fast implementation of the continuous wavelet transform with integer scales we describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m in this approach the input signal and the analyzing wavelet are both represented by polynomial splines the algorithm uses a combination of moving sum and zero padded filters and its complexity per scale is o n where n is the signal length the computation is exact and the implementation is noniterative across scales we also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection first and second derivative operators or gabor like time frequency signal analysis

a review of geometric transformations for nonrigid body registration this paper provides a comprehensive and quantitative review of spatial transformations models for nonrigid image registration it explains the theoretical foundation of the models and classifies them according to this basis this results in two categories physically based models described by partial differential equations of continuum mechanics e g linear elasticity and fluid flow and basis function expansions derived from interpolation and approximation theory e g radial basis functions b splines and wavelets recent work on constraining the transformation so that it preserves the topology or is diffeomorphic is also described the final section reviews some recent evaluation studies the paper concludes by explaining under what conditions a particular transformation model is appropriate

boundary conditions for young van vliet recursive filtering young and van vliet have designed computationally efficient methods for approximating gaussian based convolutions by running a recursive infinite impulse response iir filter forward over the input signal then running a second iir filter backward over the first filter s output to transition between the two filters they use a suboptimal heuristic that produces significant amplitude and phase distortion for all points within about three standard deviations of the right hand boundary in this correspondence a simple linear transition rule that eliminates this distortion is derived

unwarping of unidirectionally distorted epi images echo planar imaging epi is a fast nuclear magnetic resonance imaging mri method unfortunately local magnetic field inhomogeneities induced mainly by the subject s presence cause significant geometrical distortion predominantly along the phase encoding direction which must be undone to allow for meaningful further processing so far this aspect has been too often neglected in this paper the authors suggest a new approach using an algorithm specifically developed for the automatic registration of distorted epi images with corresponding anatomically correct mri images they model the deformation field with splines which gives us a great deal of flexibility while comprising the affine transform as a special case the registration criterion is least squares interestingly the complexity of its evaluation does not depend on the resolution of the control grid the spline model gives the authors good accuracy thanks to its high approximation order the short support of splines leads to a fast algorithm a multiresolution approach yields robustness and additional speedup the algorithm was tested on real as well as synthetic data and the results were compared with a manual method a wavelet based sobolev type random deformation generator was developed for testing purposes a blind test indicates that the proposed automatic method is faster more reliable and more precise than the manual one

towards statistical mechanics of a 2d random cellular structure using the methods of statistical mechanics the authors construct the partition function of a system of a large number of 2d cells that covers without pores or overlaps a flat surface each cell is defined by its position area perimeter and number of sides besides topological and space filling constraints they impose new ones regarding the energy and maximum area of the cells the entropy and free energy of the system are calculated and the mean values of the macroscopic parameters are determined in the equilibrium configuration the results are compared with experimental ones for soap froths and metallurgical aggregates

a comment on rivier s maximum entropy method of statistical crystallography in the literature rivier s maximum entropy method was used to prove lewis law and a linear aboav s law in this paper we show that the functional forms of these two laws for a statistically equilibrated cellular network even if such a network really exists cannot be derived or proved by this method for example within the maximum entropy method we demonstrate that a quadratic aboav s or lewis s law is as probable as a linear one

model for coarsening froths and foams the coarsening process of froths and foams is described statistically in two dimensions by a random neighbor model this nonlinear dynamical model contains no free parameters all rates are determined dynamically it describes normal growth agrees with experimental results throws light on old data and hopefully inspires new simulations and measurements lewis s law anat rec 38 341 1928 am j bot 30 74 1943 and some other properties can be derived analytically from the model

diffusion tensor analysis with invariant gradients and rotation tangents guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation tensor shape and orientation both have three degrees of freedom spanned by invariant gradients and rotation tangents respectively as an initial demonstration of the framework we create a tunable measure of tensor difference that can selectively respond to shape and orientation second to analyze the spatial gradient in a tensor volume a third order tensor our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented finally we apply the framework to decompose the fourth order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance including a direct approximation for the variance of tensor invariants such as fractional anisotropy

comparison of interpolating methods for image resampling when resampling an image to a new set of coordinates for example when rotating an image there is often a noticeable loss in image quality to preserve image quality the interpolating function used for the resampling should be an ideal low pass filter to determine which limited extent convolving functions would provide the best interpolation five functions were compared a nearest neighbor b linear c cubic b spline d high resolution cubic spline with edge enhancement a 1 and e high resolution cubic spline a 0 5 the functions which extend over four picture elements c d e were shown to have a better frequency response than those which extend over one a or two b pixels the nearest neighbor function shifted the image up to one half a pixel linear and cubic b spline interpolation tended to smooth the image the best response was obtained with the high resolution cubic spline functions the location of the resampled points with respect to the initial coordinate system has a dramatic effect on the response of the sampled interpolating function the data are exactly reproduced when the points are aligned and the response has the most smoothing when the resampled points are equidistant from the original coordinate points thus at the expense of some increase in computing time image quality can be improved by resampled using the high resolution cubic spline function as compared to the nearest neighbor linear or cubic b spline functions

dynamic graph cuts for efficient inference in markov random fields in this paper we present a fast new fully dynamic algorithm for the st mincut max flow problem we show how this algorithm can be used to efficiently compute map solutions for certain dynamically changing mrf models in computer vision such as image segmentation specifically given the solution of the max flow problem on a graph the dynamic algorithm efficiently computes the maximum flow in a modified version of the graph the time taken by it is roughly proportional to the total amount of change in the edge weights of the graph our experiments show that when the number of changes in the graph is small the dynamic algorithm is significantly faster than the best known static graph cut algorithm we test the performance of our algorithm on one particular problem the object background segmentation problem for video it should be noted that the application of our algorithm is not limited to the above problem the algorithm is generic and can be used to yield similar improvements in many other cases that involve dynamic change

recursive anisotropic 2 d gaussian filtering based on a triple axis decomposition we describe a recursive algorithm for anisotropic 2 d gaussian filtering based on separating the filter into the cascade of three rather two 1 d filters the filters operate along axes obtained by integer horizontal and or vertical pixel shifts this eliminates interpolation which removes spatial inhomogeneity in the filter and produces more elliptically shaped kernels it also results in a more regular filter structure which facilitates implementation in dsp chips finally it improves matching between filters with the same eccentricity and width but different orientations our analysis and experiments indicate that the computational complexity is similar to an algorithm that operates along two axes 11ms for a 512times512 image using a 3 2 ghz pentium 4 pc on the other hand given a limited set of basis filter axes there is an orientation dependent lower bound on the achievable aspect ratios

two dimensional spline interpolation for image reconstruction a scheme of two dimensional 2 d spline interpolation for image reconstruction is presented it is aimed to be applied in image data compression an impulse response derived from 2 d quadratic spline functions is theoretically analysed then it is approximated to be a finite duration one fir for practical implementation with this scheme compression rate of greater than 12 is achieved for x ray left ventricular cineangiogram image data compression

optimized compact support interpolation kernels in this paper we investigate the problem of designing compact support interpolation kernels for a given class of signals by using calculus of variations we simplify the optimization problem from an nonlinear infinite dimensional problem to a linear finite dimensional case and then find the optimum compact support function that best approximates a given filter in the least square sense l2 norm the benefit of compact support interpolants is the low computational complexity in the interpolation process while the optimum compact support interpolant guarantees the highest achievable signal to noise ratio snr our simulation results confirm the superior performance of the proposed kernel compared to other conventional compact support interpolants such as cubic spline

coarsening of three dimensional grains in crystals or bubbles in dry foams tends towards a universal statistically scale invariant regime we perform extensive potts model simulations of three dimensional dry foam coarsening starting with 2 25 million bubbles we have enough statistics to fulfill the three constraints required for the study of statistical scale invariance first enough time for the transient to end and reach the scaling state then enough time in the scaling state itself to characterize its properties and finally enough bubbles at the end to avoid spurious finite size effects in the scaling state we find that the average surface area of the bubbles increases linearly with time the geometry bubble shape and size and topology number of faces and edges as well as their correlations become constant in time their distributions agree with the data of the literature we present an analytical model universal up to parameters extracted from the simulations for a disordered foam minimizing its free energy which agrees with the simulations we discuss the limitations of the simulations and of the model

stochastic geometry of polygonal networks an alternative approach to the hexagon square transition in benard convection we apply stochastic geometry to the transition from hexagonal to square cells recently observed in surface tension driven benard convection in particular we study the metric and topological evolution of benard patterns as a function of the temperature difference dt across the layer the preference of square benard cells at higher dt is a consequence of both a higher efficiency in heat transfer and more favorable metric properties most notably the perimeter area ratio of a square cell exceeds that of a hexagonal cell by an unexpectedly high value from a topological point of view the benard pattern obeys the aboav weaire law at all times even in the presence of threefold and fourfold vertices the regimes above and below the transition are characterized by different topological correlations between neighboring cells with the appearance of fourfold vertices the topological correlation changes from negative to positive s1063 651x 98 14709 8

estimating intensity variance due to noise in registered images applications to diffusion tensor mri image registration techniques which require image interpolation are widely used in neuroimaging research we show that signal variance in interpolated images differs significantly from the signal variance of the original images in native space we describe a simple approach to compute the signal variance in registered images based on the signal variance and covariance of the original images the spatial transformations computed by the registration procedure and the interpolation or approximation kernel chosen the method is general and could handle various sources of signal variability such as thermal noise and physiological noise provided that their effects can be assessed in the original images our approach is applied to diffusion tensor dt mri data assuming only thermal noise as the source of variability in the data we show that incorrect noise variance estimates in registered diffusion weighted images can affect dt parameters as well as indices of goodness of fit such as chi square maps in addition to dt mri we believe that this methodology would be useful any time parameter extraction methods are applied to registered or interpolated data such as in relaxometry and functional mri studies

support and approximation properties of hermite splines abstract in this paper we formally investigate two mathematical aspects of hermite splines that are relevant to practical applications we first demonstrate that hermite splines are maximally localized in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties then we precisely quantify the approximation power of hermite splines for the reconstruction of functions and their derivatives it is known that the hermite and b spline approximation schemes have the same approximation order more precisely their approximation error vanishes as o t 4 when the step size t goes to zero in this work we show that they actually have the same asymptotic approximation error constants too therefore they have identical asymptotic approximation properties hermite splines combine optimal localization and excellent approximation power while retaining interpolation properties and closed form expression in contrast to existing similar functions these findings shed a new light on the convenience of hermite splines in the context of computer graphics and geometrical design

applicability of the lewis and aboav weaire laws to 2d and 3d cellular structures based on poisson partitions two and three dimensional networks of a columnar type are described which result from partitions based on poisson point distributions the metric and topological properties of such laminated poisson networks are derived and the applicability of the lewis and aboav weaire laws to them is tested the 2d poisson network contains cells with i or 4 i is the number of sides and both laws are obeyed the 3d poisson networks are of various types and have f or 6 f is the number of faces in a cell and f 14 in one particular type of 3d poisson network the two laws are again exactly obeyed in another type the laws show large deviations at low f but are asymptotically obeyed when f tends to infinite

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

pegasos primal estimated sub gradient solver for svm we describe and analyze a simple and effective stochastic sub gradient descent algorithm for solving the optimization problem cast by support vector machines svm we prove that the number of iterations required to obtain a solution of accuracy epsilon is tilde o 1 epsilon where each iteration operates on a single training example in contrast previous analyses of stochastic gradient descent methods for svms require omega 1 epsilon 2 iterations as in previously devised svm solvers the number of iterations also scales linearly with 1 where is the regularization parameter of svm for a linear kernel the total run time of our method is tilde o d lambda epsilon where d is a bound on the number of non zero features in each example since the run time does not depend directly on the size of the training set the resulting algorithm is especially suited for learning from large datasets our approach also extends to non linear kernels while working solely on the primal objective function though in this case the runtime does depend linearly on the training set size our algorithm is particularly well suited for large text classification problems where we demonstrate an order of magnitude speedup over previous svm learning methods

dynamic hybrid algorithms for map inference in discrete mrfs in this paper we present novel techniques that improve the computational and memory efficiency of algorithms for solving multi label energy functions arising from discrete mrfs or crfs these methods are motivated by the observations that the performance of minimization algorithms depends on 1 the initialization used for the primal and dual variables and 2 the number of primal variables involved in the energy function our first method dynamic expansion works by recycling results from previous problem instances the second method simplifies the energy function by reducing the number of unknown variables present in the problem further we show that it can also be used to generate a good initialization for the dynamic expansion algorithm by reusing dual variables we test the performance of our methods on energy functions encountered in the problems of stereo matching and color and object based segmentation experimental results show that our methods achieve a substantial improvement in the performance of expansion as well as other popular algorithms such as sequential tree re weighted message passing and max product belief propagation we also demonstrate the applicability of our schemes for certain higher order energy functions such as the one described for interactive texture based image and video segmentation in most cases we achieve a 10 15 times speed up in the computation time our modified expansion algorithm provides similar performance to fast pd but is conceptually much simpler both expansion and fast pd can be made orders of magnitude faster when used in conjunction with the reduce scheme proposed in this paper

recursive regularization filters design properties and applications least squares approximation problems that are regularized with specified highpass stabilizing kernels are discussed for each problem there is a family of discrete regularization filters r filters which allow an efficient determination of the solutions these operators are stable symmetric lowpass filters with an adjustable scale factor two decomposition theorems for the z transform of such systems are presented one facilitates the determination of their impulse response while the other allows an efficient implementation through successive causal and anticausal recursive filtering a case of special interest is the design of r filters for the first and second order difference operators these results are extended for two dimensional signals and for illustration purposes are applied to the problem of edge detection this leads to a very efficient implementation 8 multiplies 10 adds per pixel of the optimal canny edge detector based on the use of a separable second order r filter

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

on the correlation between sizes and shapes of cells in epithelial mosaics it is shown that lewis s empirical linear relationship between the average area of a cell and the number of its sides in two dimensional mosaics corresponds to maximal arbitrariness in the cellular distribution an expression for the distribution is given in the general case

fast migration inversion with multivalued ray fields part 1 method validation test and application in 2d to marmousi we address the problem of building an efficient algorithm for ray based migration inversion in complex media characterized by multivalued traveltime since the multivalued ray maps can be computed by wavefront construction the key point lies in the interpolation of these multivalued coarse maps for this goal we propose two new strategies the first one is based on an interpolation of the rayfield after identifying the ray branches on the elementary cells of the coarse grid the second is based on an extrapolation of the arrivals from the coarse grid points both strategies are evaluated in two dimensions with the marmousi data set demonstrating the cpu efficiency and accuracy of both approaches since they both consider multiarrivals locally the associated overhead cost with respect to a single arrival implementation is quite low twice the computing time of the marmousi test case the application also reveals that the new interpolation strategy is more efficient in terms of cpu time compared to the extrapolation strategy but at the expense of a more complex numerical implementation and less accurate results

multiscale curvature based shape representation using b spline wavelets this paper presents a new multiscale curvature based shape representation technique with application to curve data compression using b spline wavelets the evolution of the curve is implemented in the b spline scale space which enjoys a number of advantages over the classical gaussian scale space for instance the availability of fast algorithms the b spline wavelet transforms are used to efficiently estimate the multiscale curvature functions based on the curvature scale space image we introduce a coarse to fine matching algorithm which automatically detects the dominant points and uses them as knots for curve interpolation

soft computing approach to feature extraction based on both wavelet theory and fuzzy theory a soft computing system scs is proposed for feature extraction of signals the proposed scs approach possesses the advantages of soft decision making on wavelet coefficients for feature extraction adaptive selectivity of mapping factors to coarse to fine resolution and compact form of feature representation with the scs feature extractor fuzzy sets are used to provide a robust representation for signal information and wavelet transform is used to decompose a signal into detail and approximation signals at a given resolution the detail and approximation signals are inputted to the proposed scs to extract signal features at that resolution level the sensitivity in feature extraction of the proposed approach can be adapted by tuning the fuzzy sets for the detail and approximation signals at different resolutions the signal can be examined and suitable features can be extracted examples of both one dimensional signals and two dimensional fingerprint images are used to illustrate the proposed soft computing approach for feature extraction and pattern recognition the results show that the extracted features are sensitive enough to distinguish the similar signal from the different ones and are robust enough to tolerate noise corruption

scaling properties of three dimensional foams we obtain the scaling states of three dimensional soap foams by using equilibrium statistical mechanics techniques we consider a phase space where each bubble is characterized by its center of mass position volume surface and number of faces a gibbs entropy function is then defined and as done in standard statistical mechanics we obtain the probability density function defined in the phase space by maximizing the entropy subject to convenient constraints the froth is supposed to completely fill the available volume and we consider energy terms accounting for surface tension and the thermal energy of the gas inside the bubbles we find that the volume distribution presents an exponential decay with cell size and the correlation between cell volume and number of faces fits very well the available numerical data additional distribution functions and average values are also in good agreement with numerical simulations

simple algorithms and architectures for b spline interpolation it is proved that the toeplitz binary value matrix inversion associated with mth order b spline interpolation can be implemented using only 2 m 1 additions pipelined architectures are developed for real time b spline interpolation based on simple running average filters it is shown that an ideal interpolating function which is approximated by a truncated sinc function with m half cycles can be implemented using b splines with m 2 multiplies with insignificant loss of performance the coefficients at the knots of the truncated sinc function can be approximated using coefficients which are powers of two the resulting implementation requires only m 4m 6 additions it is believed that the truncated sinc function approximated by zero order b spline functions actually achieves the best visual performance

polynomial spline approximation of clarke s model we investigate polynomial spline approximation of stationary random processes on a uniform grid applied to clarke s model of time variations of path amplitudes in multipath fading channels with doppler scattering the integral mean square error mse for optimal and interpolation splines is presented as a series of spectral moments the optimal splines outperform the interpolation splines however as the sampling factor increases the optimal and interpolation splines of even order tend to provide the same accuracy to build such splines the process to be approximated needs to be known for all time which is impractical local splines on the other hand may be used where the process is known only over a finite interval we first consider local splines with quasioptimal spline coefficients then we derive optimal spline coefficients and investigate the error for different sets of samples used for calculating the spline coefficients in practice approximation with a low processing delay is of interest we investigate local spline extrapolation with a zero processing delay the results of our investigation show that local spline approximation is attractive for implementation from viewpoints of both low processing delay and small approximation error the error can be very close to the minimum error provided by optimal splines thus local splines can be effectively used for channel estimation in multipath fast fading channels

addendum b spline interpolation in medical image processing analyzes b spline interpolation techniques of degree 2 4 and 5 with respect to all criteria that have been applied to evaluate various interpolation schemes in a recently published survey on image interpolation in medical imaging lehmann et al 1999 it is shown that high degree b spline interpolation has superior fourier properties smallest interpolation error and reasonable computing times therefore high degree b splines are preferable interpolators for numerous applications in medical image processing particularly if high precision is required if no aliasing occurs this result neither depends on the geometric transform applied for the tests nor the actual content of images

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

adaptive morphological filters based on a multiple orientation vector field dependent on image local features this paper addresses the formulation of adaptive morphological filters based on spatially variant structuring elements the adaptivity of these filters is achieved by modifying the shape and orientation of the structuring elements according to a multiple orientation vector field this vector field is provided by means of a bank of directional openings which can take into account the possible multiple orientations of the contours in the image after reviewing and formalizing the definition of the spatially variant dilation erosion opening and closing the proposed structuring elements are described these spatially variant structuring elements are based on ellipses which vary over the image domain adapting locally their orientation according to the multiple orientation vector field and their shape the eccentricity of the ellipses according to the distance to relevant contours of the objects the proposed adaptive morphological filters are used on gray level images and are compared with spatially invariant filters with spatially variant filters based on a single orientation vector field and with adaptive morphological bilateral filters results show that the morphological filters based on a multiple orientation vector field are more adept at enhancing and preserving structures which contains more than one orientation

a study of parts based object class detection using complete graphs object detection is one of the key components in modern computer vision systems while the detection of a specific rigid object under changing viewpoints was considered hard just a few years ago current research strives to detect and recognize classes of non rigid articulated objects hampered by the omnipresent confusing information due to clutter and occlusion the focus has shifted from holistic approaches for object detection to representations of individual object parts linked by structural information along with richer contextual descriptions of object configurations along this line of research we present a practicable and expandable probabilistic framework for parts based object class representation enabling the detection of rigid and articulated object classes in arbitrary views we investigate learning of this representation from labelled training images and infer globally optimal solutions to the contextual map detection problem using a search with a novel lower bound as admissible heuristic an assessment of the inference performance of belief propagation and tree reweighted belief propagation is obtained as a by product the generality of our approach is demonstrated on four different datasets utilizing domain dependent information cues

b spline snakes a flexible tool for parametric contour detection we present a novel formulation for b spline snakes that can be used as a tool for fast and intuitive contour outlining we start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal curvature constrained snake is a cubic spline irrespective of the form of the external energy field unfortunately such regularized snakes suffer from slow convergence speed because of a large number of control points as well as from difficulties in determining the weight factors associated to the internal energies of the curve we therefore propose an alternative formulation in which the intrinsic scale of the spline model is adjusted a priori this leads to a reduction of the number of parameters to be optimized and eliminates the need for internal energies i e the regularization term in other words we are now controlling the elasticity of the spline implicitly and rather intuitively by varying the spacing between the spline knots the theory is embedded into a multiresolution formulation demonstrating improved stability in noisy image environments validation results are presented comparing the traditional snake using internal energies and the proposed approach without internal energies showing the similar performance of the latter several biomedical examples of applications are included to illustrate the versatility of the method

an elastic deformation field model for object detection and tracking deformable parts models dpm are the current state of the art for object detection nevertheless they seem sub optimal in the representation of deformations object deformations are often continuous and not confined to big parts therefore we propose to replace the dpm star model based on big parts by a deformation field this consists of a grid of small parts connected with pairwise constraints which can better handle continuous deformations the naive application of this model for object detection would consist of a bounded sliding window approach for each possible location of the image the best part configuration within a limited bound around this location is found this is computationally very expensive instead we propose a different inference procedure where an iterative image level search finds the best object hypothesis we show that this approach is faster than bounded sliding windows yet produces comparable accuracy experiments further show that the deformation field can better approximate real object deformations and therefore for certain classes produces even better detection accuracy than state of the art dpm finally the same approach is adapted to model free tracking showing improved accuracy also in this case

fast b spline transforms for continuous image representation and interpolation efficient algorithms for the continuous representation of a discrete signal in terms of b splines direct b spline transform and for interpolative signal reconstruction indirect b spline transform with an expansion factor m are described expressions for the z transforms of the sampled b spline functions are determined and a convolution property of these kernels is established it is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering fast computational algorithms based on the recursive implementations of these filters are proposed a b spline interpolator can also be characterized in terms of its transfer function and its global impulse response cardinal spline of order n the case of the cubic spline is treated in greater detail the present approach is compared with previous methods that are reexamined from a critical point of view it is concluded that b spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner

b spline design of maximally flat and prolate spheroidal type fir filters a digital fir filter is described that offers excellent passband and stopband characteristics for general applications design formulae include parameters that adjust the magnitude response from one having characteristics like the maximally flat designs of hermann 1971 and kaiser 1975 1979 to one having characteristics like the minimum sidelobe energy approximations of kaiser and saramaki 1989 the impulse response coefficients are more straightforward to obtain than these filter designs while offering preferable response characteristics in many instances unlike fir filters designed by window or frequency sampling methods the filter coefficients are determined from the inverse fourier transform in closed form once b splines have been used to replace sharp transition edges of the magnitude response although the filters are developed in the frequency domain a convergence window is identified in the convolution series and compared with windows of popular fir filters by means of example adjustment of the transitional parameter is shown to produce a filter response that rivals the stopband attenuation and transition width of prolate spheroidal designs the design technique is extended to create additional transitional filters from prototype window functions such as the transitional hann window filter the filters are particularly suitable for precision filtering and reconstruction of sampled physiologic and acoustic signals common to the health sciences but will also be useful in other applications requiring low passband and stopband errors

multiresolution approximation using shifted splines we consider the construction of least squares pyramids using shifted polynomial spline basis functions we derive the pre and post filters as a function of the degree n and the shift parameter spl delta we show that the underlying projection operator is entirely specified by two transfer functions acting on the even and odd signal samples respectively we introduce a measure of shift invariance and show that the most favorable configuration is obtained when the knots of the splines are centered with respect to the grid points i e spl delta 1 2 when n is odd and spl delta 0 when n is even the worst case corresponds to the standard multiresolution setting where the spline spaces are nested

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

stable spins in the zero temperature spinodal decomposition of 2d potts models we present the results of zero temperature monte carlo simulations of the q state potts model on a square lattice with either four or eight neighbors and for the triangular lattice with six neighbors in agreement with previous works we observe that the domain growth process gets blocked for the nearest neighbor square lattice when q is large enough whereas for the eight neighbor square lattice and for the triangular lattice no blocking is observed our simulations indicate that the number of spins which never flipped from the beginning of the simulation up to time t follows a power law as a function of the energy even in the case of blocking the exponent of this power law varies from less than 1 2 for the ising case q 2 to 2 for q o0 and seems to be universal the effect of blocking on this exponent is invisible at least up to q 7

a continuous time linear system identification method for slowly sampled data both direct and indirect methods exist for identifying continuous time linear systems a direct method estimates continuous time input and output signals from their samples and then use them to obtain a continuous time model whereas an indirect method estimates a discrete time model first both methods rely on fast sampling to ensure good accuracy in this paper we propose a more direct method where a continuous time linear model is directly fitted to the available samples this method produces an exact model asymptotically modulo some possible aliasing ambiguity even when the sampling rate is relatively slow we also state conditions under which the aliasing ambiguity can be resolved and we provide experiments showing that the proposed method is a valid option when a slow sampling frequency must be used but a large number of samples is available

phase vector incompressible registration algorithm for motion estimation from tagged magnetic resonance images tagged magnetic resonance imaging has been used for decades to observe and quantify motion and strain of deforming tissue it is challenging to obtain 3 d motion estimates due to a tradeoff between image slice density and acquisition time typically interpolation methods are used either to combine 2 d motion extracted from sparse slice acquisitions into 3 d motion or to construct a dense volume from sparse acquisitions before image registration methods are applied this paper proposes a new phase based 3 d motion estimation technique that first computes harmonic phase volumes from interpolated tagged slices and then matches them using an image registration framework the approach uses several concepts from diffeomorphic image registration with a key novelty that defines a symmetric similarity metric on harmonic phase volumes from multiple orientations the material property of harmonic phase solves the aperture problem of optical flow and intensity based methods and is robust to tag fading a harmonic magnitude volume is used in enforcing incompressibility in the tissue regions the estimated motion fields are dense incompressible diffeomorphic and inverse consistent at a 3 d voxel level the method was evaluated using simulated phantoms human brain data in mild head accelerations human tongue data during speech and an open cardiac data set the method shows comparable accuracy to three existing methods while demonstrating low computation time and robustness to tag fading and noise

choquet integrals as projection operators for quantified tomographic reconstruction in this paper we propose to investigate and analyze a new method for performing quantified projection and back projection in emission tomography this method is based on using non summative kernels capacities and asymmetric choquet integral to obtain imprecise projected values i e intervals instead of usual reconstructed pixel values validation studies using numerical and physical single photon computed emission tomography spect phantoms were used to demonstrate links between the length of these reconstructed intervals and the stochastic noise level in reconstructed slices

an optimal nonorthogonal separation of the anisotropic gaussian convolution filter we give an analytical and geometrical treatment of what it means to separate a gaussian kernel along arbitrary axes in ropfn and we present a separation scheme that allows us to efficiently implement anisotropic gaussian convolution filters for data of arbitrary dimensionality based on our previous analysis we show that this scheme is optimal with regard to the number of memory accesses and interpolation operations needed the proposed method relies on nonorthogonal convolution axes and works completely in image space thus it avoids the need for a fast fourier transform fft subroutine depending on the accuracy and speed requirements different interpolation schemes and methods to implement the one dimensional gaussian finite impulse response and infinite impulse response can be integrated special emphasis is put on analyzing the performance and accuracy of the new method in particular we show that without any special optimization of the source code it can perform anisotropic gaussian filtering faster than methods relying on the fft

structure of random cellular networks and their evolution abstract the methods of statistical mechanics are applied to the structure of random space filling cellular structures foams metallurgical grain aggregates biological tissues microreversibility of the structural properties under elementary transformations is demonstrated maximum entropy inference under a few constraints yields structural equations of state relating the size of cells to their topological shape these equations of state classify the structures and are criteria for their randomness quasistatic evolution is obtained if the structure remains in statistical equilibrium at all times in 2d it is governed by von neumann s law

scaling laws and topological exponents in voronoi tessellations of intermittent point distributions voronoi tessellations of scale invariant fractal sets are characterized by topological and metrical properties that are significantly different from those of natural cellular structures as an example we analyse voronoi diagrams of intermittent particle distributions generated by a directed percolation process in 2 1 dimensions we observe that the average area of a cell increases much faster with the number of its neighbours than in natural cellular structures where lewis law predicts a linear behaviour we propose and numerically verify a universal scaling law that relates shape and size of the cells in scale invariant tessellations an exponent related to the topological properties of the tessellation is introduced and estimated numerically

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

grain growth in three dimensions depends on grain topology while the growth rate of the average volume in a three dimensional polycrystal is well understood the growth rates of individual grains which determine material properties through the topological and volume distributions are not known using a three dimensional q state potts model simulation i find that the average canonical growth rate of a grain depends linearly on its number of faces and is independent of its volume a surprisingly simple and unexplained result

time evolution of three dimensional cellular systems computer modeling based on vertex type models abstract an effective computer modeling of time evolution of three dimensional cellular systems like soap froths and crystalline grain aggregates has been devised which captures the essence of difficult correlation effects of neighboring cells this can be achieved by eliminating the continuous degrees of freedom besides the immediate vicinity of the center of a singular region of space that is an intersection of interfaces from the original full curvature drien equation of motion of interfaces thus leaving a set of equations of motion for such intersections i e vertices to actually carry out this projection operation each interface is divided intoa set of two dimensional simplexes a derivation of the model equations is given in the most general possible form various results including topological characteristics of three dimensional cellular patterns were obtained using the simpler version of these vertex equations among which the result for the average growth rate of f sided cells is presented an application to some specific cellular systems is also discussed

modified b splines for the sampling of bandlimited functions it is shown that there is an optimal finite linear combination of b splines denominated modified b splines such that a pertinent low frequency condition called m flatness is satisfied a profound relationship of the modified b splines with the beta distribution implies an asymptotic sampling theorem with exact reconstruction requiring only small oversampling

biorthogonal partners and applications two digital filters h z and f z are said to be biorthogonal partners of each other if their cascade h z f z satisfies the nyquist or zero crossing property biorthogonal partners arise in many different contexts such as filterbank theory exact and least squares digital interpolation and multiresolution theory they also play a central role in the theory of equalization especially fractionally spaced equalizers in digital communications we first develop several theoretical properties of biorthogonal partners we also develop conditions for the existence of biorthogonal partners and fir biorthogonal pairs and establish the connections to the riesz basis property we then explain how these results play a role in many of the above mentioned applications

evaluation of optimization methods for nonrigid medical image registration using mutual information and b splines a popular technique for nonrigid registration of medical images is based on the maximization of their mutual information in combination with a deformation field parameterized by cubic b splines the coordinate mapping that relates the two images is found using an iterative optimization procedure this work compares the performance of eight optimization methods gradient descent with two different step size selection algorithms quasi newton nonlinear conjugate gradient kiefer wolfowitz simultaneous perturbation robbins monro and evolution strategy special attention is paid to computation time reduction by using fewer voxels to calculate the cost function and its derivatives the optimization methods are tested on manually deformed ct images of the heart on follow up ct chest scans and on mr scans of the prostate acquired using a bffe tl and t2 protocol registration accuracy is assessed by computing the overlap of segmented edges precision and convergence properties are studied by comparing deformation fields the results show that the robbins monro method is the best choice in most applications with this approach the computation time per iteration can be lowered approximately 500 times without affecting the rate of convergence by using a small subset of the image randomly selected in every iteration to compute the derivative of the mutual information from the other methods the quasi newton and the nonlinear conjugate gradient method achieve a slightly higher precision at the price of larger computation times

lewis law versus feltham s law in soap froth abstract the scaling of area and perimeter of n sided cells versus n in a two dimensional cellular pattern are investigated lewis law states that a n is linear in n while feltham s law states that the perimeter e n is linear in n we have tested these two different predictions on soap froth for large samples starting with approximately 5000 cells at different times and different gap thicknesses we conclude that feltham s law has a smaller statistical error than lewis law we also compared the results with several two dimensional voronoi constructions of random points and perturbed triangular lattice we found that the error bars of the lewis s law and feltham s law are comparable in the voronoi constructions using geometric constraints we argue and verified by experimental data that there is a universal law obeyed by the slopes and intercepts of these laws on all samples

on causal estimation from bandlimited stationary sequences this paper considers the problem of estimating a stationary sequence y from the observation of a stationary correlated sequence x by means of a causal linear filter thereby it is assumed that the spectral density x of x vanishes on a subset of the unit circle of positive lebesgue measure such that the classical derivation of the estimation filter based on the spectral factorization of x can not be applied the paper derives the transfer function of such an estimation filter discusses its stability behavior and applies the result to the causal reconstruction of deterministic signals from its samples

adaptive fuzzy segmentation of magnetic resonance images an algorithm is presented for the fuzzy segmentation of two dimensional 2 d and three dimensional 3 d multispectral magnetic resonance mr images that have been corrupted by intensity inhomogeneities also known as shading artifacts the algorithm is an extension of the 2 d adaptive fuzzy c means algorithm 2 d afcm presented in previous work by the authors this algorithm models the intensity inhomogeneities as a gain field that causes image intensities to smoothly and slowly vary through the image space it iteratively adapts to the intensity inhomogeneities and is completely automated in this paper the authors fully generalize 2 d afcm to three dimensional 3 d multispectral images because of the potential size of 3 d image data they also describe a new faster multigrid based algorithm for its implementation they show using simulated mr data that 3 d afcm yields lower error rates than both the standard fuzzy c means fcm algorithm and two other competing methods when segmenting corrupted images its efficacy is further demonstrated using real 3 d scalar and multispectral mr brain images

bubble size topology correlations in two dimensional foams derived from surface energy minimization we obtain the number fractions xi the average areas a i and the average perimeters p i of i sided bubbles in a two dimensional foam by minimizing the total surface energy and assuming a simple relation between p i and a i calculations for linear and weibull distributions of the square root of the bubble area yield large deviations particularly at small i from lewis s and desch s laws which linearly relate a i and p i respectively to i nevertheless we find good agreement with experimental results for xi p i and a i in foams

correlations in topological models of 2d random cellular structures general relations and constraints which must be satisfied by the topological correlations in 2d space filling random cellular structures are discussed and a topological short range order coefficient is defined topological models of 2d structures are associated with planar tessellations with topologically unstable sites which belong to z 3 polygons the stable configurations called states are obtained by replacing every vertex by z 3 added sides the topological properties of the latter models are calculated exactly for a distribution of independent and equiprobable states on the various sites and for any value of z the case of the structures associated with tilings by triangles is thoroughly considered the calculated correlations are compared with the correlations in alumina cuts and in random voronoi froths the variability of the topological properties of 2d random cellular structures is discussed

interpolation artifacts in sub pixel image registration we consider the problem of registering aligning two images to sub pixel accuracy by optimization of objective functions constructed from the images intensity values we show that some widely used interpolation methods can introduce multiple local optima in the energy of the interpolated image which if not counter balanced by other terms can cause local optima in registration objective functions including the sum of squared differences cross correlation and mutual information we discuss different solutions to address the problem based on high degree b spline interpolation low pass filtering the images and stochastic integration numerical examples using synthetic and real signals and images are shown