Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification
Citations
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3,830 citations
2,649 citations
Cites methods or result from "Curve evolution implementation of t..."
...Most of the models need three level set functions, as in Zhao et al. (1996) and Samson et al....
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...A standard rescaling can be made, as in Zhao et al. (1996), by replacing δε(φ) by |∇φ|, giving the following equations, already introduced in Osher and Sethian (1988) in the context of the level set theory:...
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...We would like to mention that ideas very similar with those from the above case, have been also developed by Tsai et al. (2001), independently and contemporaneously....
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...14 and 15 have been obtained independently and contemporaneously by Tsai et al. (2001). Finally, we show in Fig....
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...14 and 15 have been obtained independently and contemporaneously by Tsai et al. (2001)....
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1,630 citations
Cites background from "Curve evolution implementation of t..."
...For an image on the image domain , they propose to minimize the following energy: (2) where and represent the regions outside and inside the contour , respectively, and and are two constants that approximate the image intensity in and ....
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...In [29], Vese and Chan introduced an energy functional on a level set function and two smooth functions and that are defined on the regions outside and inside the zero level contour of a level set function , respectively....
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...Obviously, the involved computation in PS model is expensive, which limits its applications in practice....
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1,201 citations
Cites background from "Curve evolution implementation of t..."
...Z. Ding and J. C. Gore are with the Institute of Imaging Science, Vanderbilt University, Nashville, TN 37232 USA (e-mail: zhaohua.ding@vanderbilt.edu; john.gore@vanderbilt.edu)....
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...J. C. Gatenby was with the Institute of Imaging Science, Vanderbilt University, Nashville, TN 37232 USA....
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...C. Li was with the Institute of Imaging Science, Vanderbilt University, Nashville, TN 37232 USA....
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1,117 citations
Cites background or methods from "Curve evolution implementation of t..."
...The probabilistic formulation of the segmentation problem presented in the following extends the statistical approaches pioneered in Leclerc (1989), Zhu and Yuille (1996), Paragios and Deriche (2002) and Tsai et al. (2001)....
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...Tsai et al. (2001, 2003) proposed a very efficient implementation of shape-driven level set segmentation by directly optimizing in the linear subspace spanned by the principal components....
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References
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"Curve evolution implementation of t..." refers methods in this paper
...For image smoothing, the technique of anisotropic diffusion has become a widespread field of research ranging from techniques based upon the original formulation of Perona and Malik [29], [30] to curve and surface evolution methods based upon geometric scale spaces [13], [16], [17], [35] and to a number of recent techniques for color imagery and other forms of vector-valued data [39], [40], [44]–[46], [50]....
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10,404 citations
"Curve evolution implementation of t..." refers background or methods or result in this paper
...Obviously, this is not possible ith lower dimensional models (such as ones based on mean ntensities [9])....
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...In particular, it naturally generalizes the recent work of Chan and Vese in [9] who consider piecewise constant generalization of the Mumford–Shah functional within a level set framework.2 We note that region-based approaches in general, enjoy a number of attractive properties including greater robustness to noise (by avoiding derivatives of the image intensity) and initial contour placement (by being less local than most edge-based approaches)....
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...Consider the following vector version of the Mumford–Shah functional: 4Chan and Vese, who have considered the piecewise constant version of the Mumford–Shah functional [9], have also extended their framework to vectorvalued data in “Active Contours Without Edges for Vector-Valued Images” (see http://www.math.ucla.edu/applied/cam). where and denote theth component of the-dimensional vector-valued observed data and its smooth estimate, respectively....
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...In contrast to most other region based techniques however (including our own previous work [48], [49] and that of Chan-Vese [9] and Paragios-Deriche [27]), which assume highly constrained parametric models for pixel intensities within each region, our approach employs the statistical model directly implied by the Mumford–Shah functional....
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...This development may be regarded as an extension of several recent region-based approaches to curve evolution [9], [27], [48]....
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5,516 citations
"Curve evolution implementation of t..." refers background or methods in this paper
...We note that one of the most widely studied mathematical models in image processing and computer vision addresses both goals simultaneously, namely that of Mumford and Shah [22], [23] who presented the variational problem of minimizing a functional involving a piecewise smooth representation of an image....
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...This follows from the theory of junctions as presented in [22]....
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...The estimates and that minimize (2) satisfy (decoupled) PDEs which can be obtained using standard variational methods [22]....
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...in which denotes the smooth, closed segmenting curve, denotes the observed data, denotes the piecewise smooth approximation to with discontinuities only along , and denotes the image domain [22], [23]....
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...For the rest of the paper, we will refer to this gradient flow, which is also derived in [22], as the Mumford–Shah flow....
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