a variable selection heuristic for k means clustering one of the most vexing problems in cluster analysis is the selection and or weighting of variables in order to include those that truly define cluster structure while eliminating those that might mask such structure this paper presents a variable selection heuristic for nonhierarchical k means cluster analysis based on the adjusted rand index for measuring cluster recovery the heuristic was subjected to monte carlo testing across more than 2200 datasets with known cluster structure the results indicate the heuristic is extremely effective at eliminating masking variables a cluster analysis of real world financial services data revealed that using the variable selection heuristic prior to the k means algorithm resulted in greater cluster stability

four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences a class of four simultaneous component models for the exploratory analysis of multivariate time series collected from more than one subject simultaneously is discussed in each of the models the multivariate time series of each subject is decomposed into a few series of component scores and a loading matrix the component scores series reveal the latent data structure in the course of time the interpretation of the components is based on the loading matrix the simultaneous component models model not only intraindividual variability but interindividual variability as well the four models can be ordered hierarchically from weakly to severely constrained thus allowing for big to small interindividual differences in the model the use of the models is illustrated by an empirical example

derivation of optimal filters for the detection of coronary arteries optimal filters for the detection of coronary arteries with a diameter range of 0 5 6 0 mm in digital x ray images are derived using a computational approach this approach is based on the two requirements for optimal detection first the filter should maximize the number of detected true edges and minimize the number of detected false edges second if an edge has been detected its position should be as close as possible to the true edge position in the image since the grey value profile in a digital x ray image associated with an arterial vessel is asymmetric the theory on edge detection derived by canny has been expanded with two additional boundary constraints to make it suitable for the derivation of filters for asymmetric edges it is demonstrated that it is possible to derive optimal filters for coronary segments the localization error defined by the square root of the sum of the squared systematic and random errors in the assessment of the arterial diameter depends on the size of the coronary artery and the amount of noise in the image here an evaluation study is described to assess the relationship between localization error and the amount of noise upon the vessel profile for that purpose an analytical description of the vessel profile in an angiographic image was derived for the larger arteries the relation between noise and localization error was found to be linear and no systematic over or underestimations were observed even if the noise level was very high however it can be shown that the smallest diameter that can be measured depends on the amount of noise present in the data even for images that contain only a low amount of noise arterial diameters below 0 7 mm cannot be measured accurately if the noise in the image increases the lowest measurable arterial diameter value also increases also the random error increases rapidly for vessel diameters below 1 2 mm but with a limited amount of noise and a diameter value above 0 7 mm the random error is still acceptable 0 15 mm 21 for 0 7 mm vessels 0 06 mm 6 for 1 mm vessels

a luminance and contrast invariant edge similarity measure a novel similarity measure for edge detection that is robust to varying luminance and contrast is presented it incorporates a regularization term and employs directional fir edge filters with hyperbolic tangent profiles to ensure improved noise performance and edge localization compared to classical methods

fast algorithms for low level vision a recursive filtering structure is proposed that drastically reduces the computational effort required for smoothing performing the first and second directional derivatives and carrying out the laplacian of an image these operations are done with a fixed number of multiplications and additions per output point independently of the size of the neighborhood considered the key to the approach is first the use of an exponentially based filter family and second the use of the recursive filtering applications to edge detection problems and multiresolution techniques are considered and an edge detector allowing the extraction of zero crossings of an image with only 14 operations per output element at any resolution is proposed various experimental results are shown

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

hierarchical classes models for three way three mode binary data interrelations and model selection several hierarchical classes models can be considered for the modeling of three way three mode binary data including the indclas model leenen van mechelen de boeck and rosenberg 1999 the tucker3 hiclas model ceulemans van mechelen and leenen 2003 the tucker2 hiclas model ceulemans and van mechelen 2004 and the tucker1 hiclas model that is introduced in this paper two questions then may be raised 1 how are these models interrelated and 2 given a specific data set which of these models should be selected and in which rank in the present paper we deal with these questions by 1 showing that the distinct hierarchical classes models for three way three mode binary data can be organized into a partially ordered hierarchy and 2 by presenting model selection strategies based on extensions of the well known scree test and on the akaike information criterion the latter strategies are evaluated by means of an extensive simulation study and are illustrated with an application to interpersonal emotion data finally the presented hierarchy and model selection strategies are related to corresponding work by kiers 1991 for principal component models for three way three mode real valued data

k means type algorithms a generalized convergence theorem and characterization of local optimality the k means algorithm is a commonly used technique in cluster analysis in this paper several questions about the algorithm are addressed the clustering problem is first cast as a nonconvex mathematical program then a rigorous proof of the finite convergence of the k means type algorithm is given for any metric it is shown that under certain conditions the algorithm may fail to converge to a local minimum and that it converges under differentiability conditions to a kuhn tucker point finally a method for obtaining a local minimum solution is given

gradient estimation using wide support operators one of the fastest methods of localizing edges in images is based on small gradient kernels such as sobel prewitt and roberts although small gradient kernels provide a fast way of computing the gradients they have little control over noise edge location and edge orientation they are known to be only sensitive to step edges and fail to detect smooth boundaries on the other hand large kernels provide superior noise suppression characteristics but they suffer from wide response area around edges they cause edges of neighboring objects to merge due to their wide support problems associated with large gradient kernels prevent their widespread usage this paper presents a fuzzy topology based method to facilitate the use of larger gradient kernels the new method effectively limits the response area around the edge and prevents neighboring objects to affect each other synthetic images are used to show the superior noise suppression properties and response characteristics to both step and ramp edges natural images are also used to assess the performance of the newly proposed topological gradient estimation qualitatively

noise robust edge detector combining isotropic and anisotropic gaussian kernels a new noise robust edge detector is proposed which combines a small scaled isotropic gaussian kernel and large scaled anisotropic gaussian kernels angks to obtain edge maps of images its main advantage is that noise reduction is attained while maintaining high edge resolution from the angks anisotropic directional derivatives andds are derived to capture the locally directional variation of an image the andd based edge strength map esm is constructed its noise robustness is determined by the scale alone and its edge resolution by the ratio of the scale to the anisotropic factor moreover the edge stretch effect in anisotropic smoothing is revealed the andd based esm and the gradient based esm with a small scaled isotropic gaussian kernel are fused into a noise robust esm with high edge resolution and little edge stretch embedding the fused esm into the routine of canny detector a noise robust edge detector is developed which includes two additional modifications contrast equalization and noise dependent lower threshold the aggregate test receiver operating characteristic roc curves and the pratt s figure of merit fom are used to evaluate the proposed detector by abundant experiments the experimental results show that the proposed detector can obtain high quality edge maps for noise free and noisy images

finite mixtures in confirmatory factor analysis models in this paper various types of finite mixtures of confirmatory factor analysis models are proposed for handling data heterogeneity under the proposed mixture approach observations are assumed to be drawn from mixtures of distinct confirmatory factor analysis models but each observation does not need to be identified to a particular model prior to model fitting several classes of mixture models are proposed these models differ by their unique representations of data heterogeneity three different sampling schemes for these mixture models are distinguished a mixed type of the these three sampling schemes is considered throughout this article the proposed mixture approach reduces to regular multiple group confirmatory factor analysis under a restrictive sampling scheme in which the structural equation model for each observation is assumed to be known by assuming a mixture of multivariate normals for the data maximum likelihood estimation using the em expectation maximization algorithm and the as approximate scoring method are developed respectively some mixture models were fitted to a real data set for illustrating the application of the theory although the em algorithm and the as method gave similar sets of parameter estimates the as method was found computationally more efficient than the em algorithm some comments on applying the mixture approach to structural equation modeling are made

factorization of the eighth fermat number we describe a monte carlo factorization algorithm which was used to factorize the fermat number f8 2256 1 previously f8 was known to be composite but its factors were unknown comments only the abstract is given here the full paper appeared as 2 for a succinct proof of primality of the larger factor q8 of f8 see 3 at the time of this paper lenstra s elliptic curve method ecm had not been invented thus a modification of pollard s rho method 1 was used to factor f8 and several mersenne numbers 4 for the factorization of a larger fermat number by ecm see 5 the smaller factor of f8 is 1238926361552897 the epigram i am now entirely persuaded to employ the method a handy trick on gigantic composite numbers may appeal to readers who wish to memorize this factor

robust extrema features for time series data analysis the extraction of robust features for comparing and analyzing time series is a fundamentally important problem research efforts in this area encompass dimensionality reduction using popular signal analysis tools such as the discrete fourier and wavelet transforms various distance metrics and the extraction of interest points from time series recently extrema features for analysis of time series data have assumed increasing significance because of their natural robustness under a variety of practical distortions their economy of representation and their computational benefits invariably the process of encoding extrema features is preceded by filtering of the time series with an intuitively motivated filter e g for smoothing and subsequent thresholding to identify robust extrema we define the properties of robustness uniqueness and cardinality as a means to identify the design choices available in each step of the feature generation process unlike existing methods which utilize filters x201c inspired x201d from either domain knowledge or intuition we explicitly optimize the filter based on training time series to optimize robustness of the extracted extrema features we demonstrate further that the underlying filter optimization problem reduces to an eigenvalue problem and has a tractable solution an encoding technique that enhances control over cardinality and uniqueness is also presented experimental results obtained for the problem of time series subsequence matching establish the merits of the proposed algorithm

optimal edge detectors for ramp edges it is argued that the best way to model an edge is by assuming all ideal mathematical function passed through a low pass filter and and immersed in noise using techniques similar to those developed by j canny 1983 1986 and l a spacek 1986 optimal filters are derived for ramp edges of various slopes the optimal nonrecursive filter for ideal step edges is then derived as a limiting case of the filters for ramp edges because there are no step edges in images edge detection is improved when the ramp filter is used instead of the filters developed for step edges for practical purposes some convolution masks are given which can be used directly for edge detection without the need to go into the details of the subject

laplacian operator based edge detectors laplacian operator is a second derivative operator often used in edge detection compared with the first derivative based edge detectors such as sobel operator the laplacian operator may yield better results in edge localization unfortunately the laplacian operator is very sensitive to noise in this paper based on the laplacian operator a model is introduced for making some edge detectors also the optimal threshold is introduced for obtaining a maximum a posteriori map estimate of edges

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

using canny s criteria to derive a recursively implemented optimal edge detector a highly efficient recursive algorithm for edge detection is presented using canny s design 1 we show that a solution to his precise formulation of detection and localization for an infinite extent filter leads to an optimal operator in one dimension which can be efficiently implemented by two recursive filters moving in opposite directions in addition to the noise truncature immunity which results the recursive nature of the filtering operations leads with sequential machines to a substantial saving in computational effort five multiplications and five additions for one pixel independent of the size of the neighborhood the extension to the two dimensional case is considered and the resulting filtering structures are implemented as two dimensional recursive filters hence the filter size can be varied by simply changing the value of one parameter without affecting the time execution of the algorithm performance measures of this new edge detector are given and compared to canny s filters various experimental results are shown

factors of fermat numbers and large primes of the form 2ⁿ 1 a new factor is given for each of the fermat numbers f52 f931 f6835 and f9448 in addition a factor of f75 discovered by gary gostin is presented the current status for all f is shown in a table primes of the form k 2 1 k odd are listed for 31 k 149 1500 ii 4000 and for 151 k 199 1000 ii 4000 some primes for even larger values of ii are included the largest one being 5 213165 1 also a survey of several related questions is given in particular values of k such that k 2 i is composite for every ii are considered as well as odd values of h such that 3h 2 1 never yields a twin prime pair

multi edge detection by isotropical 2 d isef cascade most research in edge detection has been based on the classical mono edge model in the present paper we propose an optimal edge detector for complicated noisy multi edge images and the isotropical 2 d realization of the optimal filter we introduce at first the complicated multi edge model and show that the optimal filter as a preparation for multi edge detection is a cascade of infinite size symmetrical exponential filters isef we then propose a fast realization of the isotropical 2 d optimal filter experimental results are also presented

quantitative error measures for edge detection how to measure the performance of an edge detection method is an open problem many performance measures have been presented in the literature however there seems to be no agreement on the best option and there exist no practical studies of the features of some of the most recent measures in this work we make a comprehensive overview of the different proposals followed by a practical comparison of the most representative measures on synthetic as well as natural edge images

directional filtering in edge detection two dimensional 2 d edge detection can be performed by applying a suitably selected optimal edge half filter in n directions computationally such a two dimensional n directional filter can be represented by a pair of real masks that is by one complex number matrix regardless of the number of filtering directions n specific calculations of the edge strength were conducted using a 2 d tridirectional filter based on a petrou kittler 1991 one dimensional 1 d detector optimized for the ramp edges which are characteristic of posterior eye capsule images that were used here as a test set in applications to image segmentation tridirectional filtering results in co occurrence arrays of low dimensionality

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

variable structuring element based fuzzy morphological operations for single viewpoint omnidirectional images morphological tools can provide transformations suitable for real projective images but the camera and objects to be analyzed have to be positioned in such a manner that a regular mesh on the objects projects a regular mesh on the image a morphological modification of the image is thus the projection of an equivalent operation on the object otherwise due to perspective effects a morphological operation on the image is not the projection of an equivalent operation on the objects to be analyzed with catadioptric omnidirectional images it is almost impossible to place the sensor such that a regular mesh on the scene projects a regular mesh on the image nevertheless with proper calibration of a central catadioptric system the projection of a regular structuring element in a scene can be determined for each point on the image the aim of this paper is to present new morphological operators that use this projective property these operators make use of a structuring element of varying shape since this varying shape cannot be represented as a binary union of pixels we propose to use a fuzzy extension of the classical gray level morphology to account for this phenomenon this fuzzy extension is performed via fuzzy integrals

gabor filter based edge detection abstract it is common practice to utilize evidence from biological and psychological vision experiments to develop computational models for low level feature extraction the receptive profiles of simple cells in mammalian visual systems have been found to closely resemble gabor filters daugman proved that gabor filters achieve joint minimal joint uncertainty these results led researchers to develop computational models based on gabor filters for several low level vision applications such as edge detection texture classification optical flow estimation and data compression in this paper the performance of a gabor odd filter based edge detector is investigated using the measures proposed by canny based on this performance analysis a design criterion for one dimensional 1d gabor filter based edge detector is derived it is shown that this design criterion also holds good for a two dimensional 2d gabor filter based edge detector experimental results are presented to demonstrate the performance of the gabor filter based edge detector

a discrete expression of canny s criteria for step edge detector performances evaluation optimal filters for edge detection are usually developed in the continuous domain and then transposed by sampling to the discrete domain simpler filters are directly defined in the discrete domain we define criteria to compare filter performances in the discrete domain canny has defined 1983 1986 three criteria to derive the equation of an optimal filter for step edge detection good detection good localization and low responses multiplicity these criteria seem to be good candidates for filter comparison unfortunately they have been developed in the continuous domain and their analytical expressions cannot be used in the discrete domain we establish three criteria with the same meaning as canny s some comparisons with experimental results confirm the validity of our approach this study highlights the existence of two classes of derivative operators that are distinguished by whether or not the impulse response of the filter in continuous space domain is continuous on its center these classes exhibit very different properties for the second and third criteria we extend the use of the first and third criteria to the smoothing filters we also define an optimal continuous filter according to the continuous third criterion and an optimal discrete filter according to the discrete third criterion we compare the performances of the sampled version of the continuous filter to those of the optimal discrete filter

the effect of cluster size dimensionality and the number of clusters on recovery of true cluster structure an evaluation of four clustering methods and four external criterion measures was conducted with respect to the effect of the number of clusters dimensionality and relative cluster sizes on the recovery of true cluster structure the four methods were the single link complete link group average upgma and ward s minimum variance algorithms the results indicated that the four criterion measures were generally consistent with each other of which two highly similar pairs were identified the tirst pair consisted of the rand and corrected rand statistics and the second pair was the jaccard and the fowlkes and mallows indexes with respect to the methods recovery was found to improve as the number of clusters increased and as the number of dimensions increased the relative cluster size factor produced differential performance effects with ward s procedure providing the best recovery when the clusters were of equal size the group average method gave equivalent or better recovery when the clusters were of unequal size

a nonlinear derivative scheme applied to edge detection this paper presents a nonlinear derivative approach to addressing the problem of discrete edge detection this edge detection scheme is based on the nonlinear combination of two polarized derivatives its main property is a favorable signal to noise ratio snr at a very low computation cost and without any regularization a 2d extension of the method is presented and the benefits of the 2d localization are discussed the performance of the localization and snr are compared to that obtained using classical edge detection schemes tests of the regularized versions and a theoretical estimation of the snr improvement complete this work

on detecting edges an edge in an image corresponds to a discontinuity in the intensity surface of the underlying scene it can be approximated by a piecewise straight curve composed of edgels i e short linear edge elements each characterized by a direction and a position the approach to edgel detection here is to fit a series of one dimensional surfaces to each window kernel of the operator and accept the surface description which is adequate in the least squares sense and has the fewest parameters a one dimensional surface is one which is constant along some direction the tanh is an adequate basis for the stepedge and its combinations are adequate for the roofedge and the line edge the proposed method of step edgel detection is robust with respect to noise for step size noise 2 5 it has subpixel position localization position and an angular localization better than 10 further it is designed to be insensitive to smooth shading these results are demonstrated by some simple analysis statistical data and edgelimages also included is a comparison of performance on a real image with a typical operator difference of gaussians the results indicate that the proposed operator is superior with respect to detection localization and resolution

close form solution and parameter selection for convex minimization based edge preserving smoothing this work presents a new approach for the analysis of convex minimization based edge preserving image smoothing and the parameter selection therein the global solution that is the response of a convex smoothing model to the ideal step edge is derived in close form by analyzing the close form solution insights are drawn into how the optimal solution responds to edges in the data and how the parameter values affect resultant edges in the solution based on this a scheme is proposed for selecting parameters to achieve desirable responses at edges the theoretic results are substantiated by experiments

an edge detection technique using the facet model and parameterized relaxation labeling we present a method for detecting and labeling the edge structures in digital gray scale images in two distinct stages first a variant of the cubic facet model is applied to detect the location orientation and curvature of the putative edge points next a relaxation labeling network is used to reinforce meaningful edge structures and suppress noisy edges each node label of this network is a 3d vector parameterizing the orientation and curvature information of the corresponding edge point a hysteresis step in the relaxation process maximizes connected contours for certain types of images prefiltering by adaptive smoothing improves robustness against noise and spatial blurring

on optimal linear filtering for edge detection in this paper we revisit the analytical expressions of the three canny s 1983 criteria for edge detection quality good detection good localization and low multiplicity of false detections our work differs from canny s work on two essential points here the criteria are given for discrete sampled signals i e for the real implemented filters instead of a single step edge as input signal we use pulses of various width the proximity of other edges affects the quality of the detection process this is taken into account in the new expressions of these criteria we derive optimal filters for each of the criteria and for any combination of them in particular we define an original filter which maximizes detection and localization and a simple approximation of the optimal filter for the simultaneous maximization of the three criteria the upper bounds of the criteria are computed which allow users to measure the absolute and relative performance of any filter exponential deriche 1987 and first derivative of gaussian filters are evaluated our criteria can also be used to compute the optimal value of the scale parameter of a given filter when the resolution of the detection is fixed

a new generalised α scale spaces quadrature filters abstract the scale spaces is a recent theory that open new possibilities of phase based image processing it is a parameterised class 0 1 of linear scale space representations which allows a continuous connection beyond the well known gaussian scale space 1 in this paper we make use of this unified representation to derive new families of band pass quadrature filters built from derivatives and difference of the scale space generating kernels this construction leads to generalised kernel filters including the commonly known families derived from the gaussian and the poisson kernels the properties of each family are first studied and then experiments on one and two dimensional signals are shown to exemplify how the suggested filters can be used for edge detection this work is complemented by an experimental evaluation which demonstrates that the new proposed filters are a good alternative to the commonly used log gabor filter

edge and line feature extraction based on covariance models image segmentation based on contour extraction usually involves three stages of image operations feature extraction edge detection and edge linking this paper is devoted to the first stage a method to design feature extractors used to detect edges from noisy and or blurred images the method relies on a model that describes the existence of image discontinuities e g edges in terms of covariance functions the feature extractor transforms the input image into a log likelihood ratio image such an image is a good starting point of the edge detection stage since it represents a balanced trade off between signal to noise ratio and the ability to resolve detailed structures for 1 d signals the performance of the edge detector based on this feature extractor is quantitatively assessed by the so called average risk measure the results are compared with the performances of 1 d edge detectors known from literature generalizations to 2 d operators are given applications on real world images are presented showing the capability of the covariance model to build edge and line feature extractors finally it is shown that the covariance model can be coupled to a mrf model of edge configurations so as to arrive at a maximum a posteriori estimate of the edges or lines in the image

scale invariant filtering design and analysis for edge detection existing edge detection filters work well on straight edges but make significant errors near sharp corners by producing rounded corners this is due to the fact that the edge maps produced by these filters are scale variant we enhance canny s optimality criteria to incorporate detection performance near corners as an explicit design objective the resulting optimal filter termed bessel integral filter can be derived analytically and exhibits superior performance over recent alternatives both in terms of numerical accuracy and experimental fidelity a noise free localization index is also derived here to account for the detection accuracy of discontinuities forming sharp corners in the absence of noise we prove here that edges detected by the filters which are not optimal with respect to this noise free localization index are scale variant however the bessel integral filter proposed here is optimal with respect to the noise free localization index and therefore it is a scale invariant filter

design of steerable filters for feature detection using canny like criteria we propose a general approach for the design of 2d feature detectors from a class of steerable functions based on the optimization of a canny like criterion in contrast with previous computational designs our approach is truly 2d and provides filters that have closed form expressions it also yields operators that have a better orientation selectivity than the classical gradient or hessian based detectors we illustrate the method with the design of operators for edge and ridge detection we present some experimental results that demonstrate the performance improvement of these new feature detectors we propose computationally efficient local optimization algorithms for the estimation of feature orientation we also introduce the notion of shape adaptable feature detection and use it for the detection of image corners

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

six new factors of fermat numbers in the past few years several investigators 1 11 have found many new prime factors of the fermat numbers fm 22 1 the increased availability and speed of computing equipment have allowed the search limits for factors of these numbers to be extended significantly in tables below we include the known search limits and all of the results that have been obtained since h c williams 1978 paper 12 was published using the method outlined by hallyburton and brillhart 8 we tested integers of the form k 2r 1 k odd as factors of the fermat numbers fm 9 m n 2 for the ranges shown in table 1 during our search the following new prime factors were found 1120049 23 1jf29

canny edge detection enhancement by scale multiplication the technique of scale multiplication is analyzed in the framework of canny edge detection a scale multiplication function is defined as the product of the responses of the detection filter at two scales edge maps are constructed as the local maxima by thresholding the scale multiplication results the detection and localization criteria of the scale multiplication are derived at a small loss in the detection criterion the localization criterion can be much improved by scale multiplication the product of the two criteria for scale multiplication is greater than that for a single scale which leads to better edge detection performance experimental results are presented

on component analyses principal components analysis can be redefined in terms of the regression of observed variables upon component variables two criteria for the adequacy of a component representation in this context are developed and are shown to lead to different component solutions both criteria are generalized to allow weighting the choice of weights determining the scale invariance properties of the resulting solution a theorem is presented giving necessary and sufficient conditions for equivalent component solutions under different choices of weighting applications of the theorem are discussed that involve the components analysis of linearly derived variables and of external variables

optimal edge detection using expansion matching and restoration discusses the application of a newly developed expansion matching method for edge detection expansion matching optimizes a novel matching criterion called the discriminative signal to noise ratio dsnr and has been shown to robustly recognize templates under conditions of noise severe occlusion and superposition the dsnr criterion is better suited to evaluate matching in practical conditions than the traditional snr since it considers as noise even the off center response of the filter to the template itself we introduce a family of optimal dsnr edge detectors based on the expansion filter for several edge models for step edges the optimal dsnr step expansion filter sef is compared with the widely used canny edge detector ced experimental comparisons show that our edge detector yields better performance than the ced in terms of dsnr even under very adverse noise conditions as for boundary detection the sef consistently yields higher figures of merit than the ced on a synthetic binary image over a wide range of noise levels results also show that the design parameters of size or width of the sef are less critical than the ced variance this means that a single scale of the sef spans a larger range of input noise than a single scale of the ced experiments on a noisy image reveal that the sef yields less noisy edge elements and preserves structural details more accurately on the other hand the ced output has better suppression of multiple responses than the corresponding sef output

on the localization performance measure and optimal edge detection the localization performance measure of edge detectors is addressed a one dimensional formulation of the problem is considered a linear space invariant filter is used for the detection the locations of local maxima in the thresholded output of the filter are declared to be the edges the limitations of conventional performance measures are shown and a localization performance measure for edge detection is suggested this performance measure is based on the theory of zero crossings of stochastic processes it is shown that the derivative of a gaussian is the optimal edge detector for the measure

on optimal infinite impulse response edge detection filters the authors outline the design of an optimal computationally efficient infinite impulse response edge detection filter the optimal filter is computed based on canny s high signal to noise ratio good localization criteria and a criterion on the spurious response of the filter to noise an expression for the width of the filter which is appropriate for infinite length filters is incorporated directly in the expression for spurious responses the three criteria are maximized using the variational method and nonlinear constrained optimization the optimal filter parameters are tabulated for various values of the filter performance criteria a complete methodology for implementing the optimal filter using approximating recursive digital filtering is presented the approximating recursive digital filter is separable into two linear filters operating in two orthogonal directions the implementation is very simple and computationally efficient has a constant time of execution for different sizes of the operator and is readily amenable to real time hardware implementation

modeling differences in the dimensionality of multiblock data by means of clusterwise simultaneous component analysis given multivariate multiblock data e g subjects nested in groups are measured on multiple variables one may be interested in the nature and number of dimensions that underlie the variables and in differences in dimensional structure across data blocks to this end clusterwise simultaneous component analysis sca was proposed which simultaneously clusters blocks with a similar structure and performs an sca per cluster however the number of components was restricted to be the same across clusters which is often unrealistic in this paper this restriction is removed the resulting challenges with respect to model estimation and selection are resolved