367 research outputs found
Efficient Algorithms for Image and High Dimensional Data Processing Using Eikonal Equation on Graphs
International audienceIn this paper we propose an adaptation of the static eikonal equation over weighted graphs of arbitrary structure using a framework of discrete operators. Based on this formulation, we provide explicit solu- tions for the L1,L2 and L∞ norms. Efficient algorithms to compute the explicit solution of the eikonal equation on graphs are also described. We then present several applications of our methodology for image processing such as superpixels decomposition, region based segmentation or patch- based segmentation using non-local configurations. By working on graphs, our formulation provides an unified approach for the processing of any data that can be represented by a graph such as high-dimensional data
Limits and consistency of non-local and graph approximations to the Eikonal equation
In this paper, we study a non-local approximation of the time-dependent
(local) Eikonal equation with Dirichlet-type boundary conditions, where the
kernel in the non-local problem is properly scaled. Based on the theory of
viscosity solutions, we prove existence and uniqueness of the viscosity
solutions of both the local and non-local problems, as well as regularity
properties of these solutions in time and space. We then derive error bounds
between the solution to the non-local problem and that of the local one, both
in continuous-time and Backward Euler time discretization. We then turn to
studying continuum limits of non-local problems defined on random weighted
graphs with vertices. In particular, we establish that if the kernel scale
parameter decreases at an appropriate rate as grows, then almost surely,
the solution of the problem on graphs converges uniformly to the viscosity
solution of the local problem as the time step vanishes and the number vertices
grows large
Graph-based skin lesion segmentation of multispectral dermoscopic images
International audienceAccurate skin lesion segmentation is critical for automated early skin cancer detection and diagnosis. We present a novel method to detect skin lesion borders in multispectral der-moscopy images. First, hairs are detected on infrared images and removed by inpainting visible spectrum images. Second, skin lesion is pre-segmented using a clustering of a superpixel partition. Finally, the pre-segmentation is globally regular-ized at the superpixel level and locally regularized in a narrow band at the pixel level
A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)
We present a new flexible wavefront propagation algorithm for the boundary
value problem for sub-Riemannian (SR) geodesics in the roto-translation group
with a metric tensor depending on a smooth
external cost , , computed from
image data. The method consists of a first step where a SR-distance map is
computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system
derived via Pontryagin's Maximum Principle (PMP). Subsequent backward
integration, again relying on PMP, gives the SR-geodesics. For
we show that our method produces the global minimizers. Comparison with exact
solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics.
We present numerical computations of Maxwell points and cusp points, which we
again verify for the uniform cost case . Regarding image
analysis applications, tracking of elongated structures in retinal and
synthetic images show that our line tracking generically deals with crossings.
We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven
Sub-Riemannian Geodesics in SE(2)
A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow
In this paper we present a Semi-Lagrangian scheme for a regularized version
of the Hughes model for pedestrian flow. Hughes originally proposed a coupled
nonlinear PDE system describing the evolution of a large pedestrian group
trying to exit a domain as fast as possible. The original model corresponds to
a system of a conservation law for the pedestrian density and an Eikonal
equation to determine the weighted distance to the exit. We consider this model
in presence of small diffusion and discuss the numerical analysis of the
proposed Semi-Lagrangian scheme. Furthermore we illustrate the effect of small
diffusion on the exit time with various numerical experiments
PDEs level sets on weighted graphs
International audienceIn this paper we propose an adaptation of PDEs level sets over weighted graphs of arbitrary structure, based on PdEs and using a framework of discrete operators. A general PDEs level sets formulation is presented and an algorithm to solve such equation is described. Some transcriptions of well-known models under this formalism, as the mean-curvature-motion or active contours, are also provided. Then, we present several applications of our formalism, including image segmentation with active contours, using weighted graphs of arbitrary topologies
Geodesic Models with Convexity Shape Prior
The minimal geodesic models based on the Eikonal equations are capable of
finding suitable solutions in various image segmentation scenarios. Existing
geodesic-based segmentation approaches usually exploit image features in
conjunction with geometric regularization terms, such as Euclidean curve length
or curvature-penalized length, for computing geodesic curves. In this paper, we
take into account a more complicated problem: finding curvature-penalized
geodesic paths with a convexity shape prior. We establish new geodesic models
relying on the strategy of orientation-lifting, by which a planar curve can be
mapped to an high-dimensional orientation-dependent space. The convexity shape
prior serves as a constraint for the construction of local geodesic metrics
encoding a particular curvature constraint. Then the geodesic distances and the
corresponding closed geodesic paths in the orientation-lifted space can be
efficiently computed through state-of-the-art Hamiltonian fast marching method.
In addition, we apply the proposed geodesic models to the active contours,
leading to efficient interactive image segmentation algorithms that preserve
the advantages of convexity shape prior and curvature penalization.Comment: This paper has been accepted by TPAM
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