model driven parameterization of the cortical surface for localization and inter subject matching in this paper we present a generic and organized model of cortical folding and a way to implement this model on any given cortical surface this results in a model driven parameterization providing an anatomically meaningful coordinate system for cortical localization and implicitly defining inter subject surface matching without any deformation of surfaces we present our cortical folding model and show how it naturally defines a parameterization of the cortex the mapping of the model to any given cortical surface is detailed leading to an anatomically invariant coordinate system the process is evaluated on real data in terms of both anatomical and functional localization and shows improved performance compared to a traditional volume based normalization it is fully automatic and available with the brainvisa software platform

global attractor for degenerate damped hyperbolic equations abstract the aim of this paper is to consider a class of degenerate damped hyperbolic equations with the critical nonlinearity involving an operator l that is x elliptic with respect to a family of vector fields x we prove the global existence of solutions and characterize their long time behavior in particular we show the semigroup generated by the equation has a compact connected and invariant attractor

global attractors for the wave equation with nonlinear damping based on a new a priori estimate method so called asymptotic a priori estimate the existence of a global attractor is proved for the wave equation utt kg ut u f u 0 on a bounded domain r3 with dirichlet boundary conditions the nonlinear damping term g is supposed to satisfy the growth condition c1 s c2 g s c3 1 s p where 1 p 0 is arbitrary the nonlinear term f is supposed to satisfy the growth condition f s c4 1 s q where q 2 it is remarkable that when 2 p 5 we positively answer an open problem in chueshov and lasiecka i chueshov i lasiecka long time behavior of second evolution equations with nonlinear damping math scuola norm sup 2004 and improve the corresponding results in feireisl e feireisl global attractors for damped wave equations with supercritical exponent j differential equations 116 1995 431 447

on semilinear δλ laplace equation abstract we prove some existence nonexistence and regularity results for the boundary value problem u f u 0 in u 0 where is a bounded subset of r n n 2 and is a laplacian i e a degenerate elliptic operator of the kind i 1 n x i i 2 x x i 1 n together with some assumptions made in franchi and lanconelli 1984 1 the family is supposed to verify a condition making homogeneous of degree two with respect to a group of dilations in r n

smoothing and matching of 3 d space curves we present a new approach to the problem of matching 3 d curves the approach has a low algorithmic complexity in the number of models and can operate in the presence of noise and partial occlusions our method builds upon the seminal work of kishon et al 1990 where curves are first smoothed using b splines with matching based on hashing using curvature and torsion measures however we introduce two enhancements we present experimental results using synthetic data and also using characteristic curves extracted from 3 d medical images an earlier version of this article was presented at the 2nd european conference on computer vision in italy

dynamic programming generation of curves on brain surfaces dynamic programming algorithms are presented for automated generation of length minimizing geodesics and curves of extremal curvature on the neocortex of the macaque and the visible human probabilistic models of curve variation are constructed in terms of the variability in speed curvature and torsion in the frenet representation

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

diffeomorphic brain registration under exhaustive sulcal constraints the alignment and normalization of individual brain structures is a prerequisite for group level analyses of structural and functional neuroimaging data the techniques currently available are either based on volume and or surface attributes with limited insight regarding the consistent alignment of anatomical landmarks across individuals this article details a global geometric approach that performs the alignment of the exhaustive sulcal imprints cortical folding patterns across individuals this diffeomorphic sulcal based cortical disco technique proceeds to the automatic extraction identification and simplification of sulcal features from t1 weighted magnetic resonance image mri series these features are then used as control measures for fully 3 d diffeomorphic deformations quantitative and qualitative evaluations show that disco correctly aligns the sulcal folds and gray and white matter volumes across individuals the comparison with a recent iconic diffeomorphic approach dartel highlights how the absence of explicit cortical landmarks may lead to the misalignment of cortical sulci we also feature disco in the automatic design of an empirical sulcal template from group data we also demonstrate how disco can efficiently be combined with an image based deformation dartel to further improve the consistency and accuracy of alignment performances finally we illustrate how the optimized alignment of cortical folds across subjects improves sensitivity in the detection of functional activations in a group level analysis of neuroimaging data

automatic extraction of sulcal lines on cortical surfaces based on anisotropic geodesic distance analyzing cortical sulci is important for studying cortical morphology and brain functions although sulcal lines on cortical surfaces can be defined in various ways it is critical in a neuroimaging study to define a sulcal line along the valley of a cortical surface with a high curvature within a sulcus to extract the sulcal lines automatically we present a new geometric algorithm based on the computation of anisotropic skeletons of sulcal regions because anisotropic skeletons are highly adaptive to the anisotropic nature of the surface shape the resulting sulcal lines lie accurately on the valleys of the sulcal areas our sulcal lines remain unchanged under local shape variabilities in different human brains through experiments we show that the errors of the sulcal lines for both synthetic data and real cortical surfaces were nearly as constant as the function of random noise by measuring the changes in sulcal shape in alzheimer s disease ad patients we further investigated the effectiveness of the accuracy of our sulcal lines using a large sample of mri data this study involved 70 normal controls n men women 29 41 age mean sd 71 7 4 9 years and 100 ad subjects 37 63 72 3 5 5 we observe significantly lower absolute average mean curvature and shallower sulcal depth in ad subjects where the group difference becomes more significant if we measure the quantities along the sulcal lines rather than over the entire sulcal area the most remarkable difference in the ad patients was the average sulcal depth control 11 70 and ad 11 34

measuring brain variability by extrapolating sparse tensor fields measured on sulcal lines modeling and understanding the variability of brain structures is a fundamental problem in neurosciences improved mathematical representations of structural brain variation are needed to help detect and understand genetic or disease related sources of abnormality as well as to improve statistical power when integrating functional brain mapping data across subjects in this paper we develop a new mathematical model of normal brain variation based on a large set of cortical sulcal landmarks 72 per brain delineated in each of 98 healthy human subjects scanned with 3d mri age 51 8 6 2 years we propose an original method to compute an average representation of the sulcal curves which constitutes the mean anatomy after affine alignment of the individual data across subjects the second order moment distribution of the sulcal position is modeled as a sparse field of covariance tensors symmetric positive definite matrices to extrapolate this information to the full brain one has to overcome the limitations of the standard euclidean matrix calculus we propose an affine invariant riemannian framework to perform computations with tensors in particular we generalize radial basis function rbf interpolation and harmonic diffusion partial differential equations pdes to tensor fields as a result we obtain a dense 3d variability map that agrees well with prior results on smaller subject samples moreover leave one sulcus out tests show that our model is globally able to recover the missing information on brain variation when there is a consistent neighboring pattern of variability finally we propose an innovative method to analyze the asymmetry of brain variability as expected the greatest asymmetries are found in regions that includes the primary language areas interestingly any such asymmetries in anatomical variance if it remains after anatomical normalization could explain why there may be greater power to detect group activation in one hemisphere versus the other in fmri studies

hamilton jacobi skeleton on cortical surfaces in this paper we propose a new method to construct graphical representations of cortical folding patterns by computing skeletons on triangulated cortical surfaces in our approach a cortical surface is first partitioned into sulcal and gyral regions via the solution of a variational problem using graph cuts which can guarantee global optimality after that we extend the method of hamilton jacobi skeleton to subsets of triangulated surfaces together with a geometrically intuitive pruning process that can trade off between skeleton complexity and the completeness of representing folding patterns compared with previous work that uses skeletons of 3 d volumes to represent sulcal patterns the skeletons on cortical surfaces can be easily decomposed into branches and provide a simpler way to construct graphical representations of cortical morphometry in our experiments we demonstrate our method on two different cortical surface models its ability of capturing major sulcal patterns and its application to compute skeletons of gyral regions

nonlinear damped wave equations for the sub laplacian on the heisenberg group and for rockland operators on graded lie groups abstract in this paper we study the cauchy problem for the semilinear damped wave equation for the sub laplacian on the heisenberg group in the case of the positive mass we show the global in time well posedness for small data for power like nonlinearities we also obtain similar well posedness results for the wave equations for rockland operators on general graded lie groups in particular this includes higher order operators on r n and on the heisenberg group such as powers of the laplacian or the sub laplacian in addition we establish a new family of gagliardo nirenberg inequalities on a graded lie groups that play a crucial role in the proof but which are also of interest on their own if g is a graded lie group of homogeneous dimension q and a 0 1 r q a and 1 p q r q q a r then we have the following gagliardo nirenberg type inequality u l q g u l a r g s u l p g 1 s for s 1 p 1 q a q 1 p 1 r 1 0 1 provided that a q 1 p 1 r 0 where l a r is the homogeneous sobolev space of order a over l r if a q 1 p 1 r 0 we have p q r q q a r and then the above inequality holds for any 0 s 1

the synchronization of variable length codes many variable length codes exhibit a tendency for resynchronization to occur automatically following any error however attempts to identify an underlying synchronization mechanism and to accurately predict the expected synchronization delay for even quite specific variable length codes appear to have been largely unsuccessful the present paper explores a novel method for estimating the synchronization performance for a wide variety of variable length codes based on the t codes t codes are a class of self synchronizing codes which typically synchronize within 2 3 codewords by a mechanism that derives from a recursive t augmentation construction it is observed that the t code mechanism for synchronization is followed more or less by other variable length codes wherever substantial numbers of codewords are shared with a t code set t augmentation itself provides a means for assessing the contribution individual codewords make to the overall synchronization process for a t code set thus codeword differences between sets may be specifically evaluated to estimate the synchronization performance of a variable length code set from a closely related t code set

liouville theorem for x elliptic operators abstract we prove a liouville type theorem for a class of degenerate elliptic operators of the form l u i j 1 n x i a i j x j u i 1 n b i x i u l is supposed to be x elliptic with respect to a family x x 1 x m of locally lipschitz continuous vector fields in the sense introduced in e lanconelli a e kogoj x elliptic operators and x control distances contributions in honor of the memory of ennio de giorgi ricerche di matematica 49 suppl 2000 223 243

dynamics for a stochastic degenerate parabolic equation abstract in this paper we consider the long time behavior of a class of stochastic degenerate parabolic equations involving an operator which is x elliptic with respect to a family of locally lipschitz continuous vector fields x x 1 x 2 x m the nonlinearity satisfies a dissipative condition with polynomial growth of arbitrary order p 2 the existence of the random attractor in l 2 o higher order integrability and continuity in h are established

the construction of variable length codes with good synchronization properties variable length codes offer advantages for data compression but are susceptible to loss of synchronization if a bit error occurs this can be mitigated by the use of variable length codes with specific mechanisms for resynchronization such mechanisms exist in the codes known as huffman equivalent he codes and t codes these have been extensively studied in the literature for optimal compression a length vector is obtained from the probabilities of the symbols of the alphabet used however he codes and t codes do not exist for all length vectors in this paper a new class of variable length codes with good synchronization properties is developed these are referred to as ordered termination ot codes ot codes do exist for all length vectors experimental results and some theoretical support suggest that ot codes compare favorably with he and t codes

statistical sulcal shape comparisons application to the detection of genetic encoding of the central sulcus shape principal component analysis allows a quantitative description of shape variability with a restricted num ber of parameters or modes which can be used to quantify the difference between two shapes through the computation of a modal distance a statistical test can then be applied to this set of measurements in order to detect a statistically significant difference between two groups we have applied this methodol ogy to highlight evidence of genetic encoding of the shape of neuroanatomical structures to investigate genetic constraint we studied if shapes were more similar within 10 pairs of monozygotic twins than within interpairs and compared the results with those obtained from 10 pairs of dizygotic twins the statis tical analysis was performed using a mantel permuta tion test we show using simulations that this statis tical test applied on modal distances can detect a possible genetic encoding when applied to real data this study highlighted genetic constraints on the shape of the central sulcus we found from 10 pairs of monozygotic twins that the intrapair modal distance of the central sulcus was significantly smaller than the interpair modal distance for both the left central sul cus z 52 2 66 p 0 005 and the right central sulcus z 52 2 26 p 0 05 genetic constraints on the defi nition of the central sulcus shape were confirmed by applying the same experiment to 10 pairs of normal young individuals z 52 1 39 z 52 0 63 i e values not significant at the p 0 05 level and 10 pairs of dizy gotic twins z 5 0 47 z 5 0 03 i e values not signifi cant at the p 0 05 level 2000 academic press

model driven harmonic parameterization of the cortical surface hip hop in the context of inter subject brain surface matching we present a parameterization of the cortical surface constrained by a model of cortical organization the parameterization is defined via an harmonic mapping of each hemisphere surface to a rectangular planar domain that integrates a representation of the model as opposed to previous landmark based registration methods we do not match folds between individuals but instead optimize the fit between cortical sulci and specific iso coordinate axis in the model this strategy overcomes some limitation to sulcus based registration techniques such as topological variability in sulcal landmarks across subjects experiments on 62 subjects with manually traced sulci are presented and compared with the result of the freesurfer software the evaluation involves a measure of dispersion of sulci with both angular and area distortions we show that the model based strategy can lead to a natural efficient and very fast less than 5 min per hemisphere method for defining inter subjects correspondences we discuss how this approach also reduces the problems inherent to anatomically defined landmarks and open the way to the investigation of cortical organization through the notion of orientation and alignment of structures across the cortex

attractors met x elliptic operators abstract we consider degenerate parabolic and damped hyperbolic equations involving an operator l that is x elliptic with respect to a family of locally lipschitz continuous vector fields x x 1 x m the local well posedness is established under subcritical growth restrictions on the nonlinearity f which are determined by the geometry and functional setting naturally associated with the family of vector fields x assuming additionally that f is dissipative the global existence of solutions follows and we can characterize their longtime behavior using methods from the theory of infinite dimensional dynamical systems

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

automatic sulcal line extraction on cortical surfaces using geodesic path density maps abstract we present here a method that is designed to automatically extract sulcal lines on the mesh of any cortical surface the method is based on the definition of a new function the geodesic path density map gpdm within each sulcal basin i e regions with a negative mean curvature gpdm indicates at each vertex the likelihood that a shortest path between any two points of the basins boundary goes through that vertex if the distance used to compute shortest path is anisotropic and constrained by a geometric information such as the depth the gpdm indicates the likelihood that a vertex belongs to the sulcal line in the basin an automatic gpdm adaptive thresholding procedure is proposed and sulcal lines are then defined the process has been validated on a set of 25 subjects by comparing results to the manual segmentation from an expert and showed an average error below 2 mm it is also compared to our previous reference method in the context of inter subject cortical surface registration and shows an significant improvement in performance

on the construction of statistically synchronizable codes the problem of constructing statistically synchronizable codes over arbitrary alphabets and for any finite source is considered it is shown how to efficiently construct a statistically synchronizable code whose average codeword length is within the least likely codeword probability from that of the huffman code for the same source moreover a method is given for constructing codes having a synchronizing codeword the method yields synchronous codes that exhibit high synchronizing capability and low redundancy

almost all complete binary prefix codes have a self synchronizing string the probability that a complete binary prefix code has a self synchronizing string approaches one as the number of codewords tends to infinity

dynamics of strongly damped wave equations in locally uniform spaces attractors and asymptotic regularity this paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces after proving the global well posedness we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded in h 2 lu r n x h 1 lu r n subset which attracts exponentially every initial h 1 lu r n x l 2 lu r n bounded set with respect to the h 1 lu r n x l 2 lu r n norm then we show there is a h 1 lu r n x l 2 lu r n h 1 r n x h 1 r n global attractor which reflects the strongly damped property of u t to some extent

binary huffman equivalent codes with a short synchronizing codeword for a given set of codeword lengths there are many different optimal variable length codes which are all huffman 1952 equivalent codes some of these codes may contain a synchronizing codeword which resynchronizes the code whenever it is transmitted the shorter the synchronizing codeword the quicker the code will resynchronize ferguson and rabinowitz 1984 suggest the problem of finding for a given set of codeword lengths the binary huffman equivalent code with the shortest synchronizing codeword we consider binary huffman equivalent codes whose shortest codeword has length m 1 and which contain a synchronizing codeword of length m 1 the shortest possible in this case we provide an algorithm for constructing these codes for a given set of codeword lengths if such a code exists we study further properties of these codes and show that when in m spl ges 3 the codes contain more than one synchronizing codeword finally we suggest ways of improving the synchronization properties of the codes and provide some example codes

automated sulcal segmentation using watersheds on the cortical surface the human cortical surface is a highly complex folded structure sulci the spaces between the folds define location on the cortex and provide a parcellation into anatomically distinct areas a topic that has recently received increased attention is the segmentation of these sulci from magnetic resonance images with most work focusing on extracting either the sulcal spaces between the folds or curve representations of sulci unlike these methods we propose a technique that extracts actual regions of the cortical surface that surround sulci which we call sulcal regions the method is based on a watershed algorithm applied to a geodesic depth measure on the cortical surface a well known problem with the watershed algorithm is a tendency toward oversegmentation meaning that a single region is segmented as several pieces to address this problem we propose a postprocessing algorithm that merges appropriate segments from the watershed algorithm the sulcal regions are then manually labeled by simply selecting the appropriate regions with a mouse click and a preliminary study of sulcal depth is reported finally a scheme is presented for computing a complete parcellation of the cortical surface