maximum likelihood reconstruction for magnetic resonance fingerprinting this paper introduces a statistical estimation framework for magnetic resonance mr fingerprinting a recently proposed quantitative imaging paradigm within this framework we present a maximum likelihood ml formalism to estimate multiple mr tissue parameter maps directly from highly undersampled noisy bf k space data a novel algorithm based on variable splitting the alternating direction method of multipliers and the variable projection method is developed to solve the resulting optimization problem representative results from both simulations and in vivo experiments demonstrate that the proposed approach yields significantly improved accuracy in parameter estimation compared to the conventional mr fingerprinting reconstruction moreover the proposed framework provides new theoretical insights into the conventional approach we show analytically that the conventional approach is an approximation to the ml reconstruction more precisely it is exactly equivalent to the first iteration of the proposed algorithm for the ml reconstruction provided that a gridding reconstruction is used as an initialization

branch and price algorithm for fuzzy integer programming problems with block angular structure this paper deals with fuzzy integer linear programming problems with block angular structure in which the fuzzy constraints are simplified by using possibility and necessity relations this main fuzzy problem is efficiently decomposed and is solved by a branch and price algorithm in the nodes of the branch and price tree the linear relaxation of the problem is solved by applying a column generation method also the relationship between the optimal solutions of this problem under possibility and necessity relations is derived to show the validation of the proposed algorithm some results are proved in addition the application of this algorithm is illustrated on fuzzy multicommodity flow problem for this case a new branching scheme is proposed to preserve the network structure of the subproblems which are produced in the column generation method some examples are solved and their results are compared with the previous works also the results of the proposed algorithm are reported on some large scale benchmark instances

direct estimation of the fiber orientation density function from diffusion weighted mri data using spherical deconvolution diffusion weighted magnetic resonance imaging can provide information related to the arrangement of white matter fibers the diffusion tensor is the model most commonly used to derive the orientation of the fibers within a voxel however this model has been shown to fail in regions containing several fiber populations with distinct orientations a number of alternative models have been suggested such as multiple tensor fitting q space and q ball imaging however each of these has inherent limitations in this study we propose a novel method for estimating the fiber orientation distribution directly from high angular resolution diffusion weighted mr data without the need for prior assumptions regarding the number of fiber populations present we assume that all white matter fiber bundles in the brain share identical diffusion characteristics thus implicitly assigning any differences in diffusion anisotropy to partial volume effects the diffusion weighted signal attenuation measured over the surface of a sphere can then be expressed as the convolution over the sphere of a response function the diffusion weighted attenuation profile for a typical fiber bundle with the fiber orientation density function odf the fiber odf the distribution of fiber orientations within the voxel can therefore be obtained using spherical deconvolution the properties of the technique are demonstrated using simulations and on data acquired from a volunteer using a standard 1 5 t clinical scanner the technique can recover the fiber odf in regions of multiple fiber crossing and holds promise for applications such as tractography

multi tissue constrained spherical deconvolution for improved analysis of multi shell diffusion mri data constrained spherical deconvolution csd has become one of the most widely used methods to extract white matter wm fibre orientation information from diffusion weighted mri dw mri data overcoming the crossing fibre limitations inherent in the diffusion tensor model it is routinely used to obtain high quality fibre orientation distribution function fodf estimates and fibre tractograms and is increasingly used to obtain apparent fibre density afd measures unfortunately csd typically only supports data acquired on a single shell in q space with multi shell data becoming more and more prevalent there is a growing need for csd to fully support such data furthermore csd can only provide high quality fodf estimates in voxels containing wm only in voxels containing other tissue types such as grey matter gm and cerebrospinal fluid csf the wm response function may no longer be appropriate and spherical deconvolution produces unreliable noisy fodf estimates the aim of this study is to incorporate support for multi shell data into the csd approach as well as to exploit the unique b value dependencies of the different tissue types to estimate a multi tissue odf the resulting approach is dubbed multi shell multi tissue csd msmt csd and is compared to the state of the art single shell single tissue csd ssst csd approach using both simulations and real data we show that msmt csd can produce reliable wm gm csf volume fraction maps directly from the dw data whereas ssst csd has a tendency to overestimate the wm volume in voxels containing gm and or csf in addition compared to ssst csd msmt csd can substantially increase the precision of the fodf fibre orientations and reduce the presence of spurious fodf peaks in voxels containing gm and or csf both effects translate into more reliable afd measures and tractography results with msmt csd compared to ssst csd

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

reversible joint hilbert and linear canonical transform without distortion generalized analytic signal associated with the linear canonical transform lct was proposed recently by fu and li generalized analytic signal associated with linear canonical transform opt commun vol 281 pp 1468 1472 2008 however most real signals especially for baseband real signals cannot be perfectly recovered from their generalized analytic signals therefore in this paper the conventional hilbert transform ht and analytic signal associated with the lct are concerned to transform a real signal into the lct of its ht two integral transforms i e the ht and lct are required the goal of this paper is to simplify cascades of multiple integral transforms which may be the ht analytic signal lct or inverse lct the proposed transforms can reduce the complexity when realizing the relationships among the following six kinds of signals a real signal its ht and analytic signal and the lct of these three signals most importantly all the proposed transforms are reversible and undistorted using the proposed transforms several signal processing applications are discussed and show the advantages and flexibility over simply using the analytic signal or the lct

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

regularized positive definite fourth order tensor field estimation from dw mri in diffusion weighted magnetic resonance image dw mri processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the dw mri data from this tensor approximation one can compute useful scalar quantities e g anisotropy mean diffusivity which have been clinically used for monitoring encephalopathy sclerosis ischemia and other brain disorders it is now well known that this 2nd order tensor approximation fails to capture complex local tissue structures e g crossing fibers and as a result the scalar quantities derived from these tensors are grossly inaccurate at such locations in this paper we employ a 4th order symmetric positive definite spd tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the dw mri data guaranteeing the spd property several articles have been reported in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positivity of the estimates which is a fundamental constraint since negative values of the diffusivity are not meaningful in this paper we represent the 4th order tensors as ternary quartics and then apply hilbert s theorem on ternary quartics along with the iwasawa parametrization to guarantee an spd 4th order tensor approximation from the dw mri data the performance of this model is depicted on synthetic data as well as real dw mris from a set of excised control and injured rat spinal cords showing accurate estimation of scalar quantities such as generalized anisotropy and trace as well as fiber orientations

convex and semi nonnegative matrix factorizations we present several new variations on the theme of nonnegative matrix factorization nmf considering factorizations of the form x fgt we focus on algorithms in which g is restricted to containing nonnegative entries but allowing the data matrix x to have mixed signs thus extending the applicable range of nmf methods we also consider algorithms in which the basis vectors of f are constrained to be convex combinations of the data points this is used for a kernel extension of nmf we provide algorithms for computing these new factorizations and we provide supporting theoretical analysis we also analyze the relationships between our algorithms and clustering algorithms and consider the implications for sparseness of solutions finally we present experimental results that explore the properties of these new methods

decision theoretic foundations of qualitative possibility theory abstract this paper presents a justification of two qualitative counterparts of the expected utility criterion for decision under uncertainty which only require bounded linearly ordered valuation sets for expressing uncertainty and preferences this is carried out in the style of savage starting with a set of acts equipped with a complete preordering relation conditions on acts are given that imply a possibilistic representation of the decision maker uncertainty in this framework pessimistic i e uncertainty averse as well as optimistic attitudes can be explicitly captured the approach thus proposes an operationally testable description of possibility theory

probabilistic diffusion tractography with multiple fibre orientations what can we gain we present a direct extension of probabilistic diffusion tractography to the case of multiple fibre orientations using automatic relevance determination we are able to perform online selection of the number of fibre orientations supported by the data at each voxel simplifying the problem of tracking in a multi orientation field we then apply the identical probabilistic algorithm to tractography in the multi and single fibre cases in a number of example systems which have previously been tracked successfully or unsuccessfully with single fibre tractography we show that multi fibre tractography offers significant advantages in sensitivity when tracking non dominant fibre populations but does not dramatically change tractography results for the dominant pathways

fuzzy integral representation abstract in this paper we give representation results for nonlinear functionals in arbitrary spaces as fuzzy integrals the main assumption is that of comonotonic maxitivity i e i f g if ig for comonotonic f g under the more restrictive condition of stochastic dominance we prove a more specific representation as a fuzzy integral with respect to a distorted probability measure we discuss the particular cases of possibility measures and upper fuzzy expectations some partial randon nikodym results are proved for both the choquet and the fuzzy integrals finally an example is shown where the fuzzy integral representation is useful

estimating group delay and phase delay via discrete time analytic cross correlation starting with two finite n point real valued discrete time signals frequency domain algorithm procedures are provided for estimating the group delay and phase delay between the signals these delays are extracted from the complex valued discrete time analytic cross correlation sequence computed using the two signals special adjustments of the cross correlation transform end points are shown to be necessary in order to generate a proper analytic cross correlation sequence two cases are considered for delay estimation 1 to sample interval resolution and 2 to subsample interval resolution by interpolation

fuzzy measure of fuzzy events defined by fuzzy integrals abstract we define the concept of fuzzy measure of a fuzzy event by using a general form of fuzzy integral proposed by murofushi called fuzzy t conorm integral encompassing previous definitions zadeh defined the probability measure of a fuzzy event and later the possibility measure of fuzzy event using a duality property of fuzzy t conorm integral we propose a general definition of fuzzy measure of fuzzy events which is compatible with previous definitions of zadeh and possesses all properties of a fuzzy measure in particular the duality property using our definition we examine the case of decomposable measures and belief functions a comparison with previous works is provided

reproducibility of tract segmentation between sessions using an unsupervised modelling based approach abstract this work describes a reproducibility analysis of scalar water diffusion parameters measured within white matter tracts segmented using a probabilistic shape modelling method in common with previously reported neighbourhood tractography nt work the technique optimises seed point placement for fibre tracking by matching the tracts generated using a number of candidate points against a reference tract which is derived from a white matter atlas in the present study no direct constraints are applied to the fibre tracking results an expectation maximisation algorithm is used to fully automate the procedure and make dramatically more efficient use of data than earlier nt methods within subject and between subject variances for fractional anisotropy and mean diffusivity within the tracts are then separated using a random effects model we find test retest coefficients of variation cvs similar to those reported in another study using landmark guided single seed points and subject to subject cvs similar to a constraint based multiple roi method we conclude that our approach is at least as effective as other methods for tract segmentation using tractography whilst also having some additional benefits such as its provision of a goodness of match measure for each segmentation

on the variation of interaural time differences with frequency interaural time difference itd is a major cue to sound localization in humans and animals for a given subject and position in space itd depends on frequency this variation is analyzed here using a head related transfer functions hrtfs database collected from the literature and comprising human hrtfs from 130 subjects and animal hrtfs from six specimens of different species for humans the itd is found to vary with frequency in a way that shows consistent differences with respect to a spherical head model maximal itd values were found to be about 800 s in low frequencies and 600 s in high frequencies the itd variation with frequency up to 200 s for some positions occurs within the frequency range where itd is used to judge the lateral position of a sound source in addition itd varies substantially within the bandwidth of a single auditory filter leading to systematic differences between envelope and fine structure itds because the frequency dependent pattern of itd does not display spherical symmetries it potentially provides cues to elevation and resolves front back confusion the fact that the relation between position and itds strongly depends on the sound s spectrum in turn suggests that humans and animals make use of this relationship for the localization of sounds

segmentation of brain mr images through a hidden markov random field model and the expectation maximization algorithm the finite mixture fm model is the most commonly used model for statistical segmentation of brain magnetic resonance mr images because of its simple mathematical form and the piecewise constant nature of ideal brain mr images however being a histogram based model the fm has an intrinsic limitation no spatial information is taken into account this causes the fm model to work only on well defined images with low levels of noise unfortunately this is often not the the case due to artifacts such as partial volume effect and bias field distortion under these conditions fm model based methods produce unreliable results here the authors propose a novel hidden markov random field hmrf model which is a stochastic process generated by a mrf whose state sequence cannot be observed directly but which can be indirectly estimated through observations mathematically it can be shown that the fm model is a degenerate version of the hmrf model the advantage of the hmrf model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites although mrf modeling has been employed in mr image segmentation by other researchers most reported methods are limited to using mrf as a general prior in an fm model based approach to fit the hmrf model an em algorithm is used the authors show that by incorporating both the hmrf model and the em algorithm into a hmrf em framework an accurate and robust segmentation can be achieved more importantly the hmrf em framework can easily be combined with other techniques as an example the authors show how the bias field correction algorithm of guillemaud and brady 1997 can be incorporated into this framework to achieve a three dimensional fully automated approach for brain mr image segmentation

apparent fibre density a novel measure for the analysis of diffusion weighted magnetic resonance images this article proposes a new measure called apparent fibre density afd for the analysis of high angular resolution diffusion weighted images using higher order information provided by fibre orientation distributions fods computed using spherical deconvolution afd has the potential to provide specific information regarding differences between populations by identifying not only the location but also the orientations along which differences exist in this work analytical and numerical monte carlo simulations are used to support the use of the fod amplitude as a quantitative measure i e afd for population and longitudinal analysis to perform robust voxel based analysis of afd we present and evaluate a novel method to modulate the fod to account for changes in fibre bundle cross sectional area that occur during spatial normalisation we then describe a novel approach for statistical analysis of afd that uses cluster based inference of differences extended throughout space and orientation finally we demonstrate the capability of the proposed method by performing voxel based afd comparisons between a group of motor neurone disease patients and healthy control subjects a significant decrease in afd was detected along voxels and orientations corresponding to both the corticospinal tract and corpus callosal fibres that connect the primary motor cortices in addition to corroborating previous findings in mnd this study demonstrates the clear advantage of using this type of analysis by identifying differences along single fibre bundles in regions containing multiple fibre populations

optimizing mr scan design for model based t _ 1 t _ 2 estimation from steady state sequences rapid reliable quantification of mr relaxation parameters t1 and t2 is desirable for many clinical applications steady state sequences such as spoiled gradient recalled echo spgr and dual echo steady state dess are fast and well suited for relaxometry because the signals they produce are quite sensitive to t1 and t2 variation however t1 t2 estimation with these sequences typically requires multiple scans with varied sets of acquisition parameters this paper describes a systematic framework for selecting scan types e g combinations of spgr and dess scans and optimizing their respective parameters e g flip angles and repetition times the method is based on a cramer rao bound crb inspired min max optimization that finds scan parameter combinations that robustly enable precise object parameter estimation we apply this technique to optimize combinations of spgr and dess scans for t1 t2 relaxometry in white matter wm and grey matter gm regions of the human brain at 3t field strength phantom accuracy experiments show that spgr dess scan combinations are in excellent agreement with reference measurements phantom precision experiments show that trends in t1 t2 pooled sample standard deviations reflect crb based predictions in vivo experiments show that in wm and gm t1 and t2 estimates from a pair of optimized dess scans exhibit precision but not necessarily accuracy comparable to that of optimized combinations of spgr and dess scans to our knowledge t1 maps from dess acquisitions alone are new this example application illustrates that scan optimization may help reveal new parameter mapping techniques from combinations of established pulse sequences

fast imaging of mean axial and radial diffusion kurtosis diffusion kurtosis imaging dki is being increasingly reported to provide sensitive biomarkers of subtle changes in tissue microstructure however dki also imposes larger data requirements than diffusion tensor imaging dti hence the widespread adaptation and exploration of dki would benefit from more efficient acquisition and computational methods to meet this demand we recently developed a method capable of estimating mean kurtosis with only 13 diffusion weighted images this approach was later shown to provide very accurate mean kurtosis estimates and to be more efficient in terms of contrast to noise per unit time however insofar the computation of two other critical dki parameters radial and axial kurtosis has required the estimation of all 22 variables parameterizing the full dki signal expression here we present two strategies for estimating all of dki s principal parameters mean kurtosis radial kurtosis and axial kurtosis using only 19 diffusion weighted images compared to the current state of the art acquisitions typically requiring about 60 images the first approach is based on axially symmetric diffusion and kurtosis tensors presented here for the first time and referred to as axially symmetric dki the second approach is applicable in tissues with a priori known principal diffusion direction and does not require fitting of any kind the approaches are evaluated in human brain in vivo as well as in fixed rat spinal cord and are demonstrated to provide metrics in good agreement with their full dki counterparts estimated with nonlinear least squares for small data sets and in white matter axially symmetric dki provides more accurate and robust estimates than unconstrained dki the significant acceleration achieved further paves the way to routine application of the technique

denoising of diffusion mri using random matrix theory we introduce and evaluate a post processing technique for fast denoising of diffusion weighted mr images by exploiting the intrinsic redundancy in diffusion mri using universal properties of the eigenspectrum of random covariance matrices we remove noise only principal components thereby enabling signal to noise ratio enhancements this yields parameter maps of improved quality for visual quantitative and statistical interpretation by studying statistics of residuals we demonstrate that the technique suppresses local signal fluctuations that solely originate from thermal noise rather than from other sources such as anatomical detail furthermore we achieve improved precision in the estimation of diffusion parameters and fiber orientations in the human brain without compromising the accuracy and spatial resolution

single and multiple shell uniform sampling schemes for diffusion mri using spherical codes in diffusion mri dmri a good sampling scheme is important for efficient acquisition and robust reconstruction diffusion weighted signal is normally acquired on single or multiple shells in q space signal samples are typically distributed uniformly on different shells to make them invariant to the orientation of structures within tissue or the laboratory coordinate frame the electrostatic energy minimization eem method originally proposed for single shell sampling scheme in dmri was recently generalized to multi shell schemes called generalized eem geem geem has been successfully used in the human connectome project however eem does not directly address the goal of optimal sampling i e achieving large angular separation between sampling points in this paper we propose a more natural formulation called spherical code sc to directly maximize the minimal angle between different samples in single or multiple shells we consider not only continuous problems to design single or multiple shell sampling schemes but also discrete problems to uniformly extract sub sampled schemes from an existing single or multiple shell scheme and to order samples in an existing scheme we propose five algorithms to solve the above problems including an incremental sc isc a sophisticated greedy algorithm called iterative maximum overlap construction imoc an 1 opt greedy method a mixed integer linear programming method and a constrained non linear optimization method to our knowledge this is the first work to use the sc formulation for single or multiple shell sampling schemes in dmri experimental results indicate that sc methods obtain larger angular separation and better rotational invariance than the state of the art eem and geem the related codes and a tutorial have been released in dmritool 1 1 https diffusionmritool github io tutorial qspacesampling html

parametric spherical deconvolution inferring anatomical connectivity using diffusion mr imaging the human brain forms a complex neural network with a connectional architecture that is still far from being known in full detail even at the macroscopic level the advent of diffusion mr imaging has enabled the exploration of the structural properties of white matter in vivo in this article we propose a new forward model that maps the microscopic geometry of nervous tissue onto the water diffusion process and further onto the measured mr signals our spherical deconvolution approach completely parameterizes the fiber orientation density by a finite mixture of bingham distributions in addition we define the term anatomical connectivity taking the underlying image modality into account this neurophysiological metric may represent the proportion of the nerve fibers originating in the source area which intersect a given target region the specified inverse problem is solved by bayesian statistics posterior probability maps denote the probability that the connectivity value exceeds a chosen threshold conditional upon the noisy observations these maps allow us to draw inferences about the structural organization of the cerebral cortex moreover we will demonstrate the proposed approach with diffusion weighted data sets featuring high angular resolution

diffusion tensor imaging of the corpus callosum in healthy aging investigating higher order polynomial regression modelling previous diffusion tensor imaging dti studies confirmed the vulnerability of corpus callosum cc fibers to aging however most studies employed lower order regressions to study the relationship between age and white matter microstructure the present study investigated whether higher order polynomial regression modelling can better describe the relationship between age and cc dti metrics compared to lower order models in 140 healthy participants ages 18 85 the cc was found to be non uniformly affected by aging with accelerated and earlier degradation occurring in anterior portion callosal volume fiber count fiber length mean fibers per voxel and fa decreased with age while mean axial and radial diffusivities increased half of the parameters studied also displayed significant age sex interaction or intracranial volume effects higher order models were chosen as the best fit based on bayesian information criterion minimization in 16 out of 23 significant cases when describing the relationship between dti measurements and age higher order model fits provided different estimations of aging trajectory peaks and decline onsets than lower order models however a likelihood ratio test found that higher order regressions generally did not fit the data significantly better than lower order polynomial or linear models the results contrast the modelling approaches and highlight the importance of using higher order polynomial regression modelling when investigating associations between age and cc white matter microstructure

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

relative modalities and their use in possibilistic linear programming abstract the relative modalities defined by possibility necessity impossibility and contingency of fuzzy sets in modal semantics are investigated decision procedures based on the modal concept are introduced and classified into eight classes the relation between each decision procedure and the attitude of the decision maker toward uncertainty is discussed lastly possibilistic linear programming relying on these decision procedures is formulated and a solution algorithm is proposed a numerical example is given

weighted linear least squares estimation of diffusion mri parameters strengths limitations and pitfalls abstract purpose linear least squares estimators are widely used in diffusion mri for the estimation of diffusion parameters although adding proper weights is necessary to increase the precision of these linear estimators there is no consensus on how to practically define them in this study the impact of the commonly used weighting strategies on the accuracy and precision of linear diffusion parameter estimators is evaluated and compared with the nonlinear least squares estimation approach methods simulation and real data experiments were done to study the performance of the weighted linear least squares estimators with weights defined by a the squares of the respective noisy diffusion weighted signals and b the squares of the predicted signals which are reconstructed from a previous estimate of the diffusion model parameters results the negative effect of weighting strategy a on the accuracy of the estimator was surprisingly high multi step weighting strategies yield better performance and in some cases even outperformed the nonlinear least squares estimator conclusion if proper weighting strategies are applied the weighted linear least squares approach shows high performance characteristics in terms of accuracy precision and may even be preferred over nonlinear estimation methods

towards better mr characterization of neural tissues using directional diffusion kurtosis analysis mr diffusion kurtosis imaging dki was proposed recently to study the deviation of water diffusion from gaussian distribution mean kurtosis the directionally averaged kurtosis has been shown to be useful in assessing pathophysiological changes thus yielding another dimension of information to characterize water diffusion in biological tissues in this study orthogonal transformation of the 4th order diffusion kurtosis tensor was introduced to compute the diffusion kurtoses along the three eigenvector directions of the 2nd order diffusion tensor such axial k and radial k upper left and right quadrants kurtoses measured the kurtoses along the directions parallel and perpendicular respectively to the principal diffusion direction dki experiments were performed in normal adult n 7 and formalin fixed rat brains n 5 dki estimates were documented for various white matter wm and gray matter gm tissues and compared with the conventional diffusion tensor estimates the results showed that kurtosis estimates revealed different information for tissue characterization for example k and k upper left and right quadrants under formalin fixation condition exhibited large and moderate increases in wm while they showed little change in gm despite the overall dramatic decrease of axial and radial diffusivities in both wm and gm these findings indicate that directional kurtosis analysis can provide additional microstructural information in characterizing neural tissues

deterministic and probabilistic tractography based on complex fibre orientation distributions we propose an integral concept for tractography to describe crossing and splitting fibre bundles based on the fibre orientation distribution function odf estimated from high angular resolution diffusion imaging hardi we show that in order to perform accurate probabilistic tractography one needs to use a fibre odf estimation and not the diffusion odf we use a new fibre odf estimation obtained from a sharpening deconvolution transform sdt of the diffusion odf reconstructed from q ball imaging qbi this sdt provides new insight into the relationship between the hardi signal the diffusion odf and the fibre odf we demonstrate that the sdt agrees with classical spherical deconvolution and improves the angular resolution of qbi another important contribution of this paper is the development of new deterministic and new probabilistic tractography algorithms using the full multidirectional information obtained through use of the fibre odf an extensive comparison study is performed on human brain datasets comparing our new deterministic and probabilistic tracking algorithms in complex fibre crossing regions finally as an application of our new probabilistic tracking we quantify the reconstruction of transcallosal fibres intersecting with the corona radiata and the superior longitudinal fasciculus in a group of eight subjects most current diffusion tensor imaging dti based methods neglect these fibres which might lead to incorrect interpretations of brain functions

principal invariants and inherent parameters of diffusion kurtosis tensors a diffusion kurtosis dk tensor is a fourth order three dimensional fully symmetric tensor which is used in diffusion kurtosis imaging dki a new model in medical engineering to understand the biological and clinical meaning of the dk tensor we have to measure and calculate some quantities and parameters which are independent from coordinate system choices in this paper we study such quantities and parameters they include the largest the smallest and the average apparent kurtosis coefficients akc values which are invariant from the coordinate system choices and some parameters measured in the inherent coordinate system which is formed by the eigenvector system of the second order diffusion tensor we study their computational formulas and relationships

exploring the 3d geometry of the diffusion kurtosis tensor impact on the development of robust tractography procedures and novel biomarkers diffusion kurtosis imaging dki is a diffusion weighted technique which overcomes limitations of the commonly used diffusion tensor imaging approach this technique models non gaussian behaviour of water diffusion by the diffusion kurtosis tensor kt which can be used to provide indices of tissue heterogeneity and a better characterisation of the spatial architecture of tissue microstructure in this study the geometry of the kt is elucidated using synthetic data generated from multi compartmental models where diffusion heterogeneity between intra and extra cellular media is taken into account as well as the sensitivity of the results to each model parameter and to synthetic noise furthermore based on the assumption that the maxima of the kt are distributed perpendicularly to the direction of well aligned fibres a novel algorithm for estimating fibre direction directly from the kt is proposed and compared to the fibre directions extracted from dki based orientation distribution function odf estimates previously proposed in the literature synthetic data results showed that for fibres crossing at high intersection angles direction estimates extracted directly from the kt have smaller errors than the dki based odf estimation approaches dki odf nevertheless the proposed method showed smaller angular resolution and lower stability to changes of the simulation parameters on real data tractography performed on these kt fibre estimates suggests a higher sensitivity than the dki based odf in resolving lateral corpus callosum fibres reaching the pre central cortex when diffusion acquisition is performed with five b values using faster acquisition schemes kt based tractography did not show improved performance over the dki odf procedures nevertheless it is shown that direct kt fibre estimates are more adequate for computing a generalised version of radial kurtosis maps

computing the discrete time analytic signal via fft starting with a real valued n point discrete time signal frequency domain algorithms are provided for computing 1 the complex valued standard n point discrete time analytic signal of the same sample rate 2 the complex valued decimated n 2 point discrete time analytic signal of half the original sample rate and 3 the complex valued interpolated nm point discrete time analytic signal of m times the original sample rate special adjustment of the transform end points are shown to be necessary in order to generate proper discrete time analytic signals

optimal experiment design for magnetic resonance fingerprinting cramer rao bound meets spin dynamics magnetic resonance mr fingerprinting is a new quantitative imaging paradigm which simultaneously acquires multiple mr tissue parameter maps in a single experiment in this paper we present an estimation theoretic framework to perform experiment design for mr fingerprinting specifically we describe a discrete time dynamic system to model spin dynamics and derive an estimation theoretic bound i e the cramer rao bound to characterize the signal to noise ratio snr efficiency of an mr fingerprinting experiment we then formulate an optimal experiment design problem which determines a sequence of acquisition parameters to encode mr tissue parameters with the maximal snr efficiency while respecting the physical constraints and other constraints from the image decoding reconstruction process we evaluate the performance of the proposed approach with numerical simulations phantom experiments and in vivo experiments we demonstrate that the optimized experiments substantially reduce data acquisition time and or improve parameter estimation for example the optimized experiments achieve about a factor of two improvement in the accuracy of t 2 maps while keeping similar or slightly better accuracy of t 1 maps finally as a remarkable observation we find that the sequence of optimized acquisition parameters appears to be highly structured rather than randomly pseudo randomly varying as is prescribed in the conventional mr fingerprinting experiments

improved segmentation reproducibility in group tractography using a quantitative tract similarity measure abstract the field of tractography is rapidly developing and many automatic or semiautomatic algorithms have now been devised to segment and visualize neural white matter fasciculi in vivo however these algorithms typically need to be given a starting location as input and their output can be strongly dependent on the exact location of this seed point no robust method has yet been devised for placing these seed points so as to segment a comparable tract in a group of subjects here we develop a measure of tract similarity based on the shapes and lengths of the two tracts being compared and apply it to the problem of consistent seed point placement and tract segmentation in group data we demonstrate that using a single seed point transferred from standard space to each native space produces considerable variability in tractography output between scans however by seeding in a group of nearby candidate points and choosing the output with the greatest similarity to a reference tract chosen in advance a method we refer to as neighborhood tractography this variability can be significantly reduced

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

how to correct susceptibility distortions in spin echo echo planar images application to diffusion tensor imaging diffusion tensor imaging is often performed by acquiring a series of diffusion weighted spin echo echo planar images with different direction diffusion gradients a problem of echo planar images is the geometrical distortions that obtain near junctions between tissues of differing magnetic susceptibility this results in distorted diffusion tensor maps to resolve this we suggest acquiring two images for each diffusion gradient one with bottom up and one with top down traversal of k space in the phase encode direction this achieves the simultaneous goals of providing information on the underlying displacement field and intensity maps with adequate spatial sampling density even in distorted areas the resulting dt maps exhibit considerably higher geometric fidelity as assessed by comparison to an image volume acquired using a conventional 3d mr technique

optimal experimental design for diffusion kurtosis imaging diffusion kurtosis imaging dki is a new magnetic resonance imaging mri model that describes the non gaussian diffusion behavior in tissues it has recently been shown that dki parameters such as the radial or axial kurtosis are more sensitive to brain physiology changes than the well known diffusion tensor imaging dti parameters in several white and gray matter structures in order to estimate either dti or dki parameters with maximum precision the diffusion weighting gradient settings that are applied during the acquisition need to be optimized indeed it has been shown previously that optimizing the set of diffusion weighting gradient settings can have a significant effect on the precision with which dti parameters can be estimated in this paper we focus on the optimization of dki gradients settings commonly dki data are acquired using a standard set of diffusion weighting gradients with fixed directions and with regularly spaced gradient strengths in this paper we show that such gradient settings are suboptimal with respect to the precision with which dki parameters can be estimated furthermore the gradient directions and the strengths of the diffusion weighted mr images are optimized by minimizing the crame r rao lower bound of dki parameters the impact of the optimized gradient settings is evaluated both on simulated as well as experimentally recorded datasets it is shown that the precision with which the kurtosis parameters can be estimated increases substantially by optimizing the gradient settings

variability in diffusion kurtosis imaging impact on study design statistical power and interpretation diffusion kurtosis imaging dki is an emerging technique with the potential to quantify properties of tissue microstructure that may not be observable using diffusion tensor imaging dti in order to help design dki studies and improve interpretation of dki results we employed statistical power analysis to characterize three aspects of variability in four dki parameters the mean diffusivity fractional anisotropy mean kurtosis and radial kurtosis first we quantified the variability in terms of the group size required to obtain a statistical power of 0 9 second we investigated the relative contribution of imaging and post processing noise to the total variance in order to estimate the benefits of longer scan times versus the inclusion of more subjects third we evaluated the potential benefit of including additional covariates such as the size of the structure when testing for differences in group means the analysis was performed in three major white matter structures of the brain the superior cingulum the corticospinal tract and the mid sagittal corpus callosum extracted using diffusion tensor tractography and dki data acquired in a healthy cohort the results showed heterogeneous variability across and within the white matter structures thus the statistical power varies depending on parameter and location which is important to consider if a pathogenesis pattern is inferred from dki data in the data presented inter subject differences contributed more than imaging noise to the total variability making it more efficient to include more subjects rather than extending the scan time per subject finally strong correlations between dki parameters and the structure size were found for the cingulum and corpus callosum structure size should thus be considered when quantifying dki parameters either to control for its potentially confounding effect or as a means of reducing unexplained variance

possibility and necessity measure specification using modifiers for decision making under fuzziness in this paper we propose an approach to possibility and necessity measure specification this approach is more adequate to human reasoning than direct specification of conjunction and implication functions using the proposed approach a possibility measure is specified by two modifier functions and a strong negation while a necessity measure is specified by two modifier functions only it is demonstrated that many possibility and necessity measures defined by famous conjunction and implication functions are obtained by the proposed approach conditions for specified measures to have good properties are also given finally a simple example of application of possibility and necessity measures to a decision problem is given

investigating white matter fibre density and morphology using fixel based analysis voxel based analysis of diffusion mri data is increasingly popular however most white matter voxels contain contributions from multiple fibre populations often referred to as crossing fibres and therefore voxel averaged quantitative measures e g fractional anisotropy are not fibre specific and have poor interpretability using higher order diffusion models parameters related to fibre density can be extracted for individual fibre populations within each voxel fixels and recent advances in statistics enable the multi subject analysis of such data however investigating within voxel microscopic fibre density alone does not account for macroscopic differences in the white matter morphology e g the calibre of a fibre bundle in this work we introduce a novel method to investigate the latter which we call fixel based morphometry fbm to obtain a more complete measure related to the total number of white matter axons information from both within voxel microscopic fibre density and macroscopic morphology must be combined we therefore present the fbm method as an integral piece within a comprehensive fixel based analysis framework to investigate measures of fibre density fibre bundle morphology cross section and a combined measure of fibre density and cross section we performed simulations to demonstrate the proposed measures using various transformations of a numerical fibre bundle phantom finally we provide an example of such an analysis by comparing a clinical patient group to a healthy control group which demonstrates that all three measures provide distinct and complementary information by capturing information from both sources the combined fibre density and cross section measure is likely to be more sensitive to certain pathologies and more directly interpretable

improved tractography using asymmetric fibre orientation distributions diffusion mri allows us to make inferences on the structural organisation of the brain by mapping water diffusion to white matter microstructure however such a mapping is generally ill defined for instance diffusion measurements are antipodally symmetric diffusion along x and x are equal whereas the distribution of fibre orientations within a voxel is generally not symmetric therefore different sub voxel patterns such as crossing fanning or sharp bending cannot be distinguished by fitting a voxel wise model to the signal however asymmetric fibre patterns can potentially be distinguished once spatial information from neighbouring voxels is taken into account we propose a neighbourhood constrained spherical deconvolution approach that is capable of inferring asymmetric fibre orientation distributions a fods importantly we further design and implement a tractography algorithm that utilises the estimated a fods since the commonly used streamline tractography paradigm cannot directly take advantage of the new information we assess performance using ultra high resolution histology data where we can compare true orientation distributions against sub voxel fibre patterns estimated from down sampled data finally we explore the benefits of a fods based tractography using in vivo data by evaluating agreement of tractography predictions with connectivity estimates made using different in vivo modalities the proposed approach can reliably estimate complex fibre patterns such as sharp bending and fanning which voxel wise approaches cannot estimate moreover histology based and in vivo results show that the new framework allows more accurate tractography and reconstruction of maps quantifying symmetric and asymmetric fibre complexity

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

a note on the application of the hilbert transform to time delay estimation it is shown that the hilbert transform may be employed to convert the maximum finding task of a delay estimator to one of finding a zero crossing the basic theory of the technique is shown and a few applications are described its primary application is in servo tracking and servo velocity measurement systems which employ correlation techniques a review of prior work in this field is given

a bayesian model of shape and appearance for subcortical brain segmentation automatic segmentation of subcortical structures in human brain mr images is an important but difficult task due to poor and variable intensity contrast clear well defined intensity features are absent in many places along typical structure boundaries and so extra information is required to achieve successful segmentation a method is proposed here that uses manually labelled image data to provide anatomical training information it utilises the principles of the active shape and appearance models but places them within a bayesian framework allowing probabilistic relationships between shape and intensity to be fully exploited the model is trained for 15 different subcortical structures using 336 manually labelled t1 weighted mr images using the bayesian approach conditional probabilities can be calculated easily and efficiently avoiding technical problems of ill conditioned covariance matrices even with weak priors and eliminating the need for fitting extra empirical scaling parameters as is required in standard active appearance models furthermore differences in boundary vertex locations provide a direct purely local measure of geometric change in structure between groups that unlike voxel based morphometry is not dependent on tissue classification methods or arbitrary smoothing in this paper the fully automated segmentation method is presented and assessed both quantitatively using leave one out testing on the 336 training images and qualitatively using an independent clinical dataset involving alzheimer s disease median dice overlaps between 0 7 and 0 9 are obtained with this method which is comparable or better than other automated methods an implementation of this method called first is currently distributed with the freely available fsl package

nonnegative matrix factorization for rapid recovery of constituent spectra in magnetic resonance chemical shift imaging of the brain we present an algorithm for blindly recovering constituent source spectra from magnetic resonance mr chemical shift imaging csi of the human brain the algorithm which we call constrained nonnegative matrix factorization cnmf does not enforce independence or sparsity instead only requiring the source and mixing matrices to be nonnegative it is based on the nonnegative matrix factorization nmf algorithm extending it to include a constraint on the positivity of the amplitudes of the recovered spectra this constraint enables recovery of physically meaningful spectra even in the presence of noise that causes a significant number of the observation amplitudes to be negative we demonstrate and characterize the algorithm s performance using sup 31 p volumetric brain data comparing the results with two different blind source separation methods bayesian spectral decomposition bsd and nonnegative sparse coding nnsc we then incorporate the cnmf algorithm into a hierarchical decomposition framework showing that it can be used to recover tissue specific spectra given a processing hierarchy that proceeds coarse to fine we demonstrate the hierarchical procedure on sup 1 h brain data and conclude that the computational efficiency of the algorithm makes it well suited for use in diagnostic work up

learning fuzzy rules from fuzzy samples based on rough set technique although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness its performance in dealing with initial fuzzy data is usually poor this paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment the extension is twofold one is knowledge representation and the other is knowledge reduction first we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects second we extend a number of underlying concepts of knowledge reduction such as the reduct and core to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data finally we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm one of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension

convexity constrained and nonnegativity constrained spherical factorization in diffusion weighted imaging diffusion weighted imaging dwi facilitates probing neural tissue structure non invasively by measuring its hindrance to water diffusion analysis of dwi is typically based on generative signal models for given tissue geometry and microstructural properties in this work we generalize multi tissue spherical deconvolution to a blind source separation problem under convexity and nonnegativity constraints this spherical factorization approach decomposes multi shell dwi data represented in the basis of spherical harmonics into tissue specific orientation distribution functions and corresponding response functions without assuming the latter as known thus fully unsupervised in healthy human brain data the resulting components are associated with white matter fibres grey matter and cerebrospinal fluid the factorization results are on par with state of the art supervised methods as demonstrated also in monte carlo simulations evaluating accuracy and precision of the estimated response functions and orientation distribution functions of each component in animal data and in the presence of oedema the proposed factorization is able to recover unseen tissue structure solely relying on dwi as such our method broadens the applicability of spherical deconvolution techniques to exploratory analysis of tissue structure in data where priors are uncertain or hard to define