swept regions and surfaces modeling and volumetric properties we consider swept regions w and swept hypersurfaces b in r n 1 and especially r 3 which are a disjoint union of subspaces w t w p t or b t b p t obtained from a varying family of affine subspaces p t t c we concentrate on the case where w and b are obtained from a skeletal structure m u this generalizes the blum medial axis m of a region w which consists of the centers of interior spheres tangent to the boundary b at two or more points with u denoting the vectors from the centers of the spheres to the points of tangency we extend methods developed for skeletal structures so that they can be deduced from the properties of the individual intersections w t or b t and a relative shape operator s r e l which we introduce to capture changes relative to the varying family p t we use these results to deduce modeling properties of the global b in terms of the individual b t and determine volumetric properties of regions w expressed as global integrals of functions g on w in terms of iterated integrals over the skeletal structure of w t which is then integrated over the parameter space c

transitions of the 3d medial axis under a one parameter family of deformations the instabilities of the medial axis of a shape under deformations have long been recognized as a major obstacle to its use in recognition and other applications these instabilities or transitions occur when the structure of the medial axis graph changes abruptly under deformations of shape the recent classification of these transitions in 2d for the medial axis and for the shock graph was a key factor in the development of an object recognition system where the classified instabilities were utilized to represent deformation paths the classification of generic transitions of the 3d medial axis could likewise potentially lead to a similar representation in 3d in this paper these transitions are classified by examining the order of contact of spheres with the surface leading to an enumeration of possible transitions which are then examined on a case by case basis some cases are ruled out as never occurring in any family of deformations while others are shown to be nongeneric in a one parameter family of deformations finally the remaining cases are shown to be viable by developing a specific example for each our work is inspired by that of bogaevsky who obtained the transitions as part of an investigation of viscosity solutions of hamilton jacobi equations our contribution is to give a more down to earth approach bringing this work to the attention of the computer vision community and to provide explicit constructions for the various transitions using simple surfaces we believe that the classification of these transitions is vital to the successful regularization of the medial axis in its use in real applications

electrical impedance tomography of translationally uniform cylindrical objects with general cross sectional boundaries an algorithm is developed for electrical impedance tomography eit of finite cylinders with general cross sectional boundaries and translationally uniform conductivity distributions the electrodes for data collection are assumed to be placed around a cross sectional plane therefore the axial variation of the boundary conditions and the potential field are expanded in fourier series for each fourier component a two dimensional 2 d partial differential equation is derived thus the 3 d forward problem is solved as a succession of 2 d problems and it is shown that the fourier series can be truncated to provide substantial savings in computation time the finite element method is adopted and the accuracy of the boundary potential differences gradients thus calculated is assessed by comparison to results obtained using cylindrical harmonic expansions for circular cylinders a 1016 element and 541 node mesh is found to be optimal the algorithm is applied to data collected from phantoms and the errors incurred from the several assumptions of the method are investigated

statistical analyses of brain surfaces using gaussian random fields on 2 d manifolds interest in the morphometric analysis of the brain and its subregions has recently intensified because growth or degeneration of the brain in health or illness affects not only the volume but also the shape of cortical and subcortical brain regions and new image processing techniques permit detection of small and highly localized perturbations in shape or localized volume with remarkable precision an appropriate statistical representation of the shape of a brain region is essential however for detecting localizing and interpreting variability in its surface contour and for identifying differences in volume of the underlying tissue that produce that variability across individuals and groups of individuals our statistical representation of the shape of a brain region is defined by a reference region for that region and by a gaussian random field grf that is defined across the entire surface of the region we first select a reference region from a set of segmented brain images of healthy individuals the grf is then estimated as the signed euclidean distances between points on the surface of the reference region and the corresponding points on the corresponding region in images of brains that have been coregistered to the reference correspondences between points on these surfaces are defined through deformations of each region of a brain into the coordinate space of the reference region using the principles of fluid dynamics the warped coregistered region of each subject is then unwarped into its native space simultaneously bringing into that space the map of corresponding points that was established when the surfaces of the subject and reference regions were tightly coregistered the proposed statistical description of the shape of surface contours makes no assumptions other than smoothness about the shape of the region or its grf the description also allows for the detection and localization of statistically significant differences in the shapes of the surfaces across groups of subjects at both a fine and coarse scale we demonstrate the effectiveness of these statistical methods by applying them to study differences in shape of the amygdala and hippocampus in a large sample of normal subjects and in subjects with attention deficit hyperactivity disorder adhd

sensitivity and stability analysis in dea with interval data in this paper we find radius of stability for all decision making units with interval data in this approach organization classification remains unchanged under perturbations of the interval data some numerical examples for illustration are given

three dimensional shape description using the symmetric axis transform i theory blum s two dimensional shape description method based on the symmetric axis transform sat is generalized to three dimensions the method uniquely decomposes an object into a collection of sub objects each drawn from three separate but not completely independent primitive sets defined in the paper width primitives based on radius function properties axis primitives based on symmetric axis curvatures and boundary primitives based on boundary surface curvatures width primitives are themselves comprised of two components slope districts and curvature districts visualizing the radius function as if it were the height function of some mountainous terrain each slope district corresponds to a mountain face together with the valley below it curvature districts further partition each slope district into regions that are locally convex concave or saddle like similarly axis boundary primitives are regions of the symmetric surface where the symmetric surface boundary surfaces are locally convex concave or saddle like relations among the primitive sets are discussed

hierarchical decomposition and axial shape description a method for producing a segmented axial description of a given shape together with a hierarchical decomposition of the shape into its parts is presented the novelty of this approach lies in the combination of several competing approaches and tools into a unified scheme and an efficient implementation producing natural descriptions smooth local symmetries are used for the axial description of parts which are suggested by curvature sign changes parallel symmetries are used to provide information on global relationships within the shape this information is used for parsing shape into a hierarchy of parts this approach uses both region and contour information can handle shapes with corners and addresses the issue of local versus global information the issue of scale and the notion of part the method is computationally efficient robust and stable results that show that it provides an intuitive shape description are included

imprecise data envelopment analysis idea a review and improvement with an application abstract the standard data envelopment analysis dea method requires that the values for all inputs and outputs be known exactly when some outputs and inputs are unknown decision variables such as bounded data ordinal data and ratio bounded data the dea model becomes a non linear programming problem and is called imprecise dea idea there are two different approaches in dealing with imprecise outputs and inputs one uses scale transformations and variable alternations to convert the non linear idea model into a linear program the other converts imprecise data into exact data and then uses the standard linear dea model the current paper reviews and compares the two approaches through an efficiency analysis of a set of telephone offices a simplified approach is developed to reduce the computational burden if one uses the first approach the treatment of weight restrictions in idea is discussed it is shown that weight restrictions on imprecise data are redundant new developments and improvements to both approaches are provided

on fdh efficiency analysis with interval data most of dea models make an assumption that input and output data are exact without any variation cooper et al manage sci 45 1999 597 introduced imprecise data envelopment analysis idea when some inputs and outputs are unknown decision variables such as bounded data ordinal data and ratio bounded data the dea model becomes a non linear programming problem in this paper we consider fdh model with imprecise data first we introduce linear form of fdh model j product anal 16 2001 129 then according to eur j oper res 140 2002 24 we follow efficiency analysis

principal geodesic analysis for the study of nonlinear statistics of shape a primary goal of statistical shape analysis is to describe the variability of a population of geometric objects a standard technique for computing such descriptions is principal component analysis however principal component analysis is limited in that it only works for data lying in a euclidean vector space while this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points it does not handle more complex representations of shape we have been developing representations of geometry based on the medial axis description or m rep while the medial representation provides a rich language for variability in terms of bending twisting and widening the medial parameters are not elements of a euclidean vector space they are in fact elements of a nonlinear riemannian symmetric space in this paper we develop the method of principal geodesic analysis a generalization of principal component analysis to the manifold setting we demonstrate its use in describing the variability of medially defined anatomical objects results of applying this framework on a population of hippocampi in a schizophrenia study are presented

group wise registration of point sets for statistical shape models this paper presents a novel fast group wise registration technique based on establishing soft correspondences between groups of point sets the registration approach is used to create a statistical shape model capable of learning the shape variations within a training set the shape model consists of a mean shape and its transformations to all training shapes we decouple the procedure into two steps updating the mean shape and registering it to the training shapes the algorithm alternates between these two steps until convergence following the generation of the statistical shape model we use the soft correspondence approach to register the model to a new observation we perform extensive experiments on two data sets lumbar spine and hippocampi we compare our algorithm to available state of the art group wise registration algorithms including feature based and volume based approaches we demonstrate improved generalization specificity and compactness compared to these algorithms

medial axis transformation of a planar shape the medial axis transformation is a means first proposed by blum to describe a shape in this paper we present a 0 n log n algorithm for computing the medial axis of a planar shape represented by an n edge simple polygon the algorithm is an improvement over most previously known results interms of both efficiency and exactness and has been implemented in fortran some computer plotted output of the program are also shown in the paper

from local maxima to connected skeletons in picture processing it is often convenient to deal with a stick like version skeleton of binary digital images although skeleton connectedness is not necessary for storage and retrieval purposes this property is desirable when a structural description of images is of interest in this paper a parallel procedure is described which applied to a connected image originates a connected skeleton made by the union of simple digital arcs the procedure involves a step by step propagation of the background over the image at every step contour elements either belonging to the significant convex regions of the current image or being local maxima of the original image are selected as skeleton elements since the final set so obtained is not ensured to be connected the configurations in correspondence of which disconnections appear are investigated and the procedures to avoid this shortcoming are given the presence of the whole set of local maxima among the skeleton elements ensures the possibility of recovering the original image by means of a reverse distance transform the details of the program implementing the proposed algorithm on a parallel processor are finally included

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

anatomical labeling of the circle of willis using maximum a posteriori probability estimation anatomical labeling of the cerebral arteries forming the circle of willis cow enables inter subject comparison which is required for geometric characterization and discovering risk factors associated with cerebrovascular pathologies we present a method for automated anatomical labeling of the cow by detecting its main bifurcations the cow is modeled as rooted attributed relational graph with bifurcations as its vertices whose attributes are characterized as points on a riemannian manifold the method is first trained on a set of pre labeled examples where it learns the variability of local bifurcation features as well as the variability in the topology then the labeling of the target vasculature is obtained as maximum a posteriori probability map estimate where the likelihood of labeling individual bifurcations is regularized by the prior structural knowledge of the graph they span the method was evaluated by cross validation on 50 subjects imaged with magnetic resonance angiography and showed a mean detection accuracy of 95 in addition besides providing the map the method can rank the labelings the proposed method naturally handles anatomical structural variability and is demonstrated to be suitable for labeling arterial segments of the cow

using monte carlo method for ranking efficient dmus for ranking efficient dmus some methods have been developed these methods are not able to rank non extreme efficient dmus in this paper using monte carlo method a method has been developed which is able to rank all extreme and non extreme efficient dmus

deformable prototypes for encoding shape categories in image databases abstract we describe a method for shape based image database search that uses deformable prototypes to represent categories rather than directly comparing a candidate shape with all shape entries in the database shapes are compared in terms of the types of nonrigid deformations differences that relate them to a small subset of representative prototypes to solve the shape correspondence and alignment problem we employ the technique of em modal matching an information preserving shape decomposition for matching describing and comparing shapes despite sensor variations and nonrigid deformations in modal matching shape is decomposed into an ordered basis of orthogonal principal components we demonstrate the utility of this approach for shape comparison in 2 d image databases

ranking of decision making units in data envelopment analysis a distance based approach in non parametric methods many units are evaluated as efficient several methods for prioritizing efficient units are reported in the literature andersen et al and thereafter mehrabian et al proposed two methods for ranking efficient units but both methods break down in some cases this paper describes a new dea ranking approach that uses l 2 norm in proposed dea ranking approach the difficulties of the existing methods have been removed

an unified multiscale framework for planar surface and curve skeletonization computing skeletons of 2d shapes and medial surface and curve skeletons of 3d shapes is a challenging task in particular there is no unified framework that detects all types of skeletons using a single model and also produces a multiscale representation which allows to progressively simplify or regularize all skeleton types in this paper we present such a framework we model skeleton detection and regularization by a conservative mass transport process from a shape s boundary to its surface skeleton next to its curve skeleton and finally to the shape center the resulting density field can be thresholded to obtain a multiscale representation of progressively simplified surface or curve skeletons we detail a numerical implementation of our framework which is demonstrably stable and has high computational efficiency we demonstrate our framework on several complex 2d and 3d shapes

shape matching and modeling using skeletal context shape is a significant visual clue for human perception and shape models show considerable promise as a basis for extracting objects from images this paper proposes a novel approach for shape matching and modeling using the symmetry characterization of shape interior and the spatial relationships of shape structures based on the representative skeletal features we develop a mechanism to generate a coarse segment matching between different instances of an object additionally the natural correspondence of skeletal branches to sequential segments along the shape curves is employed in the matching process to avoid false correspondences across different segments point matches within the corresponding segments are then obtained by solving a constrained assignment problem the validation of the proposed approach is illustrated on various data sets in the presence of considerable deformation and occlusion and the results are compared with those of popular approaches we also demonstrate the performance of our method on biological objects for shape modeling showing better models than those obtained by the state of the art shape modeling approaches

growth motion and 1 parameter families of symmetry sets associated to every plane curve there is the locus of centres of circles bitangent to that curve the so called symmetry set of the curve we can view this set as the spine of our curve which can be recovered by taking the envelope of circles of varying radii along this spine varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object fixing the symmetry set and varying the radius function can be considered to model growth crudely in this paper we shall describe the generic changes in the curves which take place in the process of growth and motion and outline the corresponding results for spheres centred on a space curve we also use the idea of a stratified morse function to describe the generic changes which occur in one parameter families of full bifurcation sets in the plane applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set and evolute which occur in a generic isotopy of a plane curve

characterizing the shape of anatomical structures with poisson s equation poisson s equation a fundamental partial differential equation in classical physics has a number of properties that are interesting for shape analysis in particular the equipotential sets of the solution graph become smoother as the potential increases we use the displacement map the length of the streamlines formed by the gradient field of the solution to measure the complexity or smoothness of the equipotential sets and study its behavior as the potential increases we believe that this function complexity f potential which we call the shape characteristic is a very natural way to express shape robust algorithms are presented to compute the solution to poisson s equation the displacement map and the shape characteristic we first illustrate our technique on two dimensional synthetic examples and natural silhouettes we then perform two shape analysis studies on three dimensional neuroanatomical data extracted from magnetic resonance mr images of the brain in the first study we investigate changes in the caudate nucleus in schizotypal personality disorder spd and confirm previously published results on this structure in the second study we present a data set of caudate nuclei of premature infants with asymmetric white matter injury our method shows structural shape differences that volumetric measurements were unable to detect

continuous medial representation for anatomical structures the m rep approach pioneered by pizer 2003 is a powerful morphological tool that makes it possible to employ features derived from medial loci skeletons in shape analysis this paper extends the medial representation paradigm into the continuous realm modeling skeletons and boundaries of three dimensional objects as continuous parametric manifolds while also maintaining the proper geometric relationship between these manifolds the parametric representation of the boundary medial relationship makes it possible to fit shape based coordinate systems to the interiors of objects providing a framework for combined statistical analysis of shape and appearance our approach leverages the idea of inverse skeletonization where the skeleton of an object is defined first and the object s boundary is derived analytically from the skeleton this paper derives a set of sufficient conditions ensuring that inverse skeletonization is well posed for single manifold skeletons and formulates a partial differential equation whose solutions satisfy the sufficient conditions an efficient variational algorithm for deformable template modeling using the continuous medial representation is described and used to fit a template to the hippocampus in 87 subjects from a schizophrenia study with sub voxel accuracy and 95 mean overlap

time harmonic impedance tomography using the t matrix method a time harmonic formulation for the electrical impedance tomography eit inverse problem accounting for electrodynamic effects is derived this work abandons the standard electrostatic impedance model for a full wave t matrix model the advantage of this method is an accurate physical model that includes finite frequency effects such as diffusion phenomena and electrode contact impedance effects this model offers the potential for increased resolution and larger invertible contrast objects than other methods when used on experimental data because it may represent a more realistic physical model also an accurate gradient matrix is used in the newton iterative method so the image reconstruction converges in a few iterations these advantages are realized with no increase in the computational complexity of this algorithm compared to the static finite element model a calibration technique is suggested for measurement systems to test the validity of a theoretical model that includes electrode contact impedance effects

variational curve skeletons using gradient vector flow representing a 3d shape by a set of 1d curves that are locally symmetric with respect to its boundary i e curve skeletons is of importance in several machine intelligence tasks this paper presents a fast automatic and robust variational framework for computing continuous subvoxel accurate curve skeletons from volumetric objects a reference point inside the object is considered a point source that transmits two wave fronts of different energies the first front beta front converts the object into a graph from which the object salient topological nodes are determined curve skeletons are tracked from these nodes along the cost field constructed by the second front alpha front until the point source is reached the accuracy and robustness of the proposed work are validated against competing techniques as well as a database of 3d objects unlike other state of the art techniques the proposed framework is highly robust because it avoids locating and classifying skeletal junction nodes employs a new energy that does not form medial surfaces and finally extracts curve skeletons that correspond to the most prominent parts of the shape and hence are less sensitive to noise

generic structure of two dimensional dimages under gaussian blurring for medical imaging problems computer scientists pizer and eberly introduced the core as the analogue for grayscale images of the medial axis it is defined for two dimensional images as the ridge of a medial function defined on 2 1 dimensional scale space in this paper we extend methods from singularity theory to analyze the generic properties of ridges and cores for two dimensional images including the generic transitions which occur for sequences of images varying with a parameter such as time we do so by placing the ridge construction into a larger relative critical set structure which captures properties undetected by the disjoint ridge core segments the genericity properties are established using transversality conditions for associated mappings we introduce a method for establishing transversality conditions for medial functions obtained by gaussian blurring by using a basis of local solutions to the heat equation the method applies to many other properties besides those we consid

on return to scale of fully efficient dmus in data envelopment analysis under interval data recently cooper et al manage sci 45 1999 597 proposed imprecise data envelopment analysis idea a situation in which some or all data are unknown the data in idea could be interval data ordinal data or data with bounded ratio usually in such cases the corresponding data envelopment analysis models becomes non linear when data assume to be interval decision making units dmus can divide to three classes as follows i efficient in any cases ii efficient in maximal sense iii always inefficient in this paper we investigate to the return to scale of dmus that are locate in first class also we introduce models that use exact data and present a combination if any which via that the dmu becomes a most productive scale size

a procedure for ranking efficient units in data envelopment analysis data envelopment analysis dea evaluates the relative efficiency of decision making units dmus but does not allow for a ranking of the efficient units themselves a modified version of dea based upon comparison of efficient dmus relative to a reference technology spanned by all other units is developed the procedure provides a framework for ranking efficient units and facilitates comparison with rankings based on parametric methods

shape reconstruction on a varying mesh a central class of image understanding problems is concerned with reconstructing a shape from an incomplete data set such as fitting a surface to partially given contours a new theory for solving such problems is presented unlike the current heuristic methods the method used starts from fundamental principles that should be followed by any reconstruction method regardless of its mathematical or physical implementation a mathematical procedure which conforms to these principles is presented one major advantage of the method is the ability to handle shapes containing both smooth and sharp parts without using thresholds a sharp variation such as a corner requires a high resolution mesh for adequate representation while slowly varying sections can be represented with sparser mesh points unlike current methods this procedure fits the surface on a varying mesh the mesh is constructed automatically to be more dense at parts of the image that have more rapid variation analytical examples are given in simple cases followed by numerical experiments

multiscale medial loci and their properties blum s medial axes have great strengths in principle in intuitively describing object shape in terms of a quasi hierarchy of figures but it is well known that derived from a boundary they are damagingly sensitive to detail in that boundary the development of notions of spatial scale has led to some definitions of multiscale medial axes different from the blum medial axis that considerably overcame the weakness three major multiscale medial axes have been proposed iteratively pruned trees of voronoi edges ogniewicz 1993 szekely 1996 naf 1996 shock loci of reaction diffusion equations kimia et al 1995 siddiqi and kimia 1996 and height ridges of medialness cores fritsch et al 1994 morse et al 1993 pizer et al 1998 these are different from the blum medial axis and each has different mathematical properties of generic branching and ending properties singular transitions and geometry of implied boundary and they have different strengths and weaknesses for computing object descriptions from images or from object boundaries these mathematical properties and computational abilities are laid out and compared and contrasted in this paper

the maximum and minimum number of efficient units in dea with interval data in original data envelopment analysis dea models inputs and outputs are measured by exact values on a ratio scale cooper et al manage sci 45 1999 597 recently addressed the problem of imprecise data in dea in its general form in this case units do not have constant efficiency score and their efficiency score depend on the choice of data of units therefore it is attractive to know upper and lower bounds for the number of efficient units in this paper we concentrate on dea with interval data and describe which data settings of units produce the maximum number of efficient units then we discuss about the minimum number of efficient units

ridges crests and sub parabolic lines of evolving surfaces the ridge lines on a surface can be defined either via contact of the surface with spheres or via extrema of principal curvatures along lines of curvature certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications there is a related concept of sub parabolic line on a surface also defined via extrema of principal curvatures in this paper we study in detail the structure of the ridge lines crest lines and sub parabolic lines on a generic surface and on a surface which is evolving in a generic one parameter family the mathematical details of this study are in bruce et al 1994c

modified maj model for ranking decision making units in data envelopment analysis abstract efficiency data envelopment analysis dea models assessing decision making units dmus are unable to discriminate between efficient dmus from experience we know that usually plural dmus are efficient to discriminate between these efficient dmus is an interesting subject to this aim super efficiency models are proposed in this paper we study a super efficiency model namely maj each time an efficient dmu is excluded from the set of the observed dmus a new production possibility set pps is obtained in this model ranking is done based on the position of each excluded efficient dmu in relation to its corresponding new pps if the efficient dmus and the new ppss remain unchanged it is expected that the ranking also remain unchanged but in this paper we show that the technique used for rendering this model unit invariant causes the ranking to change when some inputs of some inefficient dmus change without causing any change in the new pps in this paper we modify this model so that this problem will not occur

hierarchic voronoi skeletons abstract robust and time efficient skeletonization of a planar shape which is connectivity preserving and based on euclidean metrics can be achieved by first regularizing the voronoi diagram vd of a shape s boundary points i e by removal of noise sensitive parts of the tessellation and then by establishing a hierarchic organization of skeleton constituents each component of the vd is attributed with a measure of prominence which exhibits the expected invariance under geometric transformations and noise the second processing step a hierarchic clustering of skeleton branches leads to a multiresolution representation of the skeleton termed skeleton pyramid

interval efficiency assessment using data envelopment analysis this paper studies how to conduct efficiency assessment using data envelopment analysis dea in interval and or fuzzy input output environments a new pair of interval dea models is constructed on the basis of interval arithmetic which differs from the existing dea models handling interval data in that the former is a linear ccr model without the need of extra variable alternations and uses a fixed and unified production frontier i e the same constraint set to measure the efficiencies of decision making units dmus with interval input and output data while the latter is usually a nonlinear optimization problem with the need of extra variable alternations or scale transformations and utilizes variable production frontiers i e different constraint sets to measure interval efficiencies ordinal preference information and fuzzy data are converted into interval data through the estimation of permissible intervals and a level sets respectively and are incorporated into the interval dea models the proposed interval dea models are developed for measuring the lower and upper bounds of the best relative efficiency of each dmu with interval input and output data which are different from the interval formed by the worst and the best relative efficiencies of each dmu a minimax regret based approach mra is introduced to compare and rank the efficiency intervals of dmus two numerical examples are provided to show the applications of the proposed interval dea models and the preference ranking approach

untangling the blum medial axis transform for over 30 years blum s medial axis transform mat has proven to be an intriguing tool for analyzing and computing with form but it is one that is notoriously difficult to apply in a robust and stable way it is well documented how a tiny change to an object s boundary can cause a large change in its mat there has also been great difficulty in using the mat to decompose an object into a hierarchy of parts reflecting the natural parts hierarchy that we perceive this paper argues that the underlying cause of these problems is that medial representations embody both the substance of each part of an object and the connections between adjacent parts a small change in an object s boundary corresponds to a small change in its substance but may involve a large change in its connection information the problems with blum s mat are generated because it does not explicitly represent this dichotomy of information to use the blum mat to it s full potential this paper presents a method for separating the substance and connection information of an object this provides a natural parts hierarchy while eliminating instabilities due to small boundary changes the method also allows for graded fuzzy classifications of object parts to match the ambiguity in human perception of many objects

tree structure for contractible regions in r 3 for contractible regions in 3 with generic smooth boundary we determine the global structure of the blum medial axis m we give an algorithm for decomposing m into irreducible components which are attached to each other along fin curves the attaching cannot be described by a tree structure as in the 2d case however a simplified but topologically equivalent medial structure m with the same irreducible components can be described by a two level tree structure the top level describes the simplified form of the attaching and the second level tree structure for each irreducible component specifies how to construct the component by attaching smooth medial sheets to the network of y branch curves the conditions for these structures are complete in the sense that any region whose blum medial axis satisfies the conditions is contractible

data envelopment analysis with imprecise data abstract in original data envelopment analysis dea models inputs and outputs are measured by exact values on a ratio scale cooper et al management science 45 1999 597 607 recently addressed the problem of imprecise data in dea in its general form we develop in this paper an alternative approach for dealing with imprecise data in dea our approach is to transform a non linear dea model to a linear programming equivalent on the basis of the original data set by applying transformations only on the variables upper and lower bounds for the efficiency scores of the units are then defined as natural outcomes of our formulations it is our specific formulation that enables us to proceed further in discriminating among the efficient units by means of a post dea model and the endurance indices we then proceed still further in formulating another post dea model for determining input thresholds that turn an inefficient unit to an efficient one

perestroikas of shocks and singularities of minimum functions the shock discontinuities generically present in inviscid solutions of the forced burgers equation and their bifurcations happening in the course of time perestroikas are classified in two and three dimensions the one dimensional case is well known this classification is a result of selecting among all the perestroikas occurring for minimum functions depending generically on time the ones permitted by the convexity of the hamiltonian of the burgers dynamics topological restrictions on the admissible perestroikas of shocks are obtained the resulting classification can be extended to the so called viscosity solutions of a hamilton jacobi equation provided the hamiltonian is convex

reconstruction of two dimensional permittivity distribution using the distorted born iterative method the distorted born iterative method dbim is used to solve two dimensional inverse scattering problems thereby providing another general method to solve the two dimensional imaging problem when the born and the rytov approximations break down numerical simulations are performed using the dbim and the method proposed previously by the authors int j imaging syst technol vol 1 no 1 p 100 8 1989 called the born iterative method bim for several cases in which the conditions for the first order born approximation are not satisfied the results show that each method has its advantages the dbim shows faster convergence rate compared to the bim while the bim is more robust to noise contamination compared to the dbim

weighted least squares criteria for electrical impedance tomography methods are developed for the design of electrical impedance tomographic reconstruction algorithms with specified properties assuming a starting model with constant conductivity or some other specified background distribution an algorithm with the following properties is found 1 the optimum constant for the starting model is determined automatically 2 the weighted least squares error between the predicted and measured power dissipation data is as small as possible 3 the variance of the reconstructed conductivity from the starting model is minimized 4 potential distributions with the largest volume integral of gradient squared have the least influence on the reconstructed conductivity and therefore distributions most likely to be corrupted by contact impedance effects are deemphasized 5 cells that dissipate the most power during the current injection tests tend to deviate least from the background value for a starting model with nonconstant conductivity the reconstruction algorithm has analogous properties

determining the geometry of boundaries of objects from medial data we consider a region in r2 or r3 with generic smooth boundary b and blum medial axis m on which is defined a multivalued radial vector field u from points x on m to the points of tangency of the sphere at x with b we introduce a radial shape operator srad and an edge shape operator se which measure how u bends along m these are not traditional differential geometric shape operators nonetheless we derive all local differential geometric invariants of b from these operators this allows us to define from m u a geometric medial map on m which corresponds via a radial map from m to b to the differential geometric properties of b the geometric medial map also includes a description of the relative geometry of b this is defined using the relative critical set of the radius function r on m this set consists of a network of curves on m which describe where b is thickest and thinnest it is computed using the covariant derivative of the tangential component of the unit radial vector field we further determine how these invariants are related to the differential geometric invariants of m and how these invariants change under deforming diffeomorphisms of m

rank order data in dea a general framework in data envelopment analysis dea performance evaluation is generally assumed to be based upon a set of quantitative data in many real world settings however it is essential to take into account the presence of qualitative factors when evaluating the performance of decision making units dmus very often rankings are provided from best to worst relative to particular attributes such rank positions might better be presented in an ordinal rather than numerical sense the paper develops a general frame work for modeling and treating qualitative data in dea and provides a unified structure for embedding rank order data into the dea framework the existing techniques are discussed and their equivalence is demonstrated both continuous and discrete projection models are provided it is shown that qualitative data can be treated in conventional dea methodology

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

interval efficiency measures in data envelopment analysis with imprecise data the conventional data envelopment analysis dea measures the relative efficiencies of a set of decision making units dmus with exact values of inputs and outputs for imprecise data i e mixtures of interval data and ordinal data some methods have been developed to calculate the upper bound of the efficiency scores this paper constructs a pair of two level mathematical programming models whose objective values represent the lower bound and upper bound of the efficiency scores respectively based on the concept of productive efficiency and the application of a variable substitution technique the pair of two level nonlinear programs is transformed to a pair of ordinary one level linear programs solving the associated pairs of linear programs produces the efficiency intervals of all dmus an illustrative example verifies the idea of this paper a real case is also provided to give some interpretation of the interval efficiency interval efficiency not only describes the real situation in better detail psychologically it also eases the tension of the dmus being evaluated as well as the persons conducting the evaluation

continuous medial representation of brain structures using the biharmonic pde a new approach for constructing deformable continuous medial models for anatomical structures is presented medial models describe geometrical objects by first specifying the skeleton of the object and then deriving the boundary surface corresponding to the skeleton however an arbitrary specification of a skeleton will not be valid unless a certain set of sufficient conditions is satisfied the most challenging of these is the non linear equality constraint that must hold along the boundaries of the manifolds forming the skeleton the main contribution of this paper is to leverage the biharmonic partial differential equation as a mapping from a codimension 0 subset of euclidean space to the space of skeletons that satisfy the equality constraint the pde supports robust numerical solution on freeform triangular meshes providing additional flexibility for shape modeling the approach is evaluated by generating continuous medial models for a large dataset of hippocampus shapes generalizations to modeling more complex shapes and to representing branching skeletons are demonstrated

computing skeletons in three dimensions skeletonization will probably become as valuable a tool for shape analysis in 3d as it is in 2d we present a topology preserving 3d skeletonization method which computes both surface and curve sk

a note on some of dea models and finding efficiency and complete ranking using common set of weights efficiency evaluation in data envelopment analysis dea depends on different factors the most important factors arc the values of input and output in this paper we present an alternative proof that if one component of output or input vectors of a dmu dominates the corresponding component of other dmus whatever the value of other components of this dmu may be then that dmu is efficient in some of the dea models an important outcome of such an analysis is a set of virtual multipliers or weights accorded to each factor taken into account these sets of weights are typically different for each of the participating dmus in this paper by means of solving only one problem we can determine common set of weights csw for all dmus and their efficiencies finally a method for ranking dmus is presented in this method by solving only two problems efficient dmus are ranked

a decision model for technology selection in the existence of both cardinal and ordinal data technology selection is an important part of management of technology to select the best technologies in the existence of both cardinal and ordinal data this paper proposes an innovative approach which is based on a mathematical programming a numerical example demonstrates the application of the proposed method

modeling multi object configurations via medial skeletal linking structures we introduce a method for modeling a configuration of objects in 2d or 3d images using a mathematical skeletal linking structure which will simultaneously capture the individual shape features of the objects and their positional information relative to one another the objects may either have smooth boundaries and be disjoint from the others or share common portions of their boundaries with other objects in a piecewise smooth manner these structures include a special class of blum medial linking structures which are intrinsically associated to the configuration and build upon the blum medial axes of the individual objects we give a classification of the properties of blum linking structures for generic configurations the skeletal linking structures add increased flexibility for modeling configurations of objects by relaxing the blum conditions and they extend in a minimal way the individual skeletal structures which have been previously used for modeling individual objects and capturing their geometric properties this allows for the mathematical methods introduced for single objects to be significantly extended to the entire configuration of objects these methods not only capture the internal shape structures of the individual objects but also the external structure of the neighboring regions of the objects in the subsequent second paper damon and gasparovic in shape and positional geometry of multi object configurations we use these structures to identify specific external regions which capture positional information about neighboring objects and we develop numerical measures for closeness of portions of objects and their significance for the configuration this allows us to use the same mathematical structures to simultaneously analyze both the shape properties of the individual objects and positional properties of the configuration this provides a framework for analyzing the statistical properties of collections of similar configurations such as for applications to medical imaging

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

the multiresolution gradient vector field skeleton many algorithms suppress skeleton associated with boundary perturbation by preventing their formation or by costly branch pruning this work proposes a novel concept of structural and textural skeletons the former is associated with the general shape structure and the latter with boundary perturbations these skeletons remain disconnected to facilitate gross shape matching without the need for branch pruning they are extracted from a multiresolution gradient vector field that is efficiently generated within a pyramidal framework experimental results show that these skeletons are scale and rotation invariant they are less affected by boundary noise compared to skeletons extracted by popular iterative and non iterative techniques

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

medial based deformable models in nonconvex shape spaces for medical image segmentation we explore the application of genetic algorithms ga to deformable models through the proposition of a novel method for medical image segmentation that combines ga with nonconvex localized medial based shape statistics we replace the more typical gradient descent optimizer used in deformable models with ga and the convex implicit global shape statistics with nonconvex explicit localized ones specifically we propose ga to reduce typical deformable model weaknesses pertaining to model initialization pose estimation and local minima through the simultaneous evolution of a large number of models furthermore we constrain the evolution and thus reduce the size of the search space by using statistically based deformable models whose deformations are intuitive stretch bulge bend and are driven in terms of localized principal modes of variation instead of modes of variation across the entire shape that often fail to capture localized shape changes although ga are not guaranteed to achieve the global optima our method compares favorably to the prevalent optimization techniques convex nonconvex gradient based optimizers and to globally optimal graph theoretic combinatorial optimization techniques when applied to the task of corpus callosum segmentation in 50 mid sagittal brain magnetic resonance images

convergence of a reconstruction method for the inverse conductivity problem the inverse conductivity problem is that of reconstructing a spatially varying isotropic conductivity in the interior of some region by means of steady state measurements taken at the boundary reconstruction schemes including least squares type minimization methods have been widely studied and implemented but convergence analysis has been largely ignored this paper establishes the convergence of a well known least squares minimization scheme the levenberg marquardt method on a regularized formulation of the inverse conductivity problem

efficient shape modeling entropy adaptive coding and boundary curves vs blum s medial axis we propose efficiency of representation as a criterion for evaluating shape models then apply this criterion to compare the boundary curve representation with the medial axis we estimate the entropy of two compact classes of curves we then construct two adaptive encodings for non compact classes of shapes one using the boundary curve and the other using the medial axis and determine precise conditions for when the medial axis is more efficient finally we apply our results to databases of naturally occurring shapes determining whether the boundary or medial axis is more efficient along the way we construct explicit near optimal boundary based approximations for compact classes of shapes construct an explicit compression scheme for non compact classes of shapes based on the medial axis and derive some new results about the medial axis

three dimensional descriptions based on the analysis of the invariant and quasi invariant properties of some curved axis generalized cylinders we address the recovery of object level 3d descriptions of some classes of curved axis generalized cylinders for this the first part of the paper analyzes the projective properties of two common generic shapes planar right constant generalized cylinders prcgcs and circular planar right generalized cylinders circular prgcs the properties we analyze include new geometric invariant and quasi invariant properties of the orthographic projection of the above shapes and a useful classification of their structural properties as functions of their pose the second part of the paper describes an implemented system which detects and recovers prcgcs and circular prgcs from an intensity image in the presence of noise surface markings shadows and partial occlusion the methods exploit the projective properties to hypothesize and verify relevant curved axis objects thus explicitly using the three dimensionality of the objects and of the desired descriptions this work extends past work on the recovery of volumetric shapes from an intensity image by addressing new primitives deriving new properties and by developing a system that recovers them from an intensity image we demonstrate our method on several real intensity images

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

simulating the grassfire transform using an active contour model a method for shape description of planar objects that integrates both region and boundary features is presented the method is an implementation of a 2d dynamic grassfire that relies on a distance surface on which elastic contours minimize an energy function the method is based on an active contour model numerous implementation aspects of the shape description method were optimized a euclidean metric was used for optimal accuracy and the active contour model permits bypassing some of the discretization limitations inherent in using a digital grid noise filtering was performed on the basis of both contour feature measures and region measures that is curvature extremum significance and ridge support respectively to obtain robust shape descriptors other improvements and variations of the algorithmic implementation are proposed

surface and curve skeletonization of large 3d models on the gpu we present a gpu based framework for extracting surface and curve skeletons of 3d shapes represented as large polygonal meshes we use an efficient parallel search strategy to compute point cloud skeletons and their distance and feature transforms fts with user defined precision we regularize skeletons by a new gpu based geodesic tracing technique which is orders of magnitude faster and more accurate than comparable techniques we reconstruct the input surface from skeleton clouds using a fast and accurate image based method we also show how to reconstruct the skeletal manifold structure as a polygon mesh and the curve skeleton as a polyline compared to recent skeletonization methods our approach offers two orders of magnitude speed up high precision and low memory footprints we demonstrate our framework on several complex 3d models

the groupwise medial axis transform for fuzzy skeletonization and pruning medial representations of shapes are useful due to their use of an object centered coordinate system that directly captures intuitive notions of shape such as thickness bending and elongation however it is well known that an object s medial axis transform mat is unstable with respect to small perturbations of its boundary this instability results in additional unwanted branches in the skeletons which must be pruned in order to recover the portions of the skeletons arising purely from the uncorrupted shape information almost all approaches to skeleton pruning compute a significance measure for each branch according to some heuristic criteria and then prune the least significant branches first current approaches to branch significance computation can be classified as either local solely using information from a neighborhood surrounding each branch or global using information about the shape as a whole in this paper we propose a third groupwise approach to branch significance computation we develop a groupwise skeletonization framework that yields a fuzzy significance measure for each branch derived from information provided by the group of shapes we call this framework the groupwise medial axis transform g mat we propose and evaluate four groupwise methods for computing branch significance and report superior performance compared to a recent leading method we measure the performance of each pruning algorithm using denoising classification and within class skeleton similarity measures this research has several applications including object retrieval and shape analysis

technologies ranking in the presence of both cardinal and ordinal data ranking of technologies is an important phase for technology transfer to rank the best technologies in the presence of both cardinal and ordinal data this paper proposes an innovative approach which is based on imprecise data envelopment analysis idea the objective of this paper is to propose a comprehensive reference that discusses the use of idea in technology ranking a numerical example demonstrates the application of the proposed method

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

scaling units via the canonical correlation analysis in the dea context abstract this paper deals with the evaluation of decision making units which have multiple inputs and outputs a new method cca dea is developed where the canonical correlation analysis cca is utilized to provide a full rank scaling for all the units rather than a categorical classification for efficient and inefficient units as done by the data envelopment analysis dea the cca dea approach is an attempt to bridge the gap between the frontier approach of dea and the average tendencies of statistics econometrics nonparametric statistical tests are employed to validate the consistency between the classification from the dea and the postclassification that was generated by the cca dea

a formal classification of 3d medial axis points and their local geometry this paper proposes a novel hypergraph skeletal representation for 3d shape based on a formal derivation of the generic structure of its medial axis by classifying each skeletal point by its order of contact we show that genetically the medial axis consists of five types of points which are then organized into sheets curves and points 1 sheets manifolds with boundary which are the locus of bitangent spheres with regular tangency a sub 1 sup 2 a sub k sup n notation means n distinct k fold tangencies of the sphere of contact as explained in the text two types of curves 2 the intersection curve of three sheets and the locus of centers of tritangent spheres a sub 1 sup 3 and 3 the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principal curvature i e higher order contact a sub 3 points and two types of points 4 centers of quad tangent spheres a sub 1 sup 4 and 5 centers of spheres with one regular tangency and one higher order tangency a sub 1 a sub 3 the geometry of the 3d medial axis thus consists of sheets a sub 1 sup 2 bounded by one type of curve a sub 3 on their free end which corresponds to ridges on the surface and attached to two other sheets at another type of curve a sub 1 sup 3 which support a generalized cylinder description the a sub 3 curves can only end in a sub 1 a sub 3 points where they must meet an a sub 1 sup 3 curve the a sub 1 sup 3 curves meet together in fours at an a sub 1 sup 4 point this formal result leads to a compact representation for 3d shape referred to as the medial axis hypergraph representation consisting of nodes a sub 1 sup 4 and a sub 1 a sub 3 points links between pairs of nodes a sub 1 sup 3 and a sub 3 curves and hyperlinks between groups of links a sub 1 sup 2 sheets the description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes in analogy to 2d we derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function thus this information completely characterizes 3d shape and lays the theoretical foundation for its use in recognition morphing design and manipulation of shapes

disconnected skeleton shape at its absolute scale we present a new skeletal representation along with a matching framework to address the deformable shape recognition problem the disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale our motivation is to rely on stable properties the shape instead of inaccurately measured secondary details the new representation does not suffer from the common instability problems of the traditional connected skeletons and the matching process gives quite successful results on a diverse database of 2d shapes an important difference of our approach from the conventional use of skeleton is that we replace the local coordinate frame with a global euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations as a result we can produce descriptions that are sensitive to any combination of changes in scale position orientation and articulation as well as invariant ones

hierarchical shape description via the multiresolution symmetric axis transform a method is proposed that produces a shape description in the form of a hierarchy by scale of simple symmetric axis sequences an axis segment that is a child of another has smaller scale and is seen as a branch of its parent the scale value and parent child relationship are induced by following the symmetric axis under successive reduction of resolution the result in two or three dimensions is a figure rather than boundary oriented shape description that has natural segments and is insensitive to noise in the object description the general approach of hierarchy production by following a feature through successive resolution reduction will be presented as will methods of resolution reduction and computer implementation also the relation of this figure based shape description to those based on boundary curvature will be briefly discussed

iterative reconstruction methods using regularization and optimal current patterns in electrical impedance tomography an iterative reconstruction method which minimizes the effects of ill conditioning is discussed based on the modified newton raphson algorithm a regularization method which integrates prior information into the image reconstruction was developed this improves the conditioning of the information matrix in the modified newton raphson algorithm optimal current patterns were used to obtain voltages with maximal signal to noise ratio snr a complete finite element model fem was used for both the internal and the boundary electric fields reconstructed images from phantom data show that the use of regularization optimal current patterns and a complete fem model improves image accuracy the authors also investigated factors affecting the image quality of the iterative algorithm such as the initial guess image iteration and optimal current updating

shape based normalization of the corpus callosum for dti connectivity analysis the continuous medial representation cm rep is an approach that makes it possible to model normalize and analyze anatomical structures on the basis of medial geometry having recently presented a partial differential equation pde based approach for 3 d cm rep modeling 1 here we present an equivalent 2 d approach that involves solving an ordinary differential equation this paper derives a closed form solution of this equation and shows how pythagorean hodograph curves can be used to express the solution as a piecewise polynomial function allowing efficient and robust medial modeling the utility of the approach in medical image analysis is demonstrated by applying it to the problem of shape based normalization of the midsagittal section of the corpus callosum using diffusion tensor tractography we show that shape based normalization aligns subregions of the corpus callosum defined by connectivity more accurately than normalization based on volumetric registration furthermore shape based normalization helps increase the statistical power of group analysis in an experiment where features derived from diffusion tensor tractography are compared between two cohorts these results suggest that cm rep is an appropriate tool for normalizing the corpus callosum in white matter studies