## Robust Anisotropic Diffusion (1998)

### Cached

### Download Links

- [www.parc.xerox.com]
- [white.stanford.edu:80]
- [www.cs.brown.edu]
- [www.cs.ualberta.ca]
- [www.cns.nyu.edu]
- [webdocs.cs.ualberta.ca]
- [cs.brown.edu]
- CiteULike
- DBLP

### Other Repositories/Bibliography

Citations: | 281 - 16 self |

### BibTeX

@MISC{Black98robustanisotropic,

author = {Michael J. Black and Guillermo Sapiro and David Marimont and David Heeger},

title = {Robust Anisotropic Diffusion},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edge-stopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edge-stopping" function based on Tukey's biweight robust estimator, that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the ...

### Citations

3767 |
Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...any robust statistical techniques have been applied to standard problems in computer vision [1], [32], [44]. There are robust approaches for performing local image smoothing [5], image reconstruction =-=[17]-=-, [19], blur classification [13], surface reconstruction [47], segmentation [31], pose estimation [26], edge detection [28], structure from motion or stereo [48], [49], optical flow estimation [7], [8... |

3144 | A computational approach to edge detection
- Canny
- 1986
(Show Context)
Citation Context ...age. Middle: line process for Perona–Malik (bottom: thresholded). Right: Perona–Malik line process with spatial coherence (bottom: thresholded). We consider two kinds of interaction terms, hystere=-=sis [10]-=- and nonmaximum suppression [34]. Other common types of interactions (for example, corners) can be modeled in a similar way. The hysteresis term assists in the formation of unbroken contours while the... |

1847 |
Robust Statistics
- Huber
- 1981
(Show Context)
Citation Context ... within each constant region, the neighbor difference can be viewed as an outlier because it does not conform to the statistical assumptions. D. Robust Estimation The field of robust statistics [22], =-=[24]-=- is concerned with estimation problems in which the data contains gross errors, or outliers. Many robust statistical techniques have been applied to standard problems in computer vision [1], [32], [44... |

1401 |
Fatemi E.: Nonlinear total variation based noise removal algorithms
- Rudin, Osher
- 1992
(Show Context)
Citation Context ...g the mathematical properties of anisotropic diffusion and related variational formulations [4], [11], [25], [38], [50], developing related well-posed and stable equations [2], [3], [11], [21], [35], =-=[40]-=-, [50], extending and modifying anisotropic diffusion for fast and accurate implementations, modifying the diffusion equations for specific applications [20], and studying the relations between anisot... |

1295 | Scale-space and edge detection using anisotropic diffusion
- Perona, Malik
- 1990
(Show Context)
Citation Context ...in the continuity of edges. Index Terms—Anisotropic diffusion, line processes, robust statistics. I. INTRODUCTION SINCE THE elegant formulation of anisotropic diffusion introduced by Perona and Mali=-=k [38]-=- (see [15] for very early work in this topic), a considerable amount of research has been devoted to the theoretical and practical understanding of this and related methods for image enhancement. Rese... |

1100 |
Robust Regression and Outlier Detection
- Rousseeuw, Leroy
- 1987
(Show Context)
Citation Context ...need to determine how large the image gradient can be before we consider it to be an outlier. We appeal to tools from robust statistics to automatically estimate the “robust scale,” , of the image=-= as [39] MAD-=- median median (21) where “MAD” denotes the median absolute deviation and the constant is derived from the fact that the MAD of a zero-mean normal distribution with unit variance is 0.6745 1/1.482... |

739 | Reconstruction
- Blake, Zisserman
- 1987
(Show Context)
Citation Context ...mage. Middle: diffused images. Bottom: magnified regions of diffused images. tion between robust statistics and line processes has been explored elsewhere; see [6] for details and examples as well as =-=[9]-=-, [12], [16], [18], and [19] for recent related results. While we work with the discrete formulation here, it is easy to verify that the connections hold for the continuous formulation as well. Recall... |

652 |
Analog VLSI and Neural Systems
- Mead
- 1989
(Show Context)
Citation Context ...rious error norms they use. Moreover, some of their theoretical results, e.g., Theorems 1 and 3, are easily interpreted based on the concept of influence functions. Finally, Mead and colleagues [23], =-=[30] h-=-ave used analog VLSI (aVLSI) technology to build hardware devices that perform regularization. The aVLSI circuits behave much like a resistive grid, except that the resistors are replaced with “robu... |

590 |
Robust Statistics: The Approach Based on Influence Functions
- Hampel, Ronchetti, et al.
- 1988
(Show Context)
Citation Context ... noise within each constant region, the neighbor difference can be viewed as an outlier because it does not conform to the statistical assumptions. D. Robust Estimation The field of robust statistics =-=[22]-=-, [24] is concerned with estimation problems in which the data contains gross errors, or outliers. Many robust statistical techniques have been applied to standard problems in computer vision [1], [32... |

317 |
Image selective smoothing and edge detection by nonlin- ear diffusion
- Alvarez, Lions, et al.
- 1992
(Show Context)
Citation Context ...ted toward understanding the mathematical properties of anisotropic diffusion and related variational formulations [4], [11], [25], [38], [50], developing related well-posed and stable equations [2], =-=[3]-=-, [11], [21], [35], [40], [50], extending and modifying anisotropic diffusion for fast and accurate implementations, modifying the diffusion equations for specific applications [20], and studying the ... |

310 |
Axioms and fundamental equations of im- age processing
- Alvarez, Guichard, et al.
- 1993
(Show Context)
Citation Context ...oriented toward understanding the mathematical properties of anisotropic diffusion and related variational formulations [4], [11], [25], [38], [50], developing related well-posed and stable equations =-=[2]-=-, [3], [11], [21], [35], [40], [50], extending and modifying anisotropic diffusion for fast and accurate implementations, modifying the diffusion equations for specific applications [20], and studying... |

265 |
Constrained restoration and the recovery of discontinuities
- Geman, Reynolds
- 1992
(Show Context)
Citation Context ...fused images. Bottom: magnified regions of diffused images. tion between robust statistics and line processes has been explored elsewhere; see [6] for details and examples as well as [9], [12], [16], =-=[18]-=-, and [19] for recent related results. While we work with the discrete formulation here, it is easy to verify that the connections hold for the continuous formulation as well. Recall that the robust f... |

260 |
Coll T.: Image selective smoothing and edge detection by nonlinear diffusion
- Catte, Lions, et al.
- 1992
(Show Context)
Citation Context ... and related methods for image enhancement. Research in this area has been oriented toward understanding the mathematical properties of anisotropic diffusion and related variational formulations [4], =-=[11]-=-, [25], [38], [50], developing related well-posed and stable equations [2], [3], [11], [21], [35], [40], [50], extending and modifying anisotropic diffusion for fast and accurate implementations, modi... |

232 | Deterministic edge-preserving regularization in computed imaging
- Charbonnier, Blanc-Féraud, et al.
- 1996
(Show Context)
Citation Context ... Middle: diffused images. Bottom: magnified regions of diffused images. tion between robust statistics and line processes has been explored elsewhere; see [6] for details and examples as well as [9], =-=[12]-=-, [16], [18], and [19] for recent related results. While we work with the discrete formulation here, it is easy to verify that the connections hold for the continuous formulation as well. Recall that ... |

223 |
Robust regression methods for computer vision: a review. Int J Comput Vis 6:57–70
- Meer, Mintz, et al.
- 1991
(Show Context)
Citation Context ...22], [24] is concerned with estimation problems in which the data contains gross errors, or outliers. Many robust statistical techniques have been applied to standard problems in computer vision [1], =-=[32]-=-, [44]. There are robust approaches for performing local image smoothing [5], image reconstruction [17], [19], blur classification [13], surface reconstruction [47], segmentation [31], pose estimation... |

222 | Constructing Simple Stable Descriptions for Image Partitioning
- Leclerc
- 1989
(Show Context)
Citation Context ...lik � stopping function. Fig. 5. Tukey’s biweight. Fig. 6. Huber’s minmax estimator (modification of the vI norm). (14) The resulting -function is related to the robust error norm proposed by Le=-=clerc [27]-=-. The derivation is straightforward and is omitted here. IV. EXPLOITING THE RELATIONSHIP The above derivations demonstrate that anisotropic diffusion is the gradient descent of an estimation problem w... |

188 | A.: On the unification of line processes, outlier rejection, and robust statistics with applications in early vision
- Black, Rangarajan
- 1996
(Show Context)
Citation Context ...re considered to be “outliers” in the robust estimation framework. Edges in a smoothed image are, therefore, very simply detected as those points that are treated as outliers. We also show (follow=-=ing [6]-=-) that, for a particular class of robust error norms, anisotropic diffusion is equivalent to regularization with an explicit line process. The advantage of the line-process formulation is that we can ... |

162 | A framework for the robust estimation of optical flow - Black, Anandan - 1993 |

154 | Nonlinear anisotropic filtering of MRI data
- Gerig, Kubler, et al.
- 1992
(Show Context)
Citation Context ...able equations [2], [3], [11], [21], [35], [40], [50], extending and modifying anisotropic diffusion for fast and accurate implementations, modifying the diffusion equations for specific applications =-=[20]-=-, and studying the relations between anisotropic diffusion and other image processing operations [41], [45]. In this paper, we develop a statistical interpretation of anisotropic diffusion, specifical... |

142 |
Anisotropic diffusion of multivalued images with applications to color filtering
- Sapiro, Ringach
- 1996
(Show Context)
Citation Context ...ourage edges at a greater variety of orientations VI. VECTOR-VALUED IMAGES The extension of the results presented above to vectorvalued images is straightforward following the framework introduced in =-=[42]. -=-The basic idea is that the gradient direction and the gradient magnitude are replaced by concepts derived from the first fundamental form of the vector image. The direction of maximal change (“the g... |

139 |
Scene segmentation from visual motion using global optimization
- Murray, Buxton
- 1987
(Show Context)
Citation Context ...e line processing (25), we can add additional constraints on its spatial organization. While numerous authors have proposed spatial coherence constraints for discrete line processes [14], [17], [18], =-=[33]-=-, we need to generalize these results to the case of analog line processes [6].sBLACK et al.: ROBUST ANISOTROPIC DIFFUSION 429 Fig. 13. Anisotropic diffusion with spatial organization of outliers. Lef... |

134 | Nonlinear image recovery with half-quadratic regularization
- Geman, Yang
- 1995
(Show Context)
Citation Context ...bust statistical techniques have been applied to standard problems in computer vision [1], [32], [44]. There are robust approaches for performing local image smoothing [5], image reconstruction [17], =-=[19]-=-, blur classification [13], surface reconstruction [47], segmentation [31], pose estimation [26], edge detection [28], structure from motion or stereo [48], [49], optical flow estimation [7], [8], [43... |

123 |
I.: Feature-oriented image enhancement using shock filters
- Osher, Rudin
- 1990
(Show Context)
Citation Context ...hat stability problems will typically be solved, or at least moderated, by the spatial regularization and temporal delays introduced by the numerical methods for computing the gradient in [11], [25], =-=[37]. -=-B. Perona–Malik Discrete Formulation Perona and Malik discretized their anisotropic diffusion equation as follows: where is a discretely sampled image, denotes the pixel position in a discrete, two-... |

110 | P.: Robust dynamic motion estimation over time - Black, Anandan - 1991 |

102 | L.: A variational method in image recovery
- Aubert, Vese
- 1948
(Show Context)
Citation Context ... this and related methods for image enhancement. Research in this area has been oriented toward understanding the mathematical properties of anisotropic diffusion and related variational formulations =-=[4]-=-, [11], [25], [38], [50], developing related well-posed and stable equations [2], [3], [11], [21], [35], [40], [50], extending and modifying anisotropic diffusion for fast and accurate implementations... |

97 |
The Theory and Practice of Bayesian Image Labeling
- Chou, Brown
- 1990
(Show Context)
Citation Context ... formulation of the line processing (25), we can add additional constraints on its spatial organization. While numerous authors have proposed spatial coherence constraints for discrete line processes =-=[14]-=-, [17], [18], [33], we need to generalize these results to the case of analog line processes [6].sBLACK et al.: ROBUST ANISOTROPIC DIFFUSION 429 Fig. 13. Anisotropic diffusion with spatial organizatio... |

80 |
A.: A common framework for image segmentation
- Geiger, Yuille
- 1991
(Show Context)
Citation Context ... used to evaluate and compare different -functions and how new functions can be chosen in a principled way. See [6] for other robust -functions which could be used for anisotropic diffusion. See also =-=[16]-=- for related work connecting anisotropic diffusion, the mean-field -function, and binary line processes. It is interesting to note that common robust error norms have frequently been proposed in the l... |

80 |
Linear feature extraction and description
- Nevatia, Babu
- 1980
(Show Context)
Citation Context ...rona–Malik (bottom: thresholded). Right: Perona–Malik line process with spatial coherence (bottom: thresholded). We consider two kinds of interaction terms, hysteresis [10] and nonmaximum suppress=-=ion [34]-=-. Other common types of interactions (for example, corners) can be modeled in a similar way. The hysteresis term assists in the formation of unbroken contours while the nonmaximum suppression term inh... |

79 |
Behavioral analysis of anisotropic diffusion in image processing
- You, Xu, et al.
- 1996
(Show Context)
Citation Context ...ds for image enhancement. Research in this area has been oriented toward understanding the mathematical properties of anisotropic diffusion and related variational formulations [4], [11], [25], [38], =-=[50]-=-, developing related well-posed and stable equations [2], [3], [11], [21], [35], [40], [50], extending and modifying anisotropic diffusion for fast and accurate implementations, modifying the diffusio... |

78 |
Nonlinear image filtering with edge and corner enhancement
- Nitzberg, Shiota
- 1992
(Show Context)
Citation Context ...tanding the mathematical properties of anisotropic diffusion and related variational formulations [4], [11], [25], [38], [50], developing related well-posed and stable equations [2], [3], [11], [21], =-=[35]-=-, [40], [50], extending and modifying anisotropic diffusion for fast and accurate implementations, modifying the diffusion equations for specific applications [20], and studying the relations between ... |

71 | A common framework for curve evolution, segmentation and anisotropic diffusion
- Shah
- 1996
(Show Context)
Citation Context ...st and accurate implementations, modifying the diffusion equations for specific applications [20], and studying the relations between anisotropic diffusion and other image processing operations [41], =-=[45]. -=-In this paper, we develop a statistical interpretation of anisotropic diffusion, specifically, from the point of view of robust statistics. We show that the Perona–Malik [38] diffusion equation is e... |

62 |
Biased Anisotropic Diffusion - a Unified Regularization and Diffusion Approach to Edge Detection
- Nordstrom
- 1990
(Show Context)
Citation Context ...osed as: where is the domain of the image and where we have omitted for notational convenience. One way to minimize (8) is via gradient descent using the calculus of variations (see for example [21], =-=[36]-=-, [38], and [50]), as follows: By defining (8) div (9) (10) we obtain the straightforward relation between image reconstruction via robust estimation (8) and image reconstruction via anisotropic diffu... |

58 |
Image flow segmentation and estimation by constraint line clustering
- Schunck
- 1989
(Show Context)
Citation Context ...19], blur classification [13], surface reconstruction [47], segmentation [31], pose estimation [26], edge detection [28], structure from motion or stereo [48], [49], optical flow estimation [7], [8], =-=[43]-=-, and regularization with line processes [6]. For further details see [22] or, for a review of the applications of robust statistics in computer vision, see [32]. The problem of estimating a piecewise... |

46 | Regularization of discontinuous flow fields - Shulman, Herve - 1989 |

45 | Image processing: Flows under min/max curvature and mean curvature - Malladi, Sethian - 1996 |

40 |
Robust window operators
- Besl, Birch, et al.
- 1988
(Show Context)
Citation Context ...oss errors, or outliers. Many robust statistical techniques have been applied to standard problems in computer vision [1], [32], [44]. There are robust approaches for performing local image smoothing =-=[5]-=-, image reconstruction [17], [19], blur classification [13], surface reconstruction [47], segmentation [31], pose estimation [26], edge detection [28], structure from motion or stereo [48], [49], opti... |

36 |
A Two-Stage Algorithm for DiscontinuityPreserving Surface Reconstruction
- Sinha, Schunck
- 1992
(Show Context)
Citation Context ...rd problems in computer vision [1], [32], [44]. There are robust approaches for performing local image smoothing [5], image reconstruction [17], [19], blur classification [13], surface reconstruction =-=[47]-=-, segmentation [31], pose estimation [26], edge detection [28], structure from motion or stereo [48], [49], optical flow estimation [7], [8], [43], and regularization with line processes [6]. For furt... |

23 |
Information theory in electron microscopy
- Gabor
- 1965
(Show Context)
Citation Context ...tinuity of edges. Index Terms—Anisotropic diffusion, line processes, robust statistics. I. INTRODUCTION SINCE THE elegant formulation of anisotropic diffusion introduced by Perona and Malik [38] (se=-=e [15]-=- for very early work in this topic), a considerable amount of research has been devoted to the theoretical and practical understanding of this and related methods for image enhancement. Research in th... |

22 |
Analog hardware for detecting discontinuities in early vision
- Harris, Koch, et al.
- 1990
(Show Context)
Citation Context ...the various error norms they use. Moreover, some of their theoretical results, e.g., Theorems 1 and 3, are easily interpreted based on the concept of influence functions. Finally, Mead and colleagues =-=[23]-=-, [30] have used analog VLSI (aVLSI) technology to build hardware devices that perform regularization. The aVLSI circuits behave much like a resistive grid, except that the resistors are replaced with... |

18 | Nonlinear diffusion scale-spaces: From the continuous to discrete setting, in: M.O. Berger et al - Weickert - 1996 |

10 |
Analysis of different robust methods for pose refinement
- Kumar, Hanson
- 1990
(Show Context)
Citation Context ... [44]. There are robust approaches for performing local image smoothing [5], image reconstruction [17], [19], blur classification [13], surface reconstruction [47], segmentation [31], pose estimation =-=[26]-=-, edge detection [28], structure from motion or stereo [48], [49], optical flow estimation [7], [8], [43], and regularization with line processes [6]. For further details see [22] or, for a review of ... |

10 |
From active contours to anisotropic diffusion: connections between basic PDE's in image processing
- Sapiro
- 1996
(Show Context)
Citation Context ...for fast and accurate implementations, modifying the diffusion equations for specific applications [20], and studying the relations between anisotropic diffusion and other image processing operations =-=[41], -=-[45]. In this paper, we develop a statistical interpretation of anisotropic diffusion, specifically, from the point of view of robust statistics. We show that the Perona–Malik [38] diffusion equatio... |

7 | Robust computational vision - Schunck - 1990 |

3 |
Robust statistical methods for building classification procedures
- Chen, Schunck
- 1990
(Show Context)
Citation Context ...s have been applied to standard problems in computer vision [1], [32], [44]. There are robust approaches for performing local image smoothing [5], image reconstruction [17], [19], blur classification =-=[13]-=-, surface reconstruction [47], segmentation [31], pose estimation [26], edge detection [28], structure from motion or stereo [48], [49], optical flow estimation [7], [8], [43], and regularization with... |

2 |
framework for the robust estimation of optical flow
- “A
- 1993
(Show Context)
Citation Context ...7], [19], blur classification [13], surface reconstruction [47], segmentation [31], pose estimation [26], edge detection [28], structure from motion or stereo [48], [49], optical flow estimation [7], =-=[8]-=-, [43], and regularization with line processes [6]. For further details see [22] or, for a review of the applications of robust statistics in computer vision, see [32]. The problem of estimating a pie... |

2 |
On robust edge detection
- Lui, Schunck, et al.
- 1990
(Show Context)
Citation Context ...st approaches for performing local image smoothing [5], image reconstruction [17], [19], blur classification [13], surface reconstruction [47], segmentation [31], pose estimation [26], edge detection =-=[28]-=-, structure from motion or stereo [48], [49], optical flow estimation [7], [8], [43], and regularization with line processes [6]. For further details see [22] or, for a review of the applications of r... |

2 |
Robust recovery of piecewise polynomial image structure
- Meer, Mintz, et al.
- 1990
(Show Context)
Citation Context ...uter vision [1], [32], [44]. There are robust approaches for performing local image smoothing [5], image reconstruction [17], [19], blur classification [13], surface reconstruction [47], segmentation =-=[31]-=-, pose estimation [26], edge detection [28], structure from motion or stereo [48], [49], optical flow estimation [7], [8], [43], and regularization with line processes [6]. For further details see [22... |

2 |
Robust dynamic stereo for incremental disparity map refinement
- Tirumalai, Schunck, et al.
- 1990
(Show Context)
Citation Context ...ge smoothing [5], image reconstruction [17], [19], blur classification [13], surface reconstruction [47], segmentation [31], pose estimation [26], edge detection [28], structure from motion or stereo =-=[48]-=-, [49], optical flow estimation [7], [8], [43], and regularization with line processes [6]. For further details see [22] or, for a review of the applications of robust statistics in computer vision, s... |

1 |
Workshop on Robust Computer Vision
- Int
- 1990
(Show Context)
Citation Context ...ics [22], [24] is concerned with estimation problems in which the data contains gross errors, or outliers. Many robust statistical techniques have been applied to standard problems in computer vision =-=[1]-=-, [32], [44]. There are robust approaches for performing local image smoothing [5], image reconstruction [17], [19], blur classification [13], surface reconstruction [47], segmentation [31], pose esti... |

1 |
Multiscale analysis of movies: Theory and algorithms
- Guichard
- 1993
(Show Context)
Citation Context ...understanding the mathematical properties of anisotropic diffusion and related variational formulations [4], [11], [25], [38], [50], developing related well-posed and stable equations [2], [3], [11], =-=[21]-=-, [35], [40], [50], extending and modifying anisotropic diffusion for fast and accurate implementations, modifying the diffusion equations for specific applications [20], and studying the relations be... |