#### DMCA

## Symmetrizing Smoothing Filters (2013)

### Cached

### Download Links

Citations: | 9 - 6 self |

### Citations

7507 |
Matrix Analysis
- Horn, Johnson
- 1985
(Show Context)
Citation Context ...tochastic matrices with non-negative entries, and define 1 as the n×1 vector of ones. By definition, any A ∈ Sn satisfies A1 = 1. (2.1) The Perron-Frobenius theory of non-negative matrices (cf. [42], =-=[28]-=-, page 498) provides a comprehensive characterization of their spectrum. Denoting the eigenvalues {λi} n i=1 in descending order, we have 5 : 1. λ1 = 1 is the unique eigenvalue of A with maximum modul... |

3722 | Normalized cuts and image segmentation
- Shi, Malik
- 1997
(Show Context)
Citation Context ...ms equal to zero. We speculate that this result may in fact yield improvements in various areas of application such as dimensionality reduction, data representation [3], clustering [55], segmentation =-=[43]-=- and more. To summarize, we studied a class of smoothing filters which operate based on non-linear, shift-variant averaging which are frequently used in both signal and image processing. We provided a... |

1737 |
Exploratory data analysis
- Tukey
- 1977
(Show Context)
Citation Context ...uaranteed to be a shrinking smoother. There are numerous ways in which iterative application of smoothers come into play. Two of the most useful and widely studied are boosting, also known as Twicing =-=[51]-=-, L2-Boosting [8], Reaction-Diffusion [38], and Bregman Iteration [39]. We studied this approach in detail in [37]. The iteration is given by yk = yk−1 +A(y− yk−1) = k∑ A(I−A) l y. (3.4) l=0 12 Oth... |

1555 | The Elements of
- Hastie, Tibshirani, et al.
- 2001
(Show Context)
Citation Context ...F Grant CCF-1016018. 1 Many linear filters such as linear minimum mean-squared error smoothing filter, Savitzky-Golay filters, smoothing splines, and wavelet smoothers can be considered special cases =-=[27]-=-. 12 Milanfar only their original sample space of definition (time domain, pixel domain, etc.) Understanding the spectral behavior of these filters in an orthogonal basis is important not only for be... |

1194 | Laplacian eigenmaps for dimensionality reduction and data representation
- Belkin, Niyogi
- 2003
(Show Context)
Citation Context ...normalization we promote would likely have some impact in other areas of work well beyond the current filtering context, such as scale-space meshing in computer graphics [18], and in machine learning =-=[3]-=-. As we mentioned earlier, many popular filters are contained in the class of smoothing operators we consider. To be more specific, we highlight a few such kernels which lead to smoothing matrices A w... |

1127 | Bilateral filtering for gray and color images
- Tomasi, Manduchi
- 1998
(Show Context)
Citation Context ...trices. These filters are commonlyusedinsignalprocessingapplications becausetheykeepthemeanvalueofthesignal unchanged. In particular, moving least squares averaging filters [36], the bilateral filter =-=[50]-=-, and the nonlocal means filter [7], are all special cases. For their part, stochastic matrices find numerous applications in statistical signal processing; including in classical optimal filtering, i... |

805 | Applied Nonparametric Regression - Härdle - 1990 |

737 | Introduction to the Numerical Solution of Markov Chains - Stewart - 1994 |

503 |
Generalized iterative scaling for log-linear models
- Darroch, Ratcliff
- 1972
(Show Context)
Citation Context ...envalues, close to the original. This is the subject of this section. We note that classical results in this direction have been available since the 1970’s. Of these, the work of Darroch and Radcliff =-=[17]-=- and Csiszar [15] involving relative entropy is particularly noteworthy. Here, we prove that the set of n×n (row-) stochastic matrices and the corresponding set of doubly-stochastic matrices are asymp... |

499 | A review of image denoising algorithms, with a new one,”Multiscale Modeling and Simulation
- Buades, Coll, et al.
- 2005
(Show Context)
Citation Context ...edinsignalprocessingapplications becausetheykeepthemeanvalueofthesignal unchanged. In particular, moving least squares averaging filters [36], the bilateral filter [50], and the nonlocal means filter =-=[7]-=-, are all special cases. For their part, stochastic matrices find numerous applications in statistical signal processing; including in classical optimal filtering, image denoising [11], Markov chain t... |

494 | Non-negative Matrices and Markov Chains - Seneta - 1981 |

419 | Image denoising by sparse 3-D transform-domain collaborative filtering
- Dabov, Foi, et al.
(Show Context)
Citation Context ...vely flat, the bias component of this minimum MSE is dominant. The fact that bias in flat regions is a problem in practice is a well-known phenomenon [11] for high performance algorithms such as BM3D =-=[16]-=-.Symmetrizing Smoothing Filters 17 Using the MSE expression in (3.10) we can also ask what class of images (that is which sequences of bi) will result in the worst or largest MSE min . This question ... |

370 |
I-divergence geometry of probability distributions and minimization problems”, The Annals of Probability 3
- Csiszar
- 1975
(Show Context)
Citation Context ...o the original. This is the subject of this section. We note that classical results in this direction have been available since the 1970’s. Of these, the work of Darroch and Radcliff [17] and Csiszar =-=[15]-=- involving relative entropy is particularly noteworthy. Here, we prove that the set of n×n (row-) stochastic matrices and the corresponding set of doubly-stochastic matrices are asymptotically close i... |

310 | Spectral grouping using the Nyström method - Fowlkes, Belongie, et al. - 2004 |

309 |
Anisotropic Diffusion
- Weickert
- 1998
(Show Context)
Citation Context ...atrix A = D −1 K is therefore nonlinear and shift varying. This kernel is closely related, but somewhat more general than the Beltrami kernel of [47] and the coherence enhancing diffusion approach of =-=[53]-=-. 2. Nearness of Stochastic and Doubly-Stochastic Matrices. Our interest in this paper is to convert a smoothing operator A which is generically not symmetric into a symmetric doubly-stochastic one. A... |

253 | Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps
- Coifman, Lafon, et al.
- 2005
(Show Context)
Citation Context ...1.1. Some Background. Before we move to the details, here is a brief summary of relevant earlier work. In the context of expressing a nonlinear filtering scheme in an orthogonal basis, Coifman et al. =-=[14]-=- proposed the construction of diffusion maps and used eigenfunctions of the Laplacian on the manifold of patches derived from an image. Peyré provided an interesting spectral analysis of the graph Lap... |

203 | Boosting with the L2-Loss: Regression and Classification
- Buhlmann, Yu
- 2001
(Show Context)
Citation Context ...shrinking smoother. There are numerous ways in which iterative application of smoothers come into play. Two of the most useful and widely studied are boosting, also known as Twicing [51], L2-Boosting =-=[8]-=-, Reaction-Diffusion [38], and Bregman Iteration [39]. We studied this approach in detail in [37]. The iteration is given by yk = yk−1 +A(y− yk−1) = k∑ A(I−A) l y. (3.4) l=0 12 Other norm can be us... |

195 |
On the origin of the bilateral filter and ways to improve it
- Elad
(Show Context)
Citation Context ...cal Gaussian kernel is ( −‖xi −xj‖ kij = exp 2 h2 ) . Such kernels lead to the classical and well-worn Gaussian filters (including shift-varying versions).4 Milanfar 1.1.2. The Bilateral Filter (BL) =-=[50, 20]-=-. This filter takes into account both the spatial and data-wise distances between two samples, in separable fashion, as follows: ( −‖xi −xj‖ kij = exp 2 ) ( −(yi −yj) exp 2 ) { −‖xi −xj‖ = exp 2 h 2 x... |

190 | An Iterative Regularization Method for Total Variation-Based Image Restoration,” Multiscale Modeling
- Osher, Burger, et al.
- 2005
(Show Context)
Citation Context ... iterative application of smoothers come into play. Two of the most useful and widely studied are boosting, also known as Twicing [51], L2-Boosting [8], Reaction-Diffusion [38], and Bregman Iteration =-=[39]-=-. We studied this approach in detail in [37]. The iteration is given by yk = yk−1 +A(y− yk−1) = k∑ A(I−A) l y. (3.4) l=0 12 Other norm can be used for the definition, but we use the L2.12 Milanfar... |

181 |
A relationship between arbitrary positive matrices and doubly stochastic matrices,” The
- Sinkhorn
- 1964
(Show Context)
Citation Context ...re informative when Â is an explicit symmetric approximation of A; and this is indeed the case. The interesting practical question of how to find this nearest element was addressed by Sinkhorn et al. =-=[34, 45, 46]-=-. Specifically, a stochastic matrix (indeed any non-negative matrix withpositive diagonals) can bescaled toadoubly-stochastic matrix via aproceduresometimes known as iterative proportional scaling, or... |

170 | Kernel regression for image processing and reconstruction
- Takeda, Farsiu, et al.
- 2007
(Show Context)
Citation Context ... instead of point-wise: (1.3) ( −‖xi −xj‖ kij = exp 2 ) ( −‖yi −yj‖ exp 2 ) , (1.4) where yi and yj refer now to subsets of samples (patches) in y. h 2 x 1.1.4. LocallyAdaptiveRegression Kernel (LARK)=-=[49]-=-. Thekeyideabehindthiskernel is to robustly measure the local structure of data by making use of an estimate of the local geodesic distance between nearby samples: h 2 y { kij = exp −(xi −xj) T } Qij(... |

158 | The approximation power of moving least-squares
- Levin
- 1998
(Show Context)
Citation Context ...called (row-) stochastic matrices. These filters are commonlyusedinsignalprocessingapplications becausetheykeepthemeanvalueofthesignal unchanged. In particular, moving least squares averaging filters =-=[36]-=-, the bilateral filter [50], and the nonlocal means filter [7], are all special cases. For their part, stochastic matrices find numerous applications in statistical signal processing; including in cla... |

113 | Gossip algorithms for distributed signal processing
- Dimakis, Kar, et al.
- 2010
(Show Context)
Citation Context ..., stochastic matrices find numerous applications in statistical signal processing; including in classical optimal filtering, image denoising [11], Markov chain theory [42, 48], distributed processing =-=[19]-=-, and many more. While the smoothing operator in (1.1) has a linear appearance, the A we consider can in fact depend on the given data samples y, and the locations x of these samples. Therefore, these... |

110 | Optimal spatial adaptation for patchbased image denoising
- Kervrann, Boulanger
(Show Context)
Citation Context ...roach has several advantages. Namely, while the kernel is easy to construct, and computationally simple to calculate, it yields useful local adaptivity to the given data. 1.1.3. Non-Local Means (NLM) =-=[7, 31, 2]-=-. The non-local means algorithm, originally proposed in [7] and [2], is a generalization of the bilateral filter in which the data-dependent distance term (1.3) is measured block-wise instead of point... |

97 | Linear smoothers and additive models
- Buja, Hastie, et al.
- 1989
(Show Context)
Citation Context ...0.9992 0.9747 0.9744 dfs 12.94 29.88 75.01 54.47 88.59 371.72 217.16 115.47 3.2. Stability of Iterated Smoothing. The general class of smoothers for which ‖Ay‖ ≤ ‖y‖ are called shrinking 12 smoothers =-=[9, 27]-=-. This happens when all the singular values of A are bounded above by 1. This may seem like a minor issue at first, but it turns out to have important consequences when it comes to something we do rou... |

89 | Tukey, Exploratory Data Analysis - W - 1977 |

85 |
Deblurring and denoising of images by nonlocal functionals,” Multiscale Modeling
- Kindermann, Osher, et al.
- 2005
(Show Context)
Citation Context ...1/2 AD −1/2 −I. Therefore, the analysis we present here is directly relevant to the study of the spectrum of the Laplacian operator as well. Meanwhile, consistent with our analysis, Kindermann et al. =-=[33]-=- have proposed to directly symmetrize the NL-means or bilateral kernel matrices. But they too do not insist on maintaining the stochastic nature of the smoothing operator. Hence, our approach to makin... |

75 | Biased anisotropic diffusion - A unified regularization and diffusion approach to edge detection, Image and Vision Computing 8 - Nordström - 1990 |

63 | Concerning nonnegative matrices and doubly stochastic matrices - Knopp, Sinkhorn - 1967 |

53 |
Perturbation Bounds for Matrix Eigenvalues
- Bhatia
- 1987
(Show Context)
Citation Context ...l give a small perturbation of its eigenvalues. The stability of eigenvalues of a matrix is in fact a strong function of the condition number of the matrix of its eigenvectors (cf. Bauer-Fike Theorem =-=[4]-=-). In our filtering framework, it is important to verify that the eigenvalues of the symmetrized matrix Â are very nearby those of A, because the spectrum determines the effect of the filter on the da... |

52 | information-theoretic, adaptive image filtering for image restoration
- Awate, Whitaker, et al.
- 2006
(Show Context)
Citation Context ...roach has several advantages. Namely, while the kernel is easy to construct, and computationally simple to calculate, it yields useful local adaptivity to the given data. 1.1.3. Non-Local Means (NLM) =-=[7, 31, 2]-=-. The non-local means algorithm, originally proposed in [7] and [2], is a generalization of the bilateral filter in which the data-dependent distance term (1.3) is measured block-wise instead of point... |

49 |
Applied Nonparametric Regression Cambridge
- Härdle
- 1991
(Show Context)
Citation Context ...smoothing matrices A which are not symmetric. These are commonly used in the signal and image processing, computer vision, and graphics literature for many purposes. 1.1.1. Classical Gaussian Filters =-=[52, 24, 54]-=-. MeasuringtheEuclidean(spatial) distance between samples, the classical Gaussian kernel is ( −‖xi −xj‖ kij = exp 2 h2 ) . Such kernels lead to the classical and well-worn Gaussian filters (including ... |

48 | Is denoising dead
- Chatterjee, Milanfar
(Show Context)
Citation Context ...cal means filter [7], are all special cases. For their part, stochastic matrices find numerous applications in statistical signal processing; including in classical optimal filtering, image denoising =-=[11]-=-, Markov chain theory [42, 48], distributed processing [19], and many more. While the smoothing operator in (1.1) has a linear appearance, the A we consider can in fact depend on the given data sample... |

31 | Natural image denoising: Optimality and inherent bounds
- Levin, Nadler
- 2011
(Show Context)
Citation Context ...ms invented in the last few years appear to display comparable performance, prompting many to wonder whether we have reached a limit on the performance of such filters 2 for the denoising application =-=[11, 12, 35]-=-. The fundamental technical roadblock in the spectral analysis of smoothing filters is that in general A is not symmetric, Toeplitz, or circulant. With a symmetric A, its eigendecomposition would reve... |

26 | The Sinkhorn–Knopp algorithm: Convergence and applications
- Knight
(Show Context)
Citation Context ...re informative when Â is an explicit symmetric approximation of A; and this is indeed the case. The interesting practical question of how to find this nearest element was addressed by Sinkhorn et al. =-=[34, 45, 46]-=-. Specifically, a stochastic matrix (indeed any non-negative matrix withpositive diagonals) can bescaled toadoubly-stochastic matrix via aproceduresometimes known as iterative proportional scaling, or... |

20 | Diffusion Interpretation of Nonlocal Neighborhood Filters for Signal Denoising - Singer, Shkolnisky, et al. |

19 | An adaptive Gaussian filter for noise reduction and edge detection - Deng, Cahill - 1993 |

19 | Über die praktische auflösung von linearen integralgleichungen mit anwendungen auf randwertaufgaben der potentialtheorie - Nyström - 1928 |

18 | Biased anisotropic diffusion—A unified regularization and diffusion approach to edge detection - Nordström - 1990 |

17 | A tour of modern image filtering
- Milanfar
- 2013
(Show Context)
Citation Context ...c.) Understanding the spectral behavior of these filters in an orthogonal basis is important not only for better intuition about their properties, but also for analyzing their statistical performance =-=[37]-=-. This latter issue has become of great practical importance recently since many competing state of the art smoothing algorithms invented in the last few years appear to display comparable performance... |

16 | Image processing with non-local spectral bases - Peyré |

14 | Circular law theorem for random Markov matrices
- Bordenave, Caputo, et al.
(Show Context)
Citation Context ...k. This result by itself shows that the set of doubly-stochastic matrices Dn and the set of ordinary (row-) stochastic matrices Sn are close. But even more compelling is what happens when A is random =-=[10, 5, 22, 23, 29]-=-. In particular, it is known [23] that if the entries are drawn at random from a distribution on [0,1] such that E(Ai,j) = 1/n and Var(Ai,j) ≤ c1/n 2 , then the subdominant eigenvalue tends, in probab... |

14 | Doubly stochastic normalization for spectral clustering
- Zass, Shashua
(Show Context)
Citation Context ...e KL measure used here to do a similar or even better job? The evidence says no. In fact, other norms such as L1 and L2 for this approximation are more common in the machine learning literature (e.g. =-=[55]-=-.) Conceptually, the L2 projection would not seem to be a very good choice as it would likely push many entries to zero, which may not be desirable. We have observed experimentally that the use of eit... |

11 | and N. Sochen , "A short time beltrami kernel for smoothing images and manifolds
- Spira
- 2007
(Show Context)
Citation Context ...hedependenceofQij onthegiven datameansthatthesmoothingmatrix A = D −1 K is therefore nonlinear and shift varying. This kernel is closely related, but somewhat more general than the Beltrami kernel of =-=[47]-=- and the coherence enhancing diffusion approach of [53]. 2. Nearness of Stochastic and Doubly-Stochastic Matrices. Our interest in this paper is to convert a smoothing operator A which is generically ... |

11 | Scale-space and edge detection using anistropic diffusion - PERONA, MALIK - 1990 |

9 |
Distribution of subdominant eigenvalues of matrices with random rows
- Goldberg, Neumann
(Show Context)
Citation Context ...k. This result by itself shows that the set of doubly-stochastic matrices Dn and the set of ordinary (row-) stochastic matrices Sn are close. But even more compelling is what happens when A is random =-=[10, 5, 22, 23, 29]-=-. In particular, it is known [23] that if the entries are drawn at random from a distribution on [0,1] such that E(Ai,j) = 1/n and Var(Ai,j) ≤ c1/n 2 , then the subdominant eigenvalue tends, in probab... |

8 |
All admissible linear estimates of the mean vector
- Cohen
(Show Context)
Citation Context ...cal effects of symmetrization using the LARK smoother [49] introduced earlier. 3.1. Performance Improvement. First off, it is worth recalling an important result about the optimality of smoothers. In =-=[13]-=-, Cohen proved that asymmetric smoothers are inadmissible with respect to the mean-squared error measure. This means that for any linear smoother A there always exists a symmetric smoother Â which out... |

8 |
Note on the geometric-arithmetic mean inequalty, Lecture
- Gluskin, Milman
(Show Context)
Citation Context ...d little known fact that asymptotically with large n, the ratio of the geometric to arithmetic mean for any random sequence of numbers in (0,1] converges to the same constant 10 with high probability =-=[1, 21]-=-. Therefore asymptotically, β = O(α 1/n ). This indicates that whenthe row-stochastic A is nearly symmetric already (i.e. α ≈ 1and therefore β ≈ 1), the gain is small. Next we provide some examples of... |

8 |
Closest matrices in the space of generalized doubly stochastic matrices
- Khoury
(Show Context)
Citation Context ...tive-valued) smoothing matrices as this would facilitate their spectral analysis in orthonormal bases. As we hinted in [37], here we would be dealing with the class of generalized stochastic matrices =-=[32]-=-. • Remark 3: It is well-known [26] that when the smoother is symmetric, the estimator always has a Bayesian interpretation with a well-defined posterior density. By approximating a given smoother wit... |

7 | Scale space meshing of raw data point sets - Digne, Morel, et al. - 2011 |

6 | Concentration of the ratio between the geometric and arithmetic means
- Aldaz
- 2010
(Show Context)
Citation Context ...ic matrices Sn are close. But even more compelling is what happens when A is random [10, 5, 22, 23, 29]. In particular, it is known [23] that if the entries are drawn at random from a distribution on =-=[0,1]-=- such that E(Ai,j) = 1/n and Var(Ai,j) ≤ c1/n 2 , then the subdominant eigenvalue tends, in probability, to zero as n → ∞ at a rate of 1/ √ n. In fact, the same behavior occurs when only the rows are ... |

6 | Aspects of large random Markov kernels - Chafäı |

6 |
Distribution of subdominant eigenvalues of random matrices
- Goldberg, Okunev, et al.
(Show Context)
Citation Context ...k. This result by itself shows that the set of doubly-stochastic matrices Dn and the set of ordinary (row-) stochastic matrices Sn are close. But even more compelling is what happens when A is random =-=[10, 5, 22, 23, 29]-=-. In particular, it is known [23] that if the entries are drawn at random from a distribution on [0,1] such that E(Ai,j) = 1/n and Var(Ai,j) ≤ c1/n 2 , then the subdominant eigenvalue tends, in probab... |

5 |
Scale space meshing of raw data point sets, Computer Graphics Forum 30 (6
- Digne, Morel, et al.
- 2011
(Show Context)
Citation Context ...lly, we note that the type of normalization we promote would likely have some impact in other areas of work well beyond the current filtering context, such as scale-space meshing in computer graphics =-=[18]-=-, and in machine learning [3]. As we mentioned earlier, many popular filters are contained in the class of smoothing operators we consider. To be more specific, we highlight a few such kernels which l... |

4 | Patch-based near-optimal denoising - Chatterjee, Milanfar - 2012 |

2 | The ensemble of random markov matrices
- Horvat
- 2009
(Show Context)
Citation Context |

2 |
Aspects of large random markov kernels, Stochastics: An
- Chafai
(Show Context)
Citation Context |

2 |
An inequality for doubly stochastic matrices
- Johnson, Kellogg
- 1976
(Show Context)
Citation Context ... of view, the improvement results from the particular way in which Sinkhorn’s diagonal scaling perturbs the eigenvalues. The following result is the first step in establishing this fact. Theorem 3.1 (=-=[30]-=-).If A is row-stochastic and Â = RAC is doubly stochastic, then det(RC) ≥ 1 Furthermore, equality holds if and only if A is actually doubly stochastic. It follows as a corollary that det(A) ≤ det( Â),... |

2 |
Concerning non-negative matrices and doubly-stochastic matrices
- Sinkhorn, Knopp
- 1967
(Show Context)
Citation Context ...re informative when Â is an explicit symmetric approximation of A; and this is indeed the case. The interesting practical question of how to find this nearest element was addressed by Sinkhorn et al. =-=[34, 45, 46]-=-. Specifically, a stochastic matrix (indeed any non-negative matrix withpositive diagonals) can bescaled toadoubly-stochastic matrix via aproceduresometimes known as iterative proportional scaling, or... |

2 |
Introduction to the Numerical Soultion of Markov Chains
- Stewart
- 1994
(Show Context)
Citation Context ...all special cases. For their part, stochastic matrices find numerous applications in statistical signal processing; including in classical optimal filtering, image denoising [11], Markov chain theory =-=[42, 48]-=-, distributed processing [19], and many more. While the smoothing operator in (1.1) has a linear appearance, the A we consider can in fact depend on the given data samples y, and the locations x of th... |

1 |
Matrices of 0s and 1s with total support
- Brualdi
(Show Context)
Citation Context ...ecomposable if there does not exist permutation matrices P and Q such that PAQ is of the form ] [ A11 A12 ○ A22 with A11 and A22 being square matrices. A fully indecomposable matrix has total support =-=[6]-=-.Symmetrizing Smoothing Filters 7 Kullback-Leibler (KL) measure: ∑ i,j Âij log Âij Aij over all Â ∈ Dn. When the starting kernel kij is positive definite, the scaling procedure, which involves left a... |

1 | Boosting with the L2 loss: Regression and classification, J.Amer.Statist - Buhlmann, Yu |

1 | Bayesian backfitting, Statist - Hastie, Tibshirani - 2000 |

1 | An inequality for doubly stochastic matrices, J.Res.Nat.Bur - Johnson, Kellogg - 1976 |

1 | Coherence-enhancing diffusion, Int - Weickert - 1999 |

1 | near-optimal denoising - Patch-based - 1649 |

1 |
Image processing with non local spectral bases
- Peyré
(Show Context)
Citation Context ...d eigenfunctions of the Laplacian on the manifold of patches derived from an image. Peyré provided an interesting spectral analysis of the graph Laplacian for non-local means and bilateral kernels in =-=[41]-=-. This paper also discussed symmetrization of the operator, but rather a different one carried out element-wise that does not preserve stochasticity. Furthermore, Peyré used a non-linear thresholding ... |