Optimal approximations by piecewise smooth functions and associated variational problems

Mumford, David; Shah, Jayant

doi:10.1002/cpa.3160420503

Cited by 4,694 publications

(3,224 citation statements)

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“…The functional (1.3) is a variant of a functional which first appeared in the theory of image segmentation, in a celebrated paper by Mumford and Shah [29]. The set Γ belongs to the class Let u Γ be a minimum energy displacement relative to ψ and Γ , that is, let u Γ be a minimizer for…”

Section: γ (T) = ∅ For Every T T Imentioning

confidence: 99%

Crack Initiation in Brittle Materials

Chambolle

Giacomini

Ponsiglione

2007

Arch Rational Mech Anal

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In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → tg(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time t i > 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functionalwhere Ω ⊆ R 2 , k > 0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (u Γ , Γ ) with H 1 (Γ ) small enough are such that E(u, ∅) < E(u Γ , Γ ).

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Section: γ (T) = ∅ For Every T T Imentioning

confidence: 99%

Crack Initiation in Brittle Materials

Chambolle

Giacomini

Ponsiglione

2007

Arch Rational Mech Anal

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“…In addition, our model can be applied to a variety of images once the statistical distribution of noise in the image is known (21), with no need for a priori knowledge of the shape of the objects to be detected. Importantly, only one parameter needs to be set in the model and it allows choosing the maximum admissible boundary curvature: the regularization term that depends on this parameter and prevents the "rupture" of the interface (33). The major goal of this study was to validate the use of this approach with CMR images against the standard semiautomated methodology in a group of patients with a wide range of LV function.…”

Section: Discussionmentioning

confidence: 99%

Automated frame‐by‐frame endocardial border detection from cardiac magnetic resonance images for quantitative assessment of left ventricular function: Validation and clinical feasibility

Corsi

Veronesi

Lamberti

et al. 2009

Magnetic Resonance Imaging

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Purpose: To develop a technique based on image noise distribution for automated endocardial border detection from cardiac magnetic resonance (CMR) images throughout the cardiac cycle, validate it, and test its clinical utility. Materials and Methods:Images obtained in 36 patients were analyzed using custom software to obtain left ventricular (LV) volume throughout the cardiac cycle, end-systolic and end-diastolic LV volumes, and ejection fraction (EF). Validation against manually-traced endocardial boundaries included intertechnique comparisons of LV volumes, slice areas, and border positions. Then, the clinical feasibility of the dynamic automated analysis of LV function was tested in 14 patients with normal LV function, 12 patients with systolic dysfunction, and 10 patients with diastolic dysfunction.Results: Analysis time for one cardiac cycle was Ͻ15 minutes. Intertechnique comparisons resulted in high correlation (r Ͼ 0.96), small biases (volumes: -6 mL; EF: 4.6%) and narrow limits of agreement (volumes: 17.6 mL; EF: 9.2%). We found significant intergroup differences in multiple quantitative indices of systolic and diastolic function. Conclusion:Fast, automated, dynamic detection of LV endocardial boundaries is feasible and allows accurate quantification of LV size and function, which is potentially clinically useful for objective assessment of systolic and diastolic dysfunction. CARDIAC MAGNETIC RESONANCE (CMR) is a noninvasive imaging modality with excellent spatial and contrast resolution that has become the standard reference in the assessment of left ventricular (LV) size and function (1,2), against which other techniques are frequently validated (3-12). However, this technique relies on the detection of endocardial boundaries, which requires frame-by-frame manual tracing on multiple slices, and is thus of limited value in clinical practice. Moreover, continuous measurement of LV volume requires slice-by-slice, phase-by-phase detection of the endocardial boundaries, which with the commercially available analysis software packages, based on the analysis of image intensity gradients, requires manual corrections in most patients and thus remains semiautomated at best, time consuming, and subjective. Thus, the dynamic nature of CMR imaging is largely unutilized in the clinical assessment of LV function, which is commonly reduced to visually identifying in a small number of slices the end-systolic (ES) and end-diastolic (ED) frames as those that depict the smallest and largest cross-sectional LV cavity areas, respectively. These frames are then traced slice-by-slice and used to obtain ES and ED volumes (ESV and EDV) and calculate the ejection fraction (EF). This practical solution can be challenging in patients with severely reduced systolic function, when volume changes are minimal, and inaccurate in hearts, wherein ventricular contraction is uneven and dyssynchronized so that minimum and maximum cross-sectional areas may appear in different slices at different times. Availability of a reliable technique f...

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“…Based on the Mumford and Shah functional [5] for segmentation, Chan and Vese [2] proposed a level set method based active contour model to detect objects whose boundaries are not necessarily defined by a gradient. Let us denote a given image by I 0 : Ω → R and suppose C (C = ∂R) is a hypersurface representing a boundary of a region of interest R ⊂ Ω.…”

Section: The Chan-vese Intensity Based Segmentation Modelmentioning

confidence: 99%