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Ranksparsity incoherence for matrix decomposition
, 2010
"... Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown lowrank matrix. Our goal is to decompose the given matrix into its sparse and lowrank components. Such a problem arises in a number of applications in model and system identification, and is intractable ..."
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Cited by 229 (23 self)
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Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown lowrank matrix. Our goal is to decompose the given matrix into its sparse and lowrank components. Such a problem arises in a number of applications in model and system identification, and is intractable to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the ℓ1 norm and the nuclear norm of the components. We develop a notion of ranksparsity incoherence, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature with the tangent spaces to the algebraic varieties of sparse and lowrank matrices playing a prominent role. When the sparse and lowrank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems.
Joint MAP registration and highresolution image estimation using a sequence of undersampled images
 IEEE Transactions on Image Processing
, 1997
"... Abstract — In many imaging systems, the detector array is not sufficiently dense to adequately sample the scene with the desired field of view. This is particularly true for many infrared focal plane arrays. Thus, the resulting images may be severely aliased. This paper examines a technique for esti ..."
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Cited by 214 (2 self)
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Abstract — In many imaging systems, the detector array is not sufficiently dense to adequately sample the scene with the desired field of view. This is particularly true for many infrared focal plane arrays. Thus, the resulting images may be severely aliased. This paper examines a technique for estimating a highresolution image, with reduced aliasing, from a sequence of undersampled frames. Several approaches to this problem have been investigated previously. However, in this paper a maximum a posteriori (MAP) framework for jointly estimating image registration parameters and the highresolution image is presented. Several previous approaches have relied on knowing the registration parameters a priori or have utilized registration techniques not specifically designed to treat severely aliased images. In the proposed method, the registration parameters are iteratively updated along with the highresolution image in a cyclic coordinatedescent optimization procedure. Experimental results are provided to illustrate the performance of the proposed MAP algorithm using both visible and infrared images. Quantitative error analysis is provided and several images are shown for subjective evaluation. Index Terms—Aliased, high resolution, image registration, image sequence, MAP estimation. I.
Greedy solution of illposed problems: Error bounds and exact inversion
, 2009
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A Frequency Analysis of Light Transport
, 2005
"... We present a signalprocessing framework for light transport. We study the frequency content of radiance and how it is altered by phenomena such as shading, occlusion, and transport. This extends previous work that considered either spatial or angular dimensions, and it offers a comprehensive treatm ..."
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Cited by 110 (19 self)
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We present a signalprocessing framework for light transport. We study the frequency content of radiance and how it is altered by phenomena such as shading, occlusion, and transport. This extends previous work that considered either spatial or angular dimensions, and it offers a comprehensive treatment of both space and angle. We show that occlusion, a multiplication in the primal, amounts in the Fourier domain to a convolution by the spectrum of the blocker. Propagation corresponds to a shear in the spaceangle frequency domain, while reflection on curved objects performs a different shear along the angular frequency axis. As shown by previous work, reflection is a convolution in the primal and therefore a multiplication in the Fourier domain. Our work shows how the spatial components of lighting are affected by this angular convolution. Our framework predicts the characteristics of interactions such as caustics and the disappearance of the shadows of small features. Predictions on the frequency content can then be used to control sampling rates for rendering. Other potential applications include precomputed radiance transfer and inverse rendering.
Depth from Defocus: A Spatial Domain Approach
 International Journal of Computer Vision
, 1994
"... A new method named STM is described for determining distance of objects and rapid autofocusing of camera systems. STM uses image defocus information and is based on a new SpatialDomain Convolution/Deconvolution Transform. The method requires only two images taken with dierent camera parameters ..."
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Cited by 100 (13 self)
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A new method named STM is described for determining distance of objects and rapid autofocusing of camera systems. STM uses image defocus information and is based on a new SpatialDomain Convolution/Deconvolution Transform. The method requires only two images taken with dierent camera parameters such as lens position, focal length, and aperture diameter. Both images can be arbitrarily blurred and neither of them needs to be a focused image. Therefore STM is very fast in comparison with DepthfromFocus methods which search for the lens position or focal length of best focus. The method involves simple local operations and can be easily implemented in parallel to obtain the depthmap of a scene. STM has been implemented on an actual camera system named SPARCS. Experiments on the performance of STM and their results on realworld planar objects are presented. The results indicate that the accuracy of STM compares well with DepthfromFocus methods and is useful in practical ap...
Iterative methods for image deblurring
 Proceedings of the IEEE
, 1990
"... This tutorial paper discusses the use of iterative restoration algorithms for the removal of linear blurs from photographic Images which may also be assumed to be degraded by pointwise nonlineariries such as film saturation and additive noise. lterative algorithms are particularly attractive for ..."
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Cited by 95 (2 self)
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This tutorial paper discusses the use of iterative restoration algorithms for the removal of linear blurs from photographic Images which may also be assumed to be degraded by pointwise nonlineariries such as film saturation and additive noise. lterative algorithms are particularly attractive for this application because they allow for the incorporation of various types of prior knowledge about the class of feasible solutions, because they can be used to remove nonstationary blurs, and because they are fairly robust with respect to errors in the approximation of the blurring operator. Special attention is given to the problem of convergence of the algorithms, and classical solutions such as inverse filters, Wiener filters, and constrained leastsquares filters are shown to be limiting solutions of variations of the iterations. Regularization is introduced as a means for preventing the excessive noise magnification that is typically associated with illconditioned inverse problems such as the deblurring problem, and it is shown that noise effects can be minimized by terminating the algorithms aftera finite number of iterations. The role and choice of constraints on the class of feasible solutions are also discussed. Ringing artifacts are common with most image restoration methods. It is shown that these artifacts can be significantly reduced both by using constraints and also by making the algorithms spatially adaptive. Some variations on the basic iterations that accelerate the rate of convergence are discussed and numerous examples are presented. I.
Better Optical Triangulation through Spacetime Analysis
 In ICCV
, 1995
"... The standard methods for extracting range data from optical triangulation scanners are accurate only for planar objects of uniform reflectance illuminated by an incoherent source. Using these methods, curved surfaces, discontinuous surfaces, and surfaces of varying reflectance cause systematic disto ..."
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Cited by 91 (8 self)
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The standard methods for extracting range data from optical triangulation scanners are accurate only for planar objects of uniform reflectance illuminated by an incoherent source. Using these methods, curved surfaces, discontinuous surfaces, and surfaces of varying reflectance cause systematic distortions of the range data. Coherent light sources such as lasers introduce speckle artifacts that further degrade the data. We present a new ranging method based on analyzing the time evolution of the structured light reflections. Using our spacetime analysis, we can correct for each of these artifacts, thereby attaining significantly higher accuracy using existing technology. We present results that demonstrate the validity of our method using a commercial laser stripe triangulation scanner. 1 Introduction Active optical triangulation is one of the most common methods for acquiring range data. Although this technology has been in use for over twenty years, its speedandaccuracyhas increasedd...
Parallel Depth Recovery by Changing Camera Parameters
, 1992
"... A new method is described for recovering the distance of objects in a scene from images formed by lenses. The recovery is based on measuring the change in the scene's image due to a known change in the three intrinsic camera parameters: (i) distance between the lens and the image detector, (ii) ..."
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Cited by 86 (14 self)
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A new method is described for recovering the distance of objects in a scene from images formed by lenses. The recovery is based on measuring the change in the scene's image due to a known change in the three intrinsic camera parameters: (i) distance between the lens and the image detector, (ii) focal length of the lens, and (iii) diameter of the lens aperture. The method is parallel involving simple local computations. In comparison with stereo vision and structurefrommotion methods, the correspondence problem does not arise. This method for depthmap recovery may also be used for (i) obtaining focused images (i.e. images having large depth of field) from two images having finite depth of field, and (ii) rapid autofocusing of computer controlled video cameras. 1. Introduction Here we describe a new passive ranging method which in principle is fast and involves relatively weak assumptions that are generally valid. The method is basically a generalized version of the `depthfromfocu...
In Search of a General Picture Processing Operator
 Computer Graphics and Image Processing
, 1978
"... INTRODUCTION Pictorial pattern recognition systems are often described as consisting of three parts: a preprocessing part, a feature extraction part, and a classification part. The preprocessing is used to enhance or sharpen the image to be processed. This is usually done using linear operations or ..."
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Cited by 69 (3 self)
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INTRODUCTION Pictorial pattern recognition systems are often described as consisting of three parts: a preprocessing part, a feature extraction part, and a classification part. The preprocessing is used to enhance or sharpen the image to be processed. This is usually done using linear operations or operations on the gray scale such as thresholding [13] The classification part is fairly well understood [4,5]. The feature extractor, on the other hand, is very much dependent upon the actual problem and no general theory has emerged on how to deal with it. Feature extraction procedures so far have been ad hoc, often referred to as "a bag of tricks." The present work grew out of an interest in finding a single picture operator that could in parallel perform a number of useful operations and that could work on several levels in a hierarchy. One background to this interest is the feeling that the eyes and brains of humans and animals are likely to have such standard operators, as the micro
Trahey G. Acoustic radiation force impulse imaging of thermallyand chemicallyinduced lesions in soft tissues: preliminary ex vivo results. Ultrasound Med Biol 2004;30:321–328. [PubMed: 15063514
"... The clinical viability of a method of acoustic remote palpation, capable of imaging local variations in the mechanical properties of soft tissue using Acoustic Radiation Force Impulse (ARFI) imaging, is investigated in vivo. In this method, focused ultrasound is used to apply localized radiation fo ..."
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Cited by 57 (8 self)
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The clinical viability of a method of acoustic remote palpation, capable of imaging local variations in the mechanical properties of soft tissue using Acoustic Radiation Force Impulse (ARFI) imaging, is investigated in vivo. In this method, focused ultrasound is used to apply localized radiation force to small volumes of tissue (2 mm3) for short durations (less than 1 msec) and the resulting tissue displacements are mapped using ultrasonic correlation based methods. The tissue displacements are inversely proportional to the sti®ness of the tissue, and thus a sti®er region of tissue exhibits smaller displacements than a more compliant region. Due to the short duration of the force application, this method provides information about the mechanical impulse response of the tissue, which re°ects variations in tissue viscoelastic characteristics. In this paper, experimental results are presented demonstrating that displacements on the order of ten microns can be generated and detected in soft tissues in vivo using a single transducer on a modi¯ed diagnostic ultrasound scanner. Di®erences in the magnitude of displacement and the transient response of tissue are correlated with tissue structures in matched Bmode images. The results comprise the ¯rst in vivo ARFI images, and support the clinical feasibility of a radiation force based remote palpation imaging system.