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557
Variable Resolution Discretization in Optimal Control
 Machine Learning
, 2001
"... The problem of state abstraction is of central importance in optimal control, reinforcement learning and Markov decision processes. This paper studies the case of variable resolution state abstraction for continuous time and space, deterministic dynamic control problems in which nearoptimal policie ..."
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Cited by 101 (2 self)
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The problem of state abstraction is of central importance in optimal control, reinforcement learning and Markov decision processes. This paper studies the case of variable resolution state abstraction for continuous time and space, deterministic dynamic control problems in which nearoptimal policies are required. We begin by defining a class of variable resolution policy and value function representations based on Kuhn triangulations embedded in a kdtrie. We then consider topdown approaches to choosing which cells to split in order to generate improved policies. The core of this paper is the introduction and evaluation of a wide variety of possible splitting criteria. We begin with local approaches based on value function and policy properties that use only features of individual cells in making split choices. Later, by introducing two new nonlocal measures, inuence and variance, we derive splitting criteria that allow one cell to efficiently take into account its impact on other cells when deciding whether to split. Influence is an efficientlyalculable measure of the extent to which changes in some state effect the value function of some other states. Variance is an efficientlycalculable measure of how risky is some state in a Markov chain: a low variance state is one in which we would be very surprised if, during any one execution, the longterm reward attained from that state differed substantially from its expected value, given by the value function. The paper proceeds by graphically demonstrating the various approaches to splitting on the familiar, nonlinear, nonminimum phase, and two dimensional problem of the "Car on the hill". It then evaluates the performance of a variety of splitting criteria on many benchmark problems, paying careful attention to their number...
Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order
 in Stochastic Analysis and Related Topics VI: The Geilo Workshop
, 1996
"... The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have ..."
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Cited by 87 (11 self)
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The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have appeared long time ago, both as the equations for the adjoint process in
An Axiomatic Approach to Image Interpolation
, 1998
"... We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The ..."
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Cited by 85 (7 self)
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We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The absolute minimal Lipschitz extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range.
The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality
 J. DIFFERENTIAL GEOM
, 1998
"... In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian version of the Penrose inequality for the total mass of an asymptotically flat 3manifold of nonnegative scalar curvature, announc ..."
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Cited by 79 (0 self)
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In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian version of the Penrose inequality for the total mass of an asymptotically flat 3manifold of nonnegative scalar curvature, announced in [HI1]. Let M be a smooth Riemannian manifold of dimension n 2 with metric g = (g ij ). A classical solution of the inverse mean curvature flow is a smooth family x : N \Theta [0; T ] !M
Fast Global Minimization of the Active Contour/Snake Model
"... The active contour/snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. ..."
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Cited by 79 (7 self)
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The active contour/snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. The only drawback of this model is the existence of local minima in the active contour energy, which makes the initial guess critical to get satisfactory results. In this paper, we propose to solve this problem by determining a global minimum of the active contour model. Our approach is based on the unification of image segmentation and image denoising tasks into a global minimization framework. More precisely, we propose to unify three wellknown image variational models, namely the snake model, the RudinOsherFatemi denoising model and the MumfordShah segmentation model. We will establish theorems with proofs to determine the existence of a global minimum of the active contour model. From a numerical point of view, we propose a new practical way to solve the active contour propagation problem toward object boundaries through a dual formulation of the minimization problem. The dual formulation, easy to implement, allows us a fast global minimization of the snake energy. It avoids the usual drawback in the level set approach that consists of initializing the active contour in a distance function and reinitializing it periodically during the evolution, which is timeconsuming. We apply our segmentation algorithms on synthetic and realworld images, such as texture images and medical images, to emphasize the performances of our model compared with other segmentation models.
Level set approach to mean curvature flow in arbitrary codimension
 J. Differential Geom
, 1996
"... codimension ..."
Optimal Algorithm for Shape from Shading and Path Planning
, 2001
"... An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A cons ..."
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Cited by 68 (3 self)
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An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A consistent numerical scheme based on Sethian’s fast marching method is used to compute the reconstructed surface. The surface is a viscosity solution of an Eikonal equation for the vertical light source case. For the oblique light source case, the reconstructed surface is the viscosity solution to a different partial differential equation. A modification of the fast marching method yields a numerically consistent, computationally optimal, and practically fast algorithm for the classical shape from shading problem. Next, the fast marching method coupled with a back tracking via gradient descent along the reconstructed surface is shown to solve the path planning problem in robot navigation.
Stereoscopic Segmentation
, 2001
"... We cast the problem of multiframe stereo reconstruction of a smooth shape as the global region segmentation of a collection of images of the scene. Dually, the problem of segmenting multiple calibrated images of an object becomes that of estimating the solid shape that gives rise to such images. We ..."
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Cited by 66 (17 self)
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We cast the problem of multiframe stereo reconstruction of a smooth shape as the global region segmentation of a collection of images of the scene. Dually, the problem of segmenting multiple calibrated images of an object becomes that of estimating the solid shape that gives rise to such images. We assume that the radiance has smooth statistics. This assumption covers Lambertian scenes with smooth or constant albedo as well as fine homogeneous textures, which are known challenges to stereo algorithms based on local correspondence. We pose the segmentation problem within a variational framework, and use fast level set methods to approximate the optimal solution numerically. Our algorithm does not work in the presence of strong textures, where traditional reconstruction algorithms do. It enjoys significant robustness to noise under the assumptions it is designed for. 1
Shapes, Shocks, and Deformations I: The Components of TwoDimensional Shape and the ReactionDiffusion Space
 International Journal of Computer Vision
, 1994
"... We undertake to develop a general theory of twodimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. Th ..."
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Cited by 64 (5 self)
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We undertake to develop a general theory of twodimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. This leads us to propose an operational theory of shape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can enclose "physical" material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and HamiltonJacobi theory. These principles are based on the work of Sethian [82, 86], the OsherSethian level set formulation [65], the classical shock theory of Lax [53, 54], as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of GageHamiltonGrayson [32, 37]. The result is a characterization of th...
A Fully Global Approach to Image Segmentation via Coupled Curve Evolution Equations
 Journal of Visual Communication and Image Representation
, 2002
"... In this paper, we develop a novel regionbased approach to snakes designed to optimally separate the values of certain image statistics over a known number of region types. Multiple sets of contours deform according to a coupled set of curve evolution equations derived from a single global cost func ..."
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Cited by 61 (12 self)
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In this paper, we develop a novel regionbased approach to snakes designed to optimally separate the values of certain image statistics over a known number of region types. Multiple sets of contours deform according to a coupled set of curve evolution equations derived from a single global cost functional. The resulting active contour model, in contrast to many other edge and region based models, is fully global in that the evolution of each curve depends at all times upon every pixel in the image and is directly coupled to the evolution of every other curve regardless of their mutual proximity. As such evolving contours enjoy a very wide “field of view, ” endowing the algorithm with a robustness to initial contour placement above and beyond the significant improvement exhibited by other region based snakes over earlier edge based snakes. C ○ 2002 Elsevier Science (USA) Key Words: active contours; curve evolution; snakes; segmentation; gradient flows.