seismic data restoration via data driven tight frame restoration interpolation of missing traces plays a crucial role in the seismic data processing pipeline efficient restoration methods have been proposed based on sparse signal representation in a transform domain such as fourier wavelet curvelet and shearlet transforms most existing methods are based on transforms with a fixed basis we considered an adaptive sparse transform for restoration of data with complex structures in particular we evaluated a data driven tight frame based sparse regularization method for seismic data restoration the main idea of the data driven tight frame tf is to adaptively learn a set of framelet filters from the currently interpolated data under which the data can be more sparsely represented hence the sparsity promoting l1 norm spl1 minimization methods can produce better restoration results by using the learned filters a split inexact uzawa algorithm which can be viewed as a generalization of the alternating direction of multiplier method admm was applied to solve the presented spl1 model numerical tests were performed on synthetic and real seismic data for restoration of randomly missing traces over a regular data grid our experiments showed that our proposed method obtains the state of the art restoration results in comparison with the traditional fourier based projection onto convex sets the tight frame based method and the recent shearlet regularization admm method

interpolation and denoising of high dimensional seismic data by learning a tight frame abstractsparse transforms play an important role in seismic signal processing steps such as prestack noise attenuation and data reconstruction analytic sparse transforms so called implicit dictionaries such as the fourier radon and curvelet transforms are often used to represent seismic data there are situations however in which the complexity of the data requires adaptive sparse transform methods whose basis functions are determined via learning methods we studied an application of the data driven tight frame ddtf method to noise suppression and interpolation of high dimensional seismic data rather than choosing a model beforehand for example a family of lines parabolas or curvelets to fit the data the ddtf derives the model from the data itself in an optimum manner the process of estimating the basis function from the data can be summarized as follows first the input data are divided into small blocks to form training sets then the ddtf algorithm is applied on the training set

seismic denoising using the redundant lifting scheme abstractseparating linear coherent noise such as ground roll from reflections remains a key challenge in seismic processing by adapting the redundant lifting scheme rls a wavelet transform method to seismic data we have determined how the wavelet domain can be used to suppress coherent and random noise the rls operates on a trace by trace basis decomposing each time series into wavelet coefficient wc time series and consequently a single gather in a shot receiver or common depth point domain into a series of wc subgathers sgs the decomposition changes the relative magnitude of wcs of various events reflection head wave ground roll etc from one sg to another without affecting their moveout in sg s in which the wcs of undesired events were significantly stronger than the desired events the wcs can be surgically muted selective muting in carefully chosen sgs attenuates undesired events while having minimal effects on frequency spectra of the desired events in addition random noi

a note on binary arrays with a certain window property corresp for some rs 2 mn there exist periodic r times s binary arrays such that all m times n binary matrices will appear in it as two dimensional versions of de bruijn sequences these kinds of arrays have possible applications in cryptography and the design of mask configurations for spectrometers

seislet transform and seislet frame we introduce a digital waveletlike transform which is tailored specifically for representing seismic data the transform provides a multiscale orthogonal basis with basis functions aligned along seismic events in the input data it is defined with the help of the wavelet lifting scheme combined with local plane wave destruction in the 1d case the seislet transform is designed to follow locally sinusoidal components in the 2d case it is designed to follow local plane wave components with smoothly variable slopes if more than one component is present the transform turns into an overcomplete representation or a tight frame in these terms the classic digital wavelet transform is simply a seislet transform for a zero frequency in one dimension or zero slope in two dimensions the main objective of the new transform is an effective seismic data compression for designing efficient data analysis algorithms traditional signal processing tasks such as noise attenuation and trace interpolation become simply defined in the seislet domain when applied in the offset direction on common midpoint or common image point gathers the seislet transform finds an additional application in optimal stacking of seismic records

the heat and moisture transfer balance theory of garment simulation to establish the human body model to analyze the heat and moisture transfer on body surface a new explicit definition of rational l recursion surface is given and the l recursion surfaces in grassmann spaces are constructed by using blossom method of the homogeneous normal pyramid form based on our human body model the balance theory of garment simulation the heat and moisture transfer balance equations called icad balance equations are obtained the balance theory of garment simulation integrally studies the complex system of human body fabric environment at the same time the method of obtaining the heat and moisture transfer balance equations is also based on the mass conservation law the energy conservation law and the fish law of capillarity a finite volume method is employed to solve the icad balance equations

sparse geometric image representations with bandelets this paper introduces a new class of bases called bandelet bases which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow this geometric flow indicates directions in which the image gray levels have regular variations the image decomposition in a bandelet basis is implemented with a fast subband filtering algorithm bandelet bases lead to optimal approximation rates for geometrically regular images for image compression and noise removal applications the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion comparisons are made with wavelet image compression and noise removal algorithms

fourier reconstruction of nonuniformly sampled aliased seismic data there are numerous methods for interpolating uniformly sampled aliased seismic data but few can handle the combination of nonuniform sampling and aliasing we combine the principles of fourier reconstruction of nonaliased nonuniformly sampled data with the ideas of frequency wavenumber f k interpolation of aliased uniformly sampled data in a new two step algorithm in the first step we estimate the fourier coefficients at the lower nonaliased temporal frequencies from the nonuniformly sampled data the coefficients are then used in the second step as an a priori model to distinguish between aliased and nonaliased energy at the higher aliased temporal frequencies by using a nonquadratic model penalty in the inversion both the artifacts in the fourier domain from nonuniform sampling and the aliased energy are suppressed the underlying assumption is that events are planar therefore the algorithm is applied to seismic data in overlapping spatiotemporal windows

the contourlet transform an efficient directional multiresolution image representation the limitations of commonly used separable extensions of one dimensional transforms such as the fourier and wavelet transforms in capturing the geometry of image edges are well known in this paper we pursue a true two dimensional transform that can capture the intrinsic geometrical structure that is key in visual information the main challenge in exploring geometry in images comes from the discrete nature of the data thus unlike other approaches such as curvelets that first develop a transform in the continuous domain and then discretize for sampled data our approach starts with a discrete domain construction and then studies its convergence to an expansion in the continuous domain specifically we construct a discrete domain multiresolution and multidirection expansion using nonseparable filter banks in much the same way that wavelets were derived from filter banks this construction results in a flexible multiresolution local and directional image expansion using contour segments and thus it is named the contourlet transform the discrete contourlet transform has a fast iterated filter bank algorithm that requires an order n operations for n pixel images furthermore we establish a precise link between the developed filter bank and the associated continuous domain contourlet expansion via a directional multiresolution analysis framework we show that with parabolic scaling and sufficient directional vanishing moments contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves finally we show some numerical experiments demonstrating the potential of contourlets in several image processing applications

fourier reconstruction of marine streamer data in four spatial coordinates many methods exist for interpolation of seismic data in one and two spatial dimensions but few can interpolate properly in three or four spatial dimensions marine multi streamer data typically are sampled relatively well in the midpoint and absolute offset coordinates but not in the azimuth because the crossline shot coordinate is significantly under sampled we approach the problem of interpolation of marine streamer data in four spatial dimensions by splitting the problem into a 1d interpolation along the densely sampled streamers and a 3d fourier reconstruction for the remaining spatial coordinates in fourier reconstruction the fourier coefficients that synthesize the nonuniformly sampled seismic data are estimated in a least squares inversion the method is computationally efficient requires no subsurface information and can handle uniform grids with missing data as well as nonuniform grids or random sampling the output grid of the 1d interpolation in the first step is arbitrary when the output grid has uniform inline midpoints spacing the 3d fourier reconstruction in the second step is performed in the crossline midpoint absolute offset and azimuth coordinates when the first step outputs to uniform absolute offset the 3d fourier reconstruction handles the crossline inline midpoint and the azimuth coordinates in both cases the main innovation is the inclusion of the azimuthal coordinate in the fourier reconstruction the azimuth multiplicity must be increased for the method to be successful which means that overlap shooting is required we have tested the algorithm on synthetic streamer data for which the proposed method outperforms an approach where the azimuthal coordinate is ignored potential applications are interpolation of marine streamer data to decrease the crossline source sampling for the benefit of 3d multiple prediction and regularization to reduce sampling related differences in processing of time lapse data

rational recurrence curves and recurrence surfaces in multivariate b form on some regions in this paper we present rational w curves and surfaces study their envelope properties and give a general definition of surfaces in multivariate b form by raising the dimension of polyhedron region thereby present an equivalent representation of rational w curves and surfaces into rational bezier curves and surfaces in multivariate b form on some regions

a tensor higher order singular value decomposition for prestack seismic data noise reduction and interpolation a patch of prestack data depends on four spatial dimensions x y midpoints and x y offsets and frequency the spatial data at one temporal frequency can be represented by a fourth order tensor in ideal conditions of high signal to noise ratio and complete sampling one can assume that the seismic data can be approximated via a low rank fourth order tensor missing samples were recovered by reinserting data obtained by approximating the original noisy and incomplete data volume with new observations obtained via the rank reduction process the higher order singular value decompostion was used to reduce the rank of the prestack seismic tensor synthetic data demonstrated the ability of the proposed seismic data completion algorithm to reconstruct events with curvature the synthetic example allowed to quantify the quality of the reconstruction for different levels of noise and survey sparsity we also provided a real data example from the western canadian sedimentary basin

adaptive prediction filtering in t x y domain for random noise attenuation using regularized nonstationary autoregression abstractmany natural phenomena including geologic events and geophysical data are fundamentally nonstationary they may exhibit stationarity on a short timescale but eventually alter their behavior in time and space we developed a 2d t x adaptive prediction filter apf and further extended this to a 3d t x y version for random noise attenuation based on regularized nonstationary autoregression rna instead of patching a popular method for handling nonstationarity we obtained smoothly nonstationary apf coefficients by solving a global regularized least squares problem we used shaping regularization to control the smoothness of the coefficients of apf three dimensional space noncausal t x y apf uses neighboring traces around the target traces in the 3d seismic cube to predict noise free signal so it provided more accurate prediction results than the 2d version in comparison with other denoising methods such as frequency space deconvolution time space prediction filter and frequency space rna

directionlets anisotropic multidirectional representation with separable filtering in spite of the success of the standard wavelet transform wt in image processing in recent years the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions one dimensional 1 d discontinuities in images edges and contours that are very important elements in visual perception intersect too many wavelet basis functions and lead to a nonsparse representation to efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions a more complex multidirectional m dir and anisotropic transform is required we present a new lattice based perfect reconstruction and critically sampled anisotropic m dir wt the transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two dimensional wt unlike in the case of some other directional transform constructions e g curvelets contourlets or edgelets the corresponding anisotropic basis functions directionlets have directional vanishing moments along any two directions with rational slopes furthermore we show that this novel transform provides an efficient tool for nonlinear approximation of images achieving the approximation power o n sup 1 55 which while slower than the optimal rate o n sup 2 is much better than o n sup 1 achieved with wavelets but at similar complexity

phase and amplitude tracking for seismic event separation we have developed a method to decompose seismic records into atomic events each defined by a smooth phase function and a smooth amplitude function this decomposition is intrinsically nonlinear and calls for a nonconvex least squares optimization formulation along the lines of full waveform inversion to overcome the lack of convexity we have developed an iterative refinement expansion scheme to initialize and track the phase and amplitude for each atomic event for short we called the method phase tracking the initialization is carried out by applying multiple signal classification to a few seed traces in which events can be separated and identified by their arrival times and amplitudes we then construct the initial solution at the seed traces using linear phase functions from the arrival times and constant amplitude functions assuming the medium is mostly dispersion free we refine this initial solution to account for dispersion and imperfect knowledge of the wavelet at the seed traces by fitting the observed data using a gradient descent method the resulting phase and amplitude functions are then carefully expanded across the traces in an adequately smooth way to match the whole data record we have evaluated the proposed method on two synthetic records and a field record because the parametrization of the seismic events is physically meaningful it also enables a simple form of bandwidth extension of the observed shot record to unobserved low and high frequencies we tested this procedure on the same shot records bandwidth extension is in principle helpful to initialize full waveform inversion with frequency sweeps and enhanced its resolution

the curvelet representation of wave propagators is optimally sparse this paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations curvelets are a recently developed multiscale system 7 9 in which the elements are highly anisotropic at fine scales with effective support shaped according to the parabolic scaling principle width length2 at fine scales we prove that for a wide class of linear hyperbolic differential equations the curvelet representation of the solution operator is both optimally sparse and well organized it is sparse in the sense that the matrix entries decay nearly exponentially fast i e faster than any negative polynomial and well organized in the sense that the very few nonnegligible entries occur near a few shifted diagonals indeed we show that the wave group maps each curvelet onto a sum of curveletlike waveforms whose locations and orientations are obtained by following the different hamiltonian flows hence the diagonal shifts in the curvelet representation a physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time with enough spatial localization so that they simultaneously behave like particles

constructions for perfect maps and pseudorandom arrays a construction of perfect maps i e periodic r v binary arrays in which each n m binary matrix appears exactly once is given a similar construction leads to arrays in which only the zero n m matrix does not appear and to a construction in which only a few n m binary matrices do not appear a generalization to the nonbinary case is given the constructions involve an interesting problem in shift register theory the solution is given for almost all the case of this problem

iterative deblending of simultaneous source seismic data using seislet domain shaping regularization abstractwe used a novel iterative estimation scheme for separation of blended seismic data from simultaneous sources the scheme is based on an augmented estimation problem that can be solved by iteratively constraining the deblended data using shaping regularization in the seislet domain we formulated the forward modeling operator in the common receiver domain in which two sources were assumed to be blended using a random time shift dithering approach the nonlinear shaping regularization framework offered some freedom in designing a shaping operator to constrain the model in an underdetermined inverse problem we designed the backward operator and the shaping operator for the shaping regularization framework the backward operator can be optimally chosen as half of the identity operator in the two source case and the shaping operator can be chosen as coherency promoting operator the high performance deblending effect of the iterative framework was tested on three numerically blended synthetic data s

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

oc seislet seislet transform construction with differential offset continuation many of the geophysical data analysis problems such as signal noise separation and data regularization are conveniently formulated in a transform domain in which the signal appears sparse classic transforms such as the fourier transform or the digital wavelet transform dwt fail occasionally in processing complex seismic wavefields because of the nonstationarity of seismic data in time and space dimensions we present a sparse multiscale transform domain specifically tailored to seismic reflection data the new wavelet like transform the oc seislet transform uses a differential offset continuation oc operator that predicts prestack reflection data in offset midpoint and time coordinates it provides a high compression of reflection events its compression properties indicate the potential of oc seislets for applications such as seismic data regularization or noise attenuation results of applying the method to synthetic and field data examples demonstrate that the oc seislet transform can recon

automatic data extrapolation to zero offset along local slope abstractvelocity independent seismic data processing requires information about the local slope in the data from estimates of local time and space derivatives of the data a total least squares algorithm gives an estimate of the local slope at each data point total least squares minimizes the orthogonal distance from the data points the local time and space derivatives to the fitted straight line defining the local slope this gives a more consistent estimate of the local slope than standard least squares because it takes into account uncertainty in the temporal and spatial derivatives the total least squares slope estimate is the same as the one obtained from using the structure tensor with a rectangular window function the estimate of the local slope field is used to extrapolate all traces in a seismic gather to the smallest recorded offset without using velocity information extrapolation to zero offset is done using a hyperbolic traveltime function in which slope information replaces the knowled

the curvelet transform for image denoising we describe approximate digital implementations of two new mathematical transforms namely the ridgelet transform and the curvelet transform our implementations offer exact reconstruction stability against perturbations ease of implementation and low computational complexity a central tool is fourier domain computation of an approximate digital radon transform we introduce a very simple interpolation in the fourier space which takes cartesian samples and yields samples on a rectopolar grid which is a pseudo polar sampling set based on a concentric squares geometry despite the crudeness of our interpolation the visual performance is surprisingly good our ridgelet transform applies to the radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain our curvelet transform uses our ridgelet transform as a component step and implements curvelet subbands using a filter bank of a spl grave trous wavelet filters our philosophy throughout is that transforms should be overcomplete rather than critically sampled we apply these digital transforms to the denoising of some standard images embedded in white noise in the tests reported here simple thresholding of the curvelet coefficients is very competitive with state of the art techniques based on wavelets including thresholding of decimated or undecimated wavelet transforms and also including tree based bayesian posterior mean methods moreover the curvelet reconstructions exhibit higher perceptual quality than wavelet based reconstructions offering visually sharper images and in particular higher quality recovery of edges and of faint linear and curvilinear features existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems the empirical results reported here are in encouraging agreement

a new construction of two dimensional arrays with the window property it is shown how row complete latin squares and tuscan squares can be used to construct nonbinary 1 d and 2 d arrays that have the u v horizontal and or vertical window property for all sufficiently large windows it is shown how binary arrays with a similar property can be derived since an array can only be a perfect map for one particular size of window the arrays that are constructed are not perfect but nevertheless have several practical applications among others in connection with the coding and transmission of pictures the method is deterministic rather than stochastic and the arrays derived contain an effectively random selection of the windows of each relevant size

representation and conversion of bezier surfaces in multivariate b form in this paper we present the representation and conversion method of bezier surfaces in multivariate b form we can obtain the relationship between the surface of the multivariate b form and the bezier surface then the continuity connection conditions of the bezier surface in a triangular and a rectangular region are given

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

aperiodic and semi periodic perfect maps paterson see ibid vol 40 p 743 1993 has shown that the trivial necessary conditions are sufficient for the existence of a binary perfect map these periodic structures can be transformed very simply into corresponding aperiodic and semi periodic perfect maps however aperiodic and semi periodic perfect maps can exist for parameter sets for which the corresponding periodic perfect maps cannot in this paper it is shown by construction that binary aperiodic and semi periodic perfect maps exist for all possible parameter sets

recurrence surfaces on arbitrary quadrilateral mesh in this paper l and w surface are constructed via recurrence scheme the derivation formulae basis function properties envelope theorems of l surface and an equivalent representation of l surface into bezier surface are presented especially a c2 bicubic spline surface on a cross cut grid partition consisting of quadrilaterals is given which can be used to replace the biquintic surface in most of the applications the new method has been applied in the field of contour design of human body and aircraft

asymmetric chirplet transform for sparse representation of seismic data abstractfor a fast and accurate extraction of important information in seismic signals a sparse representation based on the physical parameters of the given data is crucial we have used the asymmetric gaussian chirplet model agcm and established a dictionary free variant of the orthogonal matching pursuit a greedy algorithm for sparse approximation the atoms of agcm so called chirplets display asymmetric oscillation attenuation properties which make the agcm very suitable for sparse representation of absorption decay seismic signals unlike the fourier transform which assumes that the seismic signals consist of plane waves the agcm assumes that the seismic signal consists of nonstationary compressed plane waves i e symmetric or asymmetric chirplets thus agcm is a general model for seismic wave simulation and its model parameters include an envelope part and a phase part we have mainly concentrated on the parameters of the envelope part such as the envelope amplitude and arrival time we

seismic data decomposition into spectral components using regularized nonstationary autoregression summary seismic data can be decomposed into nonstationary spectral components with smoothly variable frequencies and smoothly variable amplitudes to estimate local frequencies i use a nonstationary version of prony s spectral analysis method defined with the help of regularized nonstationary autoregression rnar to estimate local amplitudes of different components i fit their sum to the data using regularized nonstationary regression rnr shaping regularization ensures stability of the estimation process and provides controls on smoothness potential applications of the proposed technique include noise attenuation seismic data compression and seismic data regularization

a computing model of pressure distribution from tight underwear based on the theory of membrane of huge deformation in elasticity a computable model is presented to solve the problem of pressure distribution on a human body from tight underwear this model is steady and reliable with the help of this model it is easy to solve the problem of describing wearing dressed effect in integrated garments cad system

signal and noise separation in prestack seismic data using velocity dependent seislet transform abstractthe seislet transform is a waveletlike transform that analyzes seismic data by following varying slopes of seismic events across different scales and provides a multiscale orthogonal basis for seismic data it generalizes the discrete wavelet transform dwt in the sense that the dwt in the lateral direction is simply the seislet transform with a zero slope our earlier work used plane wave destruction pwd to estimate smoothly varying slopes however the pwd operator can be sensitive to strong noise interference which makes the seislet transform based on pwd pwd seislet transform occasionally fail in providing a sparse multiscale representation for seismic field data we adopted a new velocity dependent vd formulation of the seislet transform in which the normal moveout equation served as a bridge between local slope patterns and conventional moveout parameters in the common midpoint domain the vd slope has better resistance to strong random noise which indicated the potential of vd sei

perfect maps given positive integers r s u and spl upsi an r s u spl upsi perfect map pm is defined to be a periodic r spl times s binary array in which every u spl times spl upsi binary array appears exactly once as a periodic subarray perfect maps are the natural extension of the de bruijn sequences to two dimensions in the paper the existence question for perfect maps is settled by giving constructions for perfect maps for all parameter sets subject to certain simple necessary conditions extensive use is made of previously known constructions by finding new conditions which guarantee their repeated application these conditions are expressed as bounds on the linear complexities of the periodic sequences formed from the rows and columns of perfect maps

accelerated plane wave destruction abstractwhen plane wave destruction pwd is implemented by implicit finite differences the local slope is estimated by an iterative algorithm we propose an analytical estimator of the local slope that is based on convergence analysis of the iterative algorithm using the analytical estimator we design a noniterative method to estimate slopes by a three point pwd filter compared with the iterative estimation the proposed method needs only one regularization step which reduces computation time significantly with directional decoupling of the plane wave filter the proposed algorithm is also applicable to 3d slope estimation we present synthetic and field experiments to demonstrate that the proposed algorithm can yield a correct estimation result with shorter computational time

signal extraction using randomized order multichannel singular spectrum analysis abstractmultichannel singular spectrum analysis mssa is an effective algorithm for random noise attenuation however it cannot be used to suppress coherent noise this limitation results from the fact that the conventional mssa method cannot distinguish between useful signals and coherent noise in the singular spectrum we have developed a randomization operator to disperse the energy of the coherent noise in the time space domain furthermore we have developed a novel algorithm for the extraction of useful signals i e for simultaneous random and coherent noise attenuation by introducing a randomization operator into the conventional mssa algorithm in this method which we call randomized order mssa the traces along the trajectory of each signal component are randomly rearranged two ways to extract the trajectories of different signal components are investigated the first is based on picking the extrema of the upper envelopes a method that is also constrained by local and global gradients th

3d interpolation of irregular data with a pocs algorithm seismic surveys generally have irregular areas where data cannot be acquired these data should often be interpolated a projection onto convex sets pocs algorithm using fourier transforms allows interpolation of irregularly populated grids of seismic data with a simple iterative method that produces high quality results the original 2d image restoration method the gerchberg saxton algorithm is extended easily to higher dimensions and the 3d version of the process used here produces much better interpolations than typical 2d methods the only parameter that makes a substantial difference in the results is the number of iterations used and this number can be overestimated without degrading the quality of the results this simplicity is a significant advantage because it relieves the user of extensive parameter testing although the cost of the algorithm is several times the cost of typical 2d methods the method is easily parallelized and still completely practical