Metamorphosis of images in reproducing kernel Hilbert spaces

Richardson, Casey L.; Younes, Laurent

doi:10.1007/s10444-015-9435-y

Cited by 20 publications

(19 citation statements)

References 38 publications

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“…Bearing in mind the above, metamorphosis based image registration in [49] easily extends to the indirect setting, which in turn extends the LDDMM based indirect registration in (24): Here, L : X → R is given by (23) and V : L 2 ([0, 1], V ) × X → X is given by (45) (geometric group action). Note that the velocity field ν ∈ L 2 ([0, 1], V ) in the objective also generates, through (10), the diffeomorphism φ ν 0,t : Ω → Ω in the ODE constraint.…”

Section: 2mentioning

confidence: 84%

Indirect Image Registration with Large Diffeomorphic Deformations

Chen

Öktem

2018

SIAM J. Imaging Sci.

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The paper adapts the large deformation diffeomorphic metric mapping framework for image registration to the indirect setting where a template is registered against a target that is given through indirect noisy observations. The registration uses diffeomorphisms that transform the template through a (group) action. These diffeomorphisms are generated by solving a flow equation that is defined by a velocity field with certain regularity. The theoretical analysis includes a proof that indirect image registration has solutions (existence) that are stable and that converge as the data error tends so zero, so it becomes a well-defined regularization method. The paper concludes with examples of indirect image registration in 2D tomography with very sparse and/or highly noisy data.

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Section: 2mentioning

confidence: 84%

Indirect Image Registration with Large Diffeomorphic Deformations

Chen

Öktem

2018

SIAM J. Imaging Sci.

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“…But it can be also viewed as extending metamorphoses of classical images studied in previous works like [37,25,34]. In the fshape perspective, this is the situation where M = Ω is a bounded domain of R d and all geometrical shapes are fixed to X = q(Ω) = Ω.…”

Section: Link To Image Metamorphosismentioning

confidence: 96%

Metamorphoses of Functional Shapes in Sobolev Spaces

Charon¹,

Charlier²,

Trouvé³

2018

Found Comput Math

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In this paper, we describe in detail a model of geometric-functional variability between fshapes. These objects were introduced for the first time by the authors in [14] and are basically the combination of classical deformable manifolds with additional scalar signal map. Building on the aforementioned work, this paper's contributions are several. We first extend the original L 2 model in order to represent signals of higher regularity on their geometrical support with more regular Hilbert norms (typically Sobolev). We describe the bundle structure of such fshape spaces with their adequate geodesic distances, encompassing in one common framework usual shape comparison and image metamorphoses. We then propose a formulation of matching between any two fshapes from the optimal control perspective, study existence of optimal controls and derive Hamiltonian equations and conservation laws describing the dynamics of geodesics. Secondly, we tackle the discrete counterpart of these problems and equations through appropriate finite elements interpolation schemes on triangular meshes. At last, we show a few results of metamorphosis matchings on synthetic and several real data examples in order to highlight the key specificities of the approach.

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“…For instance, this could happen in the case of the geometric group action due to the appearance of new structures in the target image or due to a discrepancy between the image intensities of the template and the target image. A possible solution is provided by the metamorphosis framework [38,46,51,52], which is an extension to LDDMM that allows for modulations of the image intensities along characteristics of the flow. The image intensities change according to an additional flow equation with an unknown source.…”

Section: Introductionmentioning

confidence: 99%

Template-Based Image Reconstruction from Sparse Tomographic Data

et al. 2019

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We propose a variational regularisation approach for the problem of template-based image reconstruction from indirect, noisy measurements as given, for instance, in X-ray computed tomography. An image is reconstructed from such measurements by deforming a given template image. The image registration is directly incorporated into the variational regularisation approach in the form of a partial differential equation that models the registration as either mass-or intensity-preserving transport from the template to the unknown reconstruction. We provide theoretical results for the proposed variational regularisation for both cases. In particular, we prove existence of a minimiser, stability with respect to the data, and convergence for vanishing noise when either of the abovementioned equations is imposed and more general distance functions are used. Numerically, we solve the problem by extending existing Lagrangian methods and propose a multilevel approach that is applicable whenever a suitable downsampling procedure for the operator and the measured data can be provided. Finally, we demonstrate the performance of our method for template-based image reconstruction from highly undersampled and noisy Radon transform data. We compare results for massand intensity-preserving image registration, various regularisation functionals, and different distance functions. Our results show that very reasonable reconstructions can be obtained when only few measurements are available and demonstrate that the use of a normalised cross correlation-based distance is advantageous when the image intensities between the template and the unknown image differ substantially.

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Metamorphosis of images in reproducing kernel Hilbert spaces

Cited by 20 publications

References 38 publications

Indirect Image Registration with Large Diffeomorphic Deformations

Indirect Image Registration with Large Diffeomorphic Deformations

Metamorphoses of Functional Shapes in Sobolev Spaces

Template-Based Image Reconstruction from Sparse Tomographic Data

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