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x-ray projections acquired by the scanner detectors. However, this insufficient projection data which is used to reconstruct the CT image can cause streaking artifacts. Furthermore, using these insufficient projections in standard TV algorithms end up making the problem under-determined and thus leading to infinitely many possible solutions. In this method, an additional penalty weighted function is assigned to the original TV norm. This allows for easier detection of sharp discontinuities in intensity in the images and thereby adapt the weight to store the recovered edge information during the process of signal/image reconstruction. The parameter
1033:
time due to the concavity of the function. Another disadvantage is that this method tends to uniformly penalize the image gradient irrespective of the underlying image structures. This causes over-smoothing of edges, especially those of low contrast regions, subsequently leading to loss of low contrast information. The advantages of this method include: reduction of the sampling rate for sparse signals; reconstruction of the image while being robust to the removal of noise and other artifacts; and use of very few iterations. This can also help in recovering images with sparse gradients.
406:
1232:. The edge-preserving total variation term, thus, becomes sparser and this speeds up the implementation. A two-step iteration process known as forward–backward splitting algorithm is used. The optimization problem is split into two sub-problems which are then solved with the conjugate gradient least squares method and the simple gradient descent method respectively. The method is stopped when the desired convergence has been achieved or if the maximum number of iterations is reached.
2316:
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3594:, sparse sampling, and finite rate of innovation. Its broad scope and generality has enabled several innovative CS-enhanced approaches in signal processing and compression, solution of inverse problems, design of radiating systems, radar and through-the-wall imaging, and antenna characterization. Imaging techniques having a strong affinity with compressive sensing include
827:(CT) reconstruction as a method known as edge-preserving total variation. However, as gradient magnitudes are used for estimation of relative penalty weights between the data fidelity and regularization terms, this method is not robust to noise and artifacts and accurate enough for CS image/signal reconstruction and, therefore, fails to preserve smaller structures.
1241:
profiles of the reconstructed images, it can be seen that there are sharp jumps at edge points and negligible, minor fluctuation at non-edge points. Thus, this method leads to low relative error and higher correlation as compared to the TV method. It also effectively suppresses and removes any form of image noise and image artifacts such as streaking.
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421:
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the given image. In the second stage, the CS reconstruction model is presented by utilizing directional TV regularizer. More details about these TV-based approaches – iteratively reweighted l1 minimization, edge-preserving TV and iterative model using directional orientation field and TV- are provided below.
823:-minimization which uses an iterative scheme. This method, though fast, subsequently leads to over-smoothing of edges resulting in blurred image edges. TV methods with iterative re-weighting have been implemented to reduce the influence of large gradient value magnitudes in the images. This has been used in
1171:
This is an iterative CT reconstruction algorithm with edge-preserving TV regularization to reconstruct CT images from highly undersampled data obtained at low dose CT through low current levels (milliampere). In order to reduce the imaging dose, one of the approaches used is to reduce the number of
830:
Recent progress on this problem involves using an iteratively directional TV refinement for CS reconstruction. This method would have 2 stages: the first stage would estimate and refine the initial orientation field – which is defined as a noisy point-wise initial estimate, through edge-detection, of
3637:
has applied the LASSO model- for selection of sparse models- towards analog to digital converters (the current ones use a sampling rate higher than the
Nyquist rate along with the quantized Shannon representation). This would involve a parallel architecture in which the polarity of the analog signal
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minimization. One of the earliest applications of such an approach was in reflection seismology which used sparse reflected signals from band-limited data for tracking changes between sub-surface layers. When the LASSO model came into prominence in the 1990s as a statistical method for selection of
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index (SSIM) metrics and known ground-truth images for testing performance, it is concluded that iterative directional total variation has a better reconstructed performance than the non-iterative methods in preserving edge and texture areas. The orientation field refinement model plays a major role
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minimization models are used. Other approaches also include the least-squares as has been discussed before in this article. These methods are extremely slow and return a not-so-perfect reconstruction of the signal. The current CS Regularization models attempt to address this problem by incorporating
411:
In order to choose a solution to such a system, one must impose extra constraints or conditions (such as smoothness) as appropriate. In compressed sensing, one adds the constraint of sparsity, allowing only solutions which have a small number of nonzero coefficients. Not all underdetermined systems
70:
is to reconstruct a signal from a series of sampling measurements. In general, this task is impossible because there is no way to reconstruct a signal during the times that the signal is not measured. Nevertheless, with prior knowledge or assumptions about the signal, it turns out to be possible to
1023:
norm. An additional parameter, usually to avoid any sharp transitions in the penalty function curve, is introduced into the iterative equation to ensure stability and so that a zero estimate in one iteration does not necessarily lead to a zero estimate in the next iteration. The method essentially
1032:
Early iterations may find inaccurate sample estimates, however this method will down-sample these at a later stage to give more weight to the smaller non-zero signal estimates. One of the disadvantages is the need for defining a valid starting point as a global minimum might not be obtained every
1240:
Some of the disadvantages of this method are the absence of smaller structures in the reconstructed image and degradation of image resolution. This edge preserving TV algorithm, however, requires fewer iterations than the conventional TV algorithm. Analyzing the horizontal and vertical intensity
328:
of the signal in question and not its highest frequency. This is a misconception, because the sampling theorem guarantees perfect reconstruction given sufficient, not necessary, conditions. A sampling method fundamentally different from classical fixed-rate sampling cannot "violate" the sampling
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Compressed sensing addresses the issue of high scan time by enabling faster acquisition by measuring fewer
Fourier coefficients. This produces a high-quality image with relatively lower scan time. Another application (also discussed ahead) is for CT reconstruction with fewer X-ray projections.
764:
where the underlying principle is that signals with excessive details have high total variation and that removing these details, while retaining important information such as edges, would reduce the total variation of the signal and make the signal subject closer to the original signal in the
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employed the technique in a lensless single-pixel camera that takes stills using repeated snapshots of randomly chosen apertures from a grid. Image quality improves with the number of snapshots, and generally requires a small fraction of the data of conventional imaging, while eliminating
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1983:
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Compressed sensing has been used in an experimental mobile phone camera sensor. The approach allows a reduction in image acquisition energy per image by as much as a factor of 15 at the cost of complex decompression algorithms; the computation may require an off-device implementation.
1299:, is obtained. This noisy orientation field is defined so that it can be refined at a later stage to reduce the noise influences in orientation field estimation. A coarse orientation field estimation is then introduced based on structure tensor, which is formulated as:
2205:
and augmented
Lagrangian (FFT-based fast solver with a closed form solution) methods. It (Augmented Lagrangian) is considered equivalent to the split Bregman iteration which ensures convergence of this method. The orientation field, d is defined as being equal to
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with respect to these variables. The
Lagrangian multipliers are then updated and the iterative process is stopped when convergence is achieved. For the iterative directional total variation refinement model, the augmented lagrangian method involves initializing
1877:
To overcome this drawback, a refined orientation model is defined in which the data term reduces the effect of noise and improves accuracy while the second penalty term with the L2-norm is a fidelity term which ensures accuracy of initial coarse estimation.
1052:
refers to the second step of the iterative reconstruction process wherein it utilizes the edge-preserving total variation regularization term to remove noise and artifacts, and thus improve the quality of the reconstructed image/signal. The minimization of
1249:
To prevent over-smoothing of edges and texture details and to obtain a reconstructed CS image which is accurate and robust to noise and artifacts, this method is used. First, an initial estimate of the noisy point-wise orientation field of the image
517:
To enforce the sparsity constraint when solving for the underdetermined system of linear equations, one can minimize the number of nonzero components of the solution. The function counting the number of non-zero components of a vector was called the
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Compressed sensing, in this case, removes the high spatial gradient parts – mainly, image noise and artifacts. This holds tremendous potential as one can obtain high-resolution CT images at low radiation doses (through lower current-mA settings).
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1302:
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sparse models, this method was further used in computational harmonic analysis for sparse signal representation from over-complete dictionaries. Some of the other applications include incoherent sampling of radar pulses. The work by
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showed that the number of these compressive measurements can be small and still contain nearly all the useful information. Therefore, the task of converting the image back into the intended domain involves solving an underdetermined
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54:, which requires the signal to be sparse in some domain. The second one is incoherence, which is applied through the isometric property, which is sufficient for sparse signals. Compressed sensing has applications in, for example,
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matrix. ω points in the direction of the dominant orientation having the largest contrast and υ points in the direction of the structure orientation having the smallest contrast. The orientation field coarse initial estimation
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being set to high values to account for the unknown noise levels. For every pixel (i,j) in the image, the structure tensor J is a symmetric and positive semi-definite matrix. Convolving all the pixels in the image with
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1884:
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sparsity priors of the original image, one of which is the total variation (TV). Conventional TV approaches are designed to give piece-wise constant solutions. Some of these include (as discussed ahead) – constrained
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since the number of compressive measurements taken is smaller than the number of pixels in the full image. However, adding the constraint that the initial signal is sparse enables one to solve this underdetermined
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of linear equations have a sparse solution. However, if there is a unique sparse solution to the underdetermined system, then the compressed sensing framework allows the recovery of that solution.
3811:, compressive sensing combined with random scanning of the electron beam has enabled both faster acquisition and less electron dose, which allows for imaging of electron beam sensitive materials.
297:-norm was also used in signal processing, for example, in the 1970s, when seismologists constructed images of reflective layers within the earth based on data that did not seem to satisfy the
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minimization designed to more democratically penalize nonzero coefficients. An iterative algorithm is used for constructing the appropriate weights. Each iteration requires solving one
377:
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in this improvement in performance as it increases the number of directionless pixels in the flat area while enhancing the orientation field consistency in the regions with edges.
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from the previous iteration (in order to check for convergence and the subsequent optical performance, the previous iteration is used). For the two vector fields represented by
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2005:
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1456:{\displaystyle J_{\rho }(\nabla I_{\sigma })=G_{\rho }*(\nabla I_{\sigma }\otimes \nabla I_{\sigma })={\begin{pmatrix}J_{11}&J_{12}\\J_{12}&J_{22}\end{pmatrix}}}
1297:
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of the fan-beam geometry, which is constrained by the data fidelity term. This may contain noise and artifacts as no regularization is performed. The minimization of
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Zhang, Y.; Wang, S. (2015). "Exponential
Wavelet Iterative Shrinkage Thresholding Algorithm with Random Shift for Compressed Sensing Magnetic Resonance Imaging".
2964:) is calculated. And as in the field refinement model, the lagrangian multipliers are updated and the iterative process is stopped when convergence is achieved.
46:. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the
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Goldluecke, B.; Strekalovskiy, E.; Cremers, D.; Siims, P.-T. A. I. (2012). "The natural vectorial total variation which arises from geometric measure theory".
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is changed adaptively based on the values of the histogram of the gradient magnitude so that a certain percentage of pixels have gradient values larger than
4903:
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followed by their subsequent addition. These equations are reduced to a series of convex minimization problems which are then solved with a combination of
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Conventional CS reconstruction uses sparse signals (usually sampled at a rate less than the
Nyquist sampling rate) for reconstruction through constrained
5328:
4281:
514:
of the basis sampled in). However, this leads to poor results for many practical applications, for which the unknown coefficients have nonzero energy.
346:
of linear equations has more unknowns than equations and generally has an infinite number of solutions. The figure below shows such an equation system
329:
theorem. Sparse signals with high frequency components can be highly under-sampled using compressed sensing compared to classical fixed-rate sampling.
4106:
Candès, E.J., & Plan, Y. (2010). A Probabilistic and RIPless Theory of
Compressed Sensing. IEEE Transactions on Information Theory, 57, 7235–7254.
436:, that is, they contain many coefficients close to or equal to zero, when represented in some domain. This is the same insight used in many forms of
3675:
one can infer from a single hologram. It is also used for image retrieval from undersampled measurements in optical and millimeter-wave holography.
2420:{\displaystyle \min _{\mathrm {X} }\lVert \nabla \mathrm {X} \bullet d\rVert _{1}+{\frac {\lambda }{2}}\ \lVert Y-\Phi \mathrm {X} \rVert _{2}^{2}}
1978:{\displaystyle \min _{\mathrm {X} }\lVert \nabla \mathrm {X} \bullet d\rVert _{1}+{\frac {\lambda }{2}}\ \lVert Y-\Phi \mathrm {X} \rVert _{2}^{2}}
1057:
is done through a simple gradient descent method. Convergence is determined by testing, after each iteration, for image positivity, by checking if
5265:
Fernandez Cull, Christy; Wikner, David A.; Mait, Joseph N.; Mattheiss, Michael; Brady, David J. (2010). "Millimeter-wave compressive holography".
424:
Example of the retrieval of an unknown signal (gray line) from few measurements (black dots) using the knowledge that the signal is sparse in the
4059:
Donoho, David L. (2006). "For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution".
3783:, full coverage of the Fourier plane is usually absent and phase information is not obtained in most hardware configurations. In order to obtain
2007:
is the objective signal which needs to be recovered. Y is the corresponding measurement vector, d is the iterative refined orientation field and
90:. The main idea is that with prior knowledge about constraints on the signal's frequencies, fewer samples are needed to reconstruct the signal.
3763:
In 2013 one company announced shortwave-infrared cameras which utilize compressed sensing. These cameras have light sensitivity from 0.9
703:
is preferred over linear programming, since it preserves sparsity in the face of noise and can be solved faster than an exact linear program.
4307:"Atomic decomposition by basis pursuit", by Scott Shaobing Chen, David L. Donoho, Michael, A. Saunders. SIAM Journal on Scientific Computing
4117:
113:, the signal may be reconstructed with even fewer samples than the sampling theorem requires. This idea is the basis of compressed sensing.
4495:
Xuan Fei; Zhihui Wei; Liang Xiao (2013). "Iterative
Directional Total Variation Refinement for Compressive Sensing Image Reconstruction".
432:
Compressed sensing takes advantage of the redundancy in many interesting signals—they are not pure noise. In particular, many signals are
2802:
1881:
This orientation field is introduced into the directional total variation optimization model for CS reconstruction through the equation:
87:
5086:
4224:
5532:
Zhang, Y. (2015). "Exponential
Wavelet Iterative Shrinkage Thresholding Algorithm for Compressed Sensing Magnetic Resonance Imaging".
5143:
Marim, M.; Angelini, E.; Olivo-Marin, J. C.; Atlan, M. (2011). "Off-axis compressed holographic microscopy in low-light conditions".
3944:
3808:
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Marim, M. M.; Atlan, M.; Angelini, E.; Olivo-Marin, J. C. (2010). "Compressed sensing with off-axis frequency-shifting holography".
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changes at a high rate followed by digitizing the integral at the end of each time-interval to obtain the converted digital signal.
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Compressed sensing typically starts with taking a weighted linear combination of samples also called compressive measurements in a
1192:
controls the amount of smoothing applied to the pixels at the edges to differentiate them from the non-edge pixels. The value of
405:
5676:
2510:
86:
signal's highest frequency is less than half of the sampling rate, then the signal can be reconstructed perfectly by means of
75:). Over time, engineers have improved their understanding of which assumptions are practical and how they can be generalized.
4967:
1874:= Ď…. This estimate is accurate at strong edges. However, at weak edges or on regions with noise, its reliability decreases.
4383:
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321:
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79:
47:
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The field of compressive sensing is related to several topics in signal processing and computational mathematics, such as
5807:
3791:
has been in use since 1974 for the reconstruction of images obtained from radio interferometers, which is similar to the
2967:
For the orientation field refinement model, the
Lagrangian multipliers are updated in the iterative process as follows:
5837:
5688:
Stevens, Andrew; Kovarik, Libor; Abellan, Patricia; Yuan, Xin; Carin, Lawrence; Browning, Nigel D. (13 August 2015).
5842:
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For the iterative directional total variation refinement model, the Lagrangian multipliers are updated as follows:
4911:
3803:
Compressed sensing combined with a moving aperture has been used to increase the acquisition rate of images in a
996:
minimization problem by finding the local minimum of a concave penalty function that more closely resembles the
349:
761:
699:, for which efficient solution methods already exist. When measurements may contain a finite amount of noise,
506:—that is, minimize the amount of energy in the system. This is usually simple mathematically (involving only a
5318:
71:
perfectly reconstruct a signal from a series of measurements (acquiring this series of measurements is called
5655:
3825:
2134:
72:
4874:
5822:
5723:"Implementing an accurate and rapid sparse sampling approach for low-dose atomic resolution STEM imaging"
4737:
Andrea Massa; Paolo Rocca; Giacomo Oliveri (2015). "Compressive Sensing in Electromagnetics – A Review".
3780:
2731:
1155:
refers to the different x-ray linear attenuation coefficients at different voxels of the patient image).
1632:
1244:
5827:
4850:
4319:"Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Fourier Information"
3748:
3696:
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is the CS measurement matrix. This method undergoes a few iterations ultimately leading to convergence.
470:
259:
55:
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5287:
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Candes, E. J.; Wakin, M. B.; Boyd, S. P. (2008). "Enhancing sparsity by reweighted l1 minimization".
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712:
263:
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4348:
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3354:{\displaystyle (\lambda _{P})^{k}=(\lambda _{P})^{k-1}+\gamma _{P}P^{k}-\nabla (\mathrm {X} )^{k})}
2319:
Augmented Lagrangian method for orientation field and iterative directional field refinement models
2209:
1533:
738:
228:
2681:
2631:
2162:
2092:
1988:
1490:
refers to the structure tensor related with the image pixel point (i,j) having standard deviation
1099:
382:
5832:
4166:
2255:
700:
4318:
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2030:
1848:
1819:
1273:
1024:
involves using the current solution for computing the weights to be used in the next iteration.
5282:
5049:
4709:
4672:
4343:
3903:) is constant as α approaches zero. Unfortunately, authors now neglect the quotation marks and
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999:
972:
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918:
891:
771:
745:
444:
343:
43:
4936:. 2013 IEEE International Conference on Image Processing. Vol. 2393. pp. 2101–2105.
4228:
3566:
3226:{\displaystyle (\lambda _{V})^{k}=(\lambda _{V})^{k-1}+\gamma _{V}(V^{k}-\nabla (d_{v})^{k})}
3093:{\displaystyle (\lambda _{H})^{k}=(\lambda _{H})^{k-1}+\gamma _{H}(H^{k}-\nabla (d_{h})^{k})}
2868:
2573:
2059:
1757:
1592:
1215:
1195:
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507:
150:
techniques, which several other scientific fields have used historically. In statistics, the
5781:
5690:"Applying compressive sensing to TEM video: a substantial frame rate increase on any camera"
3934:
norm for the space of measurable functions (equipped with an appropriate metric) or for the
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5274:
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5101:
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4504:
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4142:
3910:
3904:
3867:
3835:
3608:
2906:
2482:
2010:
1572:
1493:
1040:
refers to the first-step of the iterative reconstruction process, of the projection matrix
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671:
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617:
588:
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522:
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273:
234:
203:
185:
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125:
39:
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different from the basis in which the signal is known to be sparse. The results found by
8:
3477:{\displaystyle (\lambda _{Q})^{k}=(\lambda _{Q})^{k-1}+\gamma _{Q}(Q^{k}-P^{k}\bullet d)}
749:
425:
5738:
5611:
5490:
5404:
5278:
5227:
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5105:
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4951:
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4595:
Combettes, P; Wajs, V (2005). "Signal recovery by proximal forward-backward splitting".
4508:
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5509:
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Lustig, M.; Donoho, D.L.; Santos, J.M.; Pauly, J.M. (2008). "Compressed Sensing MRI;".
5373:
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Brox, T.; Weickert, J.; Burgeth, B.; Mrázek, P. (2006). "Nonlinear structure tensors".
4612:
4568:
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4436:
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4294:
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4207:
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3755:, network routing matrices usually satisfy the criterion for using compressed sensing.
3744:
3736:
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2711:
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2611:
2295:
2202:
2184:
2114:
1798:
1778:
1737:
1662:
1612:
1513:
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1138:
189:
5797:
4458:
2159:
refers to the multiplication of respective horizontal and vertical vector elements of
551:
448:
94:
5623:
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5514:
5365:
5300:
5239:
5178:
5117:
5014:
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Iterative model using a directional orientation field and directional total variation
1167:
Flow diagram figure for edge-preserving total-variation method for compressed sensing
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Denis, Loic; Lorenz, Dirk; Thibaut, Eric; Fournier, Corinne; Trede, Dennis (2009).
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306:
302:
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Brady, David; Choi, Kerkil; Marks, Daniel; Horisaki, Ryoichi; Lim, Sehoon (2009).
4261:
5791:
4686:
3776:
3651:
753:
727:
465:
4780:
Taylor, H.L.; Banks, S.C.; McCoy, J.F. (1979). "Deconvolution with the 1 norm".
4585:
Lange, K.: Optimization, Springer Texts in Statistics. Springer, New York (2004)
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1158:
4425:"Low-dose CT reconstruction via edge preserving total variation regularization"
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696:
511:
452:
193:
98:
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5689:
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5438:
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Rivenson, Y.; Stern, A.; Javidi, B. (2010). "Compressive fresnel holography".
4959:
4564:
420:
50:. There are two conditions under which recovery is possible. The first one is
5816:
5063:
4758:
4516:
3840:
3751:
where the coefficient matrix is the network routing matrix. Moreover, in the
3740:
3587:
3583:
433:
325:
314:
151:
5619:
5439:"Fill in the Blanks: Using Math to Turn Lo-Res Datasets Into Hi-Res Samples"
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4357:
4203:
5627:
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scanning sessions on conventional hardware. Reconstruction methods include
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2056:
is the orientation field approximate estimation of the reconstructed image
752:(for the case of functions of several variables). For signals, especially,
731:
460:
106:
5596:"Majorization–minimization algorithms for wavelet-based image restoration"
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5235:
5174:
5113:
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4488:
4486:
742:
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102:
83:
5344:"Sparse MRI: The application of compressed sensing for rapid MR imaging"
4253:
3864:
The quotation marks served two warnings. First, the number-of-nonzeros
3731:
Compressed sensing has showed outstanding results in the application of
614:, in a technical sense: This equivalence result allows one to solve the
5721:
Kovarik, L.; Stevens, A.; Liyu, A.; Browning, N. D. (17 October 2016).
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The Optimistic Bayesian: Replica Method Analysis of Compressed Sensing
4154:
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4338:
4137:
3655:
1163:
5722:
4118:"Stable signal recovery from incomplete and inaccurate measurements"
760:
of the signal. In signal and image reconstruction, it is applied as
5595:
5481:
4699:
3846:
Verification-based message-passing algorithms in compressed sensing
3830:
3752:
2858:{\displaystyle \lambda _{H},\lambda _{V},\lambda _{P},\lambda _{Q}}
757:
122:
110:
51:
5663:
5218:
5157:
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4555:
4441:
3667:
Compressed sensing can be used to improve image reconstruction in
1734:
The structure tensor obtained is convolved with a Gaussian kernel
5264:
3939:
3935:
3892:
585:
555:
519:
477:
155:
5775:
3787:
images, various compressed sensing algorithms are employed. The
4882:
4032:{\displaystyle (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}}
3546:{\displaystyle \gamma _{H},\gamma _{V},\gamma _{P},\gamma _{Q}}
1048:
is solved through the conjugate gradient least squares method.
915:
minimization, larger coefficients are penalized heavily in the
748:(for the case of functions of one variable) or on the space of
476:
The least-squares solution to such problems is to minimize the
5677:
Compressed sensing imaging techniques for radio interferometry
4317:
Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence (2006).
4116:
Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence (2006).
1731:
refers to the tensor product obtained by using this gradient.
1724:{\displaystyle (\nabla I_{\sigma }\otimes \nabla I_{\sigma })}
5203:
5142:
3672:
1159:
Edge-preserving total variation (TV)-based compressed sensing
554:
et al. proved that for many problems it is probable that the
5776:
SigView, the IEEE Signal Processing Society Tutorial Library
4494:
3764:
2323:
The Augmented Lagrangian method for the orientation field,
5593:
5384:
4932:
Gang Huang; Hong Jiang; Kim Matthews; Paul Wilford (2013).
4632:"Methods of conjugate gradients for solving linear systems"
4536:
4534:
4423:
Tian, Z.; Jia, X.; Yuan, K.; Pan, T.; Jiang, S. B. (2011).
5720:
5561:
IEEJ Transactions on Electrical and Electronic Engineering
5087:"Inline hologram reconstruction with sparsity constraints"
5084:
2560:{\displaystyle \mathrm {X} ,P,Q,\lambda _{P},\lambda _{Q}}
706:
320:
At first glance, compressed sensing might seem to violate
5687:
4662:
1754:
to improve the accuracy of the orientation estimate with
839:
38:
technique for efficiently acquiring and reconstructing a
5594:
Figueiredo, M.; Bioucas-Dias, J.M.; Nowak, R.D. (2007).
4531:
3767:
to 1.7 ÎĽm, wavelengths invisible to the human eye.
3650:
Compressed sensing is used in single-pixel cameras from
1629:
below which the edge detection is insensitive to noise.
942:
norm. It was proposed to have a weighted formulation of
58:
where the incoherence condition is typically satisfied.
5390:
4636:
Journal of Research of the National Bureau of Standards
3895:, because it is not continuous in its scalar argument:
1609:
refers to the manually defined parameter for the image
853:
5782:
Using Math to Turn Lo-Res Datasets Into Hi-Res Samples
5465:"Energy Preserved Sampling for Compressed Sensing MRI"
4982:
4773:
1394:
863:
802:
266:. It was used by Peter J. Huber and others working on
5774:: video tutorial by Mark Davenport, Georgia Tech. at
3947:
3913:
3870:
3611:
3493:
3370:
3245:
3109:
2976:
2936:
2909:
2871:
2805:
2779:
2759:
2734:
2714:
2684:
2664:
2634:
2614:
2576:
2513:
2485:
2433:
2329:
2298:
2258:
2212:
2187:
2165:
2137:
2117:
2095:
2062:
2033:
2013:
1991:
1887:
1851:
1822:
1801:
1781:
1760:
1740:
1685:
1665:
1635:
1615:
1595:
1575:
1536:
1516:
1496:
1469:
1305:
1276:
1256:
1218:
1198:
1178:
1141:
1102:
1063:
1002:
975:
948:
921:
894:
774:
674:
647:
620:
591:
561:
525:
483:
385:
352:
276:
237:
206:
161:
128:
4382:, IEEE Signal Processing Magazine, V.21, March 2008
768:
For the purpose of signal and image reconstruction,
428:
basis (purple dots show the retrieved coefficients).
5031:
4227:from Vivek Goyal, Alyson Fletcher, Sundeep Rangan,
3798:
2903:. For each iteration, the approximate minimizer of
415:
78:An early breakthrough in signal processing was the
5469:Computational and Mathematical Methods in Medicine
4809:"Regression shrinkage and selection via the lasso"
4282:Journal of the Royal Statistical Society, Series B
4277:"Regression shrinkage and selection via the lasso"
4031:
3926:
3883:
3624:
3545:
3476:
3353:
3225:
3092:
2956:
2922:
2895:
2857:
2791:
2765:
2745:
2720:
2700:
2670:
2650:
2620:
2600:
2559:
2498:
2471:
2419:
2304:
2284:
2244:
2193:
2173:
2151:
2123:
2103:
2081:
2048:
2019:
1999:
1977:
1866:
1837:
1807:
1795:, gives orthonormal eigen vectors ω and υ of the
1787:
1766:
1746:
1723:
1671:
1651:
1621:
1601:
1581:
1561:
1522:
1502:
1482:
1455:
1291:
1262:
1224:
1204:
1184:
1147:
1127:
1088:
1015:
988:
961:
934:
907:
888:In the CS reconstruction models using constrained
876:
815:
787:
687:
660:
633:
604:
574:
538:
496:
393:
371:
289:
250:
219:
174:
141:
5319:"Engineers Test Highly Accurate Face Recognition"
3747:detection can both be modeled as underdetermined
3556:
1235:
668:problem. Finding the candidate with the smallest
5814:
4875:"Compressive Imaging: A New Single-Pixel Camera"
4779:
4656:
4588:
4316:
4115:
2331:
2292:define the horizontal and vertical estimates of
1889:
337:
4848:
4629:
4540:
3770:
1027:
4422:
4125:Communications on Pure and Applied Mathematics
4061:Communications on Pure and Applied Mathematics
2479:and then finding the approximate minimizer of
717:
5802:What's Happening in the Mathematical Sciences
5462:
4851:"New Camera Chip Captures Only What It Needs"
4594:
3758:
3695:Compressed sensing has been used to shorten
3690:
695:norm can be expressed relatively easily as a
109:proved that given knowledge about a signal's
2403:
2385:
2360:
2342:
1961:
1943:
1918:
1900:
722:
324:, because compressed sensing depends on the
4407:L1-MAGIC is a collection of MATLAB routines
4190:Donoho, D.L. (2006). "Compressed sensing".
5798:Compressed Sensing Makes Every Pixel Count
5762:"The Fundamentals of Compressive Sensing"
5525:
5335:
4806:
4274:
3907:—clashing with the established use of the
372:{\displaystyle \mathbf {y} =D\mathbf {x} }
66:A common goal of the engineering field of
5705:
5558:
5508:
5498:
5480:
5433:
5359:
5286:
5217:
5156:
5053:
5008:
4941:
4713:
4676:
4647:
4579:
4554:
4466:
4440:
4347:
4337:
4136:
4095:"The Fundamentals of Compressive Sensing"
5694:Advanced Structural and Chemical Imaging
4867:
2314:
1162:
852:
419:
4934:Lensless Imaging by Compressive Sensing
4418:
4416:
4414:
4380:An Introduction To Compressive Sampling
4192:IEEE Transactions on Information Theory
756:refers to the integral of the absolute
707:Total variation-based CS reconstruction
5815:
5341:
4739:IEEE Antennas and Propagation Magazine
4240:Hayes, Brian (2009). "The Best Bits".
4189:
4058:
834:
5531:
4800:
4239:
4183:
3726:
3678:
2608:are newly introduced variables where
2152:{\displaystyle \mathrm {X} \bullet d}
379:where we want to find a solution for
4623:
4411:
3821:Compressed sensing in speech signals
3683:Compressed sensing has been used in
1659:refers to the gradient of the image
5587:
4904:"Bell Labs Invents Lensless Camera"
2865:are the Lagrangian multipliers for
2746:{\displaystyle \nabla \mathrm {X} }
13:
5756:
4828:10.1111/j.2517-6161.1996.tb02080.x
4693:
4295:10.1111/j.2517-6161.1996.tb02080.x
3334:
3327:
3194:
3061:
2938:
2739:
2735:
2685:
2635:
2515:
2398:
2394:
2349:
2345:
2336:
2167:
2139:
2097:
2014:
1993:
1956:
1952:
1907:
1903:
1894:
1705:
1689:
1652:{\displaystyle \nabla I_{\sigma }}
1636:
1370:
1354:
1319:
641:problem, which is easier than the
14:
5854:
4378:Candès, E.J., & Wakin, M.B.,
5463:Zhang, Y.; Peterson, B. (2014).
5331:from the original on 2014-01-10.
4630:Hestenes, M; Stiefel, E (1952).
3805:transmission electron microscope
3799:Transmission electron microscopy
3659:lens/focus-related aberrations.
2957:{\displaystyle \mathrm {X} ,P,Q}
416:Solution / reconstruction method
404:
387:
365:
354:
188:. Following the introduction of
80:Nyquist–Shannon sampling theorem
48:Nyquist–Shannon sampling theorem
5714:
5681:
5670:
5648:
5642:
5552:
5456:
5427:
5393:IEEE Signal Processing Magazine
5311:
5258:
5197:
5136:
5078:
5025:
4976:
4925:
4896:
4842:
4730:
4400:
4387:
4372:
4310:
4225:List of L1 regularization ideas
3858:
3573:
2472:{\displaystyle d_{h},d_{v},H,V}
154:method was complemented by the
5348:Magnetic Resonance in Medicine
4849:David Schneider (March 2013).
4497:IEEE Signal Processing Letters
4301:
4268:
4233:
4218:
4109:
4100:
4087:
4052:
4025:
4006:
3964:
3961:
3948:
3641:
3580:underdetermined linear systems
3471:
3439:
3411:
3397:
3385:
3371:
3348:
3339:
3330:
3286:
3272:
3260:
3246:
3220:
3211:
3197:
3178:
3150:
3136:
3124:
3110:
3087:
3078:
3064:
3045:
3017:
3003:
2991:
2977:
2239:
2213:
2040:
1858:
1829:
1718:
1686:
1556:
1537:
1530:refers to the Gaussian kernel
1383:
1351:
1332:
1316:
1283:
762:total variation regularization
44:underdetermined linear systems
1:
5808:Wiki on sparse reconstruction
5788:Compressive Sensing Resources
5034:Journal of Display Technology
4046:
3826:Low-density parity-check code
3662:
2245:{\displaystyle (d_{h},d_{v})}
1562:{\displaystyle (0,\rho ^{2})}
338:Underdetermined linear system
231:. In statistical theory, the
121:Compressed sensing relies on
4910:. 2013-05-25. Archived from
4687:10.1016/j.imavis.2005.09.010
3771:Aperture synthesis astronomy
3671:by increasing the number of
3557:Advantages and disadvantages
2701:{\displaystyle \nabla d_{v}}
2651:{\displaystyle \nabla d_{h}}
2174:{\displaystyle \mathrm {X} }
2104:{\displaystyle \mathrm {X} }
2000:{\displaystyle \mathrm {X} }
1236:Advantages and disadvantages
1128:{\displaystyle f^{k-1}<0}
1028:Advantages and disadvantages
394:{\displaystyle \mathbf {x} }
7:
4459:10.1088/0031-9155/56/18/011
4275:Tibshirani, Robert (1996).
3814:
3795:algorithm mentioned above.
3781:astronomical interferometry
3749:systems of linear equations
2930:with respect to variables (
2285:{\displaystyle d_{h},d_{v}}
1036:In the figure shown below,
718:Motivation and applications
61:
16:Signal processing technique
10:
5859:
5445:. Vol. 18, no. 3
4097:, SigView, April 12, 2013.
3759:Shortwave-infrared cameras
3697:magnetic resonance imaging
3691:Magnetic resonance imaging
3563:peak signal-to-noise ratio
2792:{\displaystyle P\bullet d}
2049:{\displaystyle {\hat {d}}}
1867:{\displaystyle {\hat {d}}}
1838:{\displaystyle {\hat {d}}}
1292:{\displaystyle {\hat {d}}}
884:minimization method for CS
710:
471:system of linear equations
264:median-unbiased estimators
184:, which was introduced by
116:
42:, by finding solutions to
5838:Mathematical optimization
5707:10.1186/s40679-015-0009-3
5600:IEEE Trans. Image Process
5546:10.1016/j.ins.2015.06.017
4960:10.1109/ICIP.2013.6738433
4565:10.1007/s00041-008-9045-x
3600:computational photography
2427:, involves initializing
1483:{\displaystyle J_{\rho }}
1089:{\displaystyle f^{k-1}=0}
1016:{\displaystyle \ell _{0}}
989:{\displaystyle \ell _{1}}
962:{\displaystyle \ell _{1}}
935:{\displaystyle \ell _{1}}
908:{\displaystyle \ell _{1}}
788:{\displaystyle \ell _{1}}
723:Role of TV regularization
713:Total-variation denoising
332:
299:Nyquist–Shannon criterion
5342:Lustig, Michael (2007).
5064:10.1109/jdt.2010.2042276
4985:"Compressive holography"
4759:10.1109/MAP.2015.2397092
4517:10.1109/LSP.2013.2280571
3891:-"norm" is not a proper
3851:
3553:are positive constants.
1569:with standard deviation
741:defined on the space of
229:computational statistics
5843:Mathematics in medicine
5727:Applied Physics Letters
5620:10.1109/tip.2007.909318
5413:10.1109/MSP.2007.914728
4358:10.1109/tit.2005.862083
4326:IEEE Trans. Inf. Theory
4204:10.1109/TIT.2006.871582
2896:{\displaystyle H,V,P,Q}
2601:{\displaystyle H,V,P,Q}
2082:{\displaystyle X^{k-1}}
1767:{\displaystyle \sigma }
1602:{\displaystyle \sigma }
1225:{\displaystyle \sigma }
1205:{\displaystyle \sigma }
1185:{\displaystyle \sigma }
857:Iteratively reweighted
840:Iteratively reweighted
701:basis pursuit denoising
5784:Wired Magazine article
4807:Tibshirani, R (1996).
4597:Multiscale Model Simul
4033:
3928:
3885:
3789:Högbom CLEAN algorithm
3626:
3547:
3478:
3355:
3227:
3094:
2958:
2924:
2897:
2859:
2793:
2767:
2747:
2722:
2702:
2672:
2652:
2622:
2602:
2561:
2500:
2473:
2421:
2320:
2306:
2286:
2246:
2195:
2175:
2153:
2125:
2105:
2083:
2050:
2021:
2001:
1979:
1868:
1839:
1809:
1789:
1768:
1748:
1725:
1673:
1653:
1623:
1603:
1583:
1563:
1524:
1504:
1484:
1457:
1293:
1264:
1226:
1206:
1186:
1168:
1149:
1129:
1090:
1017:
990:
963:
936:
909:
885:
878:
877:{\textstyle \ell _{1}}
817:
816:{\textstyle \ell _{1}}
789:
689:
662:
635:
606:
576:
540:
498:
429:
395:
373:
344:underdetermined system
291:
252:
221:
176:
143:
82:. It states that if a
5800:– article in the AMS
4908:MIT Technology Review
4543:J. Fourier Anal. Appl
4034:
3929:
3927:{\displaystyle L^{0}}
3886:
3884:{\displaystyle L^{0}}
3627:
3625:{\displaystyle l_{1}}
3567:structural similarity
3548:
3479:
3356:
3228:
3095:
2959:
2925:
2923:{\displaystyle L_{2}}
2898:
2860:
2794:
2768:
2748:
2723:
2703:
2673:
2653:
2623:
2603:
2562:
2501:
2499:{\displaystyle L_{1}}
2474:
2422:
2318:
2307:
2287:
2247:
2196:
2176:
2154:
2126:
2106:
2084:
2051:
2022:
2020:{\displaystyle \Phi }
2002:
1980:
1869:
1840:
1810:
1790:
1769:
1749:
1726:
1674:
1654:
1624:
1604:
1584:
1582:{\displaystyle \rho }
1564:
1525:
1505:
1503:{\displaystyle \rho }
1485:
1458:
1294:
1265:
1227:
1207:
1187:
1166:
1150:
1130:
1091:
1018:
991:
964:
937:
910:
879:
856:
818:
790:
690:
688:{\displaystyle L^{1}}
663:
661:{\displaystyle L^{0}}
636:
634:{\displaystyle L^{1}}
607:
605:{\displaystyle L^{0}}
584:is equivalent to the
577:
575:{\displaystyle L^{1}}
541:
539:{\displaystyle L^{0}}
508:matrix multiplication
499:
497:{\displaystyle L^{2}}
423:
396:
374:
292:
290:{\displaystyle L^{1}}
262:and later writers on
253:
251:{\displaystyle L^{1}}
222:
220:{\displaystyle L^{1}}
177:
175:{\displaystyle L^{1}}
144:
142:{\displaystyle L^{1}}
5534:Information Sciences
5297:10.1364/ao.49.000e67
5236:10.1364/ol.35.000871
5175:10.1364/ol.36.000079
5114:10.1364/ol.34.003475
5010:10.1364/oe.17.013040
4649:10.6028/jres.049.044
4395:Metric Linear Spaces
3945:
3911:
3868:
3836:Sparse approximation
3609:
3491:
3368:
3243:
3107:
2974:
2934:
2907:
2869:
2803:
2777:
2757:
2732:
2712:
2682:
2662:
2632:
2612:
2574:
2511:
2483:
2431:
2327:
2296:
2256:
2210:
2185:
2163:
2135:
2115:
2093:
2060:
2031:
2011:
1989:
1885:
1849:
1820:
1799:
1779:
1758:
1738:
1683:
1663:
1633:
1613:
1593:
1573:
1534:
1514:
1494:
1467:
1303:
1274:
1254:
1216:
1196:
1176:
1139:
1100:
1061:
1000:
973:
946:
919:
892:
861:
800:
772:
750:integrable functions
672:
645:
618:
589:
559:
523:
481:
383:
350:
322:the sampling theorem
274:
235:
204:
159:
126:
28:compressive sampling
5739:2016ApPhL.109p4102K
5612:2007ITIP...16.2980F
5500:10.1155/2014/546814
5491:2015CMMM.201514104T
5405:2008ISPM...25...72L
5279:2010ApOpt..49E..67C
5228:2010OptL...35..871M
5167:2011OptL...36...79M
5106:2009OptL...34.3475D
5046:2010JDisT...6..506R
5001:2009OExpr..1713040B
4995:(15): 13040–13049.
4952:2013arXiv1305.7181H
4751:2015IAPM...57..224M
4702:SIAM J. Imaging Sci
4509:2013ISPL...20.1070F
4451:2011PMB....56.5949T
4254:10.1511/2009.79.276
4147:2005math......3066C
2416:
1974:
835:Existing approaches
825:computed tomography
426:Hermite polynomials
24:compressive sensing
5823:Information theory
4816:J. R. Stat. Soc. B
4242:American Scientist
4029:
3976:
3938:of sequences with
3924:
3905:abused terminology
3881:
3785:aperture synthesis
3745:network congestion
3737:network management
3733:network tomography
3727:Network tomography
3685:facial recognition
3679:Facial recognition
3622:
3543:
3474:
3351:
3223:
3090:
2954:
2920:
2893:
2855:
2789:
2763:
2743:
2718:
2698:
2668:
2648:
2618:
2598:
2557:
2496:
2469:
2417:
2402:
2341:
2321:
2302:
2282:
2242:
2203:variable splitting
2191:
2171:
2149:
2121:
2101:
2079:
2046:
2017:
1997:
1975:
1960:
1899:
1864:
1835:
1805:
1785:
1764:
1744:
1721:
1669:
1649:
1619:
1599:
1579:
1559:
1520:
1500:
1480:
1453:
1447:
1289:
1260:
1222:
1202:
1182:
1169:
1145:
1125:
1096:for the case when
1086:
1013:
986:
959:
932:
905:
886:
874:
813:
785:
685:
658:
631:
602:
572:
536:
494:
430:
391:
369:
301:. It was used in
287:
258:-norm was used by
248:
227:-norm was used in
217:
190:linear programming
172:
139:
88:sinc interpolation
20:Compressed sensing
5828:Signal estimation
5747:10.1063/1.4965720
5656:"InView web site"
5606:(12): 2980–2991.
5573:10.1002/tee.22059
5435:Ellenberg, Jordan
5361:10.1002/mrm.21391
5100:(22): 3475–3477.
4969:978-1-4799-2341-0
4794:10.1190/1.1440921
4724:10.1137/110823766
4665:Image Vis. Comput
4609:10.1137/050626090
4503:(11): 1070–1073.
4435:(18): 5949–5967.
4393:Stefan Rolewicz.
4155:10.1002/cpa.20124
4073:10.1002/cpa.20132
3967:
3586:, heavy hitters,
2766:{\displaystyle Q}
2721:{\displaystyle P}
2671:{\displaystyle V}
2621:{\displaystyle H}
2384:
2380:
2330:
2305:{\displaystyle d}
2194:{\displaystyle d}
2124:{\displaystyle d}
2043:
1942:
1938:
1888:
1861:
1832:
1808:{\displaystyle J}
1788:{\displaystyle G}
1747:{\displaystyle G}
1672:{\displaystyle I}
1622:{\displaystyle I}
1523:{\displaystyle G}
1286:
1263:{\displaystyle I}
1148:{\displaystyle f}
730:can be seen as a
548:by David Donoho.
438:lossy compression
311:Robert Tibshirani
268:robust statistics
198:simplex algorithm
68:signal processing
36:signal processing
5850:
5751:
5750:
5718:
5712:
5711:
5709:
5685:
5679:
5674:
5668:
5667:
5662:. Archived from
5652:
5646:
5640:
5639:
5591:
5585:
5584:
5556:
5550:
5549:
5529:
5523:
5522:
5512:
5502:
5484:
5460:
5454:
5453:
5451:
5450:
5431:
5425:
5424:
5388:
5382:
5381:
5363:
5354:(6): 1182–1195.
5339:
5333:
5332:
5315:
5309:
5308:
5290:
5288:10.1.1.1018.5231
5262:
5256:
5255:
5221:
5201:
5195:
5194:
5160:
5140:
5134:
5133:
5091:
5082:
5076:
5075:
5057:
5029:
5023:
5022:
5012:
4980:
4974:
4973:
4945:
4929:
4923:
4922:
4920:
4919:
4900:
4894:
4893:
4891:
4890:
4881:. Archived from
4871:
4865:
4864:
4862:
4861:
4846:
4840:
4839:
4813:
4804:
4798:
4797:
4777:
4771:
4770:
4734:
4728:
4727:
4717:
4697:
4691:
4690:
4680:
4660:
4654:
4653:
4651:
4627:
4621:
4620:
4592:
4586:
4583:
4577:
4576:
4558:
4549:(5–6): 877–905.
4538:
4529:
4528:
4492:
4481:
4480:
4470:
4444:
4420:
4409:
4404:
4398:
4391:
4385:
4376:
4370:
4369:
4351:
4341:
4323:
4314:
4308:
4305:
4299:
4298:
4272:
4266:
4265:
4237:
4231:
4222:
4216:
4215:
4198:(4): 1289–1306.
4187:
4181:
4180:
4178:
4177:
4171:
4165:. Archived from
4140:
4131:(8): 1207–1223.
4122:
4113:
4107:
4104:
4098:
4091:
4085:
4084:
4056:
4040:
4038:
4036:
4035:
4030:
4028:
4024:
4023:
4005:
4000:
3999:
3990:
3989:
3975:
3960:
3959:
3933:
3931:
3930:
3925:
3923:
3922:
3890:
3888:
3887:
3882:
3880:
3879:
3862:
3793:matching pursuit
3631:
3629:
3628:
3623:
3621:
3620:
3552:
3550:
3549:
3544:
3542:
3541:
3529:
3528:
3516:
3515:
3503:
3502:
3483:
3481:
3480:
3475:
3464:
3463:
3451:
3450:
3438:
3437:
3425:
3424:
3409:
3408:
3393:
3392:
3383:
3382:
3360:
3358:
3357:
3352:
3347:
3346:
3337:
3323:
3322:
3313:
3312:
3300:
3299:
3284:
3283:
3268:
3267:
3258:
3257:
3232:
3230:
3229:
3224:
3219:
3218:
3209:
3208:
3190:
3189:
3177:
3176:
3164:
3163:
3148:
3147:
3132:
3131:
3122:
3121:
3099:
3097:
3096:
3091:
3086:
3085:
3076:
3075:
3057:
3056:
3044:
3043:
3031:
3030:
3015:
3014:
2999:
2998:
2989:
2988:
2963:
2961:
2960:
2955:
2941:
2929:
2927:
2926:
2921:
2919:
2918:
2902:
2900:
2899:
2894:
2864:
2862:
2861:
2856:
2854:
2853:
2841:
2840:
2828:
2827:
2815:
2814:
2798:
2796:
2795:
2790:
2772:
2770:
2769:
2764:
2752:
2750:
2749:
2744:
2742:
2727:
2725:
2724:
2719:
2707:
2705:
2704:
2699:
2697:
2696:
2677:
2675:
2674:
2669:
2657:
2655:
2654:
2649:
2647:
2646:
2627:
2625:
2624:
2619:
2607:
2605:
2604:
2599:
2566:
2564:
2563:
2558:
2556:
2555:
2543:
2542:
2518:
2505:
2503:
2502:
2497:
2495:
2494:
2478:
2476:
2475:
2470:
2456:
2455:
2443:
2442:
2426:
2424:
2423:
2418:
2415:
2410:
2401:
2382:
2381:
2373:
2368:
2367:
2352:
2340:
2339:
2311:
2309:
2308:
2303:
2291:
2289:
2288:
2283:
2281:
2280:
2268:
2267:
2251:
2249:
2248:
2243:
2238:
2237:
2225:
2224:
2200:
2198:
2197:
2192:
2180:
2178:
2177:
2172:
2170:
2158:
2156:
2155:
2150:
2142:
2130:
2128:
2127:
2122:
2110:
2108:
2107:
2102:
2100:
2088:
2086:
2085:
2080:
2078:
2077:
2055:
2053:
2052:
2047:
2045:
2044:
2036:
2026:
2024:
2023:
2018:
2006:
2004:
2003:
1998:
1996:
1984:
1982:
1981:
1976:
1973:
1968:
1959:
1940:
1939:
1931:
1926:
1925:
1910:
1898:
1897:
1873:
1871:
1870:
1865:
1863:
1862:
1854:
1844:
1842:
1841:
1836:
1834:
1833:
1825:
1814:
1812:
1811:
1806:
1794:
1792:
1791:
1786:
1773:
1771:
1770:
1765:
1753:
1751:
1750:
1745:
1730:
1728:
1727:
1722:
1717:
1716:
1701:
1700:
1678:
1676:
1675:
1670:
1658:
1656:
1655:
1650:
1648:
1647:
1628:
1626:
1625:
1620:
1608:
1606:
1605:
1600:
1588:
1586:
1585:
1580:
1568:
1566:
1565:
1560:
1555:
1554:
1529:
1527:
1526:
1521:
1509:
1507:
1506:
1501:
1489:
1487:
1486:
1481:
1479:
1478:
1462:
1460:
1459:
1454:
1452:
1451:
1444:
1443:
1432:
1431:
1418:
1417:
1406:
1405:
1382:
1381:
1366:
1365:
1347:
1346:
1331:
1330:
1315:
1314:
1298:
1296:
1295:
1290:
1288:
1287:
1279:
1269:
1267:
1266:
1261:
1231:
1229:
1228:
1223:
1211:
1209:
1208:
1203:
1191:
1189:
1188:
1183:
1154:
1152:
1151:
1146:
1134:
1132:
1131:
1126:
1118:
1117:
1095:
1093:
1092:
1087:
1079:
1078:
1022:
1020:
1019:
1014:
1012:
1011:
995:
993:
992:
987:
985:
984:
968:
966:
965:
960:
958:
957:
941:
939:
938:
933:
931:
930:
914:
912:
911:
906:
904:
903:
883:
881:
880:
875:
873:
872:
848:
822:
820:
819:
814:
812:
811:
794:
792:
791:
786:
784:
783:
694:
692:
691:
686:
684:
683:
667:
665:
664:
659:
657:
656:
640:
638:
637:
632:
630:
629:
611:
609:
608:
603:
601:
600:
581:
579:
578:
573:
571:
570:
545:
543:
542:
537:
535:
534:
503:
501:
500:
495:
493:
492:
408:
400:
398:
397:
392:
390:
378:
376:
375:
370:
368:
357:
303:matching pursuit
296:
294:
293:
288:
286:
285:
257:
255:
254:
249:
247:
246:
226:
224:
223:
218:
216:
215:
181:
179:
178:
173:
171:
170:
148:
146:
145:
140:
138:
137:
5858:
5857:
5853:
5852:
5851:
5849:
5848:
5847:
5813:
5812:
5792:Rice University
5759:
5757:Further reading
5754:
5719:
5715:
5686:
5682:
5675:
5671:
5654:
5653:
5649:
5643:
5592:
5588:
5557:
5553:
5530:
5526:
5461:
5457:
5448:
5446:
5432:
5428:
5389:
5385:
5340:
5336:
5317:
5316:
5312:
5273:(19): E67–E82.
5263:
5259:
5202:
5198:
5141:
5137:
5089:
5083:
5079:
5055:10.1.1.391.2020
5040:(10): 506–509.
5030:
5026:
4981:
4977:
4970:
4930:
4926:
4917:
4915:
4902:
4901:
4897:
4888:
4886:
4873:
4872:
4868:
4859:
4857:
4847:
4843:
4811:
4805:
4801:
4778:
4774:
4735:
4731:
4715:10.1.1.364.3997
4698:
4694:
4678:10.1.1.170.6085
4661:
4657:
4628:
4624:
4603:(4): 1168–200.
4593:
4589:
4584:
4580:
4539:
4532:
4493:
4484:
4421:
4412:
4405:
4401:
4392:
4388:
4377:
4373:
4349:10.1.1.122.4429
4321:
4315:
4311:
4306:
4302:
4273:
4269:
4238:
4234:
4223:
4219:
4188:
4184:
4175:
4173:
4169:
4120:
4114:
4110:
4105:
4101:
4092:
4088:
4057:
4053:
4049:
4044:
4043:
4019:
4015:
4001:
3995:
3991:
3982:
3978:
3977:
3971:
3955:
3951:
3946:
3943:
3942:
3918:
3914:
3912:
3909:
3908:
3875:
3871:
3869:
3866:
3865:
3863:
3859:
3854:
3817:
3801:
3777:radio astronomy
3773:
3761:
3743:estimation and
3729:
3693:
3681:
3665:
3652:Rice University
3644:
3616:
3612:
3610:
3607:
3606:
3576:
3559:
3537:
3533:
3524:
3520:
3511:
3507:
3498:
3494:
3492:
3489:
3488:
3459:
3455:
3446:
3442:
3433:
3429:
3414:
3410:
3404:
3400:
3388:
3384:
3378:
3374:
3369:
3366:
3365:
3342:
3338:
3333:
3318:
3314:
3308:
3304:
3289:
3285:
3279:
3275:
3263:
3259:
3253:
3249:
3244:
3241:
3240:
3214:
3210:
3204:
3200:
3185:
3181:
3172:
3168:
3153:
3149:
3143:
3139:
3127:
3123:
3117:
3113:
3108:
3105:
3104:
3081:
3077:
3071:
3067:
3052:
3048:
3039:
3035:
3020:
3016:
3010:
3006:
2994:
2990:
2984:
2980:
2975:
2972:
2971:
2937:
2935:
2932:
2931:
2914:
2910:
2908:
2905:
2904:
2870:
2867:
2866:
2849:
2845:
2836:
2832:
2823:
2819:
2810:
2806:
2804:
2801:
2800:
2778:
2775:
2774:
2758:
2755:
2754:
2738:
2733:
2730:
2729:
2713:
2710:
2709:
2692:
2688:
2683:
2680:
2679:
2663:
2660:
2659:
2642:
2638:
2633:
2630:
2629:
2613:
2610:
2609:
2575:
2572:
2571:
2551:
2547:
2538:
2534:
2514:
2512:
2509:
2508:
2490:
2486:
2484:
2481:
2480:
2451:
2447:
2438:
2434:
2432:
2429:
2428:
2411:
2406:
2397:
2372:
2363:
2359:
2348:
2335:
2334:
2328:
2325:
2324:
2297:
2294:
2293:
2276:
2272:
2263:
2259:
2257:
2254:
2253:
2233:
2229:
2220:
2216:
2211:
2208:
2207:
2186:
2183:
2182:
2166:
2164:
2161:
2160:
2138:
2136:
2133:
2132:
2116:
2113:
2112:
2096:
2094:
2091:
2090:
2067:
2063:
2061:
2058:
2057:
2035:
2034:
2032:
2029:
2028:
2012:
2009:
2008:
1992:
1990:
1987:
1986:
1969:
1964:
1955:
1930:
1921:
1917:
1906:
1893:
1892:
1886:
1883:
1882:
1853:
1852:
1850:
1847:
1846:
1824:
1823:
1821:
1818:
1817:
1800:
1797:
1796:
1780:
1777:
1776:
1759:
1756:
1755:
1739:
1736:
1735:
1712:
1708:
1696:
1692:
1684:
1681:
1680:
1664:
1661:
1660:
1643:
1639:
1634:
1631:
1630:
1614:
1611:
1610:
1594:
1591:
1590:
1574:
1571:
1570:
1550:
1546:
1535:
1532:
1531:
1515:
1512:
1511:
1495:
1492:
1491:
1474:
1470:
1468:
1465:
1464:
1446:
1445:
1439:
1435:
1433:
1427:
1423:
1420:
1419:
1413:
1409:
1407:
1401:
1397:
1390:
1389:
1377:
1373:
1361:
1357:
1342:
1338:
1326:
1322:
1310:
1306:
1304:
1301:
1300:
1278:
1277:
1275:
1272:
1271:
1255:
1252:
1251:
1247:
1238:
1217:
1214:
1213:
1197:
1194:
1193:
1177:
1174:
1173:
1161:
1140:
1137:
1136:
1107:
1103:
1101:
1098:
1097:
1068:
1064:
1062:
1059:
1058:
1030:
1007:
1003:
1001:
998:
997:
980:
976:
974:
971:
970:
953:
949:
947:
944:
943:
926:
922:
920:
917:
916:
899:
895:
893:
890:
889:
868:
864:
862:
859:
858:
851:
847:
841:
837:
807:
803:
801:
798:
797:
779:
775:
773:
770:
769:
754:total variation
728:Total variation
725:
720:
715:
709:
679:
675:
673:
670:
669:
652:
648:
646:
643:
642:
625:
621:
619:
616:
615:
596:
592:
590:
587:
586:
566:
562:
560:
557:
556:
530:
526:
524:
521:
520:
488:
484:
482:
479:
478:
466:matrix equation
449:Emmanuel Candès
418:
386:
384:
381:
380:
364:
353:
351:
348:
347:
340:
335:
307:LASSO estimator
281:
277:
275:
272:
271:
260:George W. Brown
242:
238:
236:
233:
232:
211:
207:
205:
202:
201:
166:
162:
160:
157:
156:
133:
129:
127:
124:
123:
119:
95:Emmanuel Candès
64:
32:sparse sampling
22:(also known as
17:
12:
11:
5:
5856:
5846:
5845:
5840:
5835:
5833:Linear algebra
5830:
5825:
5811:
5810:
5805:
5795:
5785:
5779:
5758:
5755:
5753:
5752:
5733:(16): 164102.
5713:
5680:
5669:
5666:on 2013-03-31.
5660:inviewcorp.com
5647:
5641:
5586:
5567:(1): 116–117.
5551:
5524:
5455:
5437:(2010-03-04).
5426:
5383:
5334:
5327:. 2008-03-24.
5310:
5257:
5212:(6): 871–873.
5206:Optics Letters
5196:
5145:Optics Letters
5135:
5077:
5024:
4989:Optics Express
4975:
4968:
4924:
4895:
4866:
4841:
4822:(1): 267–288.
4799:
4772:
4745:(1): 224–238.
4729:
4708:(2): 537–563.
4692:
4655:
4622:
4587:
4578:
4530:
4482:
4410:
4399:
4386:
4371:
4332:(8): 489–509.
4309:
4300:
4289:(1): 267–288.
4267:
4232:
4217:
4182:
4108:
4099:
4093:M. Davenport,
4086:
4067:(6): 797–829.
4050:
4048:
4045:
4042:
4041:
4027:
4022:
4018:
4014:
4011:
4008:
4004:
3998:
3994:
3988:
3985:
3981:
3974:
3970:
3966:
3963:
3958:
3954:
3950:
3921:
3917:
3878:
3874:
3856:
3855:
3853:
3850:
3849:
3848:
3843:
3838:
3833:
3828:
3823:
3816:
3813:
3800:
3797:
3772:
3769:
3760:
3757:
3728:
3725:
3720:
3719:
3716:
3713:
3710:
3707:
3704:
3692:
3689:
3687:applications.
3680:
3677:
3664:
3661:
3643:
3640:
3619:
3615:
3596:coded aperture
3575:
3572:
3558:
3555:
3540:
3536:
3532:
3527:
3523:
3519:
3514:
3510:
3506:
3501:
3497:
3485:
3484:
3473:
3470:
3467:
3462:
3458:
3454:
3449:
3445:
3441:
3436:
3432:
3428:
3423:
3420:
3417:
3413:
3407:
3403:
3399:
3396:
3391:
3387:
3381:
3377:
3373:
3362:
3361:
3350:
3345:
3341:
3336:
3332:
3329:
3326:
3321:
3317:
3311:
3307:
3303:
3298:
3295:
3292:
3288:
3282:
3278:
3274:
3271:
3266:
3262:
3256:
3252:
3248:
3234:
3233:
3222:
3217:
3213:
3207:
3203:
3199:
3196:
3193:
3188:
3184:
3180:
3175:
3171:
3167:
3162:
3159:
3156:
3152:
3146:
3142:
3138:
3135:
3130:
3126:
3120:
3116:
3112:
3101:
3100:
3089:
3084:
3080:
3074:
3070:
3066:
3063:
3060:
3055:
3051:
3047:
3042:
3038:
3034:
3029:
3026:
3023:
3019:
3013:
3009:
3005:
3002:
2997:
2993:
2987:
2983:
2979:
2953:
2950:
2947:
2944:
2940:
2917:
2913:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2852:
2848:
2844:
2839:
2835:
2831:
2826:
2822:
2818:
2813:
2809:
2788:
2785:
2782:
2762:
2741:
2737:
2717:
2695:
2691:
2687:
2667:
2645:
2641:
2637:
2617:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2554:
2550:
2546:
2541:
2537:
2533:
2530:
2527:
2524:
2521:
2517:
2493:
2489:
2468:
2465:
2462:
2459:
2454:
2450:
2446:
2441:
2437:
2414:
2409:
2405:
2400:
2396:
2393:
2390:
2387:
2379:
2376:
2371:
2366:
2362:
2358:
2355:
2351:
2347:
2344:
2338:
2333:
2301:
2279:
2275:
2271:
2266:
2262:
2241:
2236:
2232:
2228:
2223:
2219:
2215:
2190:
2169:
2148:
2145:
2141:
2120:
2099:
2076:
2073:
2070:
2066:
2042:
2039:
2016:
1995:
1972:
1967:
1963:
1958:
1954:
1951:
1948:
1945:
1937:
1934:
1929:
1924:
1920:
1916:
1913:
1909:
1905:
1902:
1896:
1891:
1860:
1857:
1845:is defined as
1831:
1828:
1804:
1784:
1763:
1743:
1720:
1715:
1711:
1707:
1704:
1699:
1695:
1691:
1688:
1668:
1646:
1642:
1638:
1618:
1598:
1578:
1558:
1553:
1549:
1545:
1542:
1539:
1519:
1499:
1477:
1473:
1450:
1442:
1438:
1434:
1430:
1426:
1422:
1421:
1416:
1412:
1408:
1404:
1400:
1396:
1395:
1393:
1388:
1385:
1380:
1376:
1372:
1369:
1364:
1360:
1356:
1353:
1350:
1345:
1341:
1337:
1334:
1329:
1325:
1321:
1318:
1313:
1309:
1285:
1282:
1259:
1246:
1243:
1237:
1234:
1221:
1201:
1181:
1160:
1157:
1144:
1124:
1121:
1116:
1113:
1110:
1106:
1085:
1082:
1077:
1074:
1071:
1067:
1029:
1026:
1010:
1006:
983:
979:
956:
952:
929:
925:
902:
898:
871:
867:
850:
845:
838:
836:
833:
810:
806:
782:
778:
724:
721:
719:
716:
708:
705:
697:linear program
682:
678:
655:
651:
628:
624:
599:
595:
569:
565:
533:
529:
512:pseudo-inverse
491:
487:
453:Justin Romberg
417:
414:
389:
367:
363:
360:
356:
339:
336:
334:
331:
284:
280:
245:
241:
214:
210:
169:
165:
136:
132:
118:
115:
99:Justin Romberg
63:
60:
15:
9:
6:
4:
3:
2:
5855:
5844:
5841:
5839:
5836:
5834:
5831:
5829:
5826:
5824:
5821:
5820:
5818:
5809:
5806:
5803:
5799:
5796:
5793:
5789:
5786:
5783:
5780:
5777:
5773:
5769:
5765:
5761:
5760:
5748:
5744:
5740:
5736:
5732:
5728:
5724:
5717:
5708:
5703:
5699:
5695:
5691:
5684:
5678:
5673:
5665:
5661:
5657:
5651:
5645:
5637:
5633:
5629:
5625:
5621:
5617:
5613:
5609:
5605:
5601:
5597:
5590:
5582:
5578:
5574:
5570:
5566:
5562:
5555:
5547:
5543:
5539:
5535:
5528:
5520:
5516:
5511:
5506:
5501:
5496:
5492:
5488:
5483:
5478:
5474:
5470:
5466:
5459:
5444:
5440:
5436:
5430:
5422:
5418:
5414:
5410:
5406:
5402:
5398:
5394:
5387:
5379:
5375:
5371:
5367:
5362:
5357:
5353:
5349:
5345:
5338:
5330:
5326:
5325:
5320:
5314:
5306:
5302:
5298:
5294:
5289:
5284:
5280:
5276:
5272:
5268:
5261:
5253:
5249:
5245:
5241:
5237:
5233:
5229:
5225:
5220:
5215:
5211:
5207:
5200:
5192:
5188:
5184:
5180:
5176:
5172:
5168:
5164:
5159:
5154:
5150:
5146:
5139:
5131:
5127:
5123:
5119:
5115:
5111:
5107:
5103:
5099:
5095:
5088:
5081:
5073:
5069:
5065:
5061:
5056:
5051:
5047:
5043:
5039:
5035:
5028:
5020:
5016:
5011:
5006:
5002:
4998:
4994:
4990:
4986:
4979:
4971:
4965:
4961:
4957:
4953:
4949:
4944:
4939:
4935:
4928:
4914:on 2016-01-20
4913:
4909:
4905:
4899:
4885:on 2010-06-05
4884:
4880:
4876:
4870:
4856:
4855:IEEE Spectrum
4852:
4845:
4837:
4833:
4829:
4825:
4821:
4817:
4810:
4803:
4795:
4791:
4787:
4783:
4776:
4768:
4764:
4760:
4756:
4752:
4748:
4744:
4740:
4733:
4725:
4721:
4716:
4711:
4707:
4703:
4696:
4688:
4684:
4679:
4674:
4670:
4666:
4659:
4650:
4645:
4642:(6): 409–36.
4641:
4637:
4633:
4626:
4618:
4614:
4610:
4606:
4602:
4598:
4591:
4582:
4574:
4570:
4566:
4562:
4557:
4552:
4548:
4544:
4537:
4535:
4526:
4522:
4518:
4514:
4510:
4506:
4502:
4498:
4491:
4489:
4487:
4478:
4474:
4469:
4464:
4460:
4456:
4452:
4448:
4443:
4438:
4434:
4430:
4429:Phys Med Biol
4426:
4419:
4417:
4415:
4408:
4403:
4396:
4390:
4384:
4381:
4375:
4367:
4363:
4359:
4355:
4350:
4345:
4340:
4335:
4331:
4327:
4320:
4313:
4304:
4296:
4292:
4288:
4284:
4283:
4278:
4271:
4263:
4259:
4255:
4251:
4247:
4243:
4236:
4230:
4226:
4221:
4213:
4209:
4205:
4201:
4197:
4193:
4186:
4172:on 2012-03-11
4168:
4164:
4160:
4156:
4152:
4148:
4144:
4139:
4134:
4130:
4126:
4119:
4112:
4103:
4096:
4090:
4082:
4078:
4074:
4070:
4066:
4062:
4055:
4051:
4020:
4016:
4012:
4009:
4002:
3996:
3992:
3986:
3983:
3979:
3972:
3968:
3956:
3952:
3941:
3937:
3919:
3915:
3906:
3902:
3898:
3894:
3876:
3872:
3861:
3857:
3847:
3844:
3842:
3841:Sparse coding
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3818:
3812:
3810:
3809:scanning mode
3806:
3796:
3794:
3790:
3786:
3782:
3778:
3768:
3766:
3756:
3754:
3750:
3746:
3742:
3741:Network delay
3738:
3734:
3724:
3718:EWISTARS etc.
3717:
3714:
3711:
3708:
3705:
3702:
3701:
3700:
3698:
3688:
3686:
3676:
3674:
3670:
3660:
3657:
3653:
3648:
3639:
3636:
3617:
3613:
3603:
3601:
3597:
3593:
3589:
3588:sparse coding
3585:
3584:group testing
3581:
3571:
3568:
3564:
3554:
3538:
3534:
3530:
3525:
3521:
3517:
3512:
3508:
3504:
3499:
3495:
3468:
3465:
3460:
3456:
3452:
3447:
3443:
3434:
3430:
3426:
3421:
3418:
3415:
3405:
3401:
3394:
3389:
3379:
3375:
3364:
3363:
3343:
3324:
3319:
3315:
3309:
3305:
3301:
3296:
3293:
3290:
3280:
3276:
3269:
3264:
3254:
3250:
3239:
3238:
3237:
3215:
3205:
3201:
3191:
3186:
3182:
3173:
3169:
3165:
3160:
3157:
3154:
3144:
3140:
3133:
3128:
3118:
3114:
3103:
3102:
3082:
3072:
3068:
3058:
3053:
3049:
3040:
3036:
3032:
3027:
3024:
3021:
3011:
3007:
3000:
2995:
2985:
2981:
2970:
2969:
2968:
2965:
2951:
2948:
2945:
2942:
2915:
2911:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2850:
2846:
2842:
2837:
2833:
2829:
2824:
2820:
2816:
2811:
2807:
2786:
2783:
2780:
2760:
2715:
2693:
2689:
2665:
2643:
2639:
2615:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2568:
2552:
2548:
2544:
2539:
2535:
2531:
2528:
2525:
2522:
2519:
2491:
2487:
2466:
2463:
2460:
2457:
2452:
2448:
2444:
2439:
2435:
2412:
2407:
2391:
2388:
2377:
2374:
2369:
2364:
2356:
2353:
2317:
2313:
2299:
2277:
2273:
2269:
2264:
2260:
2234:
2230:
2226:
2221:
2217:
2204:
2188:
2146:
2143:
2118:
2074:
2071:
2068:
2064:
2037:
1970:
1965:
1949:
1946:
1935:
1932:
1927:
1922:
1914:
1911:
1879:
1875:
1855:
1826:
1802:
1782:
1761:
1741:
1732:
1713:
1709:
1702:
1697:
1693:
1666:
1644:
1640:
1616:
1596:
1576:
1551:
1547:
1543:
1540:
1517:
1497:
1475:
1471:
1448:
1440:
1436:
1428:
1424:
1414:
1410:
1402:
1398:
1391:
1386:
1378:
1374:
1367:
1362:
1358:
1348:
1343:
1339:
1335:
1327:
1323:
1311:
1307:
1280:
1257:
1242:
1233:
1219:
1199:
1179:
1165:
1156:
1142:
1122:
1119:
1114:
1111:
1108:
1104:
1083:
1080:
1075:
1072:
1069:
1065:
1056:
1051:
1047:
1043:
1039:
1034:
1025:
1008:
1004:
981:
977:
954:
950:
927:
923:
900:
896:
869:
865:
855:
844:
832:
828:
826:
808:
804:
780:
776:
766:
763:
759:
755:
751:
747:
744:
740:
736:
733:
729:
714:
704:
702:
698:
680:
676:
653:
649:
626:
622:
613:
597:
593:
583:
567:
563:
553:
549:
547:
531:
527:
515:
513:
509:
505:
489:
485:
474:
472:
467:
462:
458:
454:
450:
446:
441:
439:
435:
427:
422:
413:
409:
407:
402:
361:
358:
345:
330:
327:
323:
318:
316:
315:basis pursuit
312:
308:
305:in 1993, the
304:
300:
282:
278:
269:
265:
261:
243:
239:
230:
212:
208:
199:
195:
191:
187:
183:
167:
163:
153:
152:least squares
149:
134:
130:
114:
112:
108:
104:
100:
96:
93:Around 2004,
91:
89:
85:
81:
76:
74:
69:
59:
57:
53:
49:
45:
41:
37:
33:
29:
25:
21:
5801:
5730:
5726:
5716:
5697:
5693:
5683:
5672:
5664:the original
5659:
5650:
5644:
5603:
5599:
5589:
5564:
5560:
5554:
5537:
5533:
5527:
5472:
5468:
5458:
5447:. Retrieved
5442:
5429:
5399:(2): 72–82.
5396:
5392:
5386:
5351:
5347:
5337:
5322:
5313:
5270:
5266:
5260:
5209:
5205:
5199:
5151:(1): 79–81.
5148:
5144:
5138:
5097:
5093:
5080:
5037:
5033:
5027:
4992:
4988:
4978:
4933:
4927:
4916:. Retrieved
4912:the original
4907:
4898:
4887:. Retrieved
4883:the original
4878:
4869:
4858:. Retrieved
4854:
4844:
4819:
4815:
4802:
4788:(1): 39–52.
4785:
4781:
4775:
4742:
4738:
4732:
4705:
4701:
4695:
4671:(1): 41–55.
4668:
4664:
4658:
4639:
4635:
4625:
4600:
4596:
4590:
4581:
4546:
4542:
4500:
4496:
4432:
4428:
4402:
4394:
4389:
4379:
4374:
4339:math/0409186
4329:
4325:
4312:
4303:
4286:
4280:
4270:
4245:
4241:
4235:
4220:
4195:
4191:
4185:
4174:. Retrieved
4167:the original
4138:math/0503066
4128:
4124:
4111:
4102:
4089:
4064:
4060:
4054:
3900:
3896:
3860:
3802:
3779:and optical
3774:
3762:
3730:
3721:
3694:
3682:
3666:
3649:
3645:
3634:
3604:
3592:multiplexing
3577:
3574:Applications
3560:
3486:
3235:
2966:
2569:
2322:
1880:
1876:
1733:
1248:
1239:
1170:
1054:
1049:
1045:
1041:
1037:
1035:
1031:
887:
849:minimization
842:
829:
767:
732:non-negative
726:
550:
516:
475:
461:David Donoho
442:
431:
410:
403:
341:
319:
313:in 1996 and
120:
107:David Donoho
92:
77:
65:
31:
27:
23:
19:
18:
5540:: 115–132.
3642:Photography
3635:Boyd et al.
3565:(PSNR) and
1135:(Note that
743:real-valued
457:Terence Tao
103:Terence Tao
5817:Categories
5482:1501.03915
5475:: 546814.
5449:2024-04-20
4918:2013-06-04
4889:2013-06-04
4860:2013-03-20
4782:Geophysics
4248:(4): 276.
4176:2011-02-10
4047:References
3669:holography
3663:Holography
739:functional
711:See also:
5581:109854375
5283:CiteSeerX
5267:Appl. Opt
5219:1004.5305
5158:1101.1735
5094:Opt. Lett
5050:CiteSeerX
4943:1305.7181
4710:CiteSeerX
4673:CiteSeerX
4556:0711.1612
4442:1009.2288
4344:CiteSeerX
4212:206737254
4163:119159284
3984:−
3969:∑
3965:↦
3656:Bell Labs
3561:Based on
3535:γ
3522:γ
3509:γ
3496:γ
3466:∙
3453:−
3431:γ
3419:−
3402:λ
3376:λ
3328:∇
3325:−
3306:γ
3294:−
3277:λ
3251:λ
3195:∇
3192:−
3170:γ
3158:−
3141:λ
3115:λ
3062:∇
3059:−
3037:γ
3025:−
3008:λ
2982:λ
2847:λ
2834:λ
2821:λ
2808:λ
2784:∙
2736:∇
2686:∇
2636:∇
2549:λ
2536:λ
2404:‖
2395:Φ
2392:−
2386:‖
2375:λ
2361:‖
2354:∙
2346:∇
2343:‖
2144:∙
2072:−
2041:^
2015:Φ
1962:‖
1953:Φ
1950:−
1944:‖
1933:λ
1919:‖
1912:∙
1904:∇
1901:‖
1859:^
1830:^
1762:σ
1714:σ
1706:∇
1703:⊗
1698:σ
1690:∇
1645:σ
1637:∇
1597:σ
1577:ρ
1548:ρ
1498:ρ
1476:ρ
1379:σ
1371:∇
1368:⊗
1363:σ
1355:∇
1349:∗
1344:ρ
1328:σ
1320:∇
1312:ρ
1284:^
1220:σ
1200:σ
1180:σ
1112:−
1073:−
1005:ℓ
978:ℓ
951:ℓ
924:ℓ
897:ℓ
866:ℓ
805:ℓ
777:ℓ
765:problem.
746:functions
317:in 1998.
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5519:24971155
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