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

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

implicit brain imaging we describe how implicit surface representations can be used to solve fundamental problems in brain imaging this kind of representation is not only natural following the state of the art segmentation algorithms reported in the literature to extract the different brain tissues but it is also as shown in this paper the most appropriate one from the computational point of view examples are provided for finding constrained special curves on the cortex such as sulcal beds regularizing surface based measures such as cortical thickness and for computing warping fields between surfaces such as the brain cortex all these result from efficiently solving partial differential equations pdes and variational problems on surfaces represented in implicit form the implicit framework avoids the need to construct intermediate mappings between 3 d anatomical surfaces and parametric objects such planes or spheres a complex step that introduces errors and is required by many other cortical processing approaches

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

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

automated 3 d extraction and evaluation of the inner and outer cortical surfaces using a laplacian map and partial volume effect classification accurate reconstruction of the inner and outer cortical surfaces of the human cerebrum is a critical objective for a wide variety of neuroimaging analysis purposes including visualization morphometry and brain mapping the anatomic segmentation using proximity asp algorithm previously developed by our group provides a topology preserving cortical surface deformation method that has been extensively used for the aforementioned purposes however constraints in the algorithm to ensure topology preservation occasionally produce incorrect thickness measurements due to a restriction in the range of allowable distances between the gray and white matter surfaces this problem is particularly prominent in pediatric brain images with tightly folded gyri this paper presents a novel method for improving the conventional asp algorithm by making use of partial volume information through probabilistic classification in order to allow for topology preservation across a less restricted range of cortical thickness values the new algorithm also corrects the classification of the insular cortex by masking out subcortical tissues for 70 pediatric brains validation experiments for the modified algorithm constrained laplacian asp clasp were performed by three methods i volume matching between surface masked gray matter gm and conventional tissue classified gm ii surface matching between simulated and clasp extracted surfaces and iii repeatability of the surface reconstruction among 16 mri scans of the same subject in the volume based evaluation the volume enclosed by the clasp wm and gm surfaces matched the classified gm volume 13 more accurately than using conventional asp in the surface based evaluation using synthesized thick cortex the average difference between simulated and extracted surfaces was 4 6 t 1 4 mm for conventional asp and 0 5 t 0 4 mm for clasp in a repeatability study clasp produced a 30 lower rms error for the gm surface and a 8 lower rms error for the

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

a primal sketch of the cortex mean curvature a morphogenesis based approach to study the variability of the folding patterns in this paper we propose a new representation of the cortical surface that may be used to study the cortex folding process and to recover some putative stable anatomical landmarks called sulcal roots usually buried in the depth of adult brains this representation is a primal sketch derived from a scale space computed for the mean curvature of the cortical surface this scale space stems from a diffusion equation geodesic to the cortical surface the primal sketch is made up of objects defined from mean curvature minima and saddle points the resulting sketch aims first at highlighting significant elementary cortical folds second at representing the fold merging process during brain growth the relevance of the framework is illustrated by the study of central sulcus sulcal roots from antenatal to adult age some results are proposed for ten different brains some preliminary results are also provided for superior temporal sulcus

on the topology of the lu attractor and related systems we use well established methods of knot theory to study the topological structure of the set of periodic orbits of the l u attractor we show that for a specific set of parameters the l u attractor is topologically different from the classical lorenz attractor whose dynamics is formed by a double cover of the simple horseshoe this argues against the similarity between the l and lorenz attractors claimed for these parameter values by some authors on the basis of non topological observations however we show that the l system belongs to the lorenz like family since by changing the values of the parameters the behaviour of the system follows the behaviour of all members

semi continuity of skeletons in two manifold and discrete voronoi approximation the skeleton of a 2d shape is an important geometric structure in pattern analysis and computer vision in this paper we study the skeleton of a 2d shape in a two manifold mathcal m based on a geodesic metric we present a formal definition of the skeleton s omega for a shape omega in mathcal m and show several properties that make s omega distinct from its euclidean counterpart in mathbb r 2 we further prove that for a shape sequence lbrace omega i rbrace that converge to a shape omega in mathcal m the mapping omega rightarrow overline s omega is lower semi continuous a direct application of this result is that we can use a set p of sample points to approximate the boundary of a 2d shape omega in mathcal m and the voronoi diagram of p inside omega subset mathcal m gives a good approximation to the skeleton s omega examples of skeleton computation in topography and brain morphometry are illustrated

classification of strange attractors by integers we show how to characterize a strange attractor by a set of integers these are extracted from the chaotic time series data by first reconstructing the low period orbits and then determining the template or knot holder which supports all periodic orbits embedded in the strange attractor and the strange attractor itself the template is identified by a set of integers which therefore characterize the strange attractor this identification is explicitly demonstrated for the pirogon using a relatively small data set 5000 points

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

segmentation registration and measurement of shape variation via image object shape a model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives metrics are described that compute an object representation s prior probability of local geometry by reflecting variabilities in the net s node and link parameter values and that compute a likelihood function measuring the degree of match of an image to that object representation a paradigm for image analysis of deforming such a model to optimize a posteriori probability is described and this paradigm is shown to be usable as a uniform approach for object definition object based registration between images of the same or different imaging modalities and measurement of shape variation of an abnormal anatomical object compared with a normal anatomical object examples of applications of these methods in radiotherapy surgery and psychiatry are given

a topology preserving level set method for geometric deformable models active contour and surface models also known as deformable models are powerful image segmentation techniques geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior parameterization independence and ease of implementation however a long claimed advantage of geometric deformable models the ability to automatically handle topology changes turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved we present a new class of geometric deformable models designed using a novel topology preserving level set method which achieves topology preservation by applying the simple point concept from digital topology these new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces moreover since the topology preserving constraint is enforced efficiently through local computations the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm

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

spectral based automatic labeling and refining of human cortical sulcal curves using expert provided examples we present a spectral based method for automatically labeling and refining major sulcal curves of a human cerebral cortex given a set of input unlabeled sulcal curves automatically extracted from a cortical surface and a collection of expert provided examples labeled sulcal curves our objective is to identify the input major sulcal curves and assign their neuroanatomical labels and then refines these curves based on the expert provided example data without employing any atlas based registration scheme as preprocessing in order to construct the example data neuroanatomists manually labeled a set of 24 major sulcal curves 12 each for the left and right hemispheres for each individual subject according to a precise protocol we collected 30 sets of such curves from 30 subjects given the raw input sulcal curve set of a subject we choose the most similar example curve to each input curve in the set to label and refine the latter according to the former we adapt a spectral matching algorithm to choose the example curve by exploiting the sulcal curve features and their relationship the high dimensionality of sulcal curve data in spectral matching is addressed by using their multi resolution representations which greatly reduces time and space complexities our method provides consistent labeling and refining results even under high variability of cortical sulci across the subjects through experiments we show that the results are comparable in accuracy to those done manually most output curves exhibited accuracy values higher than 80 and the mean accuracy values of the curves in the left and the right hemispheres were 84 69 and 84 58 respectively

an approximate stochastic process for computer simulation of the ising model at equilibrium develops the stochastic models sm method which is an approximate computer simulation technique for treating many body systems in thermodynamic equilibrium suggested by alexandrowicz 1971 1972 the method is applied as a test to the critical region of the square ising lattice first the authors define the exact transition probabilities tp which are parameter independent since their calculation is impractical for a large lattice they also define approximate tp based on two parameters which can be improved systematically the saving in computer time compared with previous work where 10 parameters have been used is significant and the accuracy of the free energy is increased by a factor of 4 100 the results for chi the magnetic susceptibility are also improved in particular for t tc using the finite size scaling theory and assuming v 1 the authors estimate at tc the exponent gamma to be 1 80 or 0 12

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

adaptive reproducing kernel particle method for extraction of the cortical surface we propose a novel adaptive approach based on the reproducing kernel particle method rkpm to extract the cortical surfaces of the brain from three dimensional 3 d magnetic resonance images mris to formulate the discrete equations of the deformable model a flexible particle shape function is employed in the galerkin approximation of the weak form of the equilibrium equations the proposed support generation method ensures that support of all particles cover the entire computational domains the deformable model is adaptively adjusted by dilating the shape function and by inserting or merging particles in the high curvature regions or regions stopped by the target boundary the shape function of the particle with a dilation parameter is adaptively constructed in response to particle insertion or merging the proposed method offers flexibility in representing highly convolved structures and in refining the deformable models self intersection of the surface during evolution is prevented by tracing backward along gradient descent direction from the crest interface of the distance field which is computed by fast marching these operations involve a significant computational cost the initial model for the deformable surface is simple and requires no prior knowledge of the segmented structure no specific template is required e g an average cortical surface obtained from many subjects the extracted cortical surface efficiently localizes the depths of the cerebral sulci unlike some other active surface approaches that penalize regions of high curvature comparisons with manually segmented landmark data are provided to demonstrate the high accuracy of the proposed method we also compare the proposed method to the finite element method and to a commonly used cortical surface extraction approach the cruise method we also show that the independence of the shape functions of the rkpm from the underlying mesh enhances the convergence speed of the deformable model

a new method for simulation of real chains scanning future steps a new method for simulating real polymer chains is developed and applied to self avoiding walks saws of length 49 599 on a three choice square lattice very good results for the entropy are obtained which deviate from the series expansion estimates by 0 1 2 the author also discusses how to extend the method to models of polymer chains with both excluded volume and finite interactions attractive or repulsive the author s method is expected to be more efficient than other simulation methods for treating self interacting saws and chains which are subject to various lattice constraints

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

symmetry groups for 3d dynamical systems we present a systematic way to construct dynamical systems with a specific symmetry group g each symmetric strange attractor has a unique image attractor that is locally identical to it but different at the global topological level image attractors can be lifted to many inequivalent covering attractors these are distinguished by an index that has related topological algebraic and group theoretical interpretations these methods are used to describe dynamical systems with symmetry groups v4 s4 and s6

automatic and robust computation of 3d medial models incorporating object variability this paper presents a novel processing scheme for the automatic and robust computation of a medial shape model which represents an object population with shape variability the sensitivity of medial descriptions to object variations and small boundary perturbations are fundamental problems of any skeletonization technique these problems are approached with the computation of a model with common medial branching topology and grid sampling this model is then used for a medial shape description of individual objects via a constrained model fit the process starts from parametric 3d boundary representations with existing point to point homology between objects the voronoi skeleton of each sampled object boundary is partitioned into non branching medial sheets and simplified by a novel pruning algorithm using a volumetric contribution criterion using the surface homology medial sheets are combined to form a common medial branching topology finally the medial sheets are sampled and represented as meshes of medial primitives results on populations of up to 184 biological objects clearly demonstrate that the common medial branching topology can be described by a small number of medial sheets and that even a coarse sampling leads to a close approximation of individual objects

classification of dynamical systems based on a decomposition of their vector fields abstract we present a method for the global classification of dynamical systems based on a specific decomposition of their vector fields every differentiable vector field on r n can be decomposed uniquely in the sum of 2 systems one gradient and one that leaves invariant the spheres s n 1 we show that under some conditions the topological class of a vector field is determined by the topological classes of its summands we illustrate this method by applying it to a number of vector fields among them being some members of the so called lorenz family the advantage of such a classification is that equivalent flows exhibit qualitatively the same dynamical phenomena as their parameters are varied

joint sulcal detection on cortical surfaces with graphical models and boosted priors in this paper we propose an automated approach for the joint detection of major sulci on cortical surfaces by representing sulci as nodes in a graphical model we incorporate markovian relations between sulci and formulate their detection as a maximum a posteriori map estimation problem over the joint space of major sulci to make the inference tractable a sample space with a finite number of candidate curves is automatically generated at each node based on the hamilton jacobi skeleton of sulcal regions using the adaboost algorithm we learn both individual and pairwise shape priors of sulcal curves from training data which are then used to define potential functions in the graphical model based on the connection between adaboost and logistic regression finally belief propagation is used to perform the map inference and select the joint detection results from the sample spaces of candidate curves in our experiments we quantitatively validate our algorithm with manually traced curves and demonstrate the automatically detected curves can capture the main body of sulci very accurately a comparison with independently detected results is also conducted to illustrate the advantage of the joint detection approach