a duality principle for groups the duality principle for gabor frames states that a gabor sequence obtained by a time frequency lattice is a frame for l2 rd if and only if the associated adjoint gabor sequence is a riesz sequence we prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups we examine the existence problem of dual pairs and establish some connection with classification problems for ii1 factors while in general such a pair may not exist for some groups we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when g is an abelian infinite countable group or an amenable icc group for free groups with finitely many generators the existence problem of such a dual pair is equivalent to the well known problem about the classification of free group von neumann algebras

fast unit modulus least squares with applications in beamforming unit modulus least squares uls problems arise in many applications including phase only beamforming sensor network localization synchronization phase retrieval and radar code design uls formulations can always be recast as unit modulus quadratic programs to which semidefinite relaxation sdr can be applied and is often the state of the art approach however sdr lifts the problem dimension i e the number of variables from n to n 2 which drastically increases the memory burden and computational cost when the problem size is already large e g when designing phase only beamformer weights for massive multiple input multiple output systems this paper focuses on scalable first order algorithms for the uls problem and some of its variants it advocates using simple gradient projection gp as a starting point for solving the uls problem establishes global convergence of gp to a karush kuhn tucker point for this np hard problem and bounds its iteration complexity then it proposes uls extensions tailored to reflect practical beamformer design objectives bringing in and exploiting new degrees of freedom to improve the beampattern designs simple variants of gp are proposed to deal with these extended uls problems simulations are used to showcase the effectiveness of the proposed algorithms in both the plain uls problem and in the context of phase only beamforming

singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

spectral matrix completion by cyclic projection and application to sound source reconstruction from non synchronous measurements a fundamental limitation of the inverse acoustic problem is determined by the size of the array and the microphone density a solution to achieve large array and or high microphone density is to scan the object of interest by moving sequentially a small prototype array which is referred to as sequential measurements in comparison to a large array and or high microphone density array that can acquire simultaneously all the information of the spectral matrix in particular all cross spectra sequential measurements can only acquire a block diagonal spectral matrix while the cross spectra between the sequential measurements remain unknown due to the missing phase relationships between consecutive positions nevertheless these unknown cross spectra are necessary for sound source reconstruction the objective of this paper is to propose an algorithm to recover the missing elements of the spectral matrix in the case where the acoustical field is highly coherent this issue is shown to boil down to a matrix completion problem subject to given constraint of hermitian symmetry measurements fitting reduced rank and spatial continuity of the sound field a cyclic projection cp algorithm is proposed in this work to find an optimal solution at the intersection between three predefined sets the proposed method is analyzed through numerical simulations of diverse setups and is also validated experimentally

frame representations and parseval duals with applications to gabor frames let x n be a frame for a hilbert space h we investigate the conditions under which there exists a dual frame for x n which is also a parseval or tight frame we show that the existence of a parseval dual is equivalent to the problem whether x n can be dilated to an orthonormal basis under an oblique projection a necessary and sufficient condition for the existence of parseval duals is obtained in terms of the frame excess for a frame g g g induced by a projective unitary representation of a group g it is possible that g g g can have a parseval dual but does not have a parseval dual of the same type the primary aim of this paper is to present a complete characterization for all the projective unitary representations such that every frame g g g with a necessary lower frame bound condition has a parseval dual of the same type as an application of this characterization together with a result about lattice tiling we prove that every gabor frame g g l k again with the same necessary lower frame bound condition has a parseval dual of the same type if and only if the volume of the fundamental domain of l k is less than or equal to 1 2

mountain pass solutions for non local elliptic operators abstract the purpose of this paper is to study the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions these equations have a variational structure and we find a non trivial solution for them using the mountain pass theorem to make the nonlinear methods work some careful analysis of the fractional spaces involved is necessary we prove this result for a general integrodifferential operator of fractional type and as a particular case we derive an existence theorem for the fractional laplacian finding non trivial solutions of the equation s u f x u in u 0 in r n as far as we know all these results are new

signal reconstruction from the magnitude of subspace components we consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems in the deterministic setting a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions that require at least a quadratic number of subspaces moreover we address reconstruction under the erasure of a subset of the norms using the concepts of p fusion frames and list decoding we propose an algorithm that outputs a finite list of candidate signals one of which is the correct one in the random setting we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program

harmonic mean iteratively reweighted least squares for low rank matrix recovery we propose a new iteratively reweighted least squares irls algorithm for the recovery of a matrix x cd1 d2 of rank r min d1 d2 from incomplete linear observations solving a sequence of low complexity linear problems the easily implementable algorithm which we call harmonic mean iteratively reweighted least squares hm irls optimizes a non convex schatten p quasi norm penalization to promote low rankness and carries three major strengths in particular for the matrix completion setting first we observe a remarkable global convergence behavior of the algorithm s iterates to the low rank matrix for relevant interesting cases for which any other state of the art optimization approach fails the recovery secondly hm irls exhibits an empirical recovery probability close to 1 even for a number of measurements very close to the theoretical lower bound r d1 d2 r i e already for significantly fewer linear observations than any other tractable approach in the literature thirdly hm irls exhibits a locally superlinear rate of convergence of order 2 p if the linear observations fulfill a suitable null space property while for the first two properties we have so far only strong empirical evidence we prove the third property as our main theoretical result

paracatadioptric camera calibration catadioptric sensors refer to the combination of lens based devices and reflective surfaces these systems are useful because they may have a field of view which is greater than hemispherical providing the ability to simultaneously view in any direction configurations which have a unique effective viewpoint are of primary interest among these is the case where the reflective surface is a parabolic mirror and the camera is such that it induces an orthographic projection and which we call paracatadioptric we present an algorithm for the calibration of such a device using only the images of lines in space in fact we show that we may obtain all of the intrinsic parameters from the images of only three lines and that this is possible without any metric information we propose a closed form solution for focal length image center and aspect ratio for skewless cameras and a polynomial root solution in the presence of skew we also give a method for determining the orientation of a plane containing two sets of parallel lines from one uncalibrated view such an orientation recovery enables a rectification which is impossible to achieve in the case of a single uncalibrated view taken by a conventional camera we study the performance of the algorithm in simulated setups and compare results on real images with an approach based on the image of the mirror s bounding circle

geometry based distributed scene representation with omnidirectional vision sensors this paper addresses the problem of efficient representation of scenes captured by distributed omnidirectional vision sensors we propose a novel geometric model to describe the correlation between different views of a 3 d scene we first approximate the camera images by sparse expansions over a dictionary of geometric atoms since the most important visual features are likely to be equivalently dominant in images from multiple cameras we model the correlation between corresponding features in different views by local geometric transforms for the particular case of omnidirectional images we define the multiview transforms between corresponding features based on shape and epipolar geometry constraints we apply this geometric framework in the design of a distributed coding scheme with side information which builds an efficient representation of the scene without communication between cameras the wyner ziv encoder partitions the dictionary into cosets of dissimilar atoms with respect to shape and position in the image the joint decoder then determines pairwise correspondences between atoms in the reference image and atoms in the cosets of the wyner ziv image in order to identify the most likely atoms to decode under epipolar geometry constraints experiments demonstrate that the proposed method leads to reliable estimation of the geometric transforms between views in particular the distributed coding scheme offers similar rate distortion performance as joint encoding at low bit rate and outperforms methods based on independent decoding of the different images

reconstruction of signals from their autocorrelation and cross correlation vectors with applications to phase retrieval and blind channel estimation we consider the problem of reconstructing two signals from the autocorrelation and cross correlation measurements this inverse problem is a fundamental one in signal processing and arises in many applications including phase retrieval and blind channel estimation in a typical phase retrieval setup only the autocorrelation measurements are obtainable we show that when the measurements are obtained using three simple masks phase retrieval reduces to the aforementioned reconstruction problem the classic solution to this problem is based on finding common factors between the z transforms of the autocorrelation and cross correlation vectors this solution has enjoyed limited practical success mainly due to the fact that it is not sufficiently stable in the noisy setting in this paper inspired by the success of convex programming in provably and stably solving various quadratic constrained problems we develop a semidefinite programming based algorithm and provide theoretical guarantees in particular we show that almost all signals can be uniquely recovered by this algorithm up to a global phase comparative numerical studies demonstrate that the proposed method significantly outperforms the classic method in the noisy setting

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

confidence region of singular subspaces for low rank matrix regression low rank matrix regression refers to the instances of recovering a low rank matrix based on specially designed measurements and the corresponding noisy outcomes numerous statistical methods have been developed over the recent decade for efficiently reconstructing the unknown low rank matrices it is often interesting in certain applications to estimate the unknown singular subspaces in this paper we revisit the low rank matrix regression model and introduce a two step procedure to construct confidence regions of the singular subspaces we investigate distributions of the joint projection distance between the empirical singular subspaces and the unknown true singular subspaces we prove asymptotical normality of the joint projection distance with data dependent centering and normalization when r 3 2 m 1 m 2 3 2 o n log n where m 1 m 2 denote the matrix row and column sizes r is the rank and n is the number of independent random measurements consequently data dependent confidence regions of the true singular subspaces are established which attain pre determined confidence levels asymptotically additionally non asymptotic convergence rates are also established numerical results are presented to show the merits of our methods

wiener s lemma for twisted convolution and gabor frames as a consequence ca is invertible and bounded on all p zd for 1 p oo simultaneously in this article we study several non commutative generalizations of wiener s lemma and their application to gabor theory the paper is divided into two parts the first part sections 2 and 3 is devoted to abstract harmonic analysis and extends wiener s lemma to twisted convolution the second part section 4 is devoted to the theory of gabor frames specifically to the design of dual windows with good time frequency localization in particular we solve a conjecture of janssen feichtinger and one of us 17 18 9 these two topics appear to be completely disjoint but they are not the solution of the conjectures about gabor frames is an unexpected application of methods from non commutative harmonic analysis to application oriented mathematics it turns out that the connection between twisted convolution and the heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis to be more concrete we formulate some of our main results first and will deal with the details and the technical background later

on twisted group frames abstract in this paper we study g frames for finite dimensional hilbert spaces where g is a finite group and is a unitary schur multiplier of g we apply the characterizations for tight g frames and central g frames to investigation of g frames that have the maximal spanning property we apply the theory of twisted group frames to obtain a criterion for maximal spanning vectors that generalizes the main result of 16 theorem 1 7 moreover we prove that a projective representation does not admit any maximal spanning vector if it has an at least 2 dimensional subrepresentation that is equivalent to a central projection induced subrepresentation of the regular representation

three positive solutions for dirichlet problems involving critical sobolev exponent and sign changing weight abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the lusternik schnirelman category to prove that a semilinear elliptic equation involving a sign changing weight function has at least three positive solutions

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

multiplicative lidskii s inequalities and optimal perturbations of frames abstract in this paper we study two design problems in frame theory on the one hand given a fixed finite frame f f j j i n for c d we compute those dual frames g of f that are optimal perturbations of the canonical dual frame for f under certain restrictions on the norms of the elements of g on the other hand we compute those v f v f j j i n for invertible operators v which are close to the identity that are optimal perturbations of f that is we compute the optimal perturbations of f among frames g g j j i n that have the same linear relations as f in both cases optimality is measured with respect to submajorization of the eigenvalues of the frame operators hence our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error the key tool for these results is a multiplicative analogue of lidskii s inequality in terms of log majorization and a characterization of the case of equality

identifiability in bilinear inverse problems with applications to subspace or sparsity constrained blind gain and phase calibration bilinear inverse problems bips the resolution of two vectors given their image under a bilinear mapping arise in many applications without further constraints bips are usually ill posed in practice the properties of natural signals are exploited to solve bips for example subspace constraints or sparsity constraints are imposed to reduce the search space these approaches have shown some success in practice however there are few results on uniqueness in bips for most bips the fundamental question of under what condition the problem admits a unique solution is yet to be answered for example blind gain and phase calibration bgpc is a structured bip which arises in many applications including inverse rendering in computational relighting albedo estimation with unknown lighting blind phase and gain calibration in sensor array processing and multichannel blind deconvolution mbd it is interesting to study the uniqueness of such problems in this paper we define identifiability of a bip up to a group of transformations we derive necessary and sufficient conditions for such identifiability i e the conditions under which the solutions can be uniquely determined up to the transformation group these conditions take the form of dividing the identifiability of the pair of unknown variables into the individual identifiability of each variable although verifying these individual conditions requires problem specific procedures this framework is universally applicable to all bips applying these results to bgpc we derive sufficient conditions for unique recovery under several scenarios including subspace joint sparsity and sparsity models for bgpc with joint sparsity or sparsity constraints we develop a procedure to compute the transformation groups corresponding to inherent ambiguities we also give necessary conditions in the form of tight lower bounds on sample complexities and demonstrate the tightness of these bounds by numerical experiments the results for bgpc not only demonstrate the application of the proposed general framework for identifiability analysis but are also of interest in their own right

dilations of operator valued measures with bounded p variations and framings on banach spaces abstract the dilations for operator valued measures ovms and bounded linear maps indicate that the dilation theory is in general heavily dependent on the banach space nature of the dilation spaces this naturally led to many questions concerning special type of dilations in particular it is not known whether ultraweakly continuous normal maps can be dilated to ultraweakly continuous homomorphisms we answer this question affirmatively for the case when the domain algebra is an abelian von neumann algebra it is well known that completely bounded hilbert space operator valued measures correspond to the existence of orthogonal projection valued dilations in the sense of naimark and stinespring and ovms with bounded total variations are completely bounded but not the vice versa with the aim of classifying ovms from the dilation point of view we introduce the concept of total p variations for ovms we prove that any completely bounded ovm has finite 2 variation and any ovm with finite p variation can be dilated to a but usually non hilbertian projection valued measure of the same type with the help of framing induced ovms we prove that conventional minimal dilation space of a non trivial framing contains c 0 then does not have bounded p variation

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

symmetric approximation of frames and bases in hilbert spaces we introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in hilbert spaces we prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames even in the case of the symmetric orthogonalization of bases our techniques and results are new a crucial role is played by whether or not a certain operator related to the initial frame or basis is hilbert schmidt

catadioptric projective geometry catadioptric sensors are devices which utilize mirrors and lenses to form a projection onto the image plane of a camera central catadioptric sensors are the class of these devices having a single effective viewpoint in this paper we propose a unifying model for the projective geometry induced by these devices and we study its properties as well as its practical implications we show that a central catadioptric projection is equivalent to a two step mapping via the sphere the second step is equivalent to a stereographic projection in the case of parabolic mirrors conventional lens based perspective cameras are also central catadioptric devices with a virtual planar mirror and are thus covered by the unifying model we prove that for each catadioptric projection there exists a dual catadioptric projection based on the duality between points and line images conics it turns out that planar and parabolic mirrors build a dual catadioptric projection pair as a practical example we describe a procedure to estimate focal length and image center from a single view of lines in arbitrary position for a parabolic catadioptric system

sparse phase retrieval uniqueness guarantees and recovery algorithms the problem of signal recovery from its fourier transform magnitude is of paramount importance in various fields of engineering and has been around for more than 100 years due to the absence of phase information some form of additional information is required in order to be able to uniquely identify the signal of interest in this paper we focus our attention on discrete time sparse signals of length n we first show that if the discrete fourier transform dimension is greater than or equal to 2n then almost all signals with aperiodic support can be uniquely identified by their fourier transform magnitude up to time shift conjugate flip and global phase then we develop an efficient two stage sparse phase retrieval algorithm tspr which involves identifying the support i e the locations of the nonzero components of the signal using a combinatorial algorithm and identifying the signal values in the support using a convex algorithm we show that tspr can provably recover most o n 1 2 epsilon sparse signals up to a time shift conjugate flip and global phase we also show that for most o n 1 4 epsilon sparse signals the recovery is robust in the presence of measurement noise these recovery guarantees are asymptotic in nature numerical experiments complement our theoretical analysis and verify the effectiveness of tspr

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

statistical guarantees for the em algorithm from population to sample based analysis we develop a general framework for proving rigorous guarantees on the performance of the em algorithm and a variant known as gradient em our analysis is divided into two parts a treatment of these algorithms at the population level in the limit of infinite data followed by results that apply to updates based on a finite set of samples first we characterize the domain of attraction of any global maximizer of the population likelihood this characterization is based on a novel view of the em updates as a perturbed form of likelihood ascent or in parallel of the gradient em updates as a perturbed form of standard gradient ascent leveraging this characterization we then provide non asymptotic guarantees on the em and gradient em algorithms when applied to a finite set of samples we develop consequences of our general theory for three canonical examples of incompletedata problems mixture of gaussians mixture of regressions and linear regression with covariates missing completely at random in each case our theory guarantees that with a suitable initialization a relatively small number of em or gradient em steps will yield with high probability an estimate that is within statistical error of the mle we provide simulations to confirm this theoretically predicted behavior

variational analysis applied to the problem of optical phase retrieval we apply nonsmooth analysis to a well known optical inverse problem phase retrieval the phase retrieval problem arises in many different modalities of electromagnetic imaging and has been studied in the optics literature for over forty years the state of the art for this problem in two dimensions involves iterated projections for solving a nonconvex feasibility problem despite widespread use of these algorithms current mathematical theory cannot explain their success at the heart of projection algorithms is a nonconvex nonsmooth optimization problem we obtain some insight into these algorithms by applying techniques from nonsmooth analysis in particular we show that the weak closure of the set of directions toward the projection generate the subdifferential of the corresponding squared set distance function following a pattern of proof described in f h clarke yu s ledyaev r j stern and p r wolenski nonsmooth analysis and control theory springer verlag new york 1998 this result is generalized to provide conditions under which the subdifferential of an integral function equals the integral of the subdifferential

approximations for gabor and wavelet frames let be a frame vector under the action of a collection of unitary operators u motivated by the recent work of frank paulsen and tiballi and some application aspects of gabor and wavelet frames we consider the existence and uniqueness of the best approximation by normalized tight frame vectors we prove that for any frame induced by a projective unitary representation for a countable discrete group the best normalized tight frame ntf approximation exists and is unique therefore it applies to gabor frames including gabor frames for subspaces and frames induced by translation groups similar results hold for semi orthogonal wavelet frames

geometric aspects of frame representations of abelian groups hhwe consider frames arising from the action of a unitary representation of a discrete countable abelian group we show that the range of the analysis operator can be determined by computing which characters appear in the representation this allows one to compare the ranges of two such frames which is useful for determining similarity and also for multiplexing schemes our results then partially extend to bessel sequences arising from the action of the group we apply the results to sampling on bandlimited functions and to wavelet and weyl heisenberg frames this yields a sufficient condition for two sampling transforms to have orthogonal ranges and two analysis operators for wavelet and weyl heisenberg frames to have orthogonal ranges the sufficient condition is easy to compute in terms of the periodization of the fourier transform of the frame generators

optimal dual frames for erasures for any given frame for the purpose of encoding in a finite dimensional hilbert space we investigate its dual frames that are optimal for erasures for the purpose of decoding we show that in general the canonical dual is not necessarily optimal moreover optimal dual frames are not necessarily unique we present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem as an application we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight

on the rank minimization problem over a positive semidefinite linear matrix inequality we consider the problem of minimizing the rank of a positive semidefinite matrix subject to the constraint that an affine transformation of it is also positive semidefinite our method for solving this problem employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice this problem is of importance in many contexts for example in feedback synthesis problems and such an example is also provided

optimal dual frames for communication coding with probabilistic erasures assume that a frame is preselected for encoding in a communication system we investigate the optimal dual frames for signal reconstruction decoding which minimize the maximal error when the probabilistic erasures occur in the transmission process from the perspective of mathematical theory of frames we set up a probability model under which we define the probability optimal po dual frames for a given frame when the frame based coding involves probabilistic erasures we obtain a sufficient and necessary condition under which the canonical dual frame is the unique po dual frame additionally we derive some general sufficient conditions for which the canonical dual frame is either not optimal or it is optimal but not the unique optimal one we present two simulation examples to compare the reconstruction effects when both the po dual frames and the general optimal go dual frames are used for reconstruction

low rank phase retrieval we develop two iterative algorithms for solving the low rank phase retrieval lrpr problem lrpr refers to recovering a low rank matrix boldsymbol x from magnitude only phaseless measurements of random linear projections of its columns both methods consist of a spectral initialization step followed by an iterative algorithm to maximize the observed data likelihood we obtain sample complexity bounds for our proposed initialization approach to provide a good approximation of the true boldsymbol x when the rank is low enough these bounds are significantly lower than what existing single vector phase retrieval algorithms need via extensive experiments we show that the same is also true for the proposed complete algorithms

optical wavefront reconstruction theory and numerical methods optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in nasa s hubble space telescope hst nasa s success with reconstruction algorithms on the hst has led to an effort to develop software that can aid and in some cases replace complicated expensive and error prone hardware among the many applications is hst s replacement the next generation space telescope ngst indent this work details the theory of optical wavefront reconstruction reviews some numerical methods for this problem and presents a novel numerical technique that we call extended least squares we compare the performance of these numerical methods for potential inclusion in prototype ngst optical wavefront reconstruction software we begin with a tutorial on rayleigh sommerfeld diffraction theory

on polyhedral approximations of the positive semidefinite cone it is well known that state of the art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes this leads us to co

dilations for systems of imprimitivity acting on banach spaces abstract motivated by a general dilation theory for operator valued measures framings and bounded linear maps on operator algebras we consider the dilation theory of the above objects with special structures we show that every operator valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a banach space this completely generalizes a well known result which states that every frame representation of a countable group on a hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group we also prove that isometric group representation induced framings on a banach space can be dilated to unconditional bases with the same structure for a larger banach space this extends several known results on the dilations of frames induced by unitary group representations on hilbert spaces

on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

camera calibration from images of spheres this paper introduces a novel approach for solving the problem of camera calibration from spheres by exploiting the relationship between the dual images of spheres and the dual image of the absolute conic iac it is shown that the common pole and polar with regard to the conic images of two spheres are also the pole and polar with regard to the iac this provides two constraints for estimating the iac and hence allows a camera to be calibrated from an image of at least three spheres experimental results show the feasibility of the proposed approach

catadioptric camera calibration using geometric invariants central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint in this paper we propose a novel method for the calibration of central catadioptric cameras using geometric invariants lines and spheres in space are all projected into conics in the catadioptric image plane we prove that the projection of a line can provide three invariants whereas the projection of a sphere can only provide two from these invariants constraint equations for the intrinsic parameters of catadioptric camera are derived therefore there are two kinds of variants of this novel method the first one uses projections of lines and the second one uses projections of spheres in general the projections of two lines or three spheres are sufficient to achieve catadioptric camera calibration one important conclusion in this paper is that the method based on projections of spheres is more robust and has higher accuracy than that based on projections of lines the performances of our method are demonstrated by both the results of simulations and experiments with real images

aliasing and oblique dual pair designs for consistent sampling abstract in this paper we study some aspects of oblique duality between finite sequences of vectors f and g lying in finite dimensional subspaces w and v respectively we compute the possible eigenvalue lists of the frame operators of oblique duals to f lying in v we compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for f with norm restrictions as an application we show that these optimal duals are the closest to being tight frames and therefore have their spectrum as concentrated as possible among oblique duals with norm restrictions we obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces v and w has in oblique duality we apply this analysis to compute those rigid rotations u for w such that the canonical oblique dual of u f minimize every convex potential we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations u for w such that the canonical oblique dual pair associated to u f minimize the aliasing we point out that these two last problems are intrinsic to oblique duality within the context of consistent sampling

random phaseless sampling for causal signals in shift invariant spaces a zero distribution perspective we proved that the phaseless sampling pls in the linear phase modulated shift invariant space sis v e bf i alpha cdot varphi alpha ne 0 is impossible even though the real valued function varphi enjoys the full spark property so does e bf i alpha cdot varphi stated another way the pls in the complex generated siss is essentially different from that in the real generated ones motivated by this we first establish the condition on the complex valued generator phi such that the pls of nonseparable causal nc signals in v phi can be achieved by random sampling the condition is established from the generalized haar condition ghc perspective based on the proposed reconstruction approach it is proved that if the ghc holds then with probability 1 the random sampling density sd 3 is sufficient for the pls of nc signals in the complex generated siss for the real valued case we also prove that if the ghc holds then with probability 1 the random sd 2 is sufficient for the pls of real valued nc signals in the real generated siss for the local reconstruction of highly oscillatory signals such as chirps a great number of deterministic samples are required compared with deterministic sampling the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples to verify our results numerical simulations were conducted to reconstruct highly oscillatory nc signals in the chirp modulated siss

parseval frames for icc groups abstract we analyze parseval frames generated by the action of an icc group on a hilbert space we parametrize the set of all such parseval frames by operators in the commutant of the corresponding representation we characterize when two such frames are strongly disjoint we prove an undersampling result showing that if the representation has a parseval frame of equal norm vectors of norm 1 n the hilbert space is spanned by an orthonormal basis generated by a subgroup as applications we obtain some sufficient conditions under which a unitary representation admits a parseval frame which is spanned by a riesz sequences generated by a subgroup in particular every subrepresentation of the left regular representation of a free group has this property

phasecode fast and efficient compressive phase retrieval based on sparse graph codes we consider the problem of recovering a complex signal x in mathbb c n from m intensity measurements of the form a i mathrm h x 1 leq i leq m where a i mathrm h is the i th row of measurement matrix a in mathbb c m times n our main focus is on the case where the measurement vectors are unconstrained and where x is exactly k sparse or the so called general compressive phase retrieval problem we introduce phasecode a novel family of fast and efficient algorithms that are based on a sparse graph coding framework we show that in the noiseless case the phasecode algorithm can recover an arbitrarily close to one fraction of the k nonzero signal components using only slightly more than 4k measurements when the support of the signal is uniformly random with the order optimal time and memory complexity of theta k 1 it is known that the fundamental limit for the number of measurements in compressive phase retrieval problem is 4k o k for the more difficult problem of recovering the signal exactly and with no assumptions on its support distribution this shows that under mild relaxation of the conditions our algorithm is the first constructive capacity approaching compressive phase retrieval algorithm in fact our algorithm is also order optimal in complexity and memory furthermore we show that for any signal x phasecode can recover a random 1 p fraction of the nonzero components of x with high probability where p can be made arbitrarily close to zero with sample complexity m c p k where c p is a small constant depending on p that can be precisely calculated with optimal time and memory complexity as a result assuming that the nonzero components of x are lower bounded by theta 1 and upper bounded by theta k gamma for some positive constant gamma we are able to provide a strong ell 1 guarantee for the estimated signal hat x as follows hat x x 1 leq p x 1 1 o 1 where p can be made arbitrarily close to zero as one instance the phasecode algorithm can provably recover with high probability a random 1 10 7 fraction of the significant signal components using at most m 14k measurements next motivated by some important practical classes of optical systems we consider a fourier friendly constrained measurement setting and show that its performance matches that of the unconstrained setting when the signal is sparse in the fourier domain with uniform support in the fourier friendly setting that we consider the measurement matrix is constrained to be a cascade of fourier matrices corresponding to optical lenses and diagonal matrices corresponding to diffraction mask patterns finally we tackle the compressive phase retrieval problem in the presence of noise where measurements are in the form of y i a i mathrm h x 2 w i and w i is the additive noise to the i th measurement we assume that the signal is quantized and each nonzero component can take l m possible magnitudes and l p possible phases we consider the regime where k beta n delta delta in 0 1 we use the same architecture of phasecode for the noiseless case and robustify it using two schemes the almost linear scheme and the sublinear scheme we prove that with high probability the almost linear scheme recovers x with sample complexity theta k log n and computational complexity theta l m l p n log n and the sublinear scheme recovers x with sample complexity theta k log 3 n and computational complexity theta l m l p k log 3 n throughout we provide extensive simulation results that validate the practical power of our proposed algorithms for the sparse unconstrained and fourier friendly measurement settings for noiseless and noisy scenarios 1 here we define the notation mathcal o cdot theta cdot and omega cdot we have f mathcal o g if and only if there exists a constant c 1 0 such that left f g right f theta g if and only if there exist two constants c 1 c 2 0 such that c 1 and f omega g if and only if there exists a constant c 1 0 such that left f g right c 1

vectorial phase retrieval of 1 d signals reconstruction of signals from measurements of their spectral intensities also known as the phase retrieval problem is of fundamental importance in many scientific fields in this paper we present a novel framework denoted as vectorial phase retrieval for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference we show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal first we prove that for compactly supported signals in the absence of measurement noise this new setup admits a unique solution next we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it finally we illustrate via simulations that our algorithm can accurately reconstruct signals even at considerable noise levels

statistical analysis of compressive low rank tomography with random measurements we consider the statistical problem of compressive estimation of low rank states r d with random basis measurements where r d are the rank and dimension of the state respectively we investigate whether for a fixed sample size n the estimation error associated with a compressive measurement setup is close to that of the setting where a large number of bases are measured we generalise and extend previous results and show that the mean square error mse associated with the frobenius norm attains the optimal rate rd n with only o rlogd random basis measurements for all states an important tool in the analysis is the concentration of the fisher information matrix fim we demonstrate that although a concentration of the mse follows from a concentration of the fim for most states the fim fails to concentrate for states with eigenvalues close to zero we analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the mse about its optimal despite a lack of concentration of the fim for states close to the boundary of the bloch sphere we also consider the estimation error in terms of a different metric the quantum infidelity we show that a concentration in the mean infidelity minf does not exist uniformly over all states highlighting the importance of loss function choice specifically we show that for states that are nearly pure the minf scales as 1 n but the constant converges to zero as the number of settings is increased this demonstrates a lack of compressive recovery for nearly pure states in this metric

the nehari manifold for a fractional critical choquard equation involving sign changing weight functions abstract in this paper we prove the existence and multiplicity of nontrivial solutions for the following fractional choquard equation with critical exponent s u f x u q 2 u g x u 2 s x y d y u 2 s 2 u in u 0 in r n where is a bounded domain in r n with smooth boundary s 0 1 0 n n 2 s 1 q 2 and 2 s 2 n n 2 s is the critical exponent in the sense of hardy littlewood sobolev inequality 0 is a parameter and f g r are continuous functions but may change sign on

finding best approximation pairs relative to a convex and prox regular set in a hilbert space we study the convergence of an iterative projection reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics there are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem nonconvexity and infeasibility the algorithm that we developed called relaxed averaged alternating reflections raar was designed primarily to address infeasibility though our strategy has advantages for nonconvex problems as well in the present work we investigate the asymptotic behavior of the raar algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a hilbert space with empty intersection and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox regular set also possibly nonintersecting the nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection to place the raar algorithm in context we develop parallel statements about the standard alternating projections algorithm and gradient descent all of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings

low rank positive semidefinite matrix recovery from corrupted rank one measurements we study the problem of estimating a low rank positive semidefinite psd matrix from a set of rank one measurements using sensing vectors composed of i i d standard gaussian entries which are possibly corrupted by arbitrary outliers this problem arises from applications such as phase retrieval covariance sketching quantum space tomography and power spectrum estimation we first propose a convex optimization algorithm that seeks the psd matrix with the minimum ell 1 norm of the observation residual the advantage of our algorithm is that it is free of parameters therefore eliminating the need for tuning parameters and allowing easy implementations we establish that with high probability a low rank psd matrix can be exactly recovered as soon as the number of measurements is large enough even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes moreover the recovery is also stable against bounded noise with the additional information of an upper bound of the rank of the psd matrix we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy

gradient descent with random initialization fast global convergence for nonconvex phase retrieval this paper considers the problem of solving systems of quadratic equations namely recovering an object of interest varvec x natural in mathbb r n from m quadratic equations samples y i varvec a i top varvec x natural 2 1 le i le m this problem also dubbed as phase retrieval spans multiple domains including physical sciences and machine learning we investigate the efficacy of gradient descent or wirtinger flow designed for the nonconvex least squares problem we prove that under gaussian designs gradient descent when randomly initialized yields an epsilon accurate solution in o big log n log 1 epsilon big iterations given nearly minimal samples thus achieving near optimal computational and sample complexities at once this provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval without the need of i carefully designed initialization ii sample splitting or iii sophisticated saddle point escaping schemes all of these are achieved by exploiting the statistical models in analyzing optimization algorithms via a leave one out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data

existence of parseval oblique duals of a frame sequence abstract we present a collection of necessary and sufficient conditions for the existence of a parseval oblique dual in an appropriate subspace of a frame sequence this result extends the previous results by han and larson antezana et al and han our proof shows that the problem is closely related with that of extending an operator with norm less than or equal to 1 to an isometry on a subspace

geometric interpretations of the relation between the image of the absolute conic and sphere images a spherical object has been introduced into camera calibration for several years through utilizing the properties of an image conic which is the projection of the occluding contour of a sphere in the perspective image however in literature only an algebraic interpretation was presented for the relation between the image of the absolute conic and sphere images in this paper we propose two geometric interpretations of this relation and two novel camera calibration methods using sphere images are derived from these geometric interpretations

identifiability in blind deconvolution with subspace or sparsity constraints blind deconvolution bd the resolution of a signal and a filter given their convolution arises in many applications without further constraints bd is ill posed in practice subspace or sparsity constraints have been imposed to reduce the search space and have shown some empirical success however the existing theoretical analysis on uniqueness in bd is rather limited in an effort to address the still open question we derive sufficient conditions under which two vectors can be uniquely identified from their circular convolution subject to subspace or sparsity constraints these sufficient conditions provide the first algebraic sample complexities for bd we first derive a sufficient condition that applies to almost all bases or frames for bd of vectors in mathbb c n with two subspace constraints of dimensions m 1 and m 2 the required sample complexity is n geq m 1 m 2 then we impose a sub band structure on one basis and derive a sufficient condition that involves a relaxed sample complexity n geq m 1 m 2 1 which we show to be optimal we present the extensions of these results to bd with sparsity constraints or mixed constraints with the sparsity level replacing the subspace dimension the cost for the unknown support in this case is an extra factor of 2 in the sample complexity

on lipschitz analysis and lipschitz synthesis for the phase retrieval problem in this paper we prove two results regarding reconstruction from magnitudes of frame coefficients the so called phase retrieval problem first we show that phase retrievability as an algebraic property implies that nonlinear maps are bi lipschitz with respect to appropriate metrics on the quotient space second we prove that reconstruction can be performed using lipschitz continuous maps specifically we show that when nonlinear analysis maps h r m are injective with x hx fki m 1 and x hx fki 2 m 1 where f1 fm is a frame for a hilbert space h and

quantum tomography under prior information we provide a detailed analysis of the question how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information we show that if the prior information restricts the possible states to a set of lower dimensionality then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states conversely we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the minkowski dimension of the set we apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints in particular we show that the minimal number of measurement outcomes povm elements necessary to identify all pure states in a d dimensional hilbert space is 4d 3 c d d for some c d in 1 2 and d being the number of ones appearing in the binary expansion of d 1

phase retrievable operator valued frames and representations of quantum channels abstract we examine some connections among phase retrievable not necessarily self adjoint operator valued frames projective group representation frames and representations of quantum channels we first present some characterizations of phase retrievable frames for general operator systems acting on both finite and infinite dimensional hilbert spaces which generalize the known results for vector valued frames fusion frames and frames of hermitian matrices for an irreducible projective unitary representation of a finite group the image system is automatically phase retrievable and moreover it is a point wisely tight operator valued frames we generalize this notion to more general operator valued frames and prove that point wise tight operator valued frames are exactly the ones that are right equivalent to operator valued tight frames for an operator system that represent a quantum channel we show that phase retrievability of the system is independent of the choices of the representations of the quantum channel

a semilinear elliptic system involving nonlinear boundary condition and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e involving nonlinear boundary condition and sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

the pohozaev identity for the fractional laplacian in this paper we prove the pohozaev identity for the semilinear dirichlet problem delta s u f u in omega u equiv0 in mathbb r n backslash omega here s in 0 1 s is the fractional laplacian in mathbb r n and is a bounded c1 1 domain to establish the identity we use among other things that if u is a bounded solution then u delta s omega is c up to the boundary where x dist x in the fractional pohozaev identity the function u delta s partial omega plays the role that u plays in the classical one surprisingly from a nonlocal problem we obtain an identity with a boundary term an integral over which is completely local as an application of our identity we deduce the nonexistence of nontrivial solutions in star shaped domains for supercritical nonlinearities

the brezis nirenberg result for the fractional laplacian the aim of this paper is to deal with the non local fractional counterpart of the laplace equation involving critical non linearities studied in the famous paper of brezis and nirenberg 1983 namely our model is the equation su u u 2 2u in u 0 in rn where s is the fractional laplace operator s 0 1 is an open bounded set of rn n 2s with lipschitz boundary 0 is a real parameter and 2 2n n 2s is a fractional critical sobolev exponent in this paper we first study the problem in a general framework indeed we consider the equation lku u u 2 2u f x u 0 in u 0 in rn where lk is a general non local integrodifferential operator of order s and f is a lower order perturbation of the critical power u 2 2u in this setting we prove an existence result through variational techniques then as a concrete example we derive a brezis nirenberg type result for our model equation that is we show that if 1 s is the first eigenvalue of the non local operator s with homogeneous dirichlet boundary datum then for any 0 1 s there exists a non trivial solution of the above model equation provided n 4s in this sense the present work may be seen as the extension of the classical brezis nirenberg result to the case of non local fractional operators

wandering vector multipliers for unitary groups a wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system into itself a special case of unitary system is a discrete unitary group we prove that for many and perhaps all discrete unitary groups the set of wandering vector multipliers is itself a group we completely characterize the wandering vector multipliers for abelian and icc unitary groups some characterizations of special wandering vector multipliers are obtained for other cases in particular there are simple characterizations for diagonal and permutation wandering vector multipliers similar results remain valid for irrational rotation unitary systems we also obtain some results concerning the wandering vector multipliers for those unitary systems which are the ordered products of two unitary groups there are applications to wavelet systems

geometric properties of central catadioptric line images and their application in calibration in central catadioptric systems lines in a scene are projected to conic curves in the image this work studies the geometry of the central catadioptric projection of lines and its use in calibration it is shown that the conic curves where the lines are mapped possess several projective invariant properties from these properties it follows that any central catadioptric system can be fully calibrated from an image of three or more lines the image of the absolute conic the relative pose between the camera and the mirror and the shape of the reflective surface can be recovered using a geometric construction based on the conic loci where the lines are projected this result is valid for any central catadioptric system and generalizes previous results for paracatadioptric sensors moreover it is proven that systems with a hyperbolic elliptical mirror can be calibrated from the image of two lines if both the shape and the pose of the mirror are known then two line images are enough to determine the image of the absolute conic encoding the camera s intrinsic parameters the sensitivity to errors is evaluated and the approach is used to calibrate a real camera

singular value estimates of oblique projections abstract let w and m be two finite dimensional subspaces of a hilbert space h such that h w m and let p w m denote the oblique projection with range w and nullspace m in this article we get the following formula for the singular values of p w m 2 s k p w m 1 min f h x w m 2 where the minimum is taken over the set of all operator pairs f h on h such that r f w r h m and fh p w m and k 1 dim w we also characterize all the pairs where the minimum is attained

the existence of tight gabor duals for gabor frames and subspace gabor frames abstract let k and l be two full rank lattices in r d we give a complete characterization for all the gabor frames that admit tight dual of the same type the characterization is given in terms of the center valued trace of the von neumann algebra generated by the left regular projective unitary representations associated with the time frequency lattice k l two applications of this characterization were obtained i we are able to prove that every gabor frame has a tight dual if and only if the volume of k l is less than or equal to 1 2 ii we are able to obtain sufficient or necessary conditions for the existence of tight gabor pseudo duals for subspace gabor frames in various cases in particular we prove that every subspace gabor frame has a tight gabor pseudo dual if either the volume v k l 1 2 or v k l 2 moreover if k z d l z d with 1 then a subspace gabor frame g g l k has a tight gabor pseudo dual only when g g l k itself is already tight

identical projective geometric properties of central catadioptric line images and sphere images with applications to calibration central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint lines and spheres in space are all projected into conics in the central catadioptric image planes and such conics are called line images and sphere images respectively we discovered that there exists an imaginary conic in the central catadioptric image planes defined as the modified image of the absolute conic miac and by utilizing the miac the novel identical projective geometric properties of line images and sphere images may be exploited each line image or each sphere image is double contact with the miac which is an analogy of the discovery in pinhole camera that the image of the absolute conic iac is double contact with sphere images note that the iac also exists in the central catadioptric image plane but it does not have the double contact properties with line images or sphere images this is the main reason to propose the miac from these geometric properties with the miac two linear calibration methods for central catadioptric cameras using sphere images as well as using line images are proposed in the same framework note that there are many linear approaches to central catadioptric camera calibration using line images it seems that to use the properties that line images are tangent to the miac only leads to an alternative geometric construction for calibration however for sphere images there are only some nonlinear calibration methods in literature therefore to propose linear methods for sphere images may be the main contribution of this paper our new algorithms have been tested in extensive experiments with respect to noise sensitivity

phase retrieval using alternating minimization phase retrieval problems involve solving linear equations but with missing sign or phase for complex numbers information more than four decades after it was first proposed the seminal error reduction algorithm of gerchberg and saxton and fienup is still the popular choice for solving many variants of this problem the algorithm is based on alternating minimization i e it alternates between estimating the missing phase information and the candidate solution despite its wide usage in practice no global convergence guarantees for this algorithm are known in this paper we show that a resampling variant of this approach converges geometrically to the solution of one such problem finding a vector bf x from bf y bf a where bf y vert bf a t bf x vert and vert bf z vert denotes a vector of element wise magnitudes of bf z under the assumption that bf a is gaussian empirically we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem which are based on lifting to a convex matrix problem in sample complexity and robustness to noise however it is much more efficient and can scale to large problems analytically for a resampling version of alternating minimization we show geometric convergence to the solution and sample complexity that is off by log factors from obvious lower bounds we also establish close to optimal scaling for the case when the unknown vector is sparse our work represents the first theoretical guarantee for alternating minimization albeit with resampling for any variant of phase retrieval problems in the non convex setting

solving phaselift by low rank riemannian optimization methods for complex semidefinite constraints a framework phaselift was recently proposed to solve the phase retrieval problem in this framework the problem is solved by optimizing a cost function over the set of complex hermitian positive semidefinite matrices this approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite hermitian matrices where the desired minimizers are known to have low rank this paper considers an approach based on an alternative cost function defined on a union of appropriate manifolds it is related to the original cost function in a manner that preserves the ability to find a global minimizer and is significantly more efficient computationally a rank based optimality condition for stationary points is given and optimization algorithms based on state of the art riemannian optimization and dynamically reducing rank are proposed empirical evaluations are performed using the phaselift problem the new approach is shown to be an effective method of phase retrieval with

structure from motion with wide circular field of view cameras this paper presents a method for fully automatic and robust estimation of two view geometry autocalibration and 3d metric reconstruction from point correspondences in images taken by cameras with wide circular field of view we focus on cameras which have more than 180deg field of view and for which the standard perspective camera model is not sufficient e g the cameras equipped with circular fish eye lenses nikon fc e8 183deg sigma 8 mm f4 ex 180deg or with curved conical mirrors we assume a circular field of view and axially symmetric image projection to autocalibrate the cameras many wide field of view cameras can still be modeled by the central projection followed by a nonlinear image mapping examples are the above mentioned fish eye lenses and properly assembled catadioptric cameras with conical mirrors we show that epipolar geometry of these cameras can be estimated from a small number of correspondences by solving a polynomial eigenvalue problem this allows the use of efficient ransac robust estimation to find the image projection model the epipolar geometry and the selection of true point correspondences from tentative correspondences contaminated by mismatches real catadioptric cameras are often slightly noncentral we show that the proposed autocalibration with approximate central models is usually good enough to get correct point correspondences which can be used with accurate noncentral models in a bundle adjustment to obtain accurate 3d scene reconstruction noncentral camera models are dealt with and results are shown for catadioptric cameras with parabolic and spherical mirrors

best constants for sobolev inequalities for higher order fractional derivatives we obtain sharp constants for sobolev inequalities for higher order fractional derivatives as an application we give a new proof of a theorem of w beckner concerning conformally invariant higher order differential operators on the sphere

frame vector multipliers for finite group representations abstract a frame vector or generator for a group representation of a countable or finite group g on a hilbert space h is a vector h such that g g g is a parseval frame for h frame vector multipliers are the unitary operators on h that map frame vectors to frame vectors based on a characterization of frame vectors with respect to the standard decomposition of a group representation as the direct sums of irreducible subrepresentations with multiplicity we obtain explicit characterizations of frame generator multipliers for two basic cases for finite group representations with the help of these characterizations we obtain some necessary conditions of frame vector multipliers for general frame representations and present several examples to demonstrate how these results can be used to get explicit characterizations for frame vector multipliers

automatic line extraction in uncalibrated omnidirectional cameras with revolution symmetry revolution symmetry is a realistic assumption for modelling the majority of catadioptric and dioptric cameras in central systems it can be described by a projection model based on radially symmetric distortion in these systems straight lines are projected on curves called line images these curves have in general more than two degrees of freedom and their shape strongly depends on the particular camera configuration therefore the existing line extraction methods for this kind of omnidirectional cameras require the camera calibration by contrast with the perspective case where the calibration is not involved in the shape of the projected line image however this drawback can be considered as an advantage because the shape of the line images can be used for self calibration in this paper we present a novel method to extract line images in uncalibrated omnidirectional images which is valid for radially symmetric central systems in this method we propose using the plumb line constraint to find closed form solutions for different types of camera systems dioptric or catadioptric the inputs of the proposed method are points belonging to the line images and their intensity gradient the gradient information allows to reduce the number of points needed in the minimal solution improving the result and the robustness of the estimation the scheme is used in a line image extraction algorithm to obtain lines from uncalibrated omnidirectional images without any assumption about the scene the algorithm is evaluated with synthetic and real images showing good performance the results of this work have been implemented in an open source matlab toolbox for evaluation and research purposes

the subdifferential of measurable composite max integrands and smoothing approximation the subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization it is known that for clarke regular integrands the clarke subdifferential of the expectation equals the expectation of their clarke subdifferential in particular this holds for convex integrands however little is known about the calculation of clarke subgradients for the expectation of non regular integrands the focus of this contribution is to approximate clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand a framework for how to proceed along this path is developed and then applied to a class of measurable composite max integrands this class contains non regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning

excitation retrieval of microwave linear arrays from phaseless far field data a methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed the approach combines tools from convex optimization to solve the phase retrieval problem and a simple measurement procedure to mitigate the non uniqueness of the solution numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed they show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem in addition the proposed approach that only calls for readily available routines is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases

nonmetric calibration of wide angle lenses and polycameras images taken with wide angle cameras tend to have severe distortions which pull points towards the optical center this paper proposes a simple method for recovering the distortion parameters without the use of any calibration objects since distortions cause straight lines in the scene to appear as curves in the image our algorithm seeks to find the distortion parameters that map the image curves to straight lines the user selects a small set of points along the image curves recovery of the distortion parameters is formulated as the minimization of an objective function which is designed to explicitly account for noise in the selected image points experimental results are presented for synthetic data as well as real images we also present the idea of a polycamera which is defined as a tightly packed camera cluster possible configurations are proposed to capture very large fields of view such camera clusters tend to have a nonsingle viewpoint we therefore provide analysis of what we call the minimum working distance for such clusters finally we present results for a polycamera consisting of four wide angle sensors having a minimum working distance of about 4 m on undistorting the acquired images using our proposed technique we create real time high resolution panoramas