multimodality image registration by maximization of mutual information a new approach to the problem of multimodality medical image registration is proposed using a basic concept from information theory mutual information mi or relative entropy as a new matching criterion the method presented in this paper applies mi to measure the statistical dependence or information redundancy between the image intensities of corresponding voxels in both images which is assumed to be maximal if the images are geometrically aligned maximization of mi is a very general and powerful criterion because no assumptions are made regarding the nature of this dependence and no limiting constraints are imposed on the image content of the modalities involved the accuracy of the mi criterion is validated for rigid body registration of computed tomography ct magnetic resonance mr and photon emission tomography pet images by comparison with the stereotactic registration solution while robustness is evaluated with respect to implementation issues such as interpolation and optimization and image content including partial overlap and image degradation our results demonstrate that subvoxel accuracy with respect to the stereotactic reference solution can be achieved completely automatically and without any prior segmentation feature extraction or other preprocessing steps which makes this method very well suited for clinical applications

adding handles to the helicoid there exist two new embedded minimal surfaces asymptotic to the helicoid one is periodic with quotient by orientation preserving translations of genus one the other is nonperiodic of genus one

classification of rotational special weingarten surfaces of minimal type in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r in this paper we classify the complete rotational special weingarten surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r i e rotational surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r whose mean curvature h and extrinsic curvature ke satisfy h f h2 ke for some function f in mathcal c 1 0 infty such that f 0 0 and 4x f x 2 1 for any x 0 furthermore we show the existence of non complete examples of such surfaces

constant mean curvature surfaces with boundary on a sphere in this article we study the shape of a compact surface of constant mean curvature of euclidean space whose boundary is contained in a round sphere we consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle we study under what geometric conditions the surface must be spherical our results apply in many scenarios in physics where in absence of gravity a liquid drop is deposited on a round solid ball and the air liquid interface is a critical point for area under all variations that preserve the enclosed volume

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

sharp eigenvalue bounds and minimal surfaces in the ball we prove existence and regularity of metrics on a surface with boundary which maximize sigma 1 l where sigma 1 is the first nonzero steklov eigenvalue and l the boundary length we show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball b n for some n in the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid the unique free boundary surface of revolution in b 3 we also show that the unique solution on the mobius band is achieved by an explicit s 1 invariant embedding in b 4 as a free boundary surface the critical mobius band for oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in b 3 we characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4 pi we also prove multiplicity bounds on sigma 1 in terms of the topology and we give a lower bound on the morse index for the area functional for free boundary surfaces in the ball

index one minimal surfaces in spherical space forms abstract we prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two this confirms a conjecture of r schoen for an infinite class of 3 manifolds

on the growth of supersolutions of nonlinear pde s on exterior domains abstract we obtain a comparison principle on annuli with catenoid like functions for supersolutions of non linear elliptic pdes over exterior domains in a non positively curved manifold with a pole this result is applied to get an upper estimate on the growth of such supersolutions and in particular of exterior graphs of non negative mean curvature

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

intensity based 2 d 3 d registration of cerebral angiograms we propose a new method for aligning three dimensional 3 d magnetic resonance angiography mra with 2 d x ray digital subtraction angiograms dsa our method is developed from our algorithm to register computed tomography volumes to x ray images based on intensity matching of digitally reconstructed radiographs drrs to make the dsa and drr more similar we transform the mra images to images of the vasculature and set to zero the contralateral side of the mra to that imaged with dsa we initialize the search for a match on a user defined circular region of interest we have tested six similarity measures using both unsegmented mra and three segmentation variants of the mra registrations were carried out on images of a physical neuro vascular phantom and images obtained during four neuro vascular interventions the most accurate and robust registrations were obtained using the pattern intensity gradient difference and gradient correlation similarity measures when used in conjunction with the most sophisticated mra segmentations using these measures 95 of the phantom start positions and 82 of the clinical start positions were successfully registered the lowest root mean square reprojection errors were 1 3 mm standard deviation 0 6 for the phantom and 1 5 mm standard deviation 0 9 for the clinical data sets finally we present a novel method for the comparison of similarity measure performance using a technique borrowed from receiver operator characteristic analysis

the spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds it is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in non fixed flat three manifolds are compact in a strong sense given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the ck topology these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds some applications to the isoperimetric problem are given

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus this paper is the fifth and final in a series on embedded minimal surfaces following our earlier papers on disks we prove here two main structure theorems for itnonsimply connected embedded minimal surfaces of any given fixed genus the first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely oriented double spiral staircases the second gives a pair of pants decomposition of any such surface when there are small necks cutting the surface along a collection of short curves after the cutting we are left with graphical pieces that are defined over a disk with either one or two sub disks removed a topological disk with two sub disks removed is called a pair of pants both of these structures occur as different extremes in the two parameter family of minimal surfaces known as the riemann examples

inequivalent complexity criteria for free boundary minimal surfaces abstract we obtain a series of results in the global theory of free boundary minimal surfaces which in particular provide a rather complete picture for the way different complexity criteria such as area topology and morse index compare beyond the regime where effective estimates are at disposal

constant mean curvature graphs on exterior domains of the hyperbolic plane we prove an existence result for non rotational constant mean curvature ends in mathbb h 2 times mathbb r where mathbb h 2 is the hyperbolic real plane the value of the curvature is h in big 0 frac 1 2 big we use schauder theory and a continuity method for solutions of the prescribed mean curvature equation on exterior domains of mathbb h 2 we also prove a fine property of the asymptotic behavior of the rotational ends introduced by sa earp and toubiana

new families of fourier eigenfunctions for steerable filtering a new diadic family of eigenfunctions of the 2 d fourier transform has been discovered specifically new wavelets are derived by steering the elongated hermite gauss filters with respect to rotations thus obtaining a natural generalization of the laguerre gauss harmonics interestingly these functions are also proportional to their 2 d fourier transform their analytical expression is provided in a compact and treatable form by means of a new ad hoc matrix notation in which the cases of even and odd orders of the hermite polynomials are unified moreover these functions can be efficiently implemented by means of a recursive formula that is derived in this paper the proposed filters are applied to the problem of gradient estimation to improve the theoretical canny tradeoff of position accuracy versus noise rejection that occurs in edge detection experimental results show considerable improvements in using the new wavelets over both isotropic gaussian derivatives and other elongated steerable filters more recently introduced finally being the proposed wavelets a set of fourier eigenfunctions they can be of interest in other fields of science such as optics and quantum mechanics

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

on stability of probability laws with univariate stable marginals examples of d marcus in 2 dispel the belief that a probability measure on d is stable if and only if all its univariate marginals are stable however in d in fact in fairly general linear spaces a probability measure whose two dimensional marginals are all infinitely divisible is stable if and only if all its univariate marginals are stable

stationary isothermic surfaces and uniformly dense domains we establish a relationship between stationary isothermic surfaces and uniformly dense domains a stationary isothermic surface is a level surface of temperature which does not evolve with time a domain in the n dimensional euclidean space r n is said to be uniformly dense in a surface r n of codimension 1 if for every small r 0 the volume of the intersection of with a ball of radius r and center x does not depend on x for x we prove that the boundary of every uniformly dense domain which is bounded or whose complement is bounded must be a sphere we then examine a uniformly dense domain with unbounded boundary and we show that the principal curvatures of satisfy certain identities the case in which the surface coincides with is particularly interesting in fact we show that if the boundary of a uniformly dense domain is connected then i if n 2 it must be either a circle or a straight line and ii if n 3 it must be either a sphere a spherical cylinder or a minimal surface we conclude with a discussion on uniformly dense domains whose boundary is a minimal surface

non stable laws with all projections stable is the characterist ic function of a probabi l i ty law on ir 2 we then show that for 0 4 define functions on ir 2 as follows

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

robust gradient based 3 d 2 d registration of ct and mr to x ray images one of the most important technical challenges in image guided intervention is to obtain a precise transformation between the intrainterventional patient s anatomy and corresponding preinterventional 3 d image on which the intervention was planned this goal can be achieved by acquiring intrainterventional 2 d images and matching them to the preinterventional 3 d image via 3 d 2 d image registration a novel 3 d 2 d registration method is proposed in this paper the method is based on robustly matching 3 d preinterventional image gradients and coarsely reconstructed 3 d gradients from the intrainterventional 2 d images to improve the robustness of finding the correspondences between the two sets of gradients hypothetical correspondences are searched for along normals to anatomical structures in 3 d images while the final correspondences are established in an iterative process combining the robust random sample consensus algorithm ransac and a special gradient matching criterion function the proposed method was evaluated using the publicly available standardized evaluation methodology for 3 d 2 d registration consisting of 3 d rotational x ray computed tomography magnetic resonance mr and 2 d x ray images of two spine segments and standardized evaluation criteria in this way the proposed method could be objectively compared to the intensity gradient and reconstruction based registration methods the obtained results indicate that the proposed method performs favorably both in terms of registration accuracy and robustness the method is especially superior when just a few x ray images and when mr preinterventional images are used for registration which are important advantages for many clinical applications

triply periodic constant mean curvature surfaces abstract given a closed flat 3 torus n for each h 0 and each non negative integer g we obtain area estimates for closed surfaces with genus g and constant mean curvature h embedded in n this result contrasts with the theorem of traizet 31 who proved that every flat 3 torus admits for every positive integer g with g 2 connected closed embedded minimal surfaces of genus g with arbitrarily large area

the asymptotic behavior of properly embedded minimal surfaces of finite topology while advances have been made in recent years in the study of properly embedded minimal surfaces of finite topology in r3 3 5 8 9 11 12 13 23 progress has depended in an essential manner on the special structure of such surfaces with finite total curvature a properly immersed minimal surface with finite total curvature is conformally a compact riemann surface punctured in a finite number of points and its gauss map extends to the compact surface as a meromorphic function 22 in particular all the topological ends are conformally equivalent to a punctured disk and there is a well defined limit tangent plane at each end outside of a sufficiently large compact set such an end is a multisheeted graph over the limit tangent plane and if the end is embedded it is asymptotic to either a plane or a half catenoid see e g schoen 24 for a survey of results on properly immersed surfaces of finite total curvature see 9 1 without the assumption of finite total curvature these results are not true in general as an indication of the relative lack of information consider that it is unknown whether or not the helicoid is the only simply connected minimal surface that is properly embedded and nonplanar a question raised by osserman the helicoid has infinite total curvature and a single annular end if a surface has finite topology then all of its ends are annular any general theory of properly embedded minimal surfaces of finite topology must include an understanding of the behavior of these ends in this paper we prove

a schoen theorem for minimal surfaces in h 2 r abstract in this paper we prove that a complete minimal surface immersed in h 2 r with finite total curvature and two ends each one asymptotic to a vertical geodesic plane must be a horizontal catenoid moreover we give a geometric description of minimal ends of finite total curvature in h 2 r we also prove that a minimal complete end e with finite total curvature is properly immersed and that the gaussian curvature of e is locally bounded in terms of the geodesic distance to its boundary

cylindrical levy processes in banach spaces cylindrical probability measures are flnitely additive measures on banach spaces that have sigma additive projections to euclidean spaces of all dimensions they are naturally associated to notions of weak cylindrical random variable and hence weak cylindrical stochastic processes in this paper we focus on cylindrical levy processes these have weak levy it o decompositions and an associated levy khintchine formula if the process is weakly square integrable its covariance oper ator can be used to construct a reproducing kernel hilbert space in which the pro cess has a decomposition as an inflnite series built from a sequence of uncorrelated bona flde one dimensional levy processes this series is used to deflne cylindrical stochastic integrals from which cylindrical ornstein uhlenbeck processes may be constructed as unique solutions of the associated cauchy problem we demonstrate that such processes are cylindrical markov processes and study their cylindrical invariant measures

the space of embedded minimal surfaces of fixed genus in a 3 manifold iv locally simply connected this paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3manifold the key is to understand the structure of an embedded minimal disk in a ball in r3 this was undertaken in cm3 cm4 and the global version of it will be completed here see the discussion around figure 12 for the local case and cm15 for some more details our main results are theorem 0 1 the lamination theorem and theorem 0 2 the one sided curvature estimate the lamination theorem is stated in the global case where the lamination is in fact a foliation the first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph this is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid to prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of n valued graphs where n is a fixed number in this paper we will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase the first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase

the first steklov eigenvalue conformal geometry and minimal surfaces we consider the relationship of the geometry of compact riemannian manifolds with boundary to the first nonzero eigenvalue 1 of the dirichlet to neumann map steklov eigenvalue for surfaces with genus and k boundary components we obtain the upper bound 1l 2 k for 0 and k 1 this result was obtained by weinstock in 1954 and is sharp we attempt to find the best constant in this inequality for annular surfaces 0 and k 2 for rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball for a general class of not necessarily rotationally symmetric metrics on the annulus which we call supercritical we prove that 1 l is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary motivated by the annulus case we show that a proper submanifold of the ball is immersed by steklov eigenfunctions if and only if it is a free boundary solution we then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities we show that these bounds are only achieved when the manifold is minimally immersed by first steklov eigenfunctions we also use these ideas to show that any free boundary solution in two dimensions has area at least and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two dimensional case

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

some uniqueness and nonexistence theorems for embedded minimal surfaces thanks to the works of callahan costa hoffman karcher meeks rosenberg wei etc see for example 1 2 3 5 8 11 12 16 20 we dispose now of a large number of properly embedded minimal surfaces in the euclidean space ir other than the classical examples all those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds the most basic invariants associated to a surface of this type are its topology and its periodicity it is an interesting problem to decide if the simplest examples like the catenoid the helicoid the scherk s surfaces or the riemann example can be characterized in terms of some of these invariants in the non periodic case there are two important uniqueness theorems in this direction the first one obtained by schoen 17 characterizes the catenoid as the only complete minimal surface embedded

which ambient spaces admit isoperimetric inequalities for submanifolds we give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities to hold

the space of minimal embeddings of a surface into a three dimensional manifold of positive ricci curvature in this paper we obtain a curvature estimate for embedded minimal surfaces in a three dimensional manifold of positive ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface it should be mentioned that there are two main points in our result one is the absence of a stability assumption and the other is the requirement of being embedded most known curvature estimates require the stability assumption and once the stability assumption is dropped many of these known results cease to be valid see ss and an for another example of a curvature estimate without the assumption of stability the embeddedness condition is rather subtle because of the way it enters in our proof our proof depends on the eigenvalue estimate and the area bound due to the first author and wang cw which require embeddedness in an essential way present knowledge indicates that closed embedded minimal surfaces in s 3 are rare while immersed surfaces are more plentiful for example only a finite number of minimal embeddings of a given genus are known on the other hand otsuki o constructed infinitely many immersed minimal tori with arbitrarily large area we obtain the curvature estimate indirectly by proving the smooth compactness theorem theorem 1 theorem 1 has many interesting consequences it shows that the set of conformal structures that can be realized on a minimal embedding in s 3 is a compact subset of the moduli space this is in contrast with the result of bryant b who showed that every riemann surface is conformally and minimally immersed in s 4 in view of our compactness result and the scarcity of examples it is very tempting to conjecture that there are only finitely many embedded minimal surfaces up to rigid motion in s 3 for each fixed genus throughout this paper manifold means manifold without boundary unless explictly stated otherwise when we say a sequence m i of surfaces converges to a

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

the classification of complete minimal surfaces with total curvature greater than 12 we classify complete orientable minimal surfaces with finite total curvature 87r introduction the classification of complete minimal surfaces with finite total curvature in r3 has been an important problem in the classical differential geometry some basic properties of these surfaces were studied by r osserman see 8 9 who showed the first nontrivial result about this subject concretely he characterized the catenoid and enneper surface as the unique complete orientable minimal surfaces of total curvature 47 however until recent years no more relevant results have been obtained w h meeks 6 gave the classification of nonorientable complete minimal surfaces with total curvature greater than 87 this paper is concerned with the total classification of orientable complete minimal surfaces with total curvature 87 chen and gackstatter 1 discovered the first example of a complete minimal surface properly of genus 1 see theorem 1 the picture of chen gackstatter surface is obtained by joining a handle on enneper s surface this genus one minimal surface has total curvature 87 and no other examples of such surfaces were found so it is expected that no other genus one orientable minimal surface of total curvature 87 does exist in this paper we give a proof of this fact more precisely we prove that chen gackstatter surface is the only genus one orientable complete minimal surface with finite total curvature 87 of course it is not difficult to find genus zero minimal surfaces with total curvature 8i a geometrically interesting example described by jorge and meeks 5 is the trinoid this surface has three embedded catenoid ends moreover its normal vectors at these ends are placed symmetrically in an equator of s2 received by the editors april 27 1990 and in revised form july 20 1990 1980 mathematics subject classification 1985 revision primary 53a10 research partially supported by dgicyt grant ps87 01 15 c03 02

stochastic integration with respect to cylindrical levy processes a cylindrical levy process does not enjoy a cylindrical version of the semimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration in this work we introduce a stochastic integral for random integrands with respect to cylindrical levy processes in hilbert spaces the space of admissible integrands consists of caglad adapted stochastic processes with values in the space of hilbert schmidt operators neither the integrands nor the integrator is required to satisfy any moment or boundedness condition the integral process is characterised as an adapted hilbert space valued semimartingale with cadlag trajectories

local removable singularity theorems for minimal laminations in this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a riemannian three manifold this removable singularity theorem is the key result used in our proof that a complete embedded minimal surface in r 3 with quadratic decay of curvature has nite total curvature