Fast Marching Method for Generic Shape from Shading

Prados, Emmanuel; Soatto, Stefano

doi:10.1007/11567646_27

Cited by 21 publications

(19 citation statements)

References 26 publications

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“…the function u → S(ρ, x, t, u) is non increasing), which partly ensures that we will obtain an approximation of the viscosity solution of the considered PDE. One can easily verify that the scheme (20) is indeed monotonic (Prados and Soatto (2005)). We will take advantage of the various properties of that scheme to obtain simultaneously and consistently the approximations of the geodesic distance function and of the optimal dynamics.…”

Section: Following Prados and Soattomentioning

confidence: 90%

“…As in the classical Fast Marching method (Sethian (1996b), Sethian and Vladimirsky (2003), Prados and Soatto (2005)), the grid points are divided into the three classes: Accepted, Considered, Far. Below U , f , R and S are respectively the approximations of the (geodesic) distance function, the optimal dynamics f * x , R and S (defined in section 4.5).…”

Section: Global Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Brain Connectivity Mapping Using Riemannian Geometry, Control Theory, and PDEs

Lenglet¹,

Prados²,

Pons³

et al. 2009

SIAM J. Imaging Sci.

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We introduce an original approach for the cerebral white matter connectivity mapping from Diffusion Tensor Imaging (DTI). Our method relies on a global modeling of the acquired Magnetic Resonance Imaging (MRI) volume as a Riemannian manifold whose metric directly derives from the diffusion tensor. These tensors will be used to measure physical three-dimensional distances between different locations of a brain diffusion tensor image.The key concept is the notion of geodesic distance that will allow us to find optimal paths in the white matter. We claim that such optimal paths are reasonable approximations of neural fiber bundles. The geodesic distance function can be seen as the solution of two theoretically equivalent but, in practice, significantly different problems in the Partial Differential Equation framework: An initial value problem which is intrinsically dynamic, and a boundary value problem which is, on the contrary, intrinsically stationary. The two approaches have very different properties which make them more or less adequate for our problem and more or less computationally efficient. The dynamic formulation is quite easy to implement but has several practical drawbacks. On the contrary, the stationary formulation is much more tedious to implement, however we will show that it has many virtue which make it more suitable for our connectivity mapping problem. Finally, we will present different possible measures of connectivity, reflecting the degree of connectivity between different regions of the brain. We will illustrate these notions on synthetic and real DTI

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Section: Following Prados and Soattomentioning

confidence: 90%

Section: Global Algorithmmentioning

confidence: 99%

Brain Connectivity Mapping Using Riemannian Geometry, Control Theory, and PDEs

Lenglet¹,

Prados²,

Pons³

et al. 2009

SIAM J. Imaging Sci.

Self Cite

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show abstract

“…We use a Lagrangian approach to compute the values D 0 and C 0 at grid points in Ω adjacent to the boundary ∂ 0 Ω. At the remain grid points in Ω, we adopted a fast marching framework to compute the values D 0 and C 0 simultaneously [14]. The values of D 1 and C 1 were computed in a similar way.…”

Section: Cortical Morphometrymentioning

confidence: 99%

3-D Analysis of Cortical Morphometry in Differential Diagnosis of Parkinson’s Plus Syndromes: Mapping Frontal Lobe Cortical Atrophy in Progressive Supranuclear Palsy Patients

Tosun

Duchesne

Rolland³

et al. 2007

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2007

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Abstract.With the ability to study brain anatomy in vivo using magnetic resonance imaging, studies on regional brain atrophy suggest possible improvements for differential diagnosis of movement disorders with parkinsonian symptoms. In this study, we investigate effects of different parkinsonian syndromes on the cortical gray matter thickness and the geometric shape of the cerebral cortex. The study consists of a total of 24 patients with a diagnosis of probable progressive supranuclear palsy (PSP), multiple systems atrophy (MSA) or idiopathic Parkinson's disease (IPD). We examine dense estimates of cortical gray matter thickness, sulcal depth, and measures of the curvature in a surface-based cortical morphometry analysis framework. Group difference results indicate higher cortical atrophy rate in the frontal lobe in PSP patients when compared to either MSA or IPD. These findings are indicative of the potential use of routine MRI and cortical morphometry in performing differential diagnosis in PSP, MSA and IPD.

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“…Tankus et al perform FM at a perspective Lambertian model also not incorporating light attenuation [14]. In [15], Prados and Soatto develop a FM approach based on ideas from optimal control theory. However, while they claim that their approach holds for perspective SfS with Lambertian reflectance, they only show computational results of their scheme for the classic orthographic model also considered in [11].…”

Section: Introductionmentioning

confidence: 99%

Fast Shape from Shading for Phong-Type Surfaces

Vogel

Breuß

Leichtweis

et al. 2009

Lecture Notes in Computer Science

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Abstract. Shape from Shading (SfS) is one of the oldest problems in image analysis that is modelled by partial differential equations (PDEs). The goal of SfS is to compute from a single 2-D image a reconstruction of the depicted 3-D scene. To this end, the brightness variation in the image and the knowledge of illumination conditions are used. While the quality of models has reached maturity, there is still the need for efficient numerical methods that enable to compute sophisticated SfS processes for large images in reasonable time. In this paper we address this problem. We consider a so-called Fast Marching (FM) scheme,which is one of the most efficient numerical approaches available. However, the FM scheme is not trivial to use for modern non-linear SfS models. We show how this is done for a recent SfS model incorporating the non-Lambertian reflectance model of Phong. Numerical experiments demonstrate thatwithout compromising quality -our FM scheme is two orders of magnitude faster than standard methods.

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Fast Marching Method for Generic Shape from Shading

Cited by 21 publications

References 26 publications

Brain Connectivity Mapping Using Riemannian Geometry, Control Theory, and PDEs

Brain Connectivity Mapping Using Riemannian Geometry, Control Theory, and PDEs

3-D Analysis of Cortical Morphometry in Differential Diagnosis of Parkinson’s Plus Syndromes: Mapping Frontal Lobe Cortical Atrophy in Progressive Supranuclear Palsy Patients

Fast Shape from Shading for Phong-Type Surfaces

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