the q numerical range of a reducible matrix via a normal operator abstract let a be an n n complex matrix and 0 q 1 the q numerical range of a is the set denoted and defined by w q a x ay x y c n x y 1 x y q we show that the q numerical range of a reducible 3 3 matrix is determined by the q numerical range of the normal operator tf z zf z f l 2 d x d y for some compact convex set the result provides a performable algorithm to compute the boundary of the q numerical range of a reducible 3 3 matrix an example is also given to illustrate the detail of computations of the boundary of the range

cortical thickness analysis in autism with heat kernel smoothing we present a novel data smoothing and analysis framework for cortical thickness data defined on the brain cortical manifold gaussian kernel smoothing which weights neighboring observations according to their 3d euclidean distance has been widely used in 3d brain images to increase the signal to noise ratio when the observations lie on a convoluted brain surface however it is more natural to assign the weights based on the geodesic distance along the surface we therefore develop a framework for geodesic distance based kernel smoothing and statistical analysis on the cortical manifolds as an illustration we apply our methods in detecting the regions of abnormal cortical thickness in 16 high functioning autistic children via random field based multiple comparison correction that utilizes the new smoothing technique

statistics on diffeomorphisms via tangent space representations in this paper we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations the space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied specializing to the landmark matching problem of computational anatomy we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis pca in this setting with three dimensional face and hippocampus databases

inverse numerical range and determinantal representation abstract the numerical range w a of an n n matrix a is the collection of quadratic forms z a over unit sphere 1 the inverse numerical range problem aims to find a unit vector which corresponds to a given point z of the numerical range w a given a matrix a the helton vinnikov theorem produces a symmetric matrix s so that a and s have the same numerical range w a w s in this paper we investigate the inverse numerical range problem for the boundary points and points on the boundary generating curve of the numerical range in place of the construction of unit vectors satisfying z a we express the kernel vector function of the linear pencil x s y s z i n as a function on the abel jacobi variety of the associated elliptic curve of a the kernel function plays a key role for the inverse numerical problem we perform this process when s is a generic 3 3 symmetric matrix

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

parametric surface diffeomorphometry for low dimensional embeddings of dense segmentations and imagery in the field of computational anatomy biological form including our focus neuroanatomy is studied quantitatively through the action of the diffeomorphism group on example anatomies a technique called diffeomorphometry here we design an algorithm within this framework to pass from dense objects common in neuromaging studies binary segmentations structural images to a sparse representation defined on the surface boundaries of anatomical structures and embedded into the low dimensional coordinates of a parametric model our main new contribution is to introduce an expanded group action to simultaneously deform surfaces through direct mapping of points as well as images through functional composition with the inverse this allows us to index the diffeomorphisms with respect to two dimensional surface geometries like subcortical gray matter structures but explicitly map onto cost functions determined by noisy 3 dimensional measurements we consider models generated from empirical covariance of training data as well as bandlimited laplace beltrami eigenfunction models when no such data is available we show applications to noisy or anomalous segmentations and other typical problems in neuroimaging studies we reproduce statistical results detecting changes in alzheimer s disease despite dimensionality reduction lastly we apply our algorithm to the common problem of segmenting subcortical structures from t1 mr images

time sequence diffeomorphic metric mapping and parallel transport track time dependent shape changes abstract serial mri human brain scans have facilitated the detection of brain development and of the earliest signs of neuropsychiatric and neurodegenerative diseases monitoring disease progression and resolving drug effects in clinical trials for preventing or slowing the rate of brain degeneration to track anatomical shape changes in serial images we introduce new point based time sequence large deformation diffeomorphic metric mapping ts lddmm to infer the time flow of within subject geometric shape changes that carry known observations through a period its euler lagrange equation is generalized for anatomies whose shapes are characterized by point sets such as landmarks curves and surfaces the time dependent momentum obtained from the ts lddmm encodes within subject shape changes for the purpose of across subject shape comparison we then propose a diffeomorphic analysis framework to translate within subject deformation in a global template without incorporating across subject anatomical variations via parallel transport technique the analysis involves the retraction of the within subject time dependent momentum along the ts lddmm trajectory from each time to the baseline the translation of the momentum in a global template and the reconstruction of the ts lddmm trajectory starting from the global template

flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices abstract let t be an n n matrix the davis wielandt shell of t is defined as the set w t t t x tx x t tx x c r x x 1 in this paper we discuss the existence of flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices

the generalized numerical range of a set of matrices abstract for a given set of n n matrices f we study the union of the c numerical ranges of the matrices in the set f denoted by w c f we obtain basic algebraic and topological properties of w c f and show that there are connections between the geometric properties of w c f and the algebraic properties of c and the matrices in f furthermore we consider the starshapedness and convexity of the set w c f in particular we show that if f is the convex hull of two matrices such that w c a and w c b are convex then the set w c f is star shaped we also investigate the extensions of the results to the joint c numerical range of an m tuple of matrices

unitary similarity invariant function preservers of skew products of operators abstract let b h denote the banach algebra of all bounded linear operators on a complex hilbert space h with dim h 3 and let a and b be subsets of b h which contain all rank one operators suppose f is a unitary invariant norm the pseudo spectra the pseudo spectral radius the c numerical range or the c numerical radius for some finite rank operator c the structure is determined for surjective maps a b satisfying f a b f a b for all a b a to establish the proofs some general results are obtained for functions f f 1 h 0 0 where f 1 h is the set of rank one operators in b h satisfying a f u a u f a for a complex unit a f 1 h and unitary u b h b for any rank one operator x f 1 h the map t f t x on 0 is strictly increasing and c the set f x x f 1 h and x 1 attains its maximum and minimum

volumetric transformation of brain anatomy presents diffeomorphic transformations of three dimensional 3 d anatomical image data of the macaque occipital lobe and whole brain cryosection imagery and of deep brain structures in human brains as imaged via magnetic resonance imagery these transformations are generated in a hierarchical manner accommodating both global and local anatomical detail the initial low dimensional registration is accomplished by constraining the transformation to be in a low dimensional basis the basis is defined by the green s function of the elasticity operator placed at predefined locations in the anatomy and the eigenfunctions of the elasticity operator the high dimensional large deformations are vector fields generated via the mismatch between the template and target image volumes constrained to be the solution of a navier stokes fluid model as part of this procedure the jacobian of the transformation is tracked insuring the generation of diffeomorphisms it is shown that transformations constrained by quadratic regularization methods such as the laplacian biharmonic and linear elasticity models do not ensure that the transformation maintains topology and therefore must only be used for coarse global registration

the decomposable numerical radius and numerical radius of a compound matrix let a be an n n complex matrix with singular values 1 2 n and eigenvalues 1 2 n where 1 2 n denote by cm a 1 m n the mth compound of a and by mcn the mth grassmann space over cn in which the elements are regarded as complex row vectors with nm coordinates we have the relation i 1m i rd cm a r cm a i 1m i where rd cm a max xcm a x x is decomposable in mcn xx 1 and rd cm a max xcm a x x mcn xx 1 are the decomposable numerical radius and numerical radius of cm a respectively in this note we classify those matrices for which mi 1 i rd cm a m i 1 i r cm a rd cm a r cm a etc some of these results answer the questions raised by marcus and andresen concerning the decomposable numerical radius of a complex matrix

three observations on the determinantal range abstract let a c m n the algebra of n n complex matrices the set of complex numbers c a det a ucu u u i n is the c determinantal range of a in this note it is proved that c a is an elliptical disc for a c m 2 a necessary and sufficient condition for c a to be a line segment is given when a and c are normal matrices with pairwise distinct eigenvalues the linear operators l that satisfy the linear preserver property c a c l a for all a c m n are characterized

geodesic estimation for large deformation anatomical shape averaging and interpolation the goal of this research is to promote variational methods for anatomical averaging that operate within the space of the underlying image registration problem this approach is effective when using the large deformation viscous framework where linear averaging is not valid or in the elastic case the theory behind this novel atlas building algorithm is similar to the traditional pairwise registration problem but with single image forces replaced by average forces these group forces drive an average transport ordinary differential equation allowing one to estimate the geodesic that moves an image toward the mean shape configuration this model gives large deformation atlases that are optimal with respect to the shape manifold as defined by the data and the image registration assumptions we use the techniques in the large deformation context here but they also pertain to small deformation atlas construction furthermore a natural inherently inverse consistent image registration is gained for free as is a tool for constant arc length geodesic shape interpolation the geodesic atlas creation algorithm is quantitatively compared to the euclidean anatomical average to elucidate the need for optimized atlases the procedures generate improved average representations of highly variable anatomy from distinct populations

geometric atlas modeling the cortex as an organized surface recent atlases of the cortical surface are based on a modelization of the cerebral cortex as a topological sphere this captures effectively its organization as a regular bidimensional sheet of layers parallel to the surface and with perpendicular cortical columns yet while in the vertical direction cortices are almost the same throughout phylia in the sense of its surface the cerebral cortex is one of the most variable and distinctive parts of the nervous system indeed gyri and sulci appear to have a crucial organizing role in an architectonic connectional and functional sense this organization is not explicitly captured by the surface model of the cortex we propose a geometric model of the cortical anatomy based on flat representations of principal sulci obtained from surface reconstructions of mri data and on neuroanatomical and theoretical considerations concerning the folding patterns of the cortex the cortex is modeled by a sphere where primary sulci are included as axes the arrangement of the axes is a simplification of the arrangement of principal sulci observed in flat stereographic representations of the whole cortical surface the position of secondary and tertiary sulci is then defined by a field of orientations parallel and orthogonal to the axes we consider the use of the geometric model as a synthetic reference cortex for addressing reconstructions of cortical surfaces we present a method which establishes a bijection between the geometric model and a cortical surface reconstruction by using the axes of the model as boundary conditions for a set of partial differential equations solved over both surfaces using the geometric model as atlas provides a natural parameterization of the cortical surface that unlike angular coordinates allows for a localization based on the surface distance to its main organizing landmarks and folding patterns

multi contrast multi scale surface registration for improved alignment of cortical areas the position of cortical areas can be approximately predicted from cortical surface folding patterns however there is extensive inter subject variability in cortical folding patterns prohibiting a one to one mapping of cortical folds in certain areas in addition the relationship between cortical area boundaries and the shape of the cortex is variable and weaker for higher order cortical areas current surface registration techniques align cortical folding patterns using sulcal landmarks or cortical curvature for instance the alignment of cortical areas by these techniques is thus inherently limited by the sole use of geometric similarity metrics magnetic resonance imaging t1 maps show intra cortical contrast that reflects myelin content and thus can be used to improve the alignment of cortical areas in this article we present a new symmetric diffeomorphic multi contrast multi scale surface registration mmsr technique that works with partially inflated surfaces in the level set framework mmsr generates a more precise alignment of cortical surface curvature in comparison to two widely recognized surface registration algorithms the resulting overlap in gyrus labels is comparable to freesurfer most importantly mmsr improves the alignment of cortical areas further by including t1 maps as a first application we present a group average t1 map at a uniquely high resolution and multiple cortical depths which reflects the myeloarchitecture of the cortex mmsr can also be applied to other mr contrasts such as functional and connectivity data

whole brain diffeomorphic metric mapping via integration of sulcal and gyral curves cortical surfaces and images this paper introduces a novel large deformation diffeomorphic metric mapping algorithm for whole brain registration where sulcal and gyral curves cortical surfaces and intensity images are simultaneously carried from one subject to another through a flow of diffeomorphisms to the best of our knowledge this is the first time that the diffeomorphic metric from one brain to another is derived in a shape space of intensity images and point sets such as curves and surfaces in a unified manner we describe the euler lagrange equation associated with this algorithm with respect to momentum a linear transformation of the velocity vector field of the diffeomorphic flow the numerical implementation for solving this variational problem which involves large scale kernel convolution in an irregular grid is made feasible by introducing a class of computationally friendly kernels we apply this algorithm to align magnetic resonance brain data our whole brain mapping results show that our algorithm outperforms the image based lddmm algorithm in terms of the mapping accuracy of gyral sulcal curves sulcal regions and cortical and subcortical segmentation moreover our algorithm provides better whole brain alignment than combined volumetric and surface registration postelnicu et al 2009 and hierarchical attribute matching mechanism for elastic registration hammer shen and davatzikos 2002 in terms of cortical and subcortical volume segmentation

large deformation diffeomorphic metric curve mapping we present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping lddmm scheme for computing an optimal transformation between two curves embedded in euclidean space d curves are first represented as vector valued measures which incorporate both location and the first order geometric structure of the curves then a hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves we describe a discretized version of this in which discrete sequences of points along the curve are represented by vector valued functionals this gives a convenient and practical way to define a matching functional for curves we derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in 2 and 3 behaviors of the curve mapping are discussed using 2d curves the applications to shape classification is shown and experiments with 3d curves extracted from brain cortical surfaces are presented

symmetry properties of higher numerical ranges abstract let a be a linear operator on a finite dimensional unitary space v of dimension n the k th higher numerical range of a denoted by w k a is the totality of complex numbers tr pap where p runs over all k dimensional orthogonal projections on v we show that w k a is a polygon with the real axis as a line of symmetry k 1 n if and only if a is normal with a real characteristic polynomial we also construct several nonnormal examples to investigate the extent to which the symmetry of all of the w k a is required

mixed effects models with censored data with application to hiv rna levels summary mixed effects models are often used for estimating fixed effects and variance components in longitudinal studies of continuous data when the outcome being modelled is a laboratory measurement however it may be subject to lower and upper detection limits i e censoring in this paper the usual em estimation procedure for mixed effects models is modified to account for left and or right censoring

on the c numerical range of a matrix abstract given two n n complex matrices c and t we prove that if the differentiable mapping q u n c r 2 defined by q u tr cu tu is of rank at most 1 on a nonempty open set then the c numerical range w c t of t is a line segment the same conclusion holds whenever the interior of w c t is empty

preservers of unitary similarity functions on lie products of matrices abstract denote by m n the set of n n complex matrices let f m n 0 be a continuous map such that f u a u f a for any complex unit a m n and unitary u m n f x 0 if and only if x 0 and the induced map t f t x is monotonically increasing on 0 for any rank one nilpotent x m n characterization is given for surjective maps on m n satisfying f a b b a f a b b a the general theorem is then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius

the cone of nonnegative c numerical range and its preservers abstract let h be a complex hilbert space and b h the banach space of all bounded linear operators on h let c c 1 c k where c 1 c k 0 for each a b h the c numerical range of a is the set w c a j 1 k c j a x j x j x 1 x k is an orthonormal set in h in this note the cone p c a b h w c a 0 of operators with nonnegative c numerical ranges is considered we first determine the extreme directions of p c then we show that all bijective linear mappings t b h b h such that t p c p c are positive multiples of jordan isomorphisms as an application the result is used to obtain a description of surjective mappings on b h that preserve the c numerical radius distance

hammer hierarchical attribute matching mechanism for elastic registration a new approach is presented for elastic registration of medical images and is applied to magnetic resonance images of the brain experimental results demonstrate very high accuracy in superposition of images from different subjects there are two major novelties in the proposed algorithm first it uses an attribute vector i e a set of geometric moment invariants gmis that are defined on each voxel in an image and are calculated from the tissue maps to reflect the underlying anatomy at different scales the attribute vector if rich enough can distinguish between different parts of an image which helps establish anatomical correspondences in the deformation procedure it also helps reduce local minima by reducing ambiguity in potential matches this is a fundamental deviation of our method referred to as the hierarchical attribute matching mechanism for elastic registration hammer from other volumetric deformation methods which are typically based on maximizing image similarity second in order to avoid being trapped by local minima i e suboptimal poor matches hammer uses a successive approximation of the energy function being optimized by lower dimensional smooth energy functions which are constructed to have significantly fewer local minima this is achieved by hierarchically selecting the driving features that have distinct attribute vectors thus drastically reducing ambiguity in finding correspondence a number of experiments demonstrate that the proposed algorithm results in accurate superposition of image data from individuals with significant anatomical differences

generalized linear mixed model for binary outcomes when covariates are subject to measurement errors and detection limits longitudinal measurement of biomarkers is important in determining risk factors for binary endpoints such as infection or disease however biomarkers are subject to measurement error and some are also subject to left censoring due to a lower limit of detection statistical methods to address these issues are few we herein propose a generalized linear mixed model and estimate the model parameters using the monte carlo newton raphson mcnr method inferences regarding the parameters are made by applying louis s method and the delta method simulation studies were conducted to compare the proposed mcnr method with existing methods including the maximum likelihood ml method and the ad hoc approach of replacing the left censored values with half of the detection limit hdl the results showed that the performance of the mcnr method is superior to ml and hdl with respect to the empirical standard error as well as the coverage probability for the 95 confidence interval the hdl method uses an incorrect imputation method and the computation is constrained by the number of quadrature points while the ml method also suffers from the constrain for the number of quadrature points the mcnr method does not have this limitation and approximates the likelihood function better than the other methods the improvement of the mcnr method is further illustrated with real world data from a longitudinal study of local cervicovaginal hiv viral load and its effects on oncogenic hpv detection in hiv positive women

quantitative evaluation of lddmm freesurfer and caret for cortical surface mapping cortical surface mapping has been widely used to compensate for individual variability of cortical shape and topology in anatomical and functional studies while many surface mapping methods were proposed based on landmarks curves spherical or native cortical coordinates few studies have extensively and quantitatively evaluated surface mapping methods across different methodologies in this study we compared five cortical surface mapping algorithms including large deformation diffeomorphic metric mapping lddmm for curves lddmm curve for surfaces lddmm surface multi manifold lddmm mm lddmm freesurfer and caret using 40 mri scans and 10 simulated datasets we computed curve variation errors and surface alignment consistency for assessing the mapping accuracy of local cortical features e g gyral sulcal curves and sulcal regions and the curvature correlation for measuring the mapping accuracy in terms of overall cortical shape in addition the simulated datasets facilitated the investigation of mapping error distribution over the cortical surface when the mm lddmm freesurfer and caret mapping algorithms were applied our results revealed that the lddmm curve mm lddmm and caret approaches best aligned the local curve features with their own curves the mm lddmm approach was also found to be the best in aligning the local regions and cortical folding patterns e g curvature as compared to the other mapping approaches the simulation experiment showed that the mm lddmm mapping yielded less local and global deformation errors than the caret and freesurfer mappings

induced operators on symmetry classes of tensors let v be an n dimensional hilbert space suppose h is a subgroup of the symmetric group of degree m and x h c is a character of degree 1 on h consider the symmetrizer on the tensor space m v s v 1 v m 1 h v 1 1 x v 1 m defined by h and x the vector space m v m x h s v is a subspace of m v called the symmetry class of tensors over v associated with h and x the elements in v m x h of the form s v 1 v m are called decomposable tensors and are denoted by v 1 v m for any linear operator t acting on v there is a unique induced operator k t acting on v m x h satisfying k t v 1 v m tv 1 tv m in this paper several basic problems on induced operators are studied

perturbation of the q numerical radius of a weighted shift operator abstract we formulate the taylor series expansion for the q numerical radius of a weighted shift operator with periodic weights near q 0 coefficients up to the fourth order in the expansion are found via the perturbation theory of hermitian matrices

linear images of joint unitary orbits of hermitian matrices abstract let h n be the set of all n n hermitian matrices and h n m be the set of all m tuples of n n hermitian matrices for a a 1 a m h n m and for any linear map l h n m r l we define the l numerical range and the l numerical radius of a by w l a l u a 1 u u a m u u c n n u u i n and r l a max x x t x w l a respectively they can be regarded as a generalization of the c numerical range and the c numerical radius respectively of n n complex matrices in this paper interplay between the geometric properties of w l a and algebraic properties of linear maps l h n m r l and a h n m will be studied we shall give a characterization of l such that r l defines a norm on h n m furthermore we shall also characterize linear maps l h n m r l and a h n m in which w l a is a singleton set more generally the characterization of linear maps l h n m r l and a h n m in which w l a lies inside a k dimensional affine subspace of r l will be given

computing large deformation metric mappings via geodesic flows of diffeomorphisms this paper examine the euler lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in dupuis et al 1998 and trouve 1995 in which two images i 0 i 1 are given and connected via the diffeomorphic change of coordinates i 0 1 i 1 where 1 is the end point at t 1 of curve t t 0 1 satisfying t v t t t 0 1 with 0 id the variational problem takes the form mathop arg text m in limits upsilon dot phi t upsilon t left dot phi right left int 0 1 left upsilon t right 2 text d t left i 0 circ phi 1 1 i 1 right l 2 2 right where v t v is an appropriate sobolev norm on the velocity field v t and the second term enforces matching of the images with l 2 representing the squared error norm in this paper we derive the euler lagrange equations characterizing the minimizing vector fields v t t 0 1 assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms we describe the implementation of the euler equations using semi lagrangian method of computing particle flows and show the solutions for various examples as well we compute the metric distance on several anatomical configurations as measured by 0 1 v t v dt on the geodesic shortest paths

properties and preservers of the pseudospectrum abstract the interplay between the algebraic and analytic properties of a matrix and the geometric properties of its pseudospectrum is investigated it is shown that one can characterize hermitian matrices positive semi definite matrices orthogonal projections unitary matrices etc in terms of the pseudospectrum also characterizations are given to maps on matrices leaving invariant the pseudospectrum of the sum difference or product of matrix pairs it is shown that such a map is always a unitary similarity transform followed by some simple operations such as adding a constant matrix taking the matrix transpose or multiplying by a scalar in 1 1

dynamic programming generation of boundaries of local coordinatized submanifolds in the neocortex application to the planum temporale dynamic programming is used to define boundaries of cortical submanifolds with focus on the planum temporale pt of the superior temporal gyrus stg which has been implicated in a variety of neuropsychiatric disorders to this end automated methods are used to generate the pt manifold from 10 high resolution mri subvolumes roi masks encompassing the stg a procedure to define the subvolume roi masks from original mri brain scans is developed bayesian segmentation is then used to segment the subvolumes into cerebrospinal fluid gray matter gm and white matter wm 3d isocontouring using the intensity value at which there is equal probability of gm and wm is used to reconstruct the triangulated graph representing the stg cortical surface enabling principal curvature at each point on the graph to be computed dynamic programming is used to delineate the pt manifold by tracking principal curves from the retro insular end of the heschl s gyrus hg to the stg along the posterior stg up to the start of the ramus and back to the retro insular end of the hg a coordinate system is then defined on the pt manifold the origin is defined by the retro insular end of the hg and the y axis passes through the point on the posterior stg where the ramus begins automated labeling of gm in the stg is robust with l 1 distances between bayesian and manual segmentation in the range 0 001 0 12 n 20 pt reconstruction is also robust with 90 of the vertices of the reconstructed pt within about 1 voxel n 20 from semiautomated contours finally the reliability index based on interrater intraclass correlation for the surface area derived from repeated reconstructions is 0 96 for the left pt and 0 94 for the right pt thus demonstrating the robustness of dynamic programming in defining a coordinate system on the pt it provides a method with potential significance in the study of neuropsychiatric disorders

the numerical range of derivations abstract let p q n be integers satisfying 1 p q n the p q numerical range of an n n complex matrix a is defined by w p q a e p uau q u unitary where for an n n complex matrix x x q denotes its q q leading principal submatrix and e p x q denotes the p th elementary symmetric function of the eigenvalues of x q when 1 p q the set reduces to the classical numerical range of a which is well known to be convex many authors have used the concept of classical numerical range to study different classes of matrices in this note we extend the results to the generalized cases besides obtaining new results we collect existing ones and give alternative proofs for some of them we also study the p q numerical radius of a defined by r p q a max w p q a

the inverse q numerical range problem and connections to the davis wielandt shell and the pseudospectra of a matrix abstract numerical ranges and related sets provide insights into the behavior of algorithms involving matrices and inverse numerical range problems attempt to enhance these insights we generalize the inverse numerical range problem as proposed by uhlig to the inverse q numerical range problem and propose an algorithm for solving the problem that relies on convexity to determine an approximation to the boundary of the q numerical range as needed by our algorithm we must approximate the top of the davis wielandt shell a generalization of the numerical range we found that the davis wielandt shell is in a sense conjugate to the extreme singular values of the resolvent of a matrix knowing the davis wielandt shell allows for the approximation of the q numerical range the pseudospectra and the davis wielandt shell for any allowed mobius transformation of a matrix we provide some examples illustrating these connections as well as how to solve the inverse q numerical range problem

davis wielandt shell and q numerical range abstract we describe the boundary of the q numerical range of a square matrix using its davis wielandt shell the result is used to generate an algorithm for plotting the q numerical range of the square matrix computations of the q numerical ranges of a special class of matrices are explicitly given

the numerical range of linear pencils of 2 by 2 matrices abstract let a b m n the numerical range of a linear pencil a b is defined by w a b t c tw aw w bw 0 for some non zero w c n the numerical range of a linear pencil of n by n matrices coincides with the union of the numerical ranges of linear pencils of 2 by 2 matrices we emphasize on computation of the numerical range of linear pencils of 2 by 2 matrices let a b m 2 with constant diagonals we show that the boundary of w a b lies on a rational curve of degree at most 4 by using lagrange s multiplier and implicit function we describe the boundary of w a b for general 2 by 2 matices

the emerging discipline of computational functional anatomy computational functional anatomy cfa is the study of functional and physiological response variables in anatomical coordinates for this we focus on two things i the construction of bijections via diffeomorphisms between the coordinatized manifolds of human anatomy and ii the transfer group action and parallel transport of functional information into anatomical atlases via these bijections we review advances in the unification of the bijective comparison of anatomical submanifolds via point sets including points curves and surface triangulations as well as dense imagery we examine the transfer via these bijections of functional response variables into anatomical coordinates via group action on scalars and matrices in dti as well as parallel transport of metric information across multiple templates which preserves the inner product

the constrained bilinear form and the c numerical range abstract let v be an n dimentional unitary space with inner product and s the set x v x x 1 for any a hom v v and q c with q 1 we define w a q ax y x y s x y q if q 1 then w a q is just the classical numerical range ax x x s the convexity of which is well known another generalization of the numerical range is the c numerical range which is defined to be the set w c a tr cu au u unitary where c hom v v in this note we prove that w a q is always convex and that wc a is convex for all a if rank c 1 or n 2

maximum likelihood estimation of the multivariate normal mixture model the hessian of the multivariate normal mixture model is derived and estimators of the information matrix are obtained thus enabling consistent estimation of all parameters and their precisions the usefulness of the new theory is illustrated with two examples and some simulation experiments the newly proposed estimators appear to be superior to the existing ones

locally linear diffeomorphic metric embedding lldme for surface based anatomical shape modeling this paper presents the algorithm locally linear diffeomorphic metric embedding lldme for constructing efficient and compact representations of surface based brain shapes whose variations are characterized using large deformation diffeomorphic metric mapping lddmm our hypothesis is that the shape variations in the infinite dimensional diffeomorphic metric space can be captured by a low dimensional space to do so traditional locally linear embedding lle that reconstructs a data point from its neighbors in euclidean space is extended to lldme that requires interpolating a shape from its neighbors in the infinite dimensional diffeomorphic metric space this is made possible through the conservation law of momentum derived from lddmm it indicates that initial momentum a linear transformation of the initial velocity of diffeomorphic flows at a fixed template shape determines the geodesic connecting the template to a subject s shape in the diffeomorphic metric space and becomes the shape signature of an individual subject this leads to the compact linear representation of the nonlinear diffeomorphisms in terms of the initial momentum since the initial momentum is in a linear space a shape can be approximated by a linear combination of its neighbors in the diffeomorphic metric space in addition we provide efficient computations for the metric distance between two shapes through the first order approximation of the geodesic using the initial momentum as well as for the reconstruction of a shape given its low dimensional euclidean coordinates using the geodesic shooting with the initial momentum as the initial condition experiments are performed on the hippocampal shapes of 302 normal subjects across the whole life span 18 94years compared with principal component analysis and isomap lldme provides the most compact and efficient representation of the age related hippocampal shapes even though the hippocampal volumes among young adults are as variable as those in older adults lldme disentangles the hippocampal local shape variation from the hippocampal size and thus reveals the nonlinear relationship of the hippocampal morphometry with age

multi structure network shape analysis via normal surface momentum maps article i nfo we present a shape analysis pipeline for the assessment of anatomical variations in subcortical networks in mr images the shape analysis pipeline injects the global shape properties of the cfa subcortical template into the subcortical parcellations generated from freesurfer via large deformation diffeomorphic metric mapping lddmm examples are shown for this injection in several subcortical structures whose raw mr images were sampled from the database of open access series of imaging studies oasis the shape analysis is performed on random field representation of the template surface momentum maps that encode the shape variation of subcortical structure targets of each individual subject relative to the template the momentum maps have the optimum property that they are supported only on the boundary of the subcortical structures with the direction normal to the subcortical nuclei boundary thereby reducing the dimension of shape variation significantly a two level statistical model was built on these momentum maps to assess anatomical connectivity among the subcortical structures on the basis of similar surface deformation compression or expansion results in the study of healthy aging on the hippocampus amygdala network indicate the anatomical connectivity between the basolateral complex of the amygdala and the subiculum of the hippocampus on the basis of shape compression in healthy elders relative to young adults

diffeomorphic metric surface mapping in subregion of the superior temporal gyrus this paper describes the application of large deformation diffeomorphic metric mapping to cortical surfaces based on the shape and geometric properties of subregions of the superior temporal gyrus in the human brain the anatomical surfaces of the cortex are represented as triangulated meshes the diffeomorphic matching algorithm is implemented by defining a norm between the triangulated meshes based on the algorithms of vaillant and glaunes the diffeomorphic correspondence is defined as a flow of the extrinsic three dimensional coordinates containing the cortical surface that registers the initial and target geometry by minimizing the norm the methods are demonstrated in 40 high resolution mri cortical surfaces of planum temporale pt constructed from subsets of the superior temporal gyrus stg the effectiveness of the algorithm is demonstrated via the euclidean positional distance distance of normal vectors and curvature before and after the surface matching as well as the comparison with a landmark matching algorithm the results demonstrate that both the positional and shape variability of the anatomical configurations are being represented by the diffeomorphic maps

linear operators preserving the p q numerical range abstract let bd m be h n or c n n the real linear space of all n n hermitian matrices and the complex linear space of all n n complex matrices respectively for 1 p q n the p q numerical radius of a m is defined and denoted by r p q a e p x ax x d n q x x i q where ep y denotes the pth elementary symmetric function on the eigenvalues of y we characterize the linear operators on m that satisfy rp q a rp q a for all a m

geodesic regression on orientation distribution functions with its application to an aging study in this paper we treat orientation distribution functions odfs derived from high angular resolution diffusion imaging hardi as elements of a riemannian manifold and present a method for geodesic regression on this manifold in order to find the optimal regression model we pose this as a least squares problem involving the sum of squared geodesic distances between observed odfs and their model fitted data we derive the appropriate gradient terms and employ gradient descent to find the minimizer of this least squares optimization problem in addition we show how to perform statistical testing for determining the significance of the relationship between the manifold valued regressors and the real valued regressands experiments on both synthetic and real human data are presented in particular we examine aging effects on hardi via geodesic regression of odfs in normal adults aged 22 years old and above

trace a topological graph representation for automatic sulcal curve extraction a proper geometric representation of the cortical regions is a fundamental task for cortical shape analysis and landmark extraction however a significant challenge has arisen due to the highly variable convoluted cortical folding patterns in this paper we propose a novel topological graph representation for automatic sulcal curve extraction trace in practice the reconstructed surface suffers from noise influences introduced during image acquisition surface reconstruction in the presence of noise on the surface trace determines stable sulcal fundic regions by employing the line simplification method that prevents the sulcal folding pattern from being significantly smoothed out the sulcal curves are then traced over the connected graph in the determined regions by the dijkstra s shortest path algorithm for validation we used the state of the art surface reconstruction pipelines on a reproducibility data set the experimental results showed higher reproducibility and robustness to noise in trace than the existing method li et al 2010 with over 20 relative improvement in error for both surface reconstruction pipelines in addition the extracted sulcal curves by trace were well aligned with manually delineated primary sulcal curves we also provided a choice of parameters to control quality of the extracted sulcal curves and showed the influences of the parameter selection on the resulting curves

conditions for the generalized numerical range to be real abstract necessary and sufficient conditions are given for the c numerical range of a matrix a to be a subset of the real axis in particular it is shown that both a and c must be translates of hermitian matrices

multi manifold diffeomorphic metric mapping for aligning cortical hemispheric surfaces cortical surface based analysis has been widely used in anatomical and functional studies because it is geometrically appropriate for the cortex one of the main challenges in the cortical surface based analysis is to optimize the alignment of the cortical hemispheric surfaces across individuals in this paper we introduce a multi manifold large deformation diffeomorphic metric mapping mm lddmm algorithm that allows simultaneously carrying the cortical hemispheric surface and its sulcal curves from one to the other through a flow of diffeomorphisms we present an algorithm based on recent derivation of a law of momentum conservation for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point on curves and surfaces along the geodesic is completely determined by the momentum at the origin we solve for trajectories geodesics of the kinetic energy by computing its variation with respect to the initial momentum and by applying a gradient descent scheme the mm lddmm algorithm optimizes the initial momenta encoding the anatomical variation of each individual relative to a common coordinate system in a linear space which provides a natural scheme for shape deformation average and template or atlas generation we applied the mm lddmm algorithm for constructing the templates for the cortical surface and 14 sulcal curves of each hemisphere using a group of 40 subjects the estimated template shape reflects regions which are highly variable across these subjects compared with existing single manifold lddmm algorithms such as the lddmm curve mapping and the lddmm surface mapping the mm lddmm mapping provides better results in terms of surface to surface distances in five predefined regions

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

correlating two viral load assays with known detection limits summary a timely objective common to many hiv studies involves assessing the correlation between two different measures of viral load obtained from each of a sample of patients this correlation has scientific utility in a number of contexts including those aimed at a comparison of competing assays for quantifying virus and those aimed at determining the level of association between viral loads in two different reservoirs using the same assay a complication for the analyst seeking valid point and interval estimates of such a correlation is the fact that both variables may be subject to left censoring due to values below assay detection limits we address this problem using a bivariate normal likelihood that accounts for left censoring of two variables that may have different detection limits we provide simulation results to evaluate sampling properties of the resulting correlation estimator and compare it with ad hoc estimators in the presence of nondetects in an effort to obtain improved confidence interval properties relative to the wald approach we evaluate and compare profile likelihood based intervals we apply the methods to hiv viral load data on women and infants from a trial in bangkok thailand and we discuss an extension of the original model to accommodate interval censoring arising due to the study design

between the von neumann inequality and the crouzeix conjecture abstract a new concept of a deformed numerical range w t is introduced here t is a bounded linear operator or a matrix and 1 is a parameter each w t is a closed convex set that contains the spectrum of t furthermore w t is decreasing with respect to and w 2 t coincides with the numerical range it is also shown that w t is contained in the closed unit disc if and only if t has a unitary dilation in the sense of nagy foias the spectral constant of w t is investigated it is shown that it is monotone and continuous with respect to the parameter

surface fluid registration of conformal representation application to detect disease burden and genetic influence on hippocampus in this paper we develop a new automated surface registration system based on surface conformal parameterization by holomorphic 1 forms inverse consistent surface fluid registration and multivariate tensor based morphometry mtbm first we conformally map a surface onto a planar rectangle space with holomorphic 1 forms second we compute surface conformal representation by combining its local conformal factor and mean curvature and linearly scale the dynamic range of the conformal representation to form the feature image of the surface third we align the feature image with a chosen template image via the fluid image registration algorithm which has been extended into the curvilinear coordinates to adjust for the distortion introduced by surface parameterization the inverse consistent image registration algorithm is also incorporated in the system to jointly estimate the forward and inverse transformations between the study and template images this alignment induces a corresponding deformation on the surface we tested the system on alzheimer s disease neuroimaging initiative adni baseline dataset to study ad symptoms on hippocampus in our system by modeling a hippocampus as a 3d parametric surface we nonlinearly registered each surface with a selected template surface then we used mtbm to analyze the morphometry difference between diagnostic groups experimental results show that the new system has better performance than two publicly available subcortical surface registration tools first and spharm we also analyzed the genetic influence of the apolipoprotein e 4 allele apoe4 which is considered as the most prevalent risk factor for ad our work successfully detected statistically significant difference between apoe4 carriers and non carriers in both patients of mild cognitive impairment mci and healthy control subjects the results show evidence that the apoe genotype may be associated with accelerated brain atrophy so that our work provides a new mri analysis tool that may help presymptomatic ad research

decomposable numerical ranges on orthonormal tensors abstract let 1 m n and let h c be a degree 1 character on a subgroup h of the symmetric group of degree m the generalized matrix function on an m m matrix b b ij associated with is defined by d b h j 1 m b j j and the decomposable numerical range of an n n matrix a on orthonormal tensors associated with is defined by w a d x ax x is an n m matrix such that x x i m we study relations between the geometrical properties of w a and the algebraic properties of a and determine the structure of those linear operators l on n n complex matrices that satisfy w l a w a for all n n matrices a these results extend those of other researchers who treat the special cases of such as the principal or alternate character

maximum likelihood inference for left censored hiv rna data left censored data are characteristic of many bioassays due to inherent limit of detection and limit of quantitation loq in the assays this paper examines how the left censoring of plasma hiv rna measurements collected for the hemophilia growth and development study affects the quantification of viral load and the assessment of its association with a continuous or dichotomous outcome data analyses using maximum likelihood estimation are compared to analyses where the loq or loq 2 value is substituted for the left censored observations and also to other methods like multiple imputation a gaussian distribution is assumed for the log transformed plasma hiv rna data and simulations are used to explore the sensitivity of the results to changes in the model parameters the robustness of the estimators is also investigated when the data are generated from a mixture of two gaussian distributions maximum likelihood is in general the least biased method however multiple imputation assuming a censored gaussian imputation model and substituting the censored values with the expectation of its conditional predictive distribution are also competitive to maximum likelihood and may be appealing because of their simpler computational algorithms

a kernel class allowing for fast computations in shape spaces induced by diffeomorphisms reproducing kernel hilbert spaces play an important role in diffeomorphic matching of shapes and in which they intervene in the construction of riemannian metrics on diffeomorphisms and shape spaces in such contexts they are directly involved in the expressions of geodesic equations and in their numerical solutions via particle evolutions solving such equations however involves computing kernel sums over irregular grids which can be a big computational overhead if the number of particles is large in this paper we introduce and establish properties of a finitely generated kernel class in which the kernel is defined using a double interpolation from a discrete kernel supported by a regular grid covering the domain of the system of particles under consideration it not only speeds up the calculations by utilizing standard algorithms for faster computations over regular grids but also maintains the exactness and consistency of the system we provide experimental results in support of this comparing in particular the computation time and accuracy to similar competing methods

multi label segmentation of white matter structures application to neonatal brains accurate and consistent segmentation of brain white matter bundles at neonatal stage plays an important role in understanding brain development and detecting white matter abnormalities for the prediction of psychiatric disorders due to the complexity of white matter anatomy and the spatial resolution of diffusion weighted mr imaging multiple fiber bundles can pass through one voxel the goal of this study is to assign one or multiple anatomical labels of white matter bundles to each voxel to reflect complex white matter anatomy of the neonatal brain for this we develop a supervised multi label k nearest neighbor ml knn classification algorithm in riemannian diffusion tensor spaces our ml knn considers diffusion tensors lying on the log euclidean riemannian manifold of symmetric positive definite spd matrices and their corresponding vector space as feature space the ml knn utilizes the maximum a posteriori map principle to make the prediction of white matter labels by reasoning with the labeling information derived from the neighbors without assuming any probabilistic distribution of the features we show that our approach automatically learns the number of white matter bundles at a location and provides anatomical annotation of the neonatal white matter in addition our approach also provides the binary mask for individual white matter bundles to facilitate tract based statistical analysis in clinical studies we apply this method to automatically segment 13 white matter bundles of the neonatal brain and examine the segmentation accuracy against semi manual labels derived from tractography

tracial bounds for multilinear schur multipliers abstract we derive new types of two sided bounds for traces of multilinear schur multipliers that sharpen existing bounds in a number of situations we also characterize the tracial positivity of the n linear multipliers on symmetric tuples assuming their symbols are mildly sparse when n 5

diffeomorphic metric mapping of high angular resolution diffusion imaging based on riemannian structure of orientation distribution functions in this paper we propose a novel large deformation diffeomorphic registration algorithm to align high angular resolution diffusion images hardi characterized by orientation distribution functions odfs our proposed algorithm seeks an optimal diffeomorphism of large deformation between two odf fields in a spatial volume domain and at the same time locally reorients an odf in a manner such that it remains consistent with the surrounding anatomical structure to this end we first review the riemannian manifold of odfs we then define the reorientation of an odf when an affine transformation is applied and subsequently define the diffeomorphic group action to be applied on the odf based on this reorientation we incorporate the riemannian metric of odfs for quantifying the similarity of two hardi images into a variational problem defined under the large deformation diffeomorphic metric mapping framework we finally derive the gradient of the cost function in both riemannian spaces of diffeomorphisms and the odfs and present its numerical implementation both synthetic and real brain hardi data are used to illustrate the performance of our registration algorithm

maps preserving unitarily invariant norms of jordan product of matrices abstract surjective isometries in a given unitarily invariant norm on n by n complex matrices and with respect to jordan product are classified moreover the similar problem is considered for a much wider class of unitarily invariant functions

on generalized numerical range of the aluthge transformation abstract in this paper the authors show that the aluthge transformation t of a matrix t and a polynomial f satisfy the inclusion relation w c f t w c f t for the generalized numerical range if c is a hermitian matrix or a rank one matrix

isometries of a generalized numerical radius abstract for 0 q 1 the q numerical range is defined on the algebra m n of all n n complex matrices by w q a x ay x y c n x y 1 y x q the q numerical radius is defined by r q a max w q a we characterize isometries of the metric space m n r q i e the maps m n m n that satisfy r q a b r q a b we also characterize maps on m n that preserves the q numerical range the maps are not assumed to be linear

a fast diffeomorphic image registration algorithm abstract this paper describes dartel which is an algorithm for diffeomorphic image registration it is implemented for both 2d and 3d image registration and has been formulated to include an option for estimating inverse consistent deformations nonlinear registration is considered as a local optimisation problem which is solved using a levenberg marquardt strategy the necessary matrix solutions are obtained in reasonable time using a multigrid method a constant eulerian velocity framework is used which allows a rapid scaling and squaring method to be used in the computations dartel has been applied to intersubject registration of 471 whole brain images and the resulting deformations were evaluated in terms of how well they encode the shape information necessary to separate male and female subjects and to predict the ages of the subjects

estimation and inference on correlations between biomarkers with repeated measures and left censoring due to minimum detection levels statistical approaches for estimating and drawing inference on the correlation between two biomarkers that are repeatedly assessed over time and subject to left censoring because minimum detection levels are lacking we propose a linear mixed effects model and estimate the parameters with the monte carlo expectation maximization mcem method inferences regarding the model parameters and the correlation between the biomarkers are performed by applying louis s method and the delta method simulation studies were conducted to compare the proposed mcem method with existing methods including the maximum likelihood estimation method the multiple imputation method and two widely used ad hoc approaches replacing the censored values with the detection limit or with half of the detection limit the results show that the performance of the mcem with respect to relative bias and coverage probability for the 95 confidence interval is superior to the detection limit and half of the detection limit approaches and exceeds that of the multiple imputation method at medium to high levels of censoring and the standard error estimates from the mcem method are close to ideal the maximum likelihood estimation method can estimate the parameters accurately however a nonpositive definite information matrix can occur so that the variances are not estimable these five methods are illustrated with data from a longitudinal human immunodeficiency virus study to estimate and draw inference on the correlation between human immunodeficiency virus rna levels measured in plasma and in cervical secretions at multiple time points copyright 2012 john wiley sons ltd

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

on the c determinantal range for special classes of matrices let a and c be square complex matrices of size n the c determinantal range of a is the subset of the complex plane det a u c u u u i n if a c are both hermitian matrices then by a result of fiedler 1971 11 this set is a real line segment in our paper we study this set for the case when c is a hermitian matrix our purpose is to revisit and improve two well known results on this topic the first result is due to li concerning the c numerical range of a hermitian matrix see condition 5 1 a in li 1994 20 the second one is due to c k li y t poon and n s sze about necessary and sufficient conditions for the c determinantal range of a to be a subset of the line see li et al 2008 21 theorem 3 3

the q numerical radius of weighted shift operators with periodic weights we deal with the q numerical radius of weighted unilateral and bilateral shift operators in particular the q numerical radius of weighted shift operators with periodic weights is discussed and computed