conjugate gradient method for minimising the objective functional. We demonstrate our methodology with the 3D inv...

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Michael S. Zhdanov

Papers

Geophysical Inverse Theory and Regularization Problems

Focusing geophysical inversion images

3‐D magnetic inversion with data compression and image focusing

Three-dimensional regularized focusing inversion of gravity gradient tensor component data

3D inversion of airborne electromagnetic data using a moving footprint