a generalized depth estimation algorithm with a single image a depth estimation algorithm proposed by a p pentland 1987 is generalized in the proposed algorithm the raw image data in the vicinity of the edge is used to estimate the depth from defocus since no differentiation operation on the image data is required before the optimization process the method is less sensitive to the noise disturbance of measurements furthermore the edge orientation that was critical in pentland s approach will not be required in the case this algorithm is then applied to synthetic images containing various amounts of noise to test its performance experimental results indicate that the depth estimation errors are kept within 5 of true values on the average when it is applied to real images

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

multitask quantile regression under the transnormal model abstractwe consider estimating multitask quantile regression under the transnormal model with focus on high dimensional setting we derive a surprisingly simple closed form solution through rank based covariance regularization in particular we propose the rank based l1 penalization with positive definite constraints for estimating sparse covariance matrices and the rank based banded cholesky decomposition regularization for estimating banded precision matrices by taking advantage of the alternating direction method of multipliers nearest correlation matrix projection is introduced that inherits sampling properties of the unprojected one our work combines strengths of quantile regression and rank based covariance regularization to simultaneously deal with nonlinearity and nonnormality for high dimensional regression furthermore the proposed method strikes a good balance between robustness and efficiency achieves the oracle like convergence rate and provides the provable prediction interval und

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

shape from focus the shape from focus method presented here uses different focus levels to obtain a sequence of object images the sum modified laplacian sml operator is developed to provide local measures of the quality of image focus the operator is applied to the image sequence to determine a set of focus measures at each image point a depth estimation algorithm interpolates a small number of focus measure values to obtain accurate depth estimates a fully automated shape from focus system has been implemented using an optical microscope and tested on a variety of industrial samples experimental results are presented that demonstrate the accuracy and robustness of the proposed method these results suggest shape from focus to be an effective approach for a variety of challenging visual inspection tasks

an invariant volume mean value property if is harmonic and integrable over the open unit disc u then so is for every moebius transformation of u and therefore 1 u d a 0 for every conversely does this mean value property imply that is harmonic a more general question with the unit ball bn of c for arbitrary n 1 in place of the disc is investigated in the present paper the answer is found to be affirmative if n 11 negative if n 12

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

a moment preserving approach for depth from defocus abstract for range sensing using depth from defocus methods the distance d of a point object from the lens can be evaluated by the concise depth formula d p q d b where p and q are constants for a given camera setting and db is the diameter of the blur circle for the point object on the image detector plane the amount of defocus db is traditionally estimated from the spatial parameter of a gaussian point spread function using a complex iterative solution in this paper we use a straightforward and computationally fast method to estimate the amount of defocus from a single camera the observed gray level image is initially converted into a gradient image using the sobel edge operator for the edge point of interest the proportion of the blurred edge region pe in a small neighborhood window is then calculated using the moment preserving technique the value of pe increases as the amount of defocus increases and therefore is used as the description of degradation of the point spread function in addition to the use of the geometric depth formula for depth estimation artificial neural networks are also proposed in this study to compensate for the estimation errors from the depth formula experiments have shown promising results that the rms depth errors are within 5 for the depth formula and within 2 for the neural networks

estimating sparse precision matrix optimal rates of convergence and adaptive estimation precision matrix is of significant importance in a wide range of applications in multivariate analysis this paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting optimal rates of convergence are established for a range of matrix norm losses a fully data driven estimator based on adaptive constrained l1 minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces the estimator called aclime is easy to implement and performs well numerically a major step in establishing the minimax rate of convergence is the derivation of a rate sharp lower bound a two directional lower bound technique is applied to obtain the minimax lower bound the upper and lower bounds together yield the optimal rates of convergence for sparse precision matrix estimation and show that the aclime estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

non perturbative deformation quantization of cartan domains abstract we construct families of non commuting c algebras of quantized functions for bounded irreducible hermitian symmetric spaces for this procedure we use algebras of toeplitz operators defined with respect to a perturbation of the ordinary bergman metric we prove the deformation quantization conditions for these algebras

an investigation of methods for determining depth from focus the concept of depth from focus involves calculating distances to points in an observed scene by modeling the effect that the camera s focal parameters have on images acquired with a small depth of field this technique is passive and requires only a single camera the most difficult segment of calculating depth from focus is deconvolving the defocus operator from the scene and modeling it most current methods for determining the defocus operator employ inverse filtering the authors reveal some fundamental problems with inverse filtering inaccuracies in finding the frequency domain representation windowing effects and border effects a general matrix based method using regularization is presented which eliminates these problems the new method is confirmed experimentally with the results showing an rms error of 1 3

a forelli rudin construction and asymptotics of weighted bergman kernels let be a pseudoconvex domain in cn with smooth boundary two smooth defining functions for 0 such that log log are plurisubharmonic z a point at which log is strictly plurisubharmonic and m 0 an integer we show that as k the bergman kernels with respect to the weights k m have the asymptotic expansionk k m z z kn n z k z m j 0 bj z k j b0 det 2log zj zk for strongly pseudoconvex with real analytic boundary real analytic and log log strictly plurisubharmonic on we obtain also the analogous result for k k m x y for x y near the diagonal and discuss consequences for the asymptotics of the berezin transform and for the berezin quantization the proofs rely on fefferman s expansion for the bergman kernel in a certain forelli rudin type domain over as another application they also yield a generalization of the cited fefferman s expansion to a class of weighted bergman kernels

asymptotic normality and optimalities in estimation of large gaussian graphical models the gaussian graphical model a popular paradigm for studying relationship among variables in a wide range of applications has attracted great attention in recent years this paper considers a fundamental question when is it possible to estimate low dimensional parameters at parametric square root rate in a large gaussian graphical model a novel regression approach is proposed to obtain asymptotically efficient estimation of each entry of a precision matrix under a sparseness condition relative to the sample size when the precision matrix is not sufficiently sparse or equivalently the sample size is not sufficiently large a lower bound is established to show that it is no longer possible to achieve the parametric rate in the estimation of each entry this lower bound result which provides an answer to the delicate sample size question is established with a novel construction of a subset of sparse precision matrices in an application of le cam s lemma moreover the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved under a minimal sample requirement the proposed estimator is applied to test the presence of an edge in the gaussian graphical model or to recover the support of the entire model to obtain adaptive rate optimal estimation of the entire precision matrix as measured by the matrix ell q operator norm and to make inference in latent variables in the graphical model all of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum this significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix irrepresentability condition on the hessian tensor operator of the covariance matrix or the ell 1 constraint on the precision matrix numerical results confirm our theoretical findings the roc curve of the proposed algorithm asymptotic normal thresholding ant for support recovery significantly outperforms that of the popular glasso algorithm

integration of multiresolution image segmentation and neural networks for object depth recovery a novel technique for three dimensional depth recovery based on two coaxial defocused images of an object with added pattern illumination is presented the approach integrates object segmentation with depth estimation firstly segmentation is performed by a multiresolution based approach to isolate object regions from the background given the presence of blur and pattern illumination the segmentation has three sub procedures image pyramid formation linkage adaptation and unsupervised clustering these maximise the object recognition capability while ensuring accurate position information for depth estimation lower resolution information with a strong correlation to depth is fed into a three layered neural network as input feature vectors and processed using a back propagation algorithm the resulting depth model of object recovery is then used with higher resolution data to obtain high accuracy depth measurements experimental results are presented that show low error rates and the robustness of the model with respect to pattern variation and inaccuracy in optical settings

minimax estimation of large precision matrices with bandable cholesky factor last decade witnesses significant methodological and theoretical advances in estimating large precision matrices in particular there are scientific applications such as longitudinal data meteorology and spectroscopy in which the ordering of the variables can be interpreted through a bandable structure on the cholesky factor of the precision matrix however the minimax theory has still been largely unknown as opposed to the well established minimax results over the corresponding bandable covariance matrices in this thesis we focus on two commonly used types of parameter spaces and develop the optimal rates of convergence under both the operator norm and the frobenius norm a striking phenomenon is found two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the frobenius norm which is in sharp contrast to the equivalence of corresponding two types of bandable covariance matrices under both norms this fundamental difference is established by carefully constructing the corresponding minimax lower bounds two new estimation procedures are developed for the operator norm our optimal procedure is based on a novel local cropping estimator targeting on all principle submatrices of the precision matrix while for the frobenius norm our optimal procedure relies on a delicate regression based block thresholding rule lepski s method is considered to achieve optimal adaptation we further establish rate optimality in the nonparanormal model by applying our local cropping procedure to the rank based estimators numerical studies are carried out to confirm our theoretical findings

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

quantization of kahler manifolds ii we use berezin s dequantization procedure to define a formal product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous kahler manifold m we prove that this formal product isconvergent when m isa hermitian symmetric space 1993 american mathematical society

the berezin transform and invariant differential operators the berezin calculus is important to quantum mechanics creation annihilation operators and operator theory toeplitz operators we study the basic berezin transform linking the contravariant and covariant symbol for all bounded symmetric domains and express it in terms of invariant differential operators

rational filter design for depth from defocus the paper describes a new simple procedure to determine the rational filters that are used in the depth from defocus dfd procedure previously researched by watanabe and nayar 1998 4 their dfd uses two differently defocused images and the filters accurately model the relative defocus in the images and provide a fast calculation of distance this paper presents a simple method to determine the filter coefficients by separating the m p ratio into a linear and a cubic error correction model the method avoids the previous iterative minimisation technique and computes efficiently the model has been verified by comparison with the theoretical m p ratio the proposed filters have been compared with the previous for frequency response closeness of fit to m p rotational symmetry and measurement accuracy experiments were performed for several defocus conditions it was observed that the new filters were largely insensitive to object texture and modelled the blur more precisely than the previous experiments with real planar images demonstrated a maximum rms depth error of 1 18 for the proposed compared to 1 54 for the previous filters complicated objects were also accurately measured

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

functions invariant under the berezin transform we show that a bounded function f satisfies bf f where b is the berezin tranform on the unit disc defined in 2 below if and only if f is harmonic there is an equivalent formulation of this result s axler and cuckovic integral equations operator theory14 1991 1 12 w rudin function theory in the unit ball of cn springer verlag new york berlin 1980 if f is bounded and satifies the invariant version of the area mean value property then f is harmonic the main tool employed is fourier analysis on the lie group of all mobius transformations

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

weighted bergman kernels and quantization let be a bounded pseudoconvex domain in c n two positive functions on such that log log are plurisubharmonic and z a point at which log is smooth and strictly plurisubharmonic we show that as k the bergman kernels with respect to the weights k have an asymptotic expansion for x y near z where x y is an almost analytic extension of phi x x x and similarly for further if in addition is of finite type behave reasonably at the boundary and log log are strictly plurisubharmonic on we obtain also an analogous asymptotic expansion for the berezin transform and give applications to the berezin quantization finally for smoothly bounded and strictly pseudoconvex and a smooth strictly plurisubharmonic defining function for we also obtain results on the berezin toeplitz quantization

a new technique to extract range information from stereo images results are described for an algorithm that automatically registers two stereo images an exact area registration process which automatically removes all residual distortion due to the geometry of the stereo imaging system is demonstrated this process allows the match judgement to be based upon an analysis of the residual differences of structure and intensity existing after all differences due to geometrical distortion have been corrected the accuracy in general seems to be of the order of 1 of range since the average distance to the model was about 600 mm and the rms error in range was about 6 mm the fact that the maximum error was greater by a factor of five indicates that there were still some mismatched points and the procedures followed in the stereo ranging algorithm were not sufficient further modifications to the algorithm are needed

correspondence principle for the quantized annulus romanovski polynomials and morse potential abstract a general theory of quantization has been proposed by berezin see e g his survey in comm math phys 40 1975 153 174 in this paper we establish a weak form of the correspondence principle for the annulus quantized according to berezin more precisely we show that b i as 0 where i is the identity operator and b the berezin transform we consider also spectral analysis on the annulus in particular we express the eigenfunctions of the relevant laplacian in terms of certain romanovski polynomials finally we write down the expression for the analogue of the morse potential in this case we remark that similar considerations can be made in principle on any circular domain in the presence of a radial hermitian metric of constant curvature

optimal rates of convergence for estimating toeplitz covariance matrices toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging target detection speech recognition and communications systems in this paper we consider optimal estimation of large toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm the properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound the results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces the minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent gaussian scale model and through a careful selection of a finite collection of least favorable parameters in addition optimal rate of convergence for estimating the inverse of a toeplitz covariance matrix is also established

the nonparanormal semiparametric estimation of high dimensional undirected graphs recent methods for estimating sparse undirected graphs for real valued data in high dimensional problems rely heavily on the assumption of normality we show how to use a semiparametric gaussian copula or nonparanormal for high dimensional inference just as additive models extend linear models by replacing linear functions with a set of one dimensional smooth functions the nonparanormal extends the normal by transforming the variables by smooth functions we derive a method for estimating the nonparanormal study the method s theoretical properties and show that it works well in many examples

asymptotic behaviour of reproducing kernels of weighted bergman spaces let q be a domain in c f a nonnegative and g a positive function on q such that 1 g is locally bounded a2 the space of all holomorphic functions on q square integrable with respect to the measure f0gda where da is the 2n dimensional lebesgue measure and k x y the reproducing kernel for a2 it has been known for a long time that in some special situations such as on bounded symmetric domains q with g 1 and f the bergman kernel function the formula lim k x x 1 0 1 f x ae 00 holds true this fact even plays a crucial role in berezin s theory of quantization on curved phase spaces in this paper we discuss the validity of this formula in the general case the answer turns out to depend on loosely speaking how well the function log f can be approximated by certain pluriharmonic functions lying below it for instance holds if log f is convex and hence can be approximated from below by linear functions for any function g counterexamples are also given to show that in general may fail drastically or even be true for some x and fail for the remaining ones finally we also consider the question of convergence of k0 x y 1 0 for x y which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of f for instance if f is not real analytic at some point then kc x y must have zeroes for all at sufficiently large

depth from defocus a spatial domain approach a new method named stm is described for determining distance of objects and rapid autofocusing of camera systems stm uses image defocus information and is based on a new spatial domain convolution deconvolution transform the method requires only two images taken with different camera parameters such as lens position focal length and aperture diameter both images can be arbitrarily blurred and neither of them needs to be a focused image therefore stm is very fast in comparison with depth from focus methods which search for the lens position or focal length of best focus the method involves simple local operations and can be easily implemented in parallel to obtain the depth map of a scene stm has been implemented on an actual camera system named sparcs experiments on the performance of stm and their results on real world planar objects are presented the results indicate that the accuracy of stm compares well with depth from focus methods and is useful in practical applications the utility of the method is demonstrated for rapid autofocusing of electronic cameras