Robust anisotropic diffusion
TL;DR: It is shown that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image and the connection to the error norm and influence function in the robust estimation framework 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.
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 line processes allows us to develop new anisotropic diffusion equations that result in a qualitative improvement in the continuity of edges.
Citations
More filters
TL;DR: It is proved the convergence of a recursive mean shift procedure to the nearest stationary point of the underlying density function and, thus, its utility in detecting the modes of the density.
Abstract: A general non-parametric technique is proposed for the analysis of a complex multimodal feature space and to delineate arbitrarily shaped clusters in it. The basic computational module of the technique is an old pattern recognition procedure: the mean shift. For discrete data, we prove the convergence of a recursive mean shift procedure to the nearest stationary point of the underlying density function and, thus, its utility in detecting the modes of the density. The relation of the mean shift procedure to the Nadaraya-Watson estimator from kernel regression and the robust M-estimators; of location is also established. Algorithms for two low-level vision tasks discontinuity-preserving smoothing and image segmentation - are described as applications. In these algorithms, the only user-set parameter is the resolution of the analysis, and either gray-level or color images are accepted as input. Extensive experimental results illustrate their excellent performance.
11,727 citations
Cites background from "Robust anisotropic diffusion"
...The connection between anisotropic diffusion and M-estimators is analyzed in [5]....
[...]
TL;DR: A novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery.
Abstract: It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained l1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms l1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted l1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the l1 norm of the coefficient sequence as is common, but by reweighting the l1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing.
4,869 citations
Cites background from "Robust anisotropic diffusion"
...As demonstrated in [61], some possible guidance in these directions may come from studying connections in robust statistics....
[...]
Book•
30 Sep 2010
TL;DR: Computer Vision: Algorithms and Applications explores the variety of techniques commonly used to analyze and interpret images and takes a scientific approach to basic vision problems, formulating physical models of the imaging process before inverting them to produce descriptions of a scene.
Abstract: Humans perceive the three-dimensional structure of the world with apparent ease. However, despite all of the recent advances in computer vision research, the dream of having a computer interpret an image at the same level as a two-year old remains elusive. Why is computer vision such a challenging problem and what is the current state of the art? Computer Vision: Algorithms and Applications explores the variety of techniques commonly used to analyze and interpret images. It also describes challenging real-world applications where vision is being successfully used, both for specialized applications such as medical imaging, and for fun, consumer-level tasks such as image editing and stitching, which students can apply to their own personal photos and videos. More than just a source of recipes, this exceptionally authoritative and comprehensive textbook/reference also takes a scientific approach to basic vision problems, formulating physical models of the imaging process before inverting them to produce descriptions of a scene. These problems are also analyzed using statistical models and solved using rigorous engineering techniques Topics and features: structured to support active curricula and project-oriented courses, with tips in the Introduction for using the book in a variety of customized courses; presents exercises at the end of each chapter with a heavy emphasis on testing algorithms and containing numerous suggestions for small mid-term projects; provides additional material and more detailed mathematical topics in the Appendices, which cover linear algebra, numerical techniques, and Bayesian estimation theory; suggests additional reading at the end of each chapter, including the latest research in each sub-field, in addition to a full Bibliography at the end of the book; supplies supplementary course material for students at the associated website, http://szeliski.org/Book/. Suitable for an upper-level undergraduate or graduate-level course in computer science or engineering, this textbook focuses on basic techniques that work under real-world conditions and encourages students to push their creative boundaries. Its design and exposition also make it eminently suitable as a unique reference to the fundamental techniques and current research literature in computer vision.
4,146 citations
TL;DR: A novel method is proposed for performing multilabel, interactive image segmentation using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs.
Abstract: A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with user-defined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a high-quality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs
2,610 citations
Cites methods from "Robust anisotropic diffusion"
...This is a common feature of graph-based algorithms for image analysis and several weighting functions are commonly used in the literature [14], [20], [45]....
[...]
TL;DR: A new technique for the display of high-dynamic-range images, which reduces the contrast while preserving detail, is presented, based on a two-scale decomposition of the image into a base layer.
Abstract: We present a new technique for the display of high-dynamic-range images, which reduces the contrast while preserving detail. It is based on a two-scale decomposition of the image into a base layer,...
1,715 citations
Cites background from "Robust anisotropic diffusion"
...Figure 7: Huber’s minimax (after [Black et al. 1998])....
[...]
...Figure 5: Tukey’s biweight (after [Black et al. 1998])....
[...]
...In the plot of ψ, we see that the influence function is redescending [Black et al. 1998; Huber 1981]1....
[...]
References
More filters
TL;DR: There is a natural uncertainty principle between detection and localization performance, which are the two main goals, and with this principle a single operator shape is derived which is optimal at any scale.
Abstract: This paper describes a computational approach to edge detection. The success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points. These goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution. We define detection and localization criteria for a class of edges, and present mathematical forms for these criteria as functionals on the operator impulse response. A third criterion is then added to ensure that the detector has only one response to a single edge. We use the criteria in numerical optimization to derive detectors for several common image features, including step edges. On specializing the analysis to step edges, we find that there is a natural uncertainty principle between detection and localization performance, which are the two main goals. With this principle we derive a single operator shape which is optimal at any scale. The optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a Gaussian-smoothed image. We extend this simple detector using operators of several widths to cope with different signal-to-noise ratios in the image. We present a general method, called feature synthesis, for the fine-to-coarse integration of information from operators at different scales. Finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge.
28,073 citations
TL;DR: The analogy between images and statistical mechanics systems is made and the analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations, creating a highly parallel ``relaxation'' algorithm for MAP estimation.
Abstract: We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
18,761 citations
"Robust anisotropic diffusion" refers background or methods in this paper
...and where l are analog line processes ( 0 1) [16, 17 ]....
[...]
...[14, 17 , 18, 33], we need to generalize these results to the ca se of analog line processes [6]....
[...]
...reconstruction [19, 17 ], blur classification [13], surface reconstruction [47], segmentation [31],...
[...]
TL;DR: In this article, a constrained optimization type of numerical algorithm for removing noise from images is presented, where the total variation of the image is minimized subject to constraints involving the statistics of the noise.
15,225 citations
"Robust anisotropic diffusion" refers background in this paper
...The hysteresis term assists in the formation of unbroken contours while the nonmaximum suppression term inhibits multiple responses to a single edge present in the data....
[...]
...To extend the robust anisotropic diffusion approach to vector data, we have a number of possibilities....
[...]
...…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…...
[...]
TL;DR: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced, chosen to vary spatially in such a way as to encourage intra Region smoothing rather than interregion smoothing.
Abstract: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced. The diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing. It is shown that the 'no new maxima should be generated at coarse scales' property of conventional scale space is preserved. As the region boundaries in the approach remain sharp, a high-quality edge detector which successfully exploits global information is obtained. Experimental results are shown on a number of images. Parallel hardware implementations are made feasible because the algorithm involves elementary, local operations replicated over the image. >
12,560 citations
"Robust anisotropic diffusion" refers background or methods in this paper
...We show that the Perona–Malik [38] diffusion equation is equivalent to a robust procedure that estimates...
[...]
...The same treatment can be used to recover a-function for the other -function proposed by Perona and Malik (14) The resulting -function is related to the robust error norm proposed by Leclerc [27]....
[...]
...Perona and Malik discretized their anisotropic diffusion equation as follows: (2) where is a discretely sampled image,denotes the pixel position in a discrete, two-dimensional (2-...
[...]
...Perona and Malik [38] replaced the classical isotropic diffusion equation with div (1) where is the gradient magnitude, and is an “edge-stopping” function....
[...]
...Perona and Malik first suggested (11) for a positive constant ....
[...]
Book•
01 Jan 1987
TL;DR: This paper presents the results of a two-year study of the statistical treatment of outliers in the context of one-Dimensional Location and its applications to discrete-time reinforcement learning.
Abstract: 1. Introduction. 2. Simple Regression. 3. Multiple Regression. 4. The Special Case of One-Dimensional Location. 5. Algorithms. 6. Outlier Diagnostics. 7. Related Statistical Techniques. References. Table of Data Sets. Index.
6,955 citations
"Robust anisotropic diffusion" refers background in this paper
...Starting with the Lorentzian norm, the new error term with constraints on the line processes becomes (30) where Differentiating this equation with respect toand gives the following update equations: (31) (32) Without the additional spatial constraints, the line process formulation was identical to…...
[...]