approximate linearized inversion by optimal scaling of prestack depth migration linearized inversion provides one possible sense of image amplitude correctness an image or bandlimited model perturbation has correct amplitudes if it is an approximate inversion that is if linearized modeling demigration with the image as input reproduces the data approximately the theory of linearized acoustic inverse scattering with slowly varying background or macromodel shows that an approximate inversion may be recovered from the output of prestack depth migration by a combination of scaling and filtering the necessary filter is completely specified by the theory and the scale factor may be estimated via filtering linearized modeling a second migration and the solution of a small auxiliary inverse problem

on asymptotic volume of finsler tori minimal surfaces in normed spaces and symplectic filling volume the main unconditional result of this paper theorem 3 states that every two dimensional affine disc in a normed space that is a disc contained in a two dimensional affine subspace is an area minimizing surface among all immersed discs with the same boundary with respect to the symplectic holmes thompson surface area to emphasize that this is not at all obvious it may be worth mentioning that a similar statement with rational chains in place of immersed discs is incorrect theorem 2 and that it is not known for surfaces that may not be topological discs the result still may not sound too exciting to the reader who never looked at the problem before even though it goes back to busemann s works in the 50 s see bes th and references there and the proof heavily relies on asymptotic geometry of tori we believe that it is more important that we embed this problem into a whole area of mostly open problems as well as give some partial results and suggest certain directions of how to attack them we begin with a trivial statement

extending lipschitz functions via random metric partitions many classical problems in geometry and analysis involve the gluing together of local information to produce a coherent global picture inevitably the difficulty of such a procedure lies at the local boundary where overlapping views of the same locality must somehow be merged it is therefore desirable that the boundaries be smooth allowing a graceful transition from one viewpoint to the next for instance one may point to whitney s use of partitions of unity in studying what is now known as the whitney extension problem 36 37 in the present work we consider what is perhaps the most basic whitney type extension problem that of extending a lipschitz function so that it remains lipschitz often such a map is extended by first producing a cover of the new domain extending the mapping locally and then gluing together the individual pieces our main observation is that in many cases if one chooses a random cover from the right distribution the boundary can be made smooth on average even when the local maps are individually quite coarse this insight leads to the unification generalization and improvement of many known results as well as to new results for many interesting spaces

amplitude and kinematic corrections of migrated images for nonunitary imaging operators obtaining migrated images with meaningful amplitudes is a challenging problem when the migration operator is not unitary one possible solution to this problem is iterative inversion however inversion is an expensive process that can be rather difficult to apply especially with 3d data in this paper i propose estimating migrated images similar to the least squares inverse images by approximating the inverse hessian thus avoiding the need for iterative inversion the inverse hessian is approximated with a bank of nonstationary matching filters these filters are not exact impulse responses and are limited in their ability to mimic the full effects of least squares inversion tests on two data sets show that this filtering approach gives results similar to iterative least squares inversion at a lower cost this technique is flexible enough to be applied to images migrated from zero offset or angle domain common image point gathers

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

splitting of operators alternate directions and paraxial approximations for the three dimensional wave equation we design a family of new three dimensional 3 d paraxial equations well adapted to numerical solution by the alternate directions method these equations do not suffer from bad anisotropic effects and the cost for their numerical integration remains cheap in comparison with classical paraxial equations

geometric information criterion for model selection in building a 3 d model of the environment from image and sensor data one must fit to the data an appropriate class of models which can be regarded as a parametrized manifold or geometric model defined in the data space in this paper we present a statistical framework for detecting degeneracies of a geometric model by evaluating its predictive capability in terms of the expected residual and derive the geometric aic we show that it allows us to detect singularities in a structure from motion analysis without introducing any empirically adjustable thresholds we illustrate our approach by simulation examples we also discuss the application potential of this theory for a wide range of computer vision and robotics problems

migration of seismic data by phase shift plus interpolation under the horizontally layered velocity assumption migration is defined by a set of independent ordinary differential equations in the wavenumber frequency domain the wave components are extrapolated downward by rotating their phases this paper shows that one can generalize the concepts of the phase shift method to media having lateral velocity variations the wave extrapolation procedure consists of two steps in the first step the wave field is extrapolated by the phase shift method using l laterally uniform velocity fields the intermediate result is l reference wave fields in the second step the actual wave field is computed by interpolation from the reference wave fields the phase shift plus interpolation pspi method is unconditionally stable and lends itself conveniently to migration of three dimensional data the performance of the methods is demonstrated on synthetic examples the pspi migration results are then compared with those obtained from a finite difference method

area minimizing discs in metric spaces we solve the classical problem of plateau in the setting of proper metric spaces precisely we prove that among all disc type surfaces with prescribed jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi conformal parametrization if the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally holder continuous in the interior and continuous up to the boundary our results generalize corresponding results of douglas rado and morrey from the setting of euclidean space and riemannian manifolds to that of proper metric spaces

the kantorovich metric for probability measures on the circle in this paper we show that there exists an analytic expression for the kantorovich distance between probability measures on the circle previously such an expression was only known for measures supported on the real line in the case that the measures are discrete this formula enables us to show that the kantorovich distance can be computed in linear time this is important for applications in particular in pattern recognition where this distance is used for texture analysis as another application we see that the analytic expression found allows us to solve a minimal matching problem in linear time for which so far only n log n algorithms were known

fast least squares migration with a deblurring filter least squares migration lsm is a linearized waveform inversion for estimating a subsurface reflectivity model that relative to conventional migration improves spatial resolution of migration images the cost however is high because lsm typically requires 10 or more iterations which is at least 20 times more than the cpu cost of conventional migration to alleviate this expense we offer a deblurring filter that can be used in a regularization scheme or a preconditioning scheme to give acceptable lsm images with less than one third the cost of the standard lsm method our results in applying deblurred lsm to synthetic data and field data support this claim in particular a marmousi2 model test shows that the data residual for preconditioned deblurred lsm decreases rapidly in the first iteration which is equivalent to 10 or more iterations of lsm empirical results suggest that regularized deblurred lsm after three iterations is equivalent to about 10 iterations of lsm applying deblurred lsm to 2d marine data gives a higher resolution image compared to those from migration or lsm with three iterations these results suggest that lsm combined with a deblurring filter allows lsm to be a fast practical tool for improved imaging of complicated structures

least squares wave equation migration for avp ava inversion we present an acoustic migration inversion algorithm that uses extended double square root wave equation migration and modeling operators to minimize a constrained least squares data misfit function least squares migration we employ an imaging principle that allows for the extraction of ray parameter domain common image gathers cigs from the propagated seismic wavefield the cigs exhibit amplitude variations as a function of half offset ray parameter avp closely related to the amplitude variation with reflection angle ava our least squares wave equation migration inversion is constrained by a smoothing regularization along the ray parameter this approach is based on the idea that rapid amplitude changes or discontinuities along the ray parameter axis result from noise incomplete wavefield sampling and numerical operator artifacts these discontinuities should therefore be penalized in the inversion the performance of the proposed algorithm is examined with two synthetic examples in the first

f x adaptive seismic trace interpolation we use exponentially weighted recursive least squares to estimate adaptive prediction filters for frequency space f x seismic interpolation adaptive prediction filters can model signals where the dominant wavenumbers vary in space this concept leads to an f x interpolation method that does not require windowing strategies for optimal results in other words adaptive prediction filters can be used to interpolate waveforms that have spatially variant dips the interpolation method s performance depends on two parameters filter length and forgetting factor we pay particular attention to selection of the forgetting factor because it controls the algorithm s adaptability to changes in local dip finally we use synthetic and real data examples to illustrate the performance of the proposed adaptive f x interpolation method

an optimal true amplitude least squares prestack depth migration operator in order to define an optimal true amplitude prestack depth migration for multishot and multitrace data we develop a general methodology based on the least squares data misfit function associated with a forward model the amplitude of the migrated events are restored at best for any given geometry and any given preliminary filtering and amplitude correction of the data the migrated section is then the gradient of the cost function multiplied by a weight matrix a study of the hessian associated with this data misfit shows how efficiently to find a good weight matrix via the computation of only few elements of this hessian thanks to this matrix the resulting migration operator is optimal in the sense that it ensures the best possible restoration of the amplitudes among the large class of least squares migrations applied to a forward model based on born ray tracing and diffracting points approximation this optimal migration outperforms or at least equals the classic kirchhoff formula since the latter belongs to the class of least squares migrations and is only optimal for one shot and an infinite aperture numerical results illustrate this construction and confirm the above expectations

derivation of forward and adjoint operators for least squares shot profile split step migration the forward and adjoint operators for shot profile least squares migration are derived the forward operator is demigration and the adjoint operator is migration the demigration operator is derived from the born approximation the process begins with a green s function that allows for a laterally varying migration velocity model using the split step approximation next the earth is divided into horizontal layers and within each layer the migration velocity model is made to be constant with respect to depth for a given layer 1 the source side wavefield is propagated down to its top using the background wavefield this gives a background wavefield incident at the layer s upper boundary 2 the layer s contribution to the scattered wavefield is computed using the born approximation to the scattered wavefield and the background wavefield 3 next its scattered wavefield is propagated back up to the measurement surface using again the background wavefield the measured wavefield is approximated by the sum of scattered wavefields from each layer in the derivation of the measured wavefield the shot profile migration geometry is used for each shot the resulting wavefield modeling operator takes the form of a fredholm integral equation of the first kind and this is used to write down its adjoint the shot profile migration operator this forward adjoint pair is used for shot profile least squares migration shot profile least squares migration is illustrated with two synthetic examples the first uses data collected over a four layer acoustic model and the second uses data from the sigsbee 2a model

regularizing seismic inverse problems by model reparameterization using plane wave construction we define plane wave construction pwc an operator for generating data aligned along predefined locally variable slopes as the inverse of plane wave destruction an operator used for measuring the slopes pwc can be applied for efficient regularization of seismic estimation problems using simple examples we demonstrate how pwc enhances the coherency of seismic images improves velocity estimation methods and separates primaries and multiples with a pattern based approach

fast recursive least squares adaptive filtering by fast fourier transform based conjugate gradient iterations recursive least squares rls estimations are used extensively in many signal processing and control applications in this paper we consider rls with sliding data windows involving multiple rank k updating and downdating computations the least squares estimator can be found by solving a near toeplitz matrix system at each step our approach is to employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems here we iterate in the time domain using toeplitz matrix vector multiplications and precondition in the fourier domain so that the fast fourier transform fft is used throughout the computations the circulant preconditioners are derived from the spectral properties of the given input stochastic process when the input stochastic process is stationary we prove that with probability 1 the spectrum of the preconditioned system is clustered around 1 and the method converges superlinearly provided that a sufficient number of data samples are taken i e

block row recursive least squares migration abstractrecursive estimates of large systems of equations in the context of least squares fitting is a common practice in different fields of study for example recursive adaptive filtering is extensively used in signal processing and control applications the necessity of solving least squares problems recursively stems from the need for fast real time signal processing strategies the computational cost of least squares solvers can also limit the applicability of this technique in geophysical problems we have considered a recursive least squares solution for least squares wave equation migration with sliding windows involving several fixed rank downdating and updating computations this technique can be applied for dynamic and stationary processes if we use enough data in each windowed setup the spectrum of the preconditioned system is clustered around one and the method will converge superlinearly with probability one numerical experiments were performed to test the effectiveness of the technique

robust regression methods for computer vision a review regression analysis fitting a model to noisy data is a basic technique in computer vision robust regression methods that remain reliable in the presence of various types of noise are therefore of considerable importance we review several robust estimation techniques and describe in detail the least median of squares lmeds method the method yields the correct result even when half of the data is severely corrupted its efficiency in the presence of gaussian noise can be improved by complementing it with a weighted least squares based procedure the high time complexity of the lmeds algorithm can be reduced by a monte carlo type speed up technique we discuss the relationship of lmeds with the ransac paradigm and its limitations in the presence of noise corrupting all the data and we compare its performance with the class of robust m estimators references to published applications of robust techniques in computer vision are also given

motion from point matches multiple of solutions in this paper we study the multiplicity of solutions of the motion problem given n point matches between two frames how many solutions are there to the motion problem we show that the maximum number of solutions is 10 when 5 point matches are available this settles a question that has been around in the computer vision community for a while we follow two tracks we show that the two approaches are compatible and yield the same result we then describe a computer implementation of the second approach that uses maple a language for symbolic computation the program allows us to compute exactly the solutions for any configuration of 5 points some experiments are described

robust estimation of three dimensional motion parameters from a sequence of image frames using regularization in this paper we look at the issue of accurate estimation of the three dimensional motion parameters of a rigid body from a sequence of synthetic images and relate the effect of some parameters to the shape of an error function we first consider the case where only a small set of corresponding points is identified and suggest that a technique called regularization improves the quality and stability of a solution we then observe that if more pairs of corresponding points are available the error function becomes smooth and the solution stable finally we try to improve the quality of estimation by considering more than two consecutive frames for a moving camera looking at a stationary scene and summing the error functions obtained for any two consecutive frames surprisingly enough this technique does not improve stability unless we use regularization again