a new generalized computational framework for finding object orientation using perspective trihedral angle constraint this paper investigates a fundamental problem of determining the position and orientation of a three dimensional 3 d object using a single perspective image view the technique is focused on the interpretation of trihedral angle constraint information a new closed form solution based on kanatani s formulation is proposed the main distinguishing feature of the authors method over the original kanatani formulation is that their approach gives an effective closed form solution for a general trihedral angle constraint the method also provides a general analytic technique for dealing with a class of problem of shape from inverse perspective projection by using angle to angle correspondence information a detailed implementation of the authors technique is presented different trihedral angle configurations were generated using synthetic data for testing the authors approach of finding object orientation by angle to angle constraint the authors performed simulation experiments by adding some noise to the synthetic data for evaluating the effectiveness of their method in a real situation it has been found that the authors method worked effectively in a noisy environment which confirms that the method is robust in practical application

the monogenic signal this paper introduces a two dimensional 2 d generalization of the analytic signal this novel approach is based on the riesz transform which is used instead of the hilbert transform the combination of a 2 d signal with the riesz transformed one yields a sophisticated 2 d analytic signal the monogenic signal the approach is derived analytically from irrotational and solenoidal vector fields based on local amplitude and local phase an appropriate local signal representation that preserves the split of identity i e the invariance equivariance property of signal decomposition is presented this is one of the central properties of the one dimensional 1 d analytic signal that decomposes a signal into structural and energetic information we show that further properties of the analytic signal concerning symmetry energy allpass transfer function and orthogonality are also preserved and we compare this with the behavior of other approaches for a 2 d analytic signal as a central topic of this paper a geometric phase interpretation that is based on the relation between the 1 d analytic signal and the 2 d monogenic signal established by the radon 1986 transform is introduced possible applications of this relationship are sketched and references to other applications of the monogenic signal are given

universal configurational structure in two dimensional scalar models monte carlo methods are used to explore the universal configurational structure of two dimensional spin 1 2 spin 1 and border phi 4 models comparison of spin 1 2 and spin 1 data provides evidence that the magnetisation distribution effectively the helmholtz free energy function and its coupling derivative effectively the internal energy function constitute readily accessible signatures of a universality class it is shown that when allowance is made for relatively large corrections to scaling effects the behaviour of the border phi 4 model may be satisfactorily matched to that of the other two models substantiating the view that the border model does indeed belong to the ising universality class

view variation of point set and line segment features the variation with respect to view of 2d features defined for projections of 3d point sets and line segments is studied it is established that general case view invariants do not exist for any number of points given true perspective weak perspective or orthographic projection models feature variation under the weak perspective approximation is then addressed though there are no general case weak perspective invariants there are special case invariants of practical importance those cited in the literature are derived from linear dependence relations and the invariance of this type of relation to linear transformation the variation with respect to view is studied for an important set of 2d line segment features the relative orientation size and position of one line segment with respect to another the analysis includes an evaluation criterion for feature utility in terms of view variation this relationship is a function of both the feature and the particular configuration of 3d line segments the use of this information in objection recognition is demonstrated for difficult discrimination tasks

conformal invariance and critical dynamics the author discusses the behaviour of dynamic correlation functions under conformal transformations in particular the exact form of the dynamic scaling function at the critical point for an arbitrary isotropic two dimensional model is obtained with either conserved or nonconserved order parameter

on universality between two dimensional ising like systems questions concerning the universality between different two dimensional ising like systems have recently been raised with evidence for the existence of several different subclasses of ising like systems being presented some new facts concerning similarity between border model and low temperature spin 1 ising model series analyses are noted for the spin 1 case an alternative determination of the critical temperature determines unequivocally that this model is in the ising universality class and it is suggested that a similar independent result could lead to a similar conclusion in the border model case the possibility of an analytic correction to scaling in the spin 1 model is proposed

on the unification of line processes outlier rejection and robust statistics with applications in early vision the modeling of spatial discontinuities for problems such as surface recovery segmentation image reconstruction and optical flow has been intensely studied in computer vision while line process models of discontinuities have received a great deal of attention there has been recent interest in the use of robust statistical techniques to account for discontinuities this paper unifies the two approaches to achieve this we generalize the notion of a line process to that of an analog outlier process and show how a problem formulated in terms of outlier processes can be viewed in terms of robust statistics we also characterize a class of robust statistical problems for which an equivalent outlier process formulation exists and give a straightforward method for converting a robust estimation problem into an outlier process formulation we show how prior assumptions about the spatial structure of outliers can be expressed as constraints on the recovered analog outlier processes and how traditional continuation methods can be extended to the explicit outlier process formulation these results indicate that the outlier process approach provides a general framework which subsumes the traditional line process approaches as well as a wide class of robust estimation problems examples in surface reconstruction image segmentation and optical flow are presented to illustrate the use of outlier processes and to show how the relationship between outlier processes and robust statistics can be exploited an appendix provides a catalog of common robust error norms and their equivalent outlier process formulations

monte carlo computation of cluster free energies in the ising model a test for the validity of the capillarity approximation a new monte carlo technique is proposed to compute directly the free energy difference fn 1 fn of two clusters with size n 1 and n the algorithm is applied to the two dimensional square ising model with nearest neighbour interactions results are given for 1 9 but for smaller cluster sizes the obtained energies are larger than predicted by the capillarity model the deviation exhibits no systematic size dependence but a strong temperature dependence the existence of a size independent term in fn is confirmed by the values of the internal energy un obtained as a by product of the calculation

autocorrelation driven diffusion filtering in this paper we present a novel scheme for anisotropic diffusion driven by the image autocorrelation function we show the equivalence of this scheme to a special case of iterated adaptive filtering by determining the diffusion tensor field from an autocorrelation estimate we obtain an evolution equation that is computed from a scalar product of diffusion tensor and the image hessian we propose further a set of filters to approximate the hessian on a minimized spatial support on standard benchmarks the resulting method performs favorable in many cases in particular at low noise levels in a gpu implementation video real time performance is easily achieved

crossing preserving coherence enhancing diffusion on invertible orientation scores many image processing problems require the enhancement of crossing elongated structures these problems cannot easily be solved by commonly used coherence enhancing diffusion methods therefore we propose a method for coherence enhancing diffusion on the invertible orientation score of a 2d image in an orientation score the local orientation is represented by an additional third dimension ensuring that crossing elongated structures are separated from each other we consider orientation scores as functions on the euclidean motion group and use the group structure to apply left invariant diffusion equations on orientation scores we describe how we can calculate regularized left invariant derivatives and use the hessian to estimate three descriptive local features curvature deviation from horizontality and orientation confidence these local features are used to adapt a nonlinear coherence enhancing crossing preserving diffusion equation on the orientation score we propose two explicit finite difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence enhancing diffusion the use of curvature leads to improved enhancement of curves with high curvature furthermore the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings

series expansion analysis of corrections to scaling in the three state potts model several extant series for the d 2 q 3 potts model are analysed with a method that explicitly accounts for the effect of confluent corrections to scaling excellent agreement with the conjectured exponents for this model is found as well as confirmation of renormalisation group estimates of the sub leading eigenvalue

universality of amplitude combinations in two dimensional percolation we generated extended low concentration series for the second moment of distances between cluster points a 2 in site and bond percolation on the square lattice and used them to calculate series for the correlation length we analysed these and other series for their critical amplitudes and compared them with results from monte carlo simulations of triangular site square site and square bond percolation the values obtained from the different models for the amplitude combination b 2 0 2 2 0 2 and b are the amplitudes of the second moment of the cluster size distribution m 2 and the strength of the infinite cluster p respectively are all within the range 2 23 0 10 confirming universality

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

a fast approximation of the bilateral filter using a signal processing approach the bilateral filter is a nonlinear filter that smoothes a signal while preserving strong edges it has demonstrated great effectiveness for a variety of problems in computer vision and computer graphics and fast versions have been proposed unfortunately little is known about the accuracy of such accelerations in this paper we propose a new signal processing analysis of the bilateral filter which complements the recent studies that analyzed it as a pde or as a robust statistical estimator the key to our analysis is to express the filter in a higher dimensional space where the signal intensity is added to the original domain dimensions importantly this signal processing perspective allows us to develop a novel bilateral filtering acceleration using downsampling in space and intensity this affords a principled expression of accuracy in terms of bandwidth and sampling the bilateral filter can be expressed as linear convolutions in this augmented space followed by two simple nonlinearities this allows us to derive criteria for downsampling the key operations and achieving important acceleration of the bilateral filter we show that for the same running time our method is more accurate than previous acceleration techniques typically we are able to process a 2 megapixel image using our acceleration technique in less than a second and have the result be visually similar to the exact computation that takes several tens of minutes the acceleration is most effective with large spatial kernels furthermore this approach extends naturally to color images and cross bilateral filtering

series expansions of the percolation probability for directed square and honeycomb lattices we have derived long series expansions of the percolation probability for site and bond percolation on directed square and honeycomb lattices for the square bond problem we have extended the series from 41 terms to 54 for the square site problem from 16 terms to 37 and for the honeycomb bond problem from 13 terms to 36 analysis of the series clearly shows that the critical exponent beta is the same for all the problems confirming expectations of universality for the critical probability and exponent we find in the square bond case qc 0 3552994 or 0 0000010 beta 0 27643 or 0 00010 in the square site case qc 0 294515 or 0 000005 beta 0 2763 or 0 0003 and in the honeycomb bond case qc 0 177143 or 0 000002 beta 0 2763 or 0 0002 in addition we have obtained accurate estimates for the critical amplitudes in all cases we find that the leading correction to scaling term is analytic i e the confluent exponent delta 1

a general framework for low level vision we introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented we consider intensity images as surfaces in the x i space the image is thereby a two dimensional 2 d surface in three dimensional 3 d space for gray level images and 2 d surfaces in five dimensions for color images the new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast and results in new and efficient schemes extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms

non perturbative renormalization using dimensional regularization applications to the epsilon expansion the authors give a prescription for the one loop renormalisation of the imaginary parts of vertex functions in g phi 4 which are generated when g 0 within the dimensional regularisation scheme this involves another non perturbative renormalisation over and above the usual one loop perturbative renormalisation the authors use an extended minimal subtraction scheme so that the resulting renormalisation group functions have a simple dependence on epsilon 4 d in contrast to the usual case however both the beta function and the renormalised coupling gr are in general complex for real g and the fixed point for epsilon gr 0 is non perturbative in epsilon this fixed point determines the imaginary parts of the critical exponents which are generated when epsilon 0 and allows the determination of the high order behaviour of the perturbation series in epsilon for these exponents the generalisation of these ideas to the o n symmetric g phi 2 2 model is also given

appearance based recognition of 3 d objects by cluttered background and occlusions in this article we present a new appearance based approach for the classification and the localization of 3 d objects in complex scenes a main problem for object recognition is that the size and the appearance of the objects in the image vary for 3 d transformations for this reason we model the region of the object in the image as well as the object features themselves as functions of these transformations we integrate the model into a statistical framework and so we can deal with noise and illumination changes to handle heterogeneous background and occlusions we introduce a background model and an assignment function thus the object recognition system becomes robust and a reliable distinction which features belong to the object and which to the background is possible experiments on three large data sets that contain rotations orthogonal to the image plane and scaling with together more than 100000 images show that the approach is well suited for this task

droplet structure in ising and potts models the structure of a recently introduced droplet in the q state potts model is analysed the authors derive exact relations from which it follows that the incipient infinite droplet at the critical temperature is a self similar fractal made of links and blobs just as recently found in random percolation the number of links nlinks between two points separated by a distance of the order b is given by nlinks varies as b1 nu b q where nu b q is the connectedness length exponent when t tc q and pb is used as an independent variable this result and the available calculations of nu b q indicate that the number of links decreases as q increases an analysis of the structure of the usual clusters made of nearest neighbour sites in a given potts site configuration is also made in particular it shows that for d 2 the incipient infinite cluster is made only of blobs and no links

series expansions of the percolation probability on the directed triangular lattice we have derived long series expansions of the percolation probability for site bond and site bond percolation on the directed triangular lattice for the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35 for the site bond problem which has not been studied before we have derived the series to order 32 our estimates of the critical exponent are in full agreement with results for similar problems on the square lattice confirming expectations of universality for the critical probability and exponent we find in the site case qc d 0 404 352 8 0 000 001 0 and d 0 276 45 0 000 10 in the bond case qc d 0 521 98 0 000 01 and d 0 2769 0 0010 and in the site bond case qc d 0 264 173 0 000 003 and d 0 2766 0 0003 in addition we have obtained accurate estimates for the critical amplitudes in all cases we find that the leading correction to scaling term is analytic i e the confluent exponent 1 d 1 in an earlier paper jensen and guttmann 1995 we reported on the derivation and analysis of long series for the percolation probability of site and bond percolation on the directed square and hexagonal lattices in this paper we extend this work to site bond and site bond percolation on the directed triangular lattice we refer to our earlier paper for a more general introduction to directed percolation and its role in the modelling of physical systems in directed site percolation each site is either present with probability p or absent with probability q d 1 p independent of all other sites on the lattice similarly for bond percolation each bond is absent or present independently of other bonds finally in site bond percolation both sites and bonds may be absent or present with equal probability but again with no dependency on any other sites or bonds two sites in the various models are connected if one can find a path respecting the directions indicated in figure 1 through occupied sites bonds or sites and bonds respectively from one to the other when p is smaller than a critical value pc all clusters of connected sites remain finite while for p pc there is an infinite cluster spanning the lattice in the preferred direction the order parameter of the system is the percolation probability p p that a given site belongs to the infinite cluster this quantity is strictly zero when p p c the behaviour of p p in the vicinity of pc may be described by a critical exponent p p p pc p p c c 1

channel smoothing efficient robust smoothing of low level signal features in this paper we present a new and efficient method to implement robust smoothing of low level signal features b spline channel smoothing this method consists of three steps encoding of the signal features into channels averaging of the channels and decoding of the channels we show that linear smoothing of channels is equivalent to robust smoothing of the signal features if we make use of quadratic b splines to generate the channels the linear decoding from b spline channels allows the derivation of a robust error norm which is very similar to tukey s biweight error norm we compare channel smoothing with three other robust smoothing techniques nonlinear diffusion bilateral filtering and mean shift filtering both theoretically and on a 2d orientation data smoothing task channel smoothing is found to be superior in four respects it has a lower computational complexity it is easy to implement it chooses the global minimum error instead of the nearest local minimum and it can also be used on nonlinear spaces such as orientation space

probabilistic 3d object recognition a probabilistic 3d object recognition algorithm is presented in order to guide the recognition process the probability that match hypotheses between image features and model features are correct is computed a model is developed which uses the probabilistic peaking effect of measured angles and ratios of lengths by tracing iso angle and iso ratio curves on the viewing sphere the model also accounts for various types of uncertainty in the input such as incomplete and inexact edge detection for each match hypothesis the pose of the object and the pose uncertainty which is due to the uncertainty in vertex position are recovered this is used to find sets of hypotheses which reinforce each other by matching features of the same object with compatible uncertainty regions a probabilistic expression is used to rank these hypothesis sets the hypothesis sets with the highest rank are output the algorithm has been fully implemented and tested on real images

susan a new approach to low level image processing this paper describes a new approach to low level image processing in particular edge and corner detection and structure preserving noise reduction non linear filtering is used to define which parts of the image are closely related to each individual pixel each pixel has associated with it a local image region which is of similar brightness to that pixel the new feature detectors are based on the minimization of this local image region and the noise reduction method uses this region as the smoothing neighbourhood the resulting methods are accurate noise resistant and fast details of the new feature detectors and of the new noise reduction method are described along with test results

time coarse graining near critical points the authors investigate the statistical properties of local variables coarse grained in time in systems undergoing continuous phase transitions a general theory is developed which suggests universal scaling behaviour in the limit of large coarse graining times the predictions are substantiated with monte carlo studies of two dimensional scalar models a simple fractal model of the time profile of the ordering variable is developed which captures and illuminates its timescale invariant character the theory is used to study the critical behaviour of local resonance lineshapes in the two dimensional case studied explicitly the critical slowing down is shown to drive a crossover to a split slow motion lineshape

variational restoration of nonflat image features models and algorithms we develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features nonflat image features are those that live on riemannian manifolds instead of on the euclidean spaces familiar examples include the orientation feature from optical flows or gradient flows that lives on the unit circle s1 the alignment feature from fingerprint waves or certain texture images that lives on the real projective line mathbb rp 1 and the chromaticity feature from color images that lives on the unit sphere s2 in this paper we apply the variational method to denoise and restore general nonflat image features mathematical models for both continuous image domains and discrete domains or graphs are constructed riemannian objects such as metric distance and levi civita connection play important roles in the models computational algorithms are also developed for the resulting nonlinear equations the mathematical framework can be applied to res

orientation diffusions diffusions are useful for image processing and computer vision because they provide a convenient way of smoothing noisy data analyzing images at multiple scales and enhancing discontinuities a number of diffusions of image brightness have been defined and studied so far they may be applied to scalar and vector valued quantities that are naturally associated with intervals of either the real line or other flat manifolds some quantities of interest in computer vision and other areas of engineering that deal with images are defined on curved manifolds typical examples are orientation and hue that are defined on the circle generalizing brightness diffusions to orientation is not straightforward especially in the case where a discrete implementation is sought an example of what may go wrong is presented a method is proposed to define diffusions of orientation like quantities first a definition in the continuum is discussed then a discrete orientation diffusion is proposed the behavior of such diffusions is explored both analytically and experimentally it is shown how such orientation diffusions contain a nonlinearity that is reminiscent of edge process and anisotropic diffusion a number of open questions are proposed

droplet theory of correlations in ordered phases the explicit droplet theory of low dimensional phase transitions developed recently is extended to yield a description of the pair correlation function for the universality class of q state potts models in the ordered phase and in zero external field it is shown that in d 1 epsilon dimensions the short distance behaviour of the correlation function and in particular the exponent eta are controlled by nested nearly spherical droplets in contrast it is argued that the spatial dependence of the large distance behaviour in d 2 or above is controlled by highly anisotropic droplets thus illuminating the widom relation sigma xi d 1 approximately constant linking the dimensionless surface tension sigma and the bulk correlation length xi in the particular case of d 2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings in the spirit of recent independent arguments by abraham and by fisher a non ornstein zernike correlation function prefactor and the result of sigma xi 1 2 follow in accord with exact results for the q 2 ising case and with implications for the q to 1 percolation problem

the ising model below tc calculation of nonuniversal amplitudes using a primitive droplet model using recent exact results for the surface free energies of ising droplets on a square lattice and applying a primitive droplet theory due to langer 1967 the author obtains an expression for the principal nonuniversal amplitude appearing in the imaginary part of the analytic continuation of the free energy across the coexistence line for the square lattice ising model below tc the significance of this calculation is the known dependence on this amplitude of the large order terms in the expansion of the physical free energy in powers of h thus allowing a direct numerical comparison with existing series data to be made after carrying out a simple renormalisation of the droplet volume term the author finds excellent numerical agreement

universal local dynamics the authors suggest that near a critical point the statistical behaviour of local variables averaged over times long on microscopic scales is universal reflecting large scale partial structure the contention is supported with monte carlo studies of two dimensional scalar models

finite size scaling study of the equilibrium cluster distribution of the two dimensional ising model the authors use a very fast and efficient algorithm to study by monte carlo methods the equilibrium cluster distribution cl l and the mean number of clusters per lattice site containing l particles in a square lattice of l2 sites of the two dimensional ising model at the critical point finite size scaling theory is then used to analyse the scaling ansatz cl l l tau f ls l tau and s being critical exponents the second moment of the cluster distribution p2 l sigma l2cl is shown to behave as p2 l approximately ltheta with theta 1 895 or 0 010 the effect of corrections to scaling is also discussed

scaling at the percolation threshold above six dimensions the fractal dimensionality of the infinite cluster at the percolation threshold for dimensionalities d 6 is shown to be d 4 rather than the naive finite size scaling prediction d d 2 similarly the conductivity of a sample of size l scales as l d rather than l 6 this anomalous behaviour is related to a dangerous irrelevant variable associated with the probability to have vertices of three bonds the crossover to the homogeneous behaviour occurs at length scales which are short compared with the correlation length the links and blobs picture is confirmed for d 6 and the size of the latter is estimated

diffusion of general data on non flat manifolds viaharmonic maps theory the direction diffusion case in a number of disciplines directional data provides a fundamental source of information a novel framework for isotropic and anisotropic diffusion of directions is presented in this paper the framework can be applied both to denoise directional data and to obtain multiscale representations of it the basic idea is to apply and extend results from the theory of harmonic maps and in particular harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data we show the corresponding variational and partial differential equations formulations for isotropic diffusion obtained from an l 2 norm and edge preserving diffusion obtained from an l norm in general and an l 1 norm in particular in contrast with previous approaches the framework is valid for directions in any dimensions supports non smooth data and gives both isotropic and anisotropic formulations in addition the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non flat manifolds that is functions between two general manifolds we present a number of theoretical results open questions and examples for gradient vectors optical flow and color images

online learning of correspondences between images we propose a novel method for iterative learning of point correspondences between image sequences points moving on surfaces in 3d space are projected into two images given a point in either view the considered problem is to determine the corresponding location in the other view the geometry and distortions of the projections are unknown as is the shape of the surface given several pairs of point sets but no access to the 3d scene correspondence mappings can be found by excessive global optimization or by the fundamental matrix if a perspective projective model is assumed however an iterative solution on sequences of point set pairs with general imaging geometry is preferable we derive such a method that optimizes the mapping based on neyman s chi square divergence between the densities representing the uncertainties of the estimated and the actual locations the densities are represented as channel vectors computed with a basis function approach the mapping between these vectors is updated with each new pair of images such that fast convergence and high accuracy are achieved the resulting algorithm runs in real time and is superior to state of the art methods in terms of convergence and accuracy in a number of experiments

monte carlo and series study of corrections to scaling in the baxter wu model abstract we compare the results of extensive monte carlo simulations with the exact solution and the series expansion for the baxter wu model analysis of the series and of the monte carlo simulations show the leading corrections to the critical magnetization and energy to be nonanalytic and compatible with the exact solution an in depth discussion of the uniqueness of parameter determination for both cases is given

numerical methods for p harmonic flows and applications to image processing we propose in this paper an alternative approach for computing p harmonic maps and flows instead of solving a constrained minimization problem on sn 1 we solve an unconstrained minimization problem on the entire space of functions this is possible using the projection on the sphere of any arbitrary function then we show how this formulation can be used in practice for problems with both isotropic and anisotropic diffusion with applications to image processing using a new finite difference scheme

synergistic development of differential approximants and the finite lattice method in lattice statistics several decades of parallel developments in the calculation and analysis of series expansions for lattice statistics have led to many new insights into critical phenomena these studies have centered on the use of the finite lattice method for series expansions in lattice statistics and the use of differential approximants in analysing such series one of these strands of research ultimately led to the result that a number of unsolved lattice statistics problems cannot be expressed as d finite functions somewhat ironically given power and success of differential approximants in analysing series neither the assumed functional form nor any finite generalisation thereof can fit such cases exactly

droplet theory in low dimensions ising systems in zero field the authors develop a theory of the universal configurational physics underlying critical point phenomena in the ising universality class the theory is formally justifiable in d 1 epsilon dimensions and may be regarded as the natural continuation of the kink based theory of one dimension which it incorporates as a limiting case in d 1 epsilon the configurational building block is the droplet the typical droplet is not spherical and the many droplet assembly is not dilute the implied problems are handled with renormalisation group methods it is found that droplet shape fluctuation effects control the correlation length exponent nu while the nesting of droplets within droplets control the order parameter exponent beta the exponents nu and beta thus effectively define respectively the fractal dimensions of the droplet surface and the droplet volume the theory is used to determine and illuminate the critical behaviour of further quantities including the free energy the susceptibility the droplet number distribution and the distribution of the intra droplet order

image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the euclidean motion group inspired by the early visual system of many mammalians we consider the construction of and reconstruction from an orientation score it u f mathbb r 2 times s 1 to mathbb c as a local orientation representation of an image f mathbb r 2 to mathbb r the mapping f mapsto it u f is a wavelet transform mathcal w psi corresponding to a reducible representation of the euclidean motion group onto mathbb l 2 mathbb r 2 and oriented wavelet psi in mathbb l 2 mathbb r 2 this wavelet transform is a special case of a recently developed generalization of the standard wavelet theory and has the practical advantage over the usual wavelet approaches in image analysis constructed by irreducible representations of the similitude group that it allows a stable reconstruction from one single scale orientation score since our wavelet transform is a unitary mapping with stable inverse we directly relate operations on orientation scores to operations on images in a robust manner furthermore by geometrical examination of the euclidean motion group g mathbb r 2 mathbb r times mathbb t which is the domain of our orientation scores we deduce that an operator on orientation scores must be left invariant to ensure that the corresponding operator mathcal w psi 1 phi mathcal w psi on images is euclidean invariant as an example we consider all linear second order left invariant evolutions on orientation scores corresponding to stochastic processes on g as an application we detect elongated structures in medical images and automatically close the gaps between them finally we consider robust orientation estimates by means of channel representations where we combine robust orientation estimation and learning of wavelets resulting in an auto associative processing of orientation features here linear averaging of the channel representation is equivalent to robust orientation estimation and an adaptation of the wavelet to the statistics of the considered image class leads to an auto associative behavior of the system

the two dimensional spin 1 ising system and related models although the two dimensional spin 1 2 ising model was solved in zero field in 1944 no exact results are yet available for the spin 1 model this model should have the same dominant critical exponents as the spin 1 2 model but appears to exhibit non analytic corrections to scaling as expected for a phi 4 model and unlike the spin 1 2 model in this paper new 45 term low temperature series for the magnetisation specific heat and susceptibility of the spin 1 model on the square lattice are presented and analysed a new 24 term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model which is also in the phi 4 universality class are also considered for both models a non analytic confluent correction with exponent 1 0 delta 1 1 3 is found the validity of this result is enhanced by a comparison with the s 1 2 case

confluent singularities in directed bond percolation existing series for directed bond percolation on the square honeycomb and face centred cubic lattices are analysed with a view to detecting their confluent singularities the authors compare our results with the confluent singularities in reggeon field theory which has been shown to be in the same universality class

universality in the two dimensional continuous spin model monte carlo methods have been used to analyse the critical behaviour of the d 2 border phi 4 model which according to a recent series expansion study violates expectations based upon universality the results suggest that the model does in fact fall into the ising universality class at least as regards those parameters which characterise the critical point configurations of the ordering variable

transfer matrix calculation of the relative noise exponent in a two dimensional percolating network a new method is given for calculating the relative noise exponent describing the size dependence of macroscopic fluctuations due to microscopic resistance noise of a random resistor network at the percolation threshold pc the macroscopic fluctuation is calculated exactly using the transfer matrix idea in a given network the asymptotic exponent b is found by finite size scaling

the probabilistic peaking effect of viewed angles and distances with application to 3 d object recognition two novel probabilistic models for viewed angles and distances are derived using an observability sphere method the method which is based on the assumption that the prior probability density is isotropic for all viewing orientations can be used for the computation of observation probabilities for object s aspects features and probability densities of their quantitative attributes using the sphere it is discovered that the probability densities of viewed angles distances and even projected curvature have sharp peaks at their original values from this peaking effect it is concluded that in most cases the values of angles and distances are being altered only slightly by the imaging process and they can still serve as a strong cue for model based recognition the probabilistic models for 3 d object recognition from monocular images are used to form the angular elements that are needed the objects are represented by their linear features and specific points primitives using the joint density model of angles and distances the probabilities of initial matching hypotheses and mutual information coefficients are estimated these results are then used for object recognition by optimal matching search and stochastic labeling schemes various synthetic and real objects are recognized by this approach

droplet theory in low dimensions potts models and percolation a recently developed droplet theory of low dimensional ising systems is generalised to describe the q state potts universality class and thence the percolation problem realised in the q to 1 limit an integral equation for the potts model free energy is derived and used to obtain the generating function for the cluster size probability distribution in the percolation problem above the percolation threshold a closed equation for the distribution is obtained and shown to yield scaling behaviour the form of the distribution is determined analytically in the regimes of small and large cluster sizes yielding respectively power law decay and kunz souillard exponential decay the form of the distribution in the regime of intermediate cluster sizes is determined numerically in space dimension d 2

revisiting the relationship between adaptive smoothing and anisotropic diffusion with modified filters anisotropic diffusion has been known to be closely related to adaptive smoothing and discretized in a similar manner this paper revisits a fundamental relationship between two approaches it is shown that adaptive smoothing and anisotropic diffusion have different theoretical backgrounds by exploring their characteristics with the perspective of normalization evolution step size and energy flow based on this principle adaptive smoothing is derived from a second order partial differential equation pde not a conventional anisotropic diffusion via the coupling of fick s law with a generalized continuity equation where a source or sink exists which has not been extensively exploited we show that the source or sink is closely related to the asymmetry of energy flow as well as the normalization term of adaptive smoothing it enables us to analyze behaviors of adaptive smoothing such as the maximum principle and stability with a perspective of a pde ultimately this relationship provides new insights into application specific filtering algorithm design by modeling the source or sink in the pde we introduce two specific diffusion filters the robust anisotropic diffusion and the robust coherence enhancing diffusion as novel instantiations which are more robust against the outliers than the conventional filters

interpreting translation invariant wavelet shrinkage as a new image smoothing scale space coifman and donoho 1995 suggested translation invariant wavelet shrinkage as a way to remove noise from images basically their technique applies wavelet shrinkage to a two dimensional 2 d version of the semi discrete wavelet representation of mallat and zhong 1992 coifman and donoho also showed how the method could be implemented in o nlog n operations where there are n pixels in this paper we provide a mathematical framework for iterated translation invariant wavelet shrinkage and show using a theorem of kato and masuda 1978 that with orthogonal wavelets it is equivalent to gradient descent in l sub 2 i along the semi norm for the besov space b sub 1 sup 1 l sub 1 i which in turn can be interpreted as a new nonlinear wavelet based image smoothing scale space unlike many other scale spaces the characterization is not in terms of a nonlinear partial differential equation

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

recognizing solid objects by alignment with an image in this paper we consider the problem of recognizing solid objects from a single two dimensional image of a three dimensional scene we develop a new method for computing a transformation from a three dimensional model coordinate frame to the two dimensional image coordinate frame using three pairs of model and image points we show that this transformation always exists for three noncollinear points and is unique up to a reflective ambiguity the solution method is closed form and only involves second order equations we have implemented a recognition system that uses this transformation method to determine possible alignments of a model with an image each of these hypothesized matches is verified by comparing the entire edge contours of the aligned object with the image edges using the entire edge contours for verification rather than a few local feature points reduces the chance of finding false matches the system has been tested on partly occluded objects in highly cluttered scenes

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

edge preserving smoothers for image processing abstract classical smoothers have limited usefulness in image processing because sharp edges tend to be blurred there is a literature on edge preserving smoothers but these all require moderately large smooth stretches here we discuss an approach to this problem called sigma filtering and propose an improvement based on running m estimation both computational and theoretical aspects are developed for image processing the methods have a niche between standard filtering approaches and bayes markov random field analysis

stability of the isotropic fixed point near one dimension within the context of the l epsilon expansion and the near planar interface hamiltonian the isotropic fixed point is found to be stable to a wide class of perturbations in particular the universality class includes all non singular forms of anisotropic surface tension

two three state potts models the universality of confluent corrections to scaling the authors present a study of an extended 35 power low temperature series for the susceptibility of the ordinary two site three state potts model on the square lattice as well as reanalyses of several extant series for the three site three state potts model on the triangular lattice dominant and confluent singularities are investigated and the exponents of both are found to be universal and in agreement with known and conjectured exact values

3 d pose from 3 points using weak perspective this correspondence discusses computing the pose of a model from three matching point pairs under weak perspective projection a new approach to the problem that is motivated geometrically is described like previous methods the method here involves solving a biquadratic equation but here the biquadratic s solutions comprised of an actual and a false solution are interpreted graphically the final equations take a new form which leads to a simple expression for the image position of any unmatched model point

droplet theory of low dimensional potts models formulates a generalisation of the droplet theory of ising systems to droplet systems containing more than two phases valid near one dimension and based on the configurational energy of surface tension thermal and magnetic exponents for these multi phase systems are compared with the results for a q state potts model in the external field the potts exponents are obtained using a migdal recursion formula and are shown to be of very similar structure

tree filtering efficient structure preserving smoothing with a minimum spanning tree we present a new efficient edge preserving filter tree filter to achieve strong image smoothing the proposed filter can smooth out high contrast details while preserving major edges which is not achievable for bilateral filter like techniques tree filter is a weighted average filter whose kernel is derived by viewing pixel affinity in a probabilistic framework simultaneously considering pixel spatial distance color intensity difference as well as connectedness pixel connectedness is acquired by treating pixels as nodes in a minimum spanning tree mst extracted from the image the fact that an mst makes all image pixels connected through the tree endues the filter with the power to smooth out high contrast fine scale details while preserving major image structures since pixels in small isolated region will be closely connected to surrounding majority pixels through the tree while pixels inside large homogeneous region will be automatically dragged away from pixels outside the region the tree filter can be separated into two other filters both of which turn out to have fast algorithms we also propose an efficient linear time mst extraction algorithm to further improve the whole filtering speed the algorithms give tree filter a great advantage in low computational complexity linear to number of image pixels and fast speed it can process a 1 megapixel 8 bit image at 0 25 s on an intel 3 4 ghz core i7 cpu including the construction of mst the proposed tree filter is demonstrated on a variety of applications

critical temperatures of the d 3 s 1 2 ising model the effect of confluent corrections to scaling much attention has been paid to the d 3 s 1 2 ising model on the bcc lattice in recent years because the best high temperature series expansions were available for this lattice however in order to compare series estimates of critical exponents and temperatures with independent evaluations via the monte carlo method lattices of lower coordination number must be considered since monte carlo studies are usually made on these a study of extant susceptibility series on the fcc sc diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent monte carlo analyses of the d 3 s 1 2 ising model on the sc and diamond lattices

droplet theory in low dimensions ising systems in an ordering field the authors extend the microscopic droplet theory of ising systems developed recently to incorporate the effects of an ordering field the theory yields a free energy which is the solution of a renormalisation group equation describing droplet nesting and which has a full scaling form the behaviour near the coexistence curve is investigated and found to display the essential singularity suggested by primitive droplet models but with parameters renormalised in a physically intelligible way the droplet population function implied by the theory is parametrised in a simple way in space dimension d 2 to yield predictions for various thermodynamic properties including the equation of state which are found to be in fair accord with series based results the structure of the field dependent droplet number distributions is investigated and contrasted with the forms assumed in phenomenological droplet theories the domain of validity of the theory is assessed it is concluded that the theory fails to describe sufficiently large droplets in the presence of a finite field and is thus in principle trustworthy only in a region close to the coexistence curve

mean shift mode seeking and clustering mean shift a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper this generalization makes some k means like clustering algorithms its special cases it is shown that mean shift is a mode seeking process on the surface constructed with a shadow kernal for gaussian kernels mean shift is a gradient mapping convergence is studied for mean shift iterations cluster analysis if treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data applications in clustering and hough transform are demonstrated mean shift is also considered as an evolutionary strategy that performs multistart global optimization

finite size scaling of the free energy levels in the ising and potts q 3 universality the finite size scaling of the free energy levels of privman and fisher 1984 is tested numerically on two dimensional lattice strips in the ising and q 3 potts universality the universal amplitudes and the free energy per site are obtained

the mean size of ising clusters a confluent singularity analysis the authors re examine using a confluent singularity analysis existing low temperature series for the mean size of finite clusters for the pure spin 1 2 nearest neighbour ising model in two dimensions although the triangular lattice is problematical results for the honeycomb and square lattices may be interpreted as indicating a dominant exponent theta 1 87 or 0 04 consistent with the theoretical prediction theta gamma beta together with competing analytic and non analytic delta 1 1 35 or 0 25 correction to scaling terms

review and analysis of solutions of the three point perspective pose estimation problem in this paper the major direct solutions to the three point perspective pose estimation problems are reviewed from a unified perspective beginning with the first solution which was published in 1841 by a german mathematician continuing through the solutions published in the german and then american photogrammetry literature and most recently in the current computer vision literature the numerical stability of these three point perspective solutions are also discussed we show that even in case where the solution is not near the geometric unstable region considerable care must be exercised in the calculation depending on the order of the substitutions utilized the relative error can change over a thousand to one this difference is due entirely to the way the calculations are performed and not due to any geometric structural instability of any problem instance we present an analysis method which produces a numerically stable calculation