new algorithms for euclidean distance transformation of an n dimensional digitized picture with applications abstract in this paper we propose a new method to obtain the euclidean distance transformation and the voronoi diagram based on the exact euclidean metric for an n dimensional picture we present four algorithms to perform the transformation which are constructed by the serial composition of n dimensional filters when performed by a general purpose computer they are faster than the method by h yamada for a two dimensional picture those algorithms require only one n dimensional array for storing input output pictures and a single one dimensional array for a work area if an input picture needs not be preserved

improved estimates of partial volume coefficients from noisy brain mri using spatial context this paper addresses the problem of accurate voxel level estimation of tissue proportions in the human brain magnetic resonance imaging mri due to the finite resolution of acquisition systems mri voxels can contain contributions from more than a single tissue type the voxel level estimation of this fractional content is known as partial volume coefficient estimation in the present work two new methods to calculate the partial volume coefficients under noisy conditions are introduced and compared with current similar methods concretely a novel markov random field model allowing sharp transitions between partial volume coefficients of neighbouring voxels and an advanced non local means filtering technique are proposed to reduce the errors due to random noise in the partial volume coefficient estimation in addition a comparison was made to find out how the different methodologies affect the measurement of the brain tissue type volumes based on the obtained results the main conclusions are that 1 both markov random field modelling and non local means filtering improved the partial volume coefficient estimation results and 2 non local means filtering was the better of the two strategies for partial volume coefficient estimation

a linear time algorithm for computing exact euclidean distance transforms of binary images in arbitrary dimensions a sequential algorithm is presented for computing the exact euclidean distance transform dt of a k dimensional binary image in time linear in the total number of voxels n the algorithm which is based on dimensionality reduction and partial voronoi diagram construction can be used for computing the dt for a wide class of distance functions including the l sub p and chamfer metrics at each dimension level the dt is computed by constructing the intersection of the voronoi diagram whose sites are the feature voxels with each row of the image this construction is performed efficiently by using the dt in the next lower dimension the correctness and linear time complexity are demonstrated analytically and verified experimentally the algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast not only does it run in o n time but the time constant is small a simple modification of the algorithm computes the weighted euclidean dt which is useful for images with anisotropic voxel dimensions a parallel version of the algorithm runs in o n p time with p processors

on the generation of skeletons from discrete euclidean distance maps the skeleton is an important representation for shape analysis a common approach for generating discrete skeletons takes three steps 1 computing the distance map 2 detecting maximal disks from the distance map and 3 linking the centers of maximal disks cmds into a connected skeleton algorithms using approximate distance metrics are abundant and their theory has been well established however the resulting skeletons may be inaccurate and sensitive to rotation in this paper we study methods for generating skeletons based on the exact euclidean metric we first show that no previous algorithms identify the exact set of discrete maximal disks under the euclidean metric we then propose new algorithms and show that they are correct to link cmds into connected skeletons we examine two prevalent approaches connected thinning and steepest ascent we point out that the connected thinning approach does not work properly for euclidean distance maps only the steepest ascent algorithm produces skeletons that are truly medially placed the resulting skeletons have all the desirable properties they have the same simple connectivity as the figure they are well centered they are insensitive to rotation and they allow exact reconstruction the effectiveness of our algorithms is demonstrated with numerous examples

efficient euclidean distance transform algorithm of binary images in arbitrary dimensions in this paper we propose an efficient algorithm i e pbedt for short to compute the exact euclidean distance transform edt of a binary image in arbitrary dimensions the pbedt is based on independent scan and implemented in a recursive way i e the edt of a d dimensional image is able to be computed from the edts of its d 1 dimensional sub images in each recursion all of the rows in the current dimensional direction are processed one by one the points in the current processing row and their closest feature points in d 1 dimensional sub images can be shown in a euclidean plane by using the geometric properties of the perpendicular bisector the closest feature points of d 1 dimensional sub images are easily verified so as to lead to the edt of a d dimensional image after eliminating the invalid points the time complexity of the pbedt algorithm is linear in the amount of both image points and dimensions with a small coefficient compared with the state of the art edt algorithms the pbedt algorithm is much faster and more stable in most cases

partial volume bayesian classification of material mixtures in mr volume data using voxel histograms the authors present a new algorithm for identifying the distribution of different material types in volumetric datasets such as those produced with magnetic resonance imaging mri or computed tomography ct because the authors allow for mixtures of materials and treat voxels as regions their technique reduces errors that other classification techniques can create along boundaries between materials and is particularly useful for creating accurate geometric models and renderings from volume data it also has the potential to make volume measurements more accurately and classifies noisy low resolution data well there are two unusual aspects to the authors approach first they assume that due to partial volume effects or blurring voxels can contain more than one material e g both muscle and fat the authors compute the relative proportion of each material in the voxels second they incorporate information from neighboring voxels into the classification process by reconstructing a continuous function spl rho x from the samples and then looking at the distribution of values that spl rho x takes on within the region of a voxel this distribution of values is represented by a histogram taken over the region of the voxel the mixture of materials that those values measure is identified within the voxel using a probabilistic bayesian approach that matches the histogram by finding the mixture of materials within each voxel most likely to have created the histogram the size of regions that the authors classify is chosen to match the sparing of the samples because the spacing is intrinsically related to the minimum feature size that the reconstructed continuous function can represent

linear time euclidean distance transform algorithms two linear time and hence asymptotically optimal algorithms for computing the euclidean distance transform of a two dimensional binary image are presented the algorithms are based on the construction and regular sampling of the voronoi diagram whose sites consist of the unit feature pixels in the image the first algorithm which is of primarily theoretical interest constructs the complete voronoi diagram the second more practical algorithm constructs the voronoi diagram where it intersects the horizontal lines passing through the image pixel centers extensions to higher dimensional images and to other distance functions are also discussed

spatial classification using fuzzy membership models in the usual statistical approach to spatial classification it is assumed that each pixel belongs to precisely one of a small number of known groups this framework is extended to include mixed pixel data then only a proportion of each pixel belongs to each group two models based on multivariate gaussian random fields are proposed to model this fuzzy membership process the problems of predicting the group membership and estimating the parameters are discussed some simulations are presented to study the properties of this approach and an example is given using landsat remote sensing data

a list processing approach to compute voronoi diagrams and the euclidean distance transform we propose an efficient voronoi transform algorithm for constructing voronoi diagrams using segment lists of rows a significant feature of the algorithm is that it takes segments rather than pixels as the basic units to represent and propagate the nearest neighbor information the segment lists are dynamically updated as they are scanned a distance map can then be easily computed from the segment list representation of the voronoi diagram experimental results have demonstrated its high efficiency extension of the algorithm to higher dimensions is also discussed

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

fast and robust parameter estimation for statistical partial volume models in brain mri due to the finite spatial resolution of imaging devices a single voxel in a medical image may be composed of mixture of tissue types an effect known as partial volume effect pve partial volume estimation that is the estimation of the amount of each tissue type within each voxel has received considerable interest in recent years much of this work has been focused on the mixel model a statistical model of pve we propose a novel trimmed minimum covariance determinant tmcd method for the estimation of the parameters of the mixel pve model in this method each voxel is first labeled according to the most dominant tissue type voxels that are prone to pve are removed from this labeled set following which robust location estimators with high breakdown points are used to estimate the mean and the covariance of each tissue class comparisons between different methods for parameter estimation based on classified images as well as expectation maximization like em like procedure for simultaneous parameter and partial volume estimation are reported the robust estimators based on a pruned classification as presented here are shown to perform well even if the initial classification is of poor quality the results obtained are comparable to those obtained using the em like procedure but require considerably less computation time segmentation results of real data based on partial volume estimation are also reported in addition to considering the parameter estimation problem we discuss differences between different approximations to the complete mixel model in summary the proposed tmcd method allows for the accurate robust and efficient estimation of partial volume model parameters which is crucial to a variety of brain mri data analysis procedures such as the accurate estimation of tissue volumes and the accurate delineation of the cortical surface

distances defined by neighborhood sequences abstract this paper investigates general properties of distance functions defined over digitized space we assume that a distance between two points is defined as the length of a shortest path connecting them in the underlying graph which is defined by a given neighborhood sequence many typical distance functions can be described in this form but there are cases in which given neighborhood sequence do not define distance functions we first derive a necessary and sufficient condition for a neighborhood sequence to define a distance function we then discuss another important problem of estimating how tight such distances can approximate the euclid distance from the view point of relative error and absolute error

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

computing distance transformations in convex and non convex domains abstract it is shown that the well known two pass sequential local transformation algorithm for computing a distance transformation in rectangular domains may fail in some convex integer domains but that a four pass algorithm is sufficient in all two dimensional convex domains for non convex domains the number of passes necessary is shown to be generally greater two propagation algorithms for computing the distance transformation are described and shown theoretically and experimentally to be computationally more efficient than the sequential local transformation algorithm in non convex domains of complex shape the relationship of the distance transformation in non convex domains to some more general transformations is explored

probabilistic segmentation of brain tissue in mr imaging a new method has been developed for probabilistic segmentation of five different types of brain structures white matter gray matter cerebro spinal fluid without ventricles ventricles and white matter lesion in cranial mr imaging the algorithm is based on information from t1 weighted t1 w inversion recovery ir proton density weighted pd t2 weighted t2 w and fluid attenuation inversion recovery flair scans it uses the k nearest neighbor classification technique that builds a feature space from spatial information and voxel intensities the technique generates for each tissue type an image representing the probability per voxel being part of it by application of thresholds on these probability maps binary segmentations can be obtained a similarity index si and a probabilistic si psi were calculated for quantitative evaluation of the results the influence of each image type on the performance was investigated by alternately leaving out one of the five scan types this procedure showed that the incorporation of the t1 w pd or t2 w did not significantly improve the segmentation results further investigation indicated that the combination of ir and flair was optimal for segmentation of the five brain tissue types evaluation with respect to the gold standard showed that the si values for all tissues exceeded 0 8 and all psi values exceeded 0 7 implying an excellent agreement

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

on the stopping criteria for k nearest neighbor in positive unlabeled time series classification problems positive unlabeled time series classification has become an important area during the last decade as often vast amounts of unlabeled time series data are available but obtaining the corresponding labels is difficult in this situation positive unlabeled learning is a suitable option to mitigate the lack of labeled examples in particular self training is a widely used technique due to its simplicity and adaptability within this technique the stopping criterion i e the decision of when to stop labeling is a critical part especially in the positive unlabeled context we propose a self training method that follows the positive unlabeled approach for time series classification and a family of parameter free stopping criteria for this method our proposal uses a graphical analysis applied to the minimum distances obtained by the k nearest neighbor as the base learner to estimate the class boundary the proposed method is evaluated in an experimental study involving various time series classification datasets the results show that our method outperforms the transductive results obtained by previous models

shape based interpolation of multidimensional grey level images shape based interpolation as applied to binary images causes the interpolation process to be influenced by the shape of the object it accomplishes this by first applying a distance transform to the data this results in the creation of a grey level data set in which the value at each point represents the minimum distance from that point to the surface of the object by convention points inside the object are assigned positive values points outside are assigned negative values this distance transformed data set is then interpolated using linear or higher order interpolation and is then thresholded at a distance value of zero to produce the interpolated binary data set here the authors describe a new method that extends shape based interpolation to grey level input data sets this generalization consists of first lifting the n dimensional n d image data to represent it as a surface or equivalently as a binary image in an n 1 dimensional n 1 d space the binary shape based method is then applied to this image to create an n 1 d binary interpolated image finally this image is collapsed inverse of lifting to create the n d interpolated grey level data set the authors have conducted several evaluation studies involving patient computed tomography ct and magnetic resonance mr data as well as mathematical phantoms they all indicate that the new method produces more accurate results than commonly used grey level linear interpolation methods although at the cost of increased computation

shape based interpolation of multidimensional objects a shape based interpolation scheme for multidimensional images is presented this scheme consists of first segmenting the given image data into a binary image converting the binary image back into a gray image wherein the gray value of a point represents its shortest distance positive value for points of the object and negative for those outside from the cross sectional boundary and then interpolating the gray image the set of all points with nonnegative values associated with them in the interpolated image constitutes the interpolated object the method not only minimizes user involvement in interactive segmentation but also leads to more accurate representation and depiction of dynamic as well as static objects the general methodology and the implementation details of the method are presented and compared on a qualitative and quantitative basis to the existing methods the generality of the proposed scheme is illustrated with a number of medical imaging examples

weighted distance transforms generalized to modules and their computation on point lattices this paper presents the generalization of weighted distances to modules and their computation through the chamfer algorithm on general point lattices the first part is dedicated to formalization of definitions and properties distance metric norm of weighted distances on modules it resumes tools found in literature to express the weighted distance of any point of a module and to compute optimal weights in the general case to get rotation invariant distances the second part of this paper proves that for any point lattice the sequential two scan chamfer algorithm produces correct distance maps finally the definitions and computation of weighted distances are applied to the face centered cubic fcc and body centered cubic bcc grids

quantification of mr brain images by mixture density and partial volume modeling the problem of automatic quantification of brain tissue by utilizing single valued single echo magnetic resonance imaging mri brain scans is addressed it is shown that this problem can be solved without classification or segmentation a method that may be particularly useful in quantifying white matter lesions where the range of values associated with the lesions and the white matter may heavily overlap the general technique utilizes a statistical model of the noise and partial volume effect together with a finite mixture density description of the tissues the quantification is then formulated as a minimization problem of high order with up to six separate densities as part of the mixture this problem is solved by tree annealing with and without partial volume utilized the results compared and the sensitivity of the tree annealing algorithm to various parameters is exhibited the actual quantification is performed by two methods a classification based method called bayes quantification and parameter estimation results from each method are presented for synthetic and actual data

a euclidean distance transform using grayscale morphology decomposition a fast and exact euclidean distance transformation using decomposed grayscale morphological operators is presented applied on a binary image a distance transformation assigns each object pixel a value that corresponds to the shortest distance between the object pixel and the background pixels it is shown that the large structuring element required for the euclidean distance transformation can be easily decomposed into 3 spl times 3 windows this is possible because the square of the euclidean distance matrix changes uniformly both in the vertical and horizontal directions a simple extension for a 3d euclidean distance transformation is discussed a fast distance transform for serial computers is also presented acting like thinning algorithms the version for serial computers focuses operations only on the potential changing pixels and propagates from the boundary of objects significantly reducing execution time nonsquare pixels can also be used in this algorithm an example application shape filtering using arbitrary sized circular dilation and erosion is discussed rotation invariant basic morphological operations can be done using this example application

an efficient uniform cost algorithm applied to distance transforms the uniform cost algorithm is a special case of the a algorithm for finding the shortest paths in graphs in the uniform cost algorithm nodes are expanded in order of increasing cost an efficient version of this algorithm is developed for integer cost values nodes are sorted by storing them at predefined places bucket sort keeping the overhead low the algorithm is applied to general distance transformation a constrained distance transform is an operation which calculates at each pixel of an image the distance to the nearest pixel of a reference set distance being defined as minimum path length the uniform cost algorithm in the constrained case proves to be the best solution for distance transformation it is fast the processing time is independent of the complexity of the image and memory requirements are moderate

construction of iso contours bisectors and voronoi diagrams on triangulated surfaces in the research of computer vision and machine perception 3d objects are usually represented by 2 manifold triangular meshes m in this paper we present practical and efficient algorithms to construct iso contours bisectors and voronoi diagrams of point sites on m based on an exact geodesic metric compared to euclidean metric spaces the voronoi diagrams on m exhibit many special properties that fail all of the existing euclidean voronoi algorithms to provide practical algorithms for constructing geodesic metric based voronoi diagrams on m this paper studies the analytic structure of iso contours bisectors and voronoi diagrams on m after a necessary preprocessing of model m practical algorithms are proposed for quickly obtaining full information about iso contours bisectors and voronoi diagrams on m the complexity of the construction algorithms is also analyzed finally three interesting applications surface sampling and reconstruction 3d skeleton extraction and point pattern analysis are presented that show the potential power of the proposed algorithms in pattern analysis

magnetic resonance image tissue classification using a partial volume model abstract we describe a sequence of low level operations to isolate and classify brain tissue within t1 weighted magnetic resonance images mri our method first removes nonbrain tissue using a combination of anisotropic diffusion filtering edge detection and mathematical morphology we compensate for image nonuniformities due to magnetic field inhomogeneities by fitting a tricubic b spline gain field to local estimates of the image nonuniformity spaced throughout the mri volume the local estimates are computed by fitting a partial volume tissue measurement model to histograms of neighborhoods about each estimate point the measurement model uses mean tissue intensity and noise variance values computed from the global image and a multiplicative bias parameter that is estimated for each region during the histogram fit voxels in the intensity normalized image are then classified into six tissue types using a maximum a posteriori classifier this classifier combines the partial volume tissue measurement model with a gibbs prior that models the spatial properties of the brain we validate each stage of our algorithm on real and phantom data using data from the 20 normal mri brain data sets of the internet brain segmentation repository our method achieved average indices of 0 746 0 114 for gray matter gm and 0 798 0 089 for white matter wm compared to expert labeled data our method achieved average indices 0 893 0 041 for gm and 0 928 0 039 for wm compared to the ground truth labeling on 12 volumes from the montreal neurological institute s brainweb phantom