Determining if uncertainty quantification is worth it or not is closely related to how that uncertainty is computed and the associated computational cost. For seismic imaging, it is typically done using Markov chain Monte Carlo algorithms (McMC). Solving an inverse problem using McMC means exploring and characterizing the ensemble of all plausible models through more or less point-wise random walk in the data misfit landscape. This is typically done using Bayes’ theorem via the computation of a posterior probability density function. Even though this can sound naively simple, it can come with a significant computational burden given the dimension of the problem to be solved and the expense of the forward solver. This is because as the number of dimensions grow, there are exponentially more possible guesses the algorithm can make, while only a few of these models will be accepted as plausible. More advanced uncertainty quantification methods such as Hamiltonian Monte Carlo (HMC) could be beneficial because they can handle higher dimensions because efficient sampling of the model space through pseudo-mechanical trajectories in the data misfit landscape is expected. In order for an HMC algorithm to efficiently sample the model space of interest and provide meaningful uncertainty estimates, three hyper-parameters need to be tuned for trajectory design: the Leapfrog steps L, the Leapfrog stepsize ε, and the Mass Matrix M. There has been already work showing how one can choose L and ε; however designing the appropriate M is far more challenging. We consider a time-lapse seismic scenario and use a local acoustic solver for fast forward solutions. We then use Singular value decomposition, in the vicinity of the true model, to transform our time-lapse optimal model to a system of normal coordinates and use only a few of the eigenvalues and eigenvectors of the Hessian as oscillators. By doing so, we can efficiently understand the impact of the initial conditions and the choice of M and gain insight on how to design M in the standard system. This gives us an intuitive way to understand the mass matrix, allowing us to determine whether gains from the HMC algorithm are worth the cost of determining the parameters.

In this study, we compared multiple quantitative ultrasound metrics for the purpose of differentiating muscle in 20 healthy, 10 dystrophic and 10 obese mice. High-frequency ultrasound scans were acquired on dystrophic (D2-mdx), obese (db/db) and control mouse hindlimbs. A total of 248 image features were extracted from each scan, using brightness-mode statistics, Canny edge detection metrics, Haralick features, envelope statistics and radiofrequency statistics. Naïve Bayes and other classifiers were trained on single and pairs of features. The a parameter from the Homodyned K distribution at 40 MHz achieved the best univariate classification (accuracy = 85.3%). Maximum classification accuracy of 97.7% was achieved using a logistic regression classifier on the feature pair of a2 (K distribution) at 30 MHz and brightness-mode variance at 40MHz. Dystrophic and obese mice have muscle with distinct acoustic properties and can be classified to a high level of accuracy using a combination of multiple features.

In this article, we present the design, validation, and imaging capabilities of a mechanically discretized ultrasound scanning apparatus (MEDUSA) that supports flexible development of ultrasound tomography (UST) algorithms for complex tissue structures. Ultrasound tomography in the recent decade has shown promising results in quantitative soft-tissue imaging for clinical breast cancer diagnostics. There is growing interest in applying tomographic techniques to image broader tissue structures that include bone, where imaging is significantly more challenging due to strong impedance mismatches and complex wave propagation within the region. Changes in data acquisition strategy, algorithms, and system design are necessary to enable quantitative imaging of soft-tissue with bone inclusions. The 36 degree-of-freedom (DOF) MEDUSA system allows free space positioning of acoustic transducers around an imaging target and enables investigation of imaging strategies not available in other UST systems. We present the mechanical design, parameter calibration, and tomographic imaging results using MEDUSA. Mono-/bistatic imaging and full-waveform inversion (FWI) results on real targets are presented and validates system performance capabilities for broader UST algorithm development for more complex tissue structures.

Quantitative ultrasound (QUS) has emerged as a viable tool in diagnosing and staging the onset and progression of various diseases. Within the field of QUS, shear wave elastography (SWE) has emerged as the clinical standard for quantifying and correlating the stiffness of tissue to its underlying pathology. Despite its widespread use, SWE suffers from drawbacks that limit its widespread clinical use; among these are low-frame rates, long settling times, and high sensitivity to operating conditions. Longitudinal speed of sound (SOS) has emerged as a viable alternative to SWE. We propose a framework to obtain 2D sound speed maps using a commercial ultrasound probe. A commercial ultrasound probe is localized in space and used to scan a domain of interest from multiple vantage points; the use of a reflector at the far end of the domain allows us to measure the round trip travel times to and from it. The known locations of the probe and the measured travel times are used to estimate the depth and inclination of the reflector as well as the unknown sound speed map. The use of multiple looks increases the effective aperture of the ultrasound probe and allows for a higher fidelity reconstruction of sound speed maps. We validate the framework using simulated and experimental data and propose a rigorous framework to quantify the uncertainty of the estimated sound speed maps.

This work presents the first quantitative ultrasonic sound speed images of ex vivo limb cross-sections containing both soft tissue and bone using Full Waveform Inversion (FWI) with level set (LS) and travel time regularization. The estimated bulk sound speed of bone and soft tissue are within 10% and 1%, respectively, of ground truth estimates. The sound speed imagery shows muscle, connective tissue and bone features. Typically, ultrasound tomography (UST) using FWI is applied to imaging breast tissue properties (e.g. sound speed and density) that correlate with cancer. With further development, UST systems have the potential to deliver volumetric operator independent tissue property images of limbs with non-ionizing and portable hardware platforms. This work addresses the algorithmic challenges of imaging the sound speed of bone and soft tissue by combining FWI with LS regularization and travel time methods to recover soft tissue and bone sound speed with improved accuracy and reduced soft tissue artifacts when compared to conventional FWI. The value of leveraging LS and travel time methods is realized by evidence of improved bone geometry estimates as well as promising convergence properties and reduced risk of final model errors due to un-modeled shear wave propagation. Ex vivo bulk measurements of sound speed and MRI cross-sections validates the final inversion results.

Speed of sound (SOS) is a biomarker that aides clinicians in tracking the onset and progression of diseases such as breast cancer and fatty liver disease. In this paper, we propose a framework to generate accurate, 2D SOS maps with a commercial ultrasound probe. We simulate freehand ultrasound probe motion and use a multi-look framework for reflection travel time tomography. In these simulations, the “measured” travel times are computed using a bent-ray Eikonal solver and direct inversion for compressional speed of sound is performed. We have shown that the assumption of straight rays breaks down for large velocity perturbations (greater than 1 percent). The error increases 70 fold for a velocity perturbation increase of 1.5 percent. Moreover, the use of multiple looks greatly aides the inversion process. Simulated RMSE drops by roughly 15 dB when the maximum scanning angle is increased from 0 to 45 degrees.

Tank based ultrasound tomography systems enable the reconstruction of compressional sound speed and density maps in hard and soft tissue such as bone and muscle. Compressional sound speed can serve as a biomarker and help to identify healthy from unhealthy tissue. Tomography systems must balance trade-offs between cost, complexity, acquisition time, and the number of transducers. In order to minimize patient motion artifacts, acquisition times should be short. In order to minimize cost, fewer transducers elements are preferred. A tomography system consisting of two single element transducers that are mechanically rotated took roughly 40 minutes to acquire a single tomographic slice. We present an optimized acquisition framework that minimizes the uncertainty of the reconstructed compressional sound speed map. Using an initial estimate of the sound speed model, we predict the ray-paths for all possible source-receiver pairs. Based on the ray-paths, we determine transducer locations that minimize the uncertainty of estimated the sound speed map. Using the proposed framework we reconstruct sound speed maps of hard and soft tissue from 25% of the measurements used for a uniformly sampled acquisition strategy in which the receiver positions are equally spaced and data is recorded at all possible back aperture locations. We demonstrate how the acquisition scheme changes when minimizing soft tissue sound speed error versus hard tissue sound speed error.

Time-lapse seismic monitoring using full-wavefield methods aims to accurately and robustly image rock and fluid changes within a reservoir. These changes are typically small and localized. Quantifying the uncertainty related to these changes is crucial for decision making, but traditional methods that use pixel by pixel uncertainty quantification with large models are computationally infeasible. We exploit the structure of the time-lapse seismic problem for fast wavefield computations using a numerically exact local acoustic solver. This allows us to perform a Bayesian inversion using a Metropolis–Hastings algorithm to sample our posterior distribution. We address the well-known dimensionality problem in global optimization using an image compression technique. We run our numerical experiments using a single shot and a single frequency, however we show that various frequencies converge to different local minima. In addition, we test our framework for both uncorrelated and correlated noise, and we retrieve different histograms for each noise type. Through our numerical examples we show the importance of defining quantities of interest in order to setup an appropriate uncertainty quantification framework involving choosing the number of degrees of freedom and model parametrization that best approximate the problem. To our knowledge, there is no work in the literature studying the time-lapse problem using stochastic full-waveform inversion.

PreviousNext No AccessSEG Technical Program Expanded Abstracts 2020Time-lapse full-waveform inversion using Hamiltonian Monte Carlo: A proof of conceptAuthors: Maria KotsiAlison MalcolmGregory ElyMaria KotsiMemorial University of NewfoundlandSearch for more papers by this author, Alison MalcolmMemorial University of NewfoundlandSearch for more papers by this author, and Gregory ElyMassachusetts Institute of TechnologySearch for more papers by this authorhttps://doi.org/10.1190/segam2020-3422774.1 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail AbstractUncertainty quantification is an important aspect of time–lapse imaging and is typically done using Bayesian inference. Traditional random–walk sampling methods are slow to converge and they fail to efficiently explore the high dimensional space that must be characterized in time-lapse imaging. We propose the use of a local acoustic Helmholtz solver for an efficient time–lapse Hamiltonian Monte Carlo (HMC) inversion. Using a local acoustic solver offers the advantage of quick and local gradient computations. Our numerical models demonstrate the robustness of the method over the Metropolis–Hastings algorithm, and set up the path towards non–linear uncertainty quantification of high dimensional velocity models. To our knowledge this is the first direct comparison of Metropolis– Hastings and HMC on a seismic example.Presentation Date: Tuesday, October 13, 2020Session Start Time: 1:50 PMPresentation Time: 3:55 PMLocation: Poster Station 3Presentation Type: PosterKeywords: statistics, full-waveform inversion, global search, nonlinear, time-lapsePermalink: https://doi.org/10.1190/segam2020-3422774.1FiguresReferencesRelatedDetailsCited ByBayesian Geophysical Inversion Using Invertible Neural Networks10 July 2021 | Journal of Geophysical Research: Solid Earth, Vol. 126, No. 7 SEG Technical Program Expanded Abstracts 2020ISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 2020 Pages: 3887 publication data© 2020 Published in electronic format with permission by the Society of Exploration GeophysicistsPublisher:Society of Exploration Geophysicists HistoryPublished: 25 Sep 2020 CITATION INFORMATION MariaKotsi, AlisonMalcolm, and GregoryEly, (2020), "Time-lapse full-waveform inversion using Hamiltonian Monte Carlo: A proof of concept," SEG Technical Program Expanded Abstracts : 845-849. https://doi.org/10.1190/segam2020-3422774.1 Plain-Language Summary Keywordsstatisticsfull-waveform inversionglobal searchnonlineartime-lapsePDF DownloadLoading ...

In geophysical imaging, uncertainty quantification is crucial for decision making. 4D seismic imaging aims to accurately recover changes that take place within a reservoir. These changes are typically characterized by their magnitude and their extent. We perform a Bayesian inversion using a Metropolis Hastings algorithm to sample our posterior distribution of 4D velocity models given observed data. To model the 4D change we use a discrete cosine transformation, and attempt to recover the lowest frequency coefficients, so that we can model realistic changes with only a few degrees of freedom. Unlike most of uncertainty quantification methodologies that use expensive forward solvers, we speed up our computations by using a numerically exact local acoustic solver. Presentation Date: Tuesday, September 17, 2019 Session Start Time: 1:50 PM Presentation Start Time: 3:05 PM Location: Poster Station 13 Presentation Type: Poster

We explore the feasibility of using ultrasound to image through bone. The strong velocity and acoustic impedance contrast between bone and soft tissue like that between salt and sediments, significantly reduces the amount of energy transmitted and generates strong internal multiples. In this paper we present a novel framework for imaging the interior of long bones by using a group sparse hyperbolic radon transform to both denoise and despeckle the resulting image and suppress the internal multiples. Using this technique we demonstrate that it is possible to image the interior of bones despite large velocity contrasts and strong multiples on synthetic and invivo data.

In this paper, we use a fast Helmholtz solver with global optimization methods to estimate an initial velocity model for full-waveform inversion based on raw recorded waveforms. More specifically, we combine the field expansion method for solving the Helmholtz equation with a reduced parameterization of the velocity model allowing for extremely fast forward solves of realistic velocity models. Unlike conventional Full Waveform Inversion that uses gradient based inversion techniques, global optimization methods are less sensitive to local minima and initial starting models. However, these global optimization methods are stochastic and are not guaranteed to converge to the global minima. Because our adaptation of the field expansion method is so fast, we can study the convergence of these algorithms across parameter choice and starting model. In this paper we compare two commonly used global optimization methods, particle swarm optimization, and simulated annealing, and examine the limitations of these algorithms and their dependence on choice of parameters. We find that PSO outperforms SA and that PSO converges reliably for noisy data provided the number of parameters remains small. In addition, using this methodology we are able to estimate reasonable FWI starting models with higher frequency data. INTRODUCTION Most seismic techniques such as imaging, rely on a velocity model inverted from noisy data through a non-linear inverse problem. These inverted velocity models may be inaccurate and lead to incorrect interpretations of the subsurface. For example, an erroneously fast or slow section of a velocity model could cause a syncline structure to appear as an anticline and be incorrectly interpreted as a potential trap. Full Waveform Inversion (FWI) and other gradient based methods will converge to local minima (Warner et al., 2013) and will lead to inaccurate velocity models if initiated with an inaccurate starting model (Virieux and Operto, 2009). In recent years Uncertainty Quantification (UQ) methods have been applied to the FWI problem in order to mitigate the impact of local minima and provide a more probabilistic interpretation of the sub surface (Stuart et al., 2016; Zhu and Gibson, 2016; Ray et al., 2016). See Osypov et al. (2013) for a discussion of importance of risk assessment and UQ in seismic imaging. These methods allow for an informed decision about the reliability of the subsurface image and aid in risk estimates for drilling a potential well. However, these methods either require hundreds of thousands of wave equation forward solves to converge to accurate error estimates of subsurface parameters (Ely et al., 2018), or use local approximations of a Gaussian to approximate the uncertainty (Fang et al., 2018). The number of forward solves required for MCMC is likely infeasible for many seismic imaging problems and forward models. In addition, in low noise scenarios there many only Ely, Malcolm, Nicholls 2 Global Optimization methods be a single region of high probability and MCMC techniques may be unnecessary. Instead, global optimization methods are an attractive alternative as they require far fewer forward solves and provide an estimate of the best fitting model (maximum Likelihood) at a fraction of the number of forward solves needed for UQ and yet can still avoid being trapped by local minima. In recent years global optimization methods such as Simulated Annealing (SA) (Datta and Sen, 2016; Galuzzi et al., 2017; Sajeva et al., 2016) and Particle Swarm Optimization (PSO) (Ely et al., 2015; Shaw and Srivastava, 2007) have become popular methods of calculating an initial velocity model. These methods can be an alternative or supplement to conventional methods of building velocity models such as tomography, NMO velocity analysis or very low frequency FWI (Woodward et al., 2008). Conventional initial velocity building methods typically rely on gradient based minimization and often require the manual picking of arrival times or picks in semblance space. By contrast, global optimization methods, although not guaranteed to converge to the global minima, are far more robust to local minima and frequently do not require the calculation of a gradient, reducing memory constraints and algorithmic complexity. However, starting models, frequency, and other parameter choice can have a significant impact on the convergence of these algorithms. Although these algorithms require significantly fewer iterations than UQ algorithms, rigorous comparison requires numerous global optimization runs and thousands or tens of thousands of forward solves. In addition, due to the stochastic nature of these algorithms, two different runs with the same starting model can converge to different finals models, requiring even more simulations to understand performance. Due to the computational cost of most forward solvers, previous work has been unable to accurately characterize these impacts and benchmark these algorithms as significant simplifications were necessary to control this computational cost. For example, Sajeva et al. (2017) use analytical solutions to test problems as well as the 1D wave equation to compare global optimization algorithms. In another attempt to make the problem computationally tractable, previous work has typically shown performance with one or only a few starting models when tens or hundreds of runs are necessary to generate a statistical model of convergence and accurately compare one algorithm to another (Rios and Sahinidis, 2013). Studies in the global optimization literature use test functions that are extremely fast to evaluate but are not representative of local minima found in FWI and seismic imaging. Other work combining global optimization methods and FWI have lacked a sufficiently fast test function that accurately accounts for the physics of seismic inversion in 2D. In this paper, we use the reduced model parameterization of velocity models introduced in Frazer and Sen (1985); Zelt and Smith (1992) and Datta and Sen (2016) combined with two global optimization methods to estimate an initial velocity model. Due to the rapid speed of our forward model and we can quantitatively characterize the (1) limits on the number of degrees of freedom, (2) importance of initial guesses, (3) choice of tuning parameters, (4) comparison of different global optimization methods and (5) impact of source frequency. These comparisons, would be impossible with slower conventional finite difference solvers. From the numerical studies presented in this paper we are able to draw several conclusions about the performance of global optimization methods for FWI and what may impact the likelihood of finding a good initial model. Although the FWI problem is known to be more prone to local minima at higher frequency (Warner et al., 2013), we find that source Ely, Malcolm, Nicholls 3 Global Optimization methods frequency does not significantly decrease the likelihood of convergence and a good starting model can be found with a relatively high frequency of 5 or 8 Hz from raw waveform data. This suggests that we could build initial background velocity models with much higher frequencies than are typically used. We also find that the accuracy of the initial starting models weakly impacted the likelihood of converging to the correct model. From our results we find that only having an extremely accurate initial model improved the likelihood of convergence. This suggests that only if the initial model is in the basin of attraction of the global minima are we guaranteed to converge to the true solution and outside of this range the global optimization methods are equally likely to find an accurate solution independent of starting model. In addition, we find that our implementation of Particle Swarm Optimization (PSO) out preforms Simulated Annealing (SA) and is able to find a more accurate initial model with the same number of forward solves. The remainder of the paper is organized as follows. We first describe a reduced parameterization of the velocity model that is compatible with the field expansion forward solver. We then demonstrate that this first forward model and parametrization can yield comparable results to finite difference simulations if modifications are made to the forward solver. Second, we describe the two global optimization methods used in the paper: Particle Swarm Optimization (PSO) and Simulated Annealing (SA). Finally, we demonstrate our inversion on several synthetic models of varying complexity to determine the limitations on number of degrees of freedom, choice of initial model, and iterations needed. The field expansion method achieves significant computational saving through restricting the velocity model to consist of a series of non-overlapping layers. Although this parameterization severely restricts the velocity models we can simulate, we find that the model is sufficiently accurate to estimate initial models for FWI as demonstrated later in our paper. FORWARD MODEL Unlike gradient based minimization methods, global optimization and uncertainty quantification methods require far fewer degrees of freedom, tens versus thousands. In this section we briefly describe the reduced parametrization previously used by the authors in Ely et al. (2018) for uncertainty quantification. The reduced parameterization is compatible with the field expansion method (Malcolm and Nicholls, 2011a), allowing for extremely rapid forward modeling of the Helmholtz equation. We briefly describe the field expansion method and the modifications we make to the forward solver to mitigate several of the artifacts inherent to the field expansion method. Reduced parameterization The field expansion method is able to achieve rapid forward solves by heavily restricting the velocity model to consists of a number of non-overlapping piecewise constant layers as shown in Figure 1. Although the Earth has often been approximated by serie

Time lapse seismic monitoring usually involves looking for small changes in localized regions. Quantifying the uncertainty related to these changes is important because it affects exploration and production decisions. Traditional methods that use pixel by pixel quantification with large models are computationally infeasible. We use a local acoustic solver in the area of interest, which allows for fast computation of the wavefield solves. This allows us to use a Metropolis Hastings algorithm in a Bayesian inversion to address the uncertainty that is present in the estimation of 4D velocity changes. Presentation Date: Tuesday, October 16, 2018 Start Time: 8:30:00 AM Location: 204C (Anaheim Convention Center) Presentation Type: Oral

We typically form seismic images from noisy data and potentially inaccurate velocity models. Uncertainties in both the observed data and velocity model directly propagate into the formation of an image and its interpretation. In this paper we present a Bayesian framework for uncertainty quantification of velocity models and seismic images using low frequency waveform data. We use the field expansion method, a fast Helmholtz solver, that achieves significant computational savings through a reduced parameterization in which we restrict the velocity model to consist of a series of perturbed layers that approximate a gradient. This reduced dimensionality and efficient modeling allow us to calculate the tens of thousands of forward solves needed for non-parametric quantification of uncertainty using the adaptive Metropolis-Hastings algorithm. We then zero-offset migrate the data with this distribution of velocity models to generate uncertainty estimates of seismic images and quantities of interest such as reflector depth and potential reservoir's cross sectional area. Presentation Date: Wednesday, September 27, 2017 Start Time: 11:25 AM Location: 371A Presentation Type: ORAL

This paper presents strategies for spectral de- noising of hyperspectral images and 3-D data cube reconstruction from a limited number of tomographic measurements, arising in single snapshot imaging systems. For de-noising the main idea is to exploit the incoherency between the algebraic complexity measure, namely the low rank of the noise-free hyperspectral data cube, and the sparsity structure of the spectral noise. In particular, the non-noisy spectral data, when stacked across the spectral dimension, exhibits low-rank due to a small number of species. On the other hand, under the same representation, the spectral noise exhibits a banded structure. Motivated by this we show that the de-noised spectral data and the unknown spectral noise and the respective bands can be simultaneously estimated through the use of a low-rank and simultaneous sparse minimization operation without prior knowledge of the noisy bands. This result is novel for hyperspectral imaging applications and we compare our results with several existing methods for noisy band recovery. For recovery under limited tomographic projections we exploit both the low algebraic and structural complexity of the data cube via joint rank penalization plus Total Variation/wavelet domain sparsity, which is novel for single snapshot hyperspectral imaging systems. We combine these two approaches for simultaneous spectral de-noising and data cube recovery under limited measurements. We perform extensive simulations and our main result indicates that exploiting both low algebraic and structural complexity has a superior performance compared to exploiting only the structural complexity. To address the computational challenges associated with the resulting optimization problem we adapt several recent developments in the area of convex optimization, specifically employing splitting and proximal point based methods.

In this paper, we combine a fast wave equation solver using boundary integral methods with a global optimization method, namely Particle Swarm Optimization (PSO), to estimate an initial velocity model. Unlike finite difference methods that discretize the model space into pixels or voxels, our forward solver achieves significant computational savings by constraining the model space to a layered model with perturbations. The speed and reduced model space of the forward solve allows us to use global optimization methods that typically require numerous evaluations and few unknown variables. Our technique does not require an initial guess of a velocity model and is robust to local minima, unlike gradient descent frequently used in methods for both initial velocity model estimation and full waveform inversion. We apply our inversion algorithm to several synthetic data sets and demonstrate how prior information can be used to greatly improve the inversion.

We have developed a novel strategy for simultaneous interpolation and denoising of prestack seismic data. Most seismic surveys fail to cover all possible source-receiver combinations, leading to missing data especially in the mid-point-offset domain. This undersampling can complicate certain data processing steps such as amplitude-variation-with-offset analysis and migration. Data interpolation can mitigate the impact of missing traces. We considered the prestack data as a 5D multidimensional array or otherwise referred to as a 5D tensor. Using synthetic data sets, we first found that prestack data can be well approximated by a low-rank tensor under a recently proposed framework for tensor singular value decomposition (tSVD). Under this low-rank assumption, we proposed a complexity-penalized algorithm for the recovery of missing traces and data denoising. In this algorithm, the complexity regularization was controlled by tuning a single regularization parameter using a statistical test. We tested the performance of the proposed algorithm on synthetic and real data to show that missing data can be reliably recovered under heavy downsampling. In addition, we demonstrated that compressibility, i.e., approximation of the data by a low-rank tensor, of seismic data under tSVD depended on the velocity model complexity and shot and receiver spacing. We further found that compressibility correlated with the recovery of missing data because high compressibility implied good recovery and vice versa.

In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].