the problem of degeneracy in structure and motion recovery from uncalibrated image sequences the aim of this work is the recovery of 3d structure and camera projection matrices for each frame of an uncalibrated image sequence in order to achieve this correspondences are required throughout the sequence a significant and successful mechanism for automatically establishing these correspondences is by the use of geometric constraints arising from scene rigidity however problems arise with such geometry guided matching if general viewpoint and general structure are assumed whilst frames in the sequence and or scene structure do not conform to these assumptions such cases are termed degenerate in this paper we describe two important cases of degeneracy and their effects on geometry guided matching the cases are a motion degeneracy where the camera does not translate between frames and a structure degeneracy where the viewed scene structure is planar the effects include the loss of correspondences due to under or over fitting of geometric models estimated from image data leading to the failure of the tracking method these degeneracies are not a theoretical curiosity but commonly occur in real sequences where models are statistically estimated from image points with measurement error we investigate two strategies for tackling such degeneracies the first uses a statistical model selection test to identify when degeneracies occur the second uses multiple motion models to overcome the degeneracies the strategies are evaluated on real sequences varying in motion scene type and length from 13 to 120 frames

multiple motion scene reconstruction with uncalibrated cameras in this paper we describe a reconstruction method for multiple motion scenes which are scenes containing multiple moving objects from uncalibrated views assuming that the objects are moving with constant velocities the method recovers the scene structure the trajectories of the moving objects the camera motion and the camera intrinsic parameters except skews simultaneously we focus on the case where the cameras have unknown and varying focal lengths while the other intrinsic parameters are known the number of the moving objects is automatically detected without prior motion segmentation the method is based on a unified geometrical representation of the static scene and the moving objects it first performs a projective reconstruction using a bilinear factorization algorithm and then converts the projective solution to a euclidean one by enforcing metric constraints experimental results on synthetic and real images are presented

on projection matrices cal p k rightarrow cal p 2 k 3 ldots 6 and their applications in computer vision projection matrices from projective spaces cal p 3 to cal p 2 have long been used in multiple view geometry to model the perspective projection created by the pin hole camera in this work we introduce higher dimensional mappings cal p k rightarrow cal p 2 k 3 4 5 6 for the representation of various applications in which the world we view is no longer rigid we also describe the multi view constraints from these new projection matrices where k 3 and methods for extracting the non rigid structure and motion for each application

a paraperspective factorization method for shape and motion recovery the factorization method first developed by tomasi and kanade 1992 recovers both the shape of an object and its motion from a sequence of images using many images and tracking many feature points to obtain highly redundant feature position information the method robustly processes the feature trajectory information using singular value decomposition svd taking advantage of the linear algebraic properties of orthographic projection however an orthographic formulation limits the range of motions the method can accommodate paraperspective projection first introduced by ohta et al 1981 is a projection model that closely approximates perspective projection by modeling several effects not modeled under orthographic projection while retaining linear algebraic properties our paraperspective factorization method can be applied to a much wider range of motion scenarios including image sequences containing motion toward the camera and aerial image sequences of terrain taken from a low altitude airplane

on the asymptotic expansion of the colored jones polynomial for torus knots in the asymptotic expansion of the hyperbolic specification of the colored jones polynomial of torus knots we identify different geometric contributions in particular chern simons invariant and twisted reidemeister torsion with coefficients in the adjoint representation

lifting representations to covering groups the primary purpose of this note is to prove that if a discrete subgroup z of z s c has no 2 torsion then it lifts to s c i e there is a homomorphism z s c such that the composition with the natural projection s c z s c is the identity on k note that the absence of 2 torsion is necessary since cyclic subgroups of even order do not lift this extends a result of km 2 for which z was required to be a function group our methods are simple and straightforward requiring only elementary covering space theory

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

actions of finite groups on homotopy 3 spheres it is conjectured that the action of a finite group of diffeomorphisms of the 3 sphere is equivariantly diffeomorphic to a linear action this conjecture is verified if both of the following conditions hold i each isotropy subgroup is dihedral or cyclic ii there is a point with cyclic isotropy subgroup of order not 1 2 3 or 5 0 introduction the generalized smith conjecture says that the effective action of a finite group g of orientation preserving diffeomorphisms on a homotopy 3 sphere i is essentially linear if y is diffeomorphic to s3 then essentially linear is equivalent to equivariantly diffeomorphic to a linear action work by thurston johannson jaco shalen bass meeks and yau established this conjecture when g is generated by a diffeomorphism with fixed points this is the classical smith conjecture davis and morgan improved this in certain cases by proving that if every isotropy subgroup of g is cyclic and if at least one of these subgroups has order greater than 5 then the action is essentially linear the group g ig finite and the action is differentiable and effective thus the isotropy subgroup of a point in e acts on the tangent space at the point and so is a finite subgroup of gl r therefore this group is cyclic dihedral tetrahedral octahedral or icosahedral in this paper the result of davis and morgan will be extended to include the case where one of the isotropy subgroups has order 4 we also show that the actions is essentially linear if every isotropy subgroup is either dihedral or cyclic and one of the cyclic isotropy subgroups has order not equal to 1 2 3 or 5 the results of this paper first appeared in the author s 1981 thesis at columbia university subsequently william thurston announced a theorem that includes these results as a special case for another proof of the davis morgan extension of the smith conjecture see culler and shalen 1 i would like to express my deepest gratitude to my thesis advisor john morgan i would also like to thank michael davis i preliminaries we start with pertinent definitions from orbifold theory and recall some results of davis and morgan 2 other results of that paper are extended to include the dihedral case for more on orbifolds see thurston 5 received by the editors march 9 1982 and in revised form june 20 1983 1980 mathematics subject classification primary 57s17 1984 american mathematical society 0002 9947 84 1 00 25 per page

on culler shalen seminorms and dehn filling if f is a finitely generated discrete group and g a complex algebraic lie group the g character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in g marc culler and peter shalen developed the theory of sl2 c character varieties of finitely generated groups and applied their results to study the topology of 3 dimensional manifolds in the papers 6 7 8 consider the exterior m of a hyperbolic knot lying in a closed connected orientable 3 manifold the mostow rigidity theorem implies that the holonomy representation p iri m isom h3 psl2 c is unique up to conjugation and taking complex conjugates the orientability of m can be used to show 1 lifts to a representation p e hom ri m sl2 c whose character determines an essentially unique point of xp of x iri m the sl2 c character variety of irl m culler and shalen 8 proved that the component x0 of x x1ri m which contains x is a curve one of their major contributions was to show how x0 determines a norm on h1 wm r which encodes many topological properties of m in particular it provides information on the dehn fillings of m their construction may be applied to arbitrary curves in the sl2 c character variety of a connected compact orientable irreducible 3manifold whose boundary is a torus though in this generality one can only guarantee that it will define a seminorm the first half of this paper is devoted to the development of the general theory of culler shalen seminorms defined for curves of psl2 c characters by working over psl2 c we obtain a theory that is more generally applicable than its sl2 c counterpart while being only mildly more difficult to set up in the second half of this paper we apply the theory of culler shalen seminorms to study the dehn filling operation in particular we examine the relationship between fillings which yield manifolds having a positive dimen

regenerating singular hyperbolic structures from sol let m be a torus bundle over s 1 with an orientation preserving anosov monodromy the manifold m admits a geometric structure modeled on sol we prove that the sol structure can be deformed into singular hyperbolic cone structures whose singular locus m is the mapping torus of the fixed point of the monodromy the hyperbolic cone metrics are parametred by the cone angle in the interval 0 2 when 2 the cone manifolds collapse to the basis of the fibration s 1 and they can be rescaled in the direction of the fibers to converge to the sol manifold

chern simons invariants of 3 manifolds decomposed along tori and the circle bundle over the representation space of t 2 we describe a cut and paste method for computing chern simons invariant of flatg connections on 3 manifolds decomposed along tori especially forg su 2 andsl 2 c we use this method to make computations ofsu 2 chern simons invariants of graph manifolds which generalize fintushel and stern s computations for seifert fibered spaces we also use this technique to give a simple derivation of a formula of yoshida relating the flatsl 2 c chern simons invariant of the holonomy representation to the volume and the metric chern simons invariant for cusped hyperbolic 3 manifolds

multiscale wiener filter for the restoration of fractal signals wavelet filter bank approach a filter bank design based on orthonormal wavelets and equipped with a multiscale wiener filter is proposed in this paper for signal restoration of 1 f family of fractal signals which are distorted by the transmission channel and corrupted by external noise first the fractal signal transmission process is transformed via the analysis filter bank into multiscale convolution subsystems in time scale domain based on orthonormal wavelets some nonstationary properties e g self similarity long term dependency of fractal signals are attenuated in each subband by wavelet multiresolution decomposition so that the wiener filter bank can be applied to estimate the multiscale input signals then the estimated multiscale input signals are synthesized to obtain the estimated input signal some simulation examples are given for testing the performance of the proposed algorithm with this multiscale analysis synthesis design via the technique of the wavelet filter bank the multiscale wiener filter can be applied to treat the signal restoration problem for nonstationary 1 f fractal signals

self calibration of an affine camera from multiple views a key limitation of all existing algorithms for shape and motion from image sequences under orthographic weak perspective and para perspective projection is that they require the calibration parameters of the camera we present in this paper a new approach that allows the shape and motion to be computed from image sequences without having to know the calibration parameters this approach is derived with the affine camera model introduced by mundy and zisserman 1992 which is a more general class of projections including orthographic weak perspective and para perspective projection models the concept of self calibration introduced by maybank and faugeras 1992 for the perspective camera and by hartley 1994 for the rotating camera is then applied for the affine camera this paper introduces the 3 intrinsic parameters that the affine camera can have at most the intrinsic parameters of the affine camera are closely related to the usual intrinsic parameters of the pin hole perspective camera but are different in the general case based on the invariance of the intrinsic parameters methods of self calibration of the affine camera are proposed it is shown that with at least four views an affine camera may be self calibrated up to a scaling factor leading to euclidean similarity shape reconstruction up to a global scaling factor another consequence of the introduction of intrinsic and extrinsic parameters of the affine camera is that all existing algorithms using calibrated affine cameras can be assembled into the same framework and some of them can be easily extented to a batch solution experimental results are presented and compared with other methods using calibrated affine cameras

degenerations of ideal hyperbolic triangulations let m be a cusped 3 manifold and let mathcal t be an ideal triangulation of m the deformation variety mathfrak d mathcal t a subset of which parameterises incomplete hyperbolic structures obtained on m using mathcal t is defined and compactified by adding certain projective classes of transversely measured singular codimension one foliations of m this leads to a combinatorial and geometric variant of well known constructions by culler morgan and shalen concerning the character variety of a 3 manifold

on certain functions for deformations of knot group representations we study the twisted knot module for the universal deformation of an sl2 representation of a knot group and introduce an associated l function which may be seen as an analogue of the algebraic p adic lfunction associated to the selmer module for the universal deformation of a galois representation we then verify affirmatively mazur s problem on the order of a zero of the l function by some examples for 2 bridge knots

estimation of fractional brownian motion with multiresolution kalman filter banks a filter bank design based on orthonormal wavelets and equipped with a multiscale kalman filter was proposed for deconvolution of fractal signals we use the same scheme for estimating fractional brownian motion in noise considering 1 the effect of correlation in the sequence of wavelet coefficients 2 the approximation term in the wavelet expansion 3 aliasing effects 4 the optimal number of scales in the filter bank considerations on the minimum number of filters in the bank are made and comparisons between wiener and kalman filters are given explicit expressions of the mean square error are given and comparisons between theoretical and simulation results are shown

on the geometry of the slice of trace free sl_2 mathbb c characters of a knot group let k be a knot in an integral homology 3 sphere with exterior e k and let b 2 denote the two fold branched cover of branched along k we construct a map from the slice of trace free rm sl 2 mathbb c characters of 1 e k to the rm sl 2 mathbb c character variety of 1 b 2 when this map is surjective it describes the slice as the two fold branched cover over the rm sl 2 mathbb c character variety of b 2 with branched locus given by the abelian characters whose preimage is precisely the set of metabelian characters we show that each metabelian character can be represented as the character of a binary dihedral representation of 1 e k the map is shown to be surjective for all 2 bridge knots and all pretzel knots of type p q r an extension of this framework to n fold branched covers is also described

ergodic theory on moduli spaces let m be a compact surface with x m 0 and let g be a compact lie group whose levi factor is a product of groups locally isomorphic to su 2 for example su 2 itself then the mapping class group rm of m acts on the moduli space x m of flat g bundles over m possibly twisted by a fixed central element of g when m is closed then fm preserves a symplectic structure on x m which has finite total volume on m more generally the subspace of x m corresponding to flat bundles with fixed behavior over am carries a rm invariant symplectic structure the main result is that rm acts ergodically on x m with respect to the measure induced by the symplectic structure

invariants of six points and projective reconstruction from three uncalibrated images there are three projective invariants of a set of six points in general position in space it is well known that these invariants cannot be recovered from one image however an invariant relationship does exist between space invariants and image invariants this invariant relationship is first derived for a single image then this invariant relationship is used to derive the space invariants when multiple images are available this paper establishes that the minimum number of images for computing these invariants is three and the computation of invariants of six points from three images can have as many as three solutions algorithms are presented for computing these invariants in closed form the accuracy and stability with respect to image noise selection of the triplets of images and distance between viewing positions are studied both through real and simulated images applications of these invariants are also presented both the results of faugeras 1992 and hartley et al 1992 for projective reconstruction and sturm s method 1869 for epipolar geometry determination from two uncalibrated images with at least seven points are extended to the case of three uncalibrated images with only six points

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

finite quotients of rings and applications to subgroup separability of linear groups in this paper we apply results from algebraic number theory to subgroup separability of linear groups we then state applications to subgroup separability of free products with amalgamation of hyperbolic 3 manifold groups

so 2n c character varieties are not varieties of characters abstract we prove that for n 2 and all groups of corank 2 the coordinate rings of s o 2 n c character varieties of are not generated by trace functions nor generalized trace functions sending s o 2 n c to t r for an arbitrary representation of s o 2 n c and any furthermore we give examples of non conjugate completely reducible representations undistinguishable by generalized trace functions hence s o 2 n c character varieties are not varieties of characters however we also prove that any generic s o 2 n c representation of a free group can be distinguished from all non equivalent representations by trace functions and by a single generalized trace function

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

grafting pruning and the antipodal map on measured laminations grafting a measured lamination on a hyperbolic surface de nes a self map of teichmuller space which is a homeomorphism by a result of scannell and wolf in this paper we study the large scale behavior of pruning which is the inverse of grafting speci cally for each conformal structure x t s pruning x gives a map ml s t s we show that this map extends to the thurston compacti cation of t s and that its boundary values are the natural antipodal involution relative to x on the space of projective measured laminations we use this result to study thurston s grafting coordinates on the space of cp structures on s for each x t s we show that the boundary of the space p x of cp structures on x in the compactication of the grafting coordinates is the graph ix of the antipodal involution ix pml s pml s

approximation by maximal cusps in boundaries of deformation spaces of kleinian groups let m be a compact oriented irreducible atoroidal 3 manifold with nonempty boundary let cc0 m denote the space of convex cocompact kleinian groups uniformizing m we show that any kleinian group in the boundary of cc0 m whose limit set is the whole sphere can be approximated by maximal cusps density of maximal cusps on the boundary of schottky space is derived as a corollary we further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3 manifold with connected conformal boundary

wavelet packet transform for fractional brownian motion asymptotic decorrelation and selection of best bases our first goal in this paper is to investigate stationarization and asymptotic decorrelation for fractional brownian motion fbm using wavelet packet transform the wavelet packets are generated by the n mathrm th order daubechies scaling function and wavelet to decorrelate the wavelet packet coefficients asymptotically we take two strategies fix n and let absolute scale difference or absolute difference between time shifts get large and fix scale level and time shift and let n get large our second goal is to present the asymptotic properties of the entropy like cost functional and denoising cost functional and their impact on the selection of best wavelet packet bases when used for fbm plus or not plus independent gaussian white noise

estimation of fractal signals using wavelets and filter banks a filter bank design based on orthonormal wavelets and equipped with a multiscale wiener filter was recently proposed for signal restoration and for signal smoothing of 1 f family of fractal signals corrupted by external noise the conclusions obtained in these papers are based on the following simplificative hypotheses 1 the wavelet transformation is a whitening filter and 2 the approximation term of the wavelet expansion can be avoided when the number of octaves in the multiresolution analysis is large enough in this paper we show that the estimation of 1 f processes in noise can be improved avoiding these two hypotheses explicit expressions of the mean square error are given and numerical comparisons with previous results are shown

three dimensional quantum gravity chern simons theory and the a polynomial we study three dimensional chern simons theory with complex gauge group sl 2 which has many interesting connections with three dimensional quantum gravity and geometry of hyperbolic 3 manifolds we show that in the presence of a single knotted wilson loop in an infinite dimensional representation of the gauge group the classical and quantum properties of such theory are described by an algebraic curve called the a polynomial of a knot using this approach we find some new and rather surprising relations between the a polynomial the colored jones polynomial and other invariants of hyperbolic 3 manifolds these relations generalize the volume conjecture and the melvin morton rozansky conjecture and suggest an intriguing connection between the sl 2 partition function and the colored jones polynomial

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

non standard components of the character variety for a family of montesinos knots we show that there are montesinos knots with n 1 geq 4 tangles whose character varieties contain arbitrarily many irreducible components of dimension d for any 1 le d le n 2 moreover these irreducible components can be chosen so that the trace of the meridian is non constant

slicing skinning and grafting we prove that a bers slice is never algebraic meaning that its zariski closure in the character variety has strictly larger dimension a corollary is that skinning maps are never constant the proof uses grafting and the theory of complex projective structures

euclidean shape and motion from multiple perspective views by affine iterations we describe a method for solving the euclidean reconstruction problem with a perspective camera model by incrementally performing euclidean reconstruction with either a weak or a paraperspective camera model with respect to other methods that compute shape and motion from a sequence of images with a calibrated camera this method converges in a few iterations is computationally efficient and solves for the sign reversal ambiguity we give a detailed account of the method analyze its convergence and test it with both synthetic and real data

characteristic subsurfaces and dehn filling let m be a simple knot manifold using the characteristic submanifold theory and the combinatorics of graphs in surfaces we develop a method for bounding the distance between the boundary slope of an essential surface in m which is not a fiber or a semi fiber and the boundary slope of a certain type of singular surface applications include bounds on the distances between exceptional dehn surgery slopes it is shown that if the fundamental group of m has no non abelian free subgroup and if m is a reducible manifold which is not homeomorphic to s 1 s 2 or p 3 p 3 then 5 under the same condition on m it is shown that if m is seifert fibered then 6 moreover in the latter situation character variety techniques are used to characterize the topological types of m and m in case the bound of 6 is attained

on character varieties sets of discrete characters and nonzero degree maps a it knot manifold is a compact connected irreducible orientable 3 manifold whose boundary is an incompressible torus we first investigate virtual epimorphisms between the fundamental groups of small knot manifolds and prove minimality results for small knot manifolds with respect to nonzero degree maps these results are applied later in the paper where we fix a small knot manifold m and investigate various sets of characters of representations rho pi 1 m to rm psl 2 bbb c whose images are discrete we show that the topology of these sets is intimately related to the algebraic structure of the rm psl 2 bbb c character variety of m as well as dominations of manifolds by m and its dehn fillings we apply our results to the following question of shicheng wang it are nonzero degree maps between infinitely many distinct dehn fillings of two hyperbolic knot manifolds m and n induced by a nonzero degree map m to n we show that the answer is yes generically using this we show that if a small mathcal h minimal hyperbolic knot manifold admits non homeomorphic mathcal h minimal dehn fillings it admits infinitely many such fillings we also construct the first infinite families of small closed connected orientable manifolds which are minimal in the sense that they do not admit nonzero degree maps other than homotopy equivalences to any aspherical manifold

homotopy hyperbolic 3 manifolds are hyperbolic this paper introduces a rigorous computer assisted procedure for analyzing hyperbolic 3 manifolds this procedure is used to complete the proof of several long standing rigidity conjectures in 3 manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3 manifold

a calculation of the culler shalen seminorms associated to small seifert dehn fillings we determine the relationship between the multiplicities of the zeros of certain rational functions defined on the rm sl 2 mathbb c and rm psl 2 mathbb c character varieties of knot exteriors this leads to a formula for the culler shalen seminorms associated to small seifert dehn fillings of such manifolds 2000 mathematics subject classification 57m05 57m27 57r65

skinning maps are finite to one we show that thurston s skinning maps of teichmuller space have finite fibers the proof centers around a study of two subvarieties of the rm sl 2 mathbb c character variety of a surface one associated with complex projective structures and the other associated with a 3 manifold using the morgan shalen compactification of the character variety and author s results on holonomy limits of complex projective structures we show that these subvarieties have only a discrete set of intersections

computing boundary slopes of 2 bridge links we describe an algorithm for computing boundary slopes of 2bridge links as an example we work out the slopes of the links obtained by 1 k surgery on one component of the borromean rings a table of all boundary slopes of all 2 bridge links with 10 or less crossings is also included

limits of geometrically tame kleinian groups let f be a finitely generated geometrically tame kleinian group with h3 f m when m has a finite volume the hyperbolic structure of m is unique up to homotopy by mostow s rigidity theorem but when m has infinite volume it is meaningful to consider the deformation space of the hyperbolic structure of m let ah m denote the space of faithful representations with discrete image from nl m to psl2 c modulo conjugacy with the topology of algebraic convergence until now only a little is known about ah m when f is of second kind there exists the space of quasi conformal deformation of f denoted by qho m with the quasi conformal or quasi isometric topology there is a canonical continuous injection t qho m ah m we abuse the notation qho m for t qho m the topological type of qho m has been determined by sullivan 21 one method for investigating ah m is to consider limits of sequences in qho m boundary groups in bers 3 and double limits in thurston 24 are such examples for considering limits it is important to determine when a sequence in ah m converges it was shown by thurston 24 and morgan shalen 16 that when m is acylindrical every sequence in ah m has a converging subsequence we gave in ohshika 19 a sufficient condition of convergence for geometrically finite groups in w 2 of this paper we prove a theorem which implies the existence of converging subsequence in all of the above cases the case of boundary groups double limits acylindrical manifolds and the theorem in 19 the statement is as follows

acylindrical accessibility for groups we define the notion of acylindrical graph of groups of a group we bound the combinatorics of these graphs of groups for f g freely indecomposable groups our arguments imply the finiteness of acylindrical surfaces in closed 3 manifolds ha finiteness of isomorphism classes of small splittings of torsion free freely indecomposable hyperbolic groups as well as finiteness results for small splittings of f g kleinian and semisimple discrete groups acting on non positively curved simply connected manifolds in order to get our accessibility for f g groups we generalize parts of rips analysis of stable actions of f p groups on real trees to f g groups the concepts we present play an essential role in constructing the canonical jsj decomposition se1 ri se2 in obtaining the hopf property for hyperbolic groups se2 and in our study of sets of solutions to equations in a free group se3

dual computation of projective shape and camera positions from multiple images given multiple image data from a set of points in 3d there are two fundamental questions that can be addressed what is the structure of the set of points in 3d what are the positions of the cameras relative to the points in this paper we show that for projective views and with structure and position defined projectively these problems are dual because they can be solved using constraint equations where space points and camera positions occur in a reciprocal way more specifically by using canonical projective reference frames for all points in space and images the imaging of point sets in space by multiple cameras can be captured by constraint relations involving three different kinds of parameters only coordinates of 1 space points 2 camera positions 3 image points the duality implies that the problem of computing camera positions fromp points in q views can be solved with the same algorithm as the problem of directly reconstructing q 4 points in p 4 views this unifies different approaches to projective reconstruction methods based on external calibration and direct methods exploiting constraints that exist between shape and image invariants

orientation reversing involutions and the first betti number for finite coverings of 3 manifolds letm s 3 p 3 be a closed orientable irreducible 3 manifold which admits an orientation reversing involution m m if dim fix 0 suppose 1 m has a subgroup of even index we show thatm has finite coverm mm with 1 m 0 as an application we show that the hyperbolic dodecahedral space has a finite cover with positive 1st betti number

self calibration and metric reconstruction inspite of varying and unknown intrinsic camera parameters in this paper the theoretical and practical feasibility of self calibration in the presence of varying intrinsic camera parameters is under investigation the paper s main contribution is to propose a self calibration method which efficiently deals with all kinds of constraints on the intrinsic camera parameters within this framework a practical method is proposed which can retrieve metric reconstruction from image sequences obtained with uncalibrated zooming focusing cameras the feasibility of the approach is illustrated on real and synthetic examples besides this a theoretical proof is given which shows that the absence of skew in the image plane is sufficient to allow for self calibration a counting argument is developed which depending on the set of constraints gives the minimum sequence length for self calibration and a method to detect critical motion sequences is proposed

plane curves associated to character varieties of 3 manifolds consider a compact 3 manifold m with boundary consisting of a single torus the papers cs1 cs2 cgls discuss the variety of characters of sl2 c representations of zl m and some of the ways in which the topological structure of m is reflected in the algebraic geometry of the character variety we will describe in this paper a certain affine algebraic curve dm which is naturally associated to the character variety of m a basis b 9 1 for the peripheral subgroup of m determines an embedding p8 of dm into c x c with coordinates l and m the closure in c 2 of ps dm is a plane algebraic curve and therefore is defined by a polynomial am n0 m that after certain normalizations is uniquely determined up to multiplication by constants the polynomial am b is effectively computable and is an invariant of the manifold m together with the choice of basis b the results in this paper describe how geometric properties of the character variety and hence topological properties of m are reflected by the polynomial am b in the case that m is the complement of a knot k in a homology 3 sphere we may take the basis b to consist of the longitude and meridian of k with the usual orientation conventions this basis is well defined modulo the involution which inverts both the longitude and meridian it will follow from the construction of dm that if the basis b is obtained from b by inverting both generators then the regular maps pb and pb have the same image thus ar am s am b is an invariant of the knot the polynomial am b displays in a striking way information about the incompressible surfaces in m this involves the newton polygon of am b

the a polynomial from the noncommutative viewpoint the paper introduces a noncommutative generalization of the a polynomial of a knot this is done using the kauffman bracket skein module of the knot complement and is based on the relationship between skein modules and character varieties the construction is possible because the kauffman bracket skein algebra of the cylinder over the torus is a subalgebra of the noncommutative torus the generalized version of the a polynomial called the noncommutative a ideal consists of a finitely generated ideal of polynomials in the quantum plane some properties of the noncommutative a ideal and its relationships with the a polynomial and the jones polynomial are discussed the paper concludes with the description of the examples of the unknot and the right and left handed trefoil knots

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

appendix to the paper some uniserial representations of certain special linear groups by p sin and j g thompson abstract this appendix collects some material on the cohomology of discrete subgroups of lie groups and the theory of their representation varieties to provide background for the results of sin and thompson 2010 2013 in 22 23

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

cyclic surgery degrees of maps of character curves and volume rigidity for hyperbolic manifolds this paper proves a theorem about dehn surgery using a new theorem about psl2c character varieties confirming a conjecture of boyer and zhang this p aper shows that a small hyperbolic knot in a homotopy sphere having a non trivial cyclic slope r has an incompressible surface with non integer boundary slope strictly between r 1 and r 1 a corollary is that any small knot which has only integer boundary slopes has property p the proof uses connections between the topology of the complement of the knot m and the psl2c character variety of m that were discovered by culler and shalen the key lemma which should be of independent interest is that for certain components of the character variety of m the map on character varieties induced by m m is a birational isomorphism onto its image this in turn depends on a fancy version of mostow rigidity due to gromov thurston and goldman the proof of the theorem about dehn surgery also gives information about the diameter of the set of boundary slopes of m let me start with the theorem about dehn surgery let k be a knot in a compact closed 3manifold that is a tame embedding s 1 the complement of k m is minus an open regular neighborhood of k so m is a compact 3 manifold whose boundary is a torus suppose is a simple closed curve in m we can create a closed manifold m from m by taking a solid torus and gluing its boundary to m in such a way that bounds a disc in the solid torus m depends only on the isotopy class of the new manifold m is called a dehn filling of m or a dehn surgeryon k recently many people have studied what kinds of manifolds arise when you

the π orbifold group of a link on associe a un enlacement l dans s 3 un groupe o l appele le groupe de variete d orbite de l on etudie certaines proprietes du groupe o l reliees a la geometrie de l enlacement l

bundles and finite foliations by a hyperbolic manifold we shall always mean a complete orientable hyperbolic manifold of nite volume we recall that if is a kleinian group then it is said to be geometrically nite if there is a nite sided convex fundamental domain for the action of on hyperbolic space otherwise is geometrically in nite if happens to be a surface group then we say it is quasi fuchsian if the limit set for the group action is a jordan curve c and preserves the components of s n c the starting point for this work is the following theorem which is a combination of theorems due to marden thurston and bonahon

wavelet based estimation of 1 f type signal parameters confidence intervals using the bootstrap we propose to construct confidence intervals of parameters of 1 f type signals using a nonparametric wavelet based bootstrap method bootstrap based confidence intervals of maximum likelihood parameter estimates are compared to the confidence intervals derived from the cramer rao lower bound crlb for moderately large data sample sizes the bootstrap approach achieves the nominal coverage and may perform better than the crlb based parametric approach

trajectory triangulation 3d reconstruction of moving points from a monocular image sequence we consider the problem of reconstructing the 3d coordinates of a moving point seen from a monocular moving camera i e to reconstruct moving objects from line of sight measurements only the task is feasible only when some constraints are placed on the shape of the trajectory of the moving point we coin the family of such tasks as trajectory triangulation we investigate the solutions for points moving along a straight line and along conic section trajectories we show that if the point is moving along a straight line then the parameters of the line and hence the 3d position of the point at each time instant can be uniquely recovered and by linear methods from at least five views for the case of conic shaped trajectory we show that generally nine views are sufficient for a unique reconstruction of the moving point and fewer views when the conic is of a known type like a circle in 3d euclidean space for which seven views are sufficient the paradigm of trajectory triangulation in general pushes the envelope of processing dynamic scenes forward thus static scenes become a particular case of a more general task of reconstructing scenes rich with moving objects where an object could be a single point

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

quasi fuchsian manifolds with particles we consider 3 dimensional hyperbolic cone manifolds which are convex cocompact in a natural sense with cone singularities along infinite lines such singularities are sometimes used by physicists as models for massive spinless point particles we prove an infinitesimal rigidity statement when the angles around the singular lines are less than any infinitesimal deformation changes either these angles or the conformal structure at infinity with marked points corresponding to the endpoints of the singular lines moreover any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone manifold structure these results hold also when the singularities are along a graph i e for interacting particles

the schwarzian derivative and measured laminations on riemann surfaces a holomorphic quadratic differential on a hyperbolic rie mann surface has an associated measured foliation which can be straight ened to yield a measured geodesic lamination on the other hand a quadratic differential can be considered as the schwarzian derivative of a 1 structure to which one can naturally associate another measured geodesic lamination using grafting we compare these two relationships between quadratic differentials and measured geodesic laminations each of which yields a homeomor phism ml s q x for each conformal structure x on a compact surface s we show that these maps are nearly the same differing by a multiplicative factor of 2 and an error term of lower order than the maps themselves which we bound explicitly as an application we show that the schwarzian derivative of a 1 structure with fuchsian holonomy is close to a 2 integral jenkins strebel differential we also study compactifications of the space of 1 structures using the schwarzian derivative and grafting coordinates we show that the natural map between these extends to the boundary of each fiber over teichmuller space and we describe this extension

on character varieties of two bridge knot groups we find explicit models for the psl2 c and sl2 c character varieties of the fundamental groups of complements in s of an infinite family of two bridge knots that contains the twist knots we compute the genus of the components of these character varieties and deduce upper bounds on the degree of the associated trace fields we also show that these knot complements are fibered if and only if they are commensurable to a fibered knot complement in a z 2z homology sphere resolving a conjecture of hoste and shanahan

detection of essential surfaces in 3 manifolds with sl 2 trees according to bass serre theory se given a field f with a discrete valuation v f z there is a canonical way to construct a simplicial tree t tf v on which sl2 f acts simplicially without inversion when a group has a representation into the group sl2 f then there is an induced action of on the tree via the representation if further is the fundamental group of a compact 3 manifoldm and the action of on the tree t is nontrivial meaning that there is no point on t fixed by every element of then the action induces a splitting of m along an essential surface in the following way let m p m be the universal covering there is a 1 m equivariant map f m t which is transverse to the set e of midpoints of edges in t such that p f 1 e is an essential surface in m we say such an essential surface is associated to or detected by an sl2 tree a natural question is which essential surfaces in a compact 3 manifold can be associated to sl2 trees and how do they depend on the choice of the field and discrete valuation this is the main issue we are going to address in this paper any 3 manifold mentioned in this paper is automatically assumed orientable and connected by an essential surface in a compact 3 manifold m we mean an orientable properly embedded incompressible surface each component of which is neither boundary parallel in m nor bounds a 3 ball when it is a 2sphere also recall that a discrete valuation on a field f is a homomorphism v from the multiplicative group f onto the group of integers z such that v a b min v a v b for any a b f with a b 0 let v f z be a discrete valuation the set ov 0 a f v a 0 s schanuel x zhang department of mathematics suny at buffalo buffalo ny 14260 2900 usa e mail schanuel buffalo edu xinzhang math buffalo edu