conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

the affine polya szego principle equality cases and stability a brothers ziemer type theorem for the affine polya szego principle and a quantitative affine polya szego principle are established

affine poisson non poisson trace principles for h 1 α 12 rn 1 2 abstract this note discovers not only an affine non sharp poisson trace inequality but also its sharp non poisson version for a sobolev function with the fractional antiderivative

valuations on sobolev spaces all affinely covariant convex body valued valuations on the sobolev space w 1 1 bbb r n are completely classified it is shown that there is a unique such valuation for blaschke addition this valuation turns out to be the operator which associates with each function f in w 1 1 bbb r n the unit ball of its optimal sobolev norm

constraints on deformable models recovering 3d shape and nongrid motion abstract inferring the 3d structures of nonrigidly moving objects from images is a difficult yet basic problem in computational vision our approach makes use of dynamic elastically deformable object models that offer the geometric flexibility to satisfy a diversity of real world visual constraints we specialize these models to include intrinsic forces inducing a preference for axisymmetry image based constraints are applied as extrinsic forces that mold the symmetry seeking model into shapes consistent with image data we describe an extrinsic force that applies constraints derived from profiles of monocularly viewed objects we generalize this constraint force to incorporate profile information from multiple views and use it to exploit binocular image data for time varying images the force becomes dynamic and the model is able to infer not only depth but nonrigid motion as well we demonstrate the recovery of 3d shape and nonrigid motion from natural imagery

reconstruction of 3 d geometry using 2 d profiles and a geometric prior model a method has been developed to reconstruct three dimensional 3 d surfaces from two dimensional 2 d projection data it is used to produce individualized boundary element models consisting of thorax and lung surfaces for electro and magnetocardiographic inverse problems two orthogonal projections are utilized a geometrical prior model built using segmented magnetic resonance images is deformed according to profiles segmented from projection images in the authors method virtual x ray images of the prior model are first constructed by simulating real x ray imaging the 2 d profiles of the model are segmented from the projections and elastically matched with the profiles segmented from patient data the displacement vectors produced by the elastic 2 d matching are back projected onto the 3 d surface of the prior model finally the model is deformed using the back projected vectors two different deformation methods are proposed the accuracy of the method is validated by a simulation the average reconstruction error of a thorax and lungs was 1 22 voxels corresponding to about 5 mm

how far 3d shapes can be understood from 2d silhouettes each 2d silhouette of a 3d unknown object o constrains o inside the volume obtained by back projecting the silhouette from the corresponding viewpoint a set of silhouettes specifies a boundary volume r obtained by intersecting the volumes due to each silhouette r more or less closely approximates o depending on the viewpoints and the object itself this approach to the reconstruction of 3d objects is usually referred to as volume intersection this correspondence addresses the problem of inferring the shape of the unknown object o from the reconstructed object r for doing this the author divides the points of the surface of r into hard points which belong to the surface of any possible object originating r and soft points which may or may not belong to o the author considers two cases in the first case r is the closest approximation of o which can be obtained from its silhouettes i e its visual hull in the second case r is a generic reconstructed object in both cases the author supplies necessary and sufficient conditions for a point to be hard and gives rules for computing the hard surfaces

a quantitative sobolev inequality in bv a quantitative version of the standard sobolev inequality with sharp constant for functions u in w1 1 rn or bv rn is established in terms of a distance of u from the manifold of all multiples of characteristic functions of balls inequalities involving non euclidean norms of the gradient are discussed as well

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

asymmetric affine lp sobolev inequalities abstract a new sharp affine l p sobolev inequality for functions on r n is established this inequality strengthens and implies the previously known affine l p sobolev inequality which in turn is stronger than the classical l p sobolev inequality

reconstruction of 3 d binary tree like structures from three mutually orthogonal projections a method is developed to generate the 3d binary representation for a tree like object from three mutually orthogonal projections this is done by first backprojecting the binarized images from three directions and then iteratively removing artifacts in the backprojection three different algorithms have been developed the lagrange multiplier algorithm the conjugate gradient algorithm and the minimum voxel representation algorithm mra the performance of these algorithms under noise free conditions is evaluated using mathematically projected images of a 3d tree structure while all three algorithms are capable of producing a relatively accurate reconstruction the mra is superior not only because it requires the least amount of computation but also because it uses binary instead of gray scale information in the input images reconstruction of 3d coronary arterial structures using mra is further verified with x ray images of a human chest phantom and shows a satisfactory performance the result of this study should be valuable for 3d imaging of blood vessels

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

motion compensated mega voltage cone beam ct using the deformation derived directly from 2d projection images this paper presents a novel method for respiratory motion compensated reconstruction for cone beam computed tomography cbct the reconstruction is based on a time sequence of motion vector fields which is generated by a dynamic geometrical object shape model the dynamic model is extracted from the 2d projection images of the cbct the process of the motion extraction is converted into an optimal 3d multiple interrelated surface detection problem which can be solved by computing a maximum flow in a 4d directed graph the method was tested on 12 mega voltage mv cbct scans from three patients two sets of motion artifact free 3d volumes full exhale fe and full inhale fi phases were reconstructed for each daily scan the reconstruction was compared with three other motion compensated approaches based on quantification accuracy of motion and size contrast to noise ratio cnr was also quantified for image quality the proposed approach has the best overall performance with a relative tumor volume quantification error of 3 39 3 64 and 8 57 8 31 for fe and fi phases respectively the cnr near the tumor area is 3 85 0 42 fe and 3 58 3 33 fi these results show the clinical feasibility to use the proposed method to reconstruct motion artifact free mvcbct volumes

a sharp trace inequality for functions of bounded variation in the ball the best constant in a mean value trace inequality for functions of bounded variation on admissible domains n is shown to agree with an isoperimetric constant associated with the existence and form of extremals is also discussed this result is exploited to compute the best constant in the relevant trace inequality when is a ball the existence and the form of extremals in this special case turn out to depend on the dimension n in particular the best constant is not achieved when is a disc in 2

asymmetric blaschke santalo functional inequalities abstract in this work we establish functional asymmetric versions of the celebrated blaschke santalo inequality as consequences of these inequalities we recover their geometric counter parts with equality cases as well as another inequality with strong probabilistic flavor that was firstly obtained by lutwak yang and zhang we present a brief study on an l p functional analogue to the center of mass that is necessary for our arguments and that might be of independent interest

the affine sobolev zhang inequality on bv rn abstract the affine sobolev zhang inequality is extended to b v r n the space of functions of bounded variation on r n and the equality cases are characterized as a consequence the petty projection inequality for sets of finite perimeter which implies the isoperimetric inequality for sets of finite perimeter is established

the lp busemann petty centroid inequality abstract the ratio between the volume of the p centroid body of a convex body k in r n and the volume of k attains its minimum value if and only if k is an origin symmetric ellipsoid this result the lp busemann petty centroid inequality was recently proved by lutwak yang and zhang in this paper we show that all the intrinsic volumes of the p centroid body of k are convex functions of a time like parameter when k is moved by shifting all the chords parallel to a fixed direction the lp busemann petty centroid inequality is a consequence of this general fact

affine vs euclidean isoperimetric inequalities abstract it is shown that every even zonal measure on the euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical euclidean isoperimetric inequality the strongest member of this large family of inequalities is shown to be the only affine invariant one among them the petty projection inequality as an application a family of sharp sobolev inequalities for functions of bounded variation is obtained each of which is stronger than the classical sobolev inequality moreover corresponding families of l p isoperimetric and sobolev type inequalities are also established

lutwak petty projection inequalities for minkowski valuations and their duals abstract lutwak s volume inequalities for polar projection bodies of all orders are generalized to polarizations of minkowski valuations generated by even zonal measures on the euclidean unit sphere this is based on analogues of mixed projection bodies for such minkowski valuations and a generalization of the notion of centroid bodies a new integral representation is used to single out lutwak s inequalities as the strongest among these families of inequalities which in turn are related to a conjecture on affine quermassintegrals in the dual setting a generalization of volume inequalities for intersection bodies of all orders by leng and lu is proved these results are related to grinberg s inequalities for dual affine quermassintegrals

best constant in sobolev trace inequalities on the half space using a mass transportation method we study optimal sobolev trace inequalities on the half space and prove a conjecture made by escobar in 1988 about the minimizers

sharp convex lorentz sobolev inequalities new sharp lorentz sobolev inequalities are obtained by convexifying level sets in lorentz integrals via the lp minkowski problem new lp isocapacitary and isoperimetric inequalities are proved for lipschitz star bodies it is shown that the sharp convex lorentz sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities

recovering the position and orientation of free form objects from image contours using 3d distance maps the accurate matching of 3d anatomical surfaces with sensory data such as 2d x ray projections is a basic problem in computer and robot assisted surgery in model based vision this problem can be formulated as the estimation of the spatial pose position and orientation of a 3d smooth object from 2d video images the authors present a new method for determining the rigid body transformation that describes this match the authors method performs a least squares minimization of the energy necessary to bring the set of the camera contour projection lines tangent to the surface to correctly deal with projection lines that penetrate the surface the authors consider the minimum signed distance to the surface along each line i e distances inside the object are negative to quickly and accurately compute distances to the surface the authors introduce a precomputed distance map represented using an octree spline whose resolution increases near the surface this octree structure allows the authors to quickly find the minimum distance along each line using best first search experimental results for 3d surface to 2d projection matching are presented for both simulated and real data the combination of the authors problem formulation in 3d their computation of line to surface distances with the octree spline distance map and their simple minimization technique based on the levenberg marquardt algorithm results in a method that solves the 3d 2d matching problem for arbitrary smooth shapes accurately and quickly

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

reconstructing convex sets from support line measurements algorithms are proposed for reconstructing convex sets given noisy support line measurements it is observed that a set of measured support lines may not be consistent with any set in the plane a theory of consistent support lines which serves as a basis for reconstruction algorithms that take the form of constrained optimization algorithms is developed the formal statement of the problem and constraints reveals a rich geometry that makes it possible to include prior information about object position and boundary smoothness the algorithms which use explicit noise models and prior knowledge are based on maximum likelihood and maximum a posteriori estimation principles and are implemented using efficient linear and quadratic programming codes experimental results are presented this research sets the stage for a more general approach to the incorporation of prior information concerning the estimation of object shape

acquiring 3 d models from sequences of contours this paper explores shape from contour for acquiring 3 d graphics models in this method a continuous sequence of images is taken as an object rotates a smooth convex shape can be estimated instantaneously from its contour and by the first derivative of contour movement trace of contour or contour distribution with time we also analyze shapes that do not satisfy the conditions of smoothness and visibility which are indispensable for modeling an object a region that does not expose as contour yields a nonsmoothness in the tracked contour movement we can thus detect such a region by contour distribution filtering and extract its accurate location by computing the left and right derivatives of the distribution this has not been studied previously these unknown regions are obtained for further investigation using other visual cues a general approach for building a geometrical object model using contours is then described the entire process from silhouettes to a 3 d model is based local computation this is promising for producing shapes in real time our direct goal is to establish 3 d graphics models of human faces for the growing needs of visual communications we have obtained some good results

reconstructing the cross sections of coronary arteries from biplane angiograms an algorithm that reconstructs the cross sections of the lumens of coronary arteries from two mutually orthogonal x ray projections is described the algorithm accommodates the possibility of elliptical crescent or star shapes it represents each biplane projection of a transverse slice of the arterial lumen as a binary valued image the single coordinate moments of these two projection images are equal to those of the slice since the cross coordinate moments of the slice are not available from the projections an algorithm to estimate these moments based on assumptions of smoothness and connectivity is developed once all the missing moments are estimated the image of the slice can be estimated by inverting these moments using the uniqueness theorem governing the relation between an image and its moments preliminary tests of the algorithm on synthetic data on hardware phantoms and on a segment of a barium enhanced in vitro coronary artery are reported

asymptotic properties of least squares estimates in stochastic regression models on etablit la consistance forte des estimations des moindres carres dans des modeles de regression stochastique sous l hypothese que le modele peut etre reparametrise de telle facon que les nouveaux vecteurs soient faiblement correles

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

the sharp affine l 2 sobolev trace inequality and variants we establish a sharp affine l p sobolev trace inequality by using the l p busemann petty centroid inequality for p 2 our affine version is stronger than the famous sharp l 2 sobolev trace inequality proved independently by escobar and beckner our approach allows also to characterize all extremizers in this case for this new inequality no euclidean geometric structure is needed