A Mumford–Shah-Like Method for Limited Data Tomography with an Application to Electron Tomography

Klann, Esther

doi:10.1137/100817371

Cited by 31 publications

(42 citation statements)

References 30 publications

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“…Additionally, the number of segments does not have to be known in advance (in contrast to multiple level set methods) and the minimization with respect to the jump indicator functions is a simple elliptic problem. We want to minimize the Mumford-Shah-like functional [30], have been applied for a wide range of inverse problems, see, e.g., [20,26,27,34,35,37]). The basic idea behind the functional is as follows.…”

Section: A Mumford-shah-like Functional For Qpatmentioning

confidence: 99%

6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

Beretta¹,

Muszkieta²,

Naetar³

et al. 2016

Variational Methods

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We consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients µ(x), D(x), from a single measurement of the absorbed energy E(x) = µ(x)u(x), where u satisfies the elliptic partial differential equationThis problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. Using similar ideas as in [31], we show that when the coefficients µ, D are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of µ, D, we suggest a variational method based on an Ambrosio-Tortorelli approximation of a MumfordShah-like functional, which we implement numerically and test on simulated two-dimensional data.

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Section: A Mumford-shah-like Functional For Qpatmentioning

confidence: 99%

6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

Beretta¹,

Muszkieta²,

Naetar³

et al. 2016

Variational Methods

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“…The Mumford Shah methods in [37,39] and the total variation methods in [66] do use them. We use this assumption in our algorithm by assuming our function can be written as a finite sums of Haar basis functions (see (3.1)).…”

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confidence: 99%

Wavelet methods for a weighted sparsity penalty for region of interest tomography

2015

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We consider region of interest (ROI) tomography of piecewise constant functions. Additionally, an algorithm is developed for ROI tomography of piecewise constant functions using a Haar wavelet basis. A weighted ℓ ppenalty is used with weights that depend on the relative location of wavelets to the region of interest. We prove that the proposed method is a regularization method, i.e., that the regularized solutions converge to the exact piecewise constant solution if the noise tends to zero. Tests on phantoms demonstrate the effectiveness of the method.

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“…Meanwhile some non-convex regularizers have also been proposed recently [13,14] since they may lead to better reconstruction results for some imaging applications than classical convex regularizers. It has also been suggested that solving joint segmentation/reconstruction problems based on the Mumford-Shah regularization functional leads to better reconstruction results than performing reconstruction and segmentation successively [15][16][17]. In detail, the Mumford-Shah functional has been firstly used for image denoising and segmentation problems for a forward operator A equal to the identity operator [18].…”

Section: Introductionmentioning

confidence: 99%

Super-resolution/segmentation of 2D trabecular bone images by a Mumford-Shah approach and comparison to total variation

Toma

Sixou

et al. 2016

2016 24th European Signal Processing Conference (EUSIPCO)

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International audienceThe analysis of trabecular bone micro structure from in-vivo CT images is still limited due to insufficient spatial resolution. The goal of this work is to address both the problem of increasing the resolution of the image and of the segmenta-tion of the bone structure. To this aim, we investigate the joint super-resolution/segmentation problem by an approach based on the Mumford-Shah model. The validation of the method is performed on blurred, noisy and down-sampled images. A comparison of the reconstruction results with the Total Variation regularization is showed

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A Mumford–Shah-Like Method for Limited Data Tomography with an Application to Electron Tomography

Cited by 31 publications

References 30 publications

6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

Wavelet methods for a weighted sparsity penalty for region of interest tomography

Super-resolution/segmentation of 2D trabecular bone images by a Mumford-Shah approach and comparison to total variation

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