generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

amplitude only log radon transform for geometric invariant shape descriptor a shape descriptor combining the radon transform the amplitude extraction and the log mapping is proposed in this paper it is invariant to shape rotation scaling and translation invariance to translation is achieved by amplitude extraction on the radial coordinate rotation and scaling are log mapped into two dimensional translations and recovered with the phase only correlation function in addition all transformation parameters rotating angle scaling factor and position shift can be determined also the efficiency of the proposed descriptor compared to existing methods is shown experimentally on different kind of datasets highlightswe propose a new shape descriptor based on radon transformthis descriptor needs no normalizations it is invariant to geometric transformations rotation scale and translation and robust to noise the rotation angle the scaling factor and the position shift can be easily determined

discretization of the radon transform and of its inverse by spline convolutions we present an explicit formula for b spline convolution kernels these are defined as the convolution of several b splines of variable widths h sub i and degrees n sub i we apply our results to derive spline convolution based algorithms for two closely related problems the computation of the radon transform and of its inverse first we present an efficient discrete implementation of the radon transform that is optimal in the least squares sense we then consider the reverse problem and introduce a new spline convolution version of the filtered back projection algorithm for tomographic reconstruction in both cases our explicit kernel formula allows for the use of high degree splines these offer better approximation performance than the conventional lower degree formulations e g piecewise constant or piecewise linear models we present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity in particular we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate

on board selection of relevant images an application to linear feature recognition we propose an on board selection scheme for aerial and space images based on linear feature detection in a feature hyperspace the detection task is performed by means of the radon transform rt and the wavelet transform a fast algorithm for the rt computation is described and counteractions against the discretization errors are proposed a new wavelet based algorithm is introduced which performs a fine analysis of the waveforms of the rt peaks yielding a possibly error free detection in images corrupted by a high level of noise a technique based on the feature hyperspace is proposed able to significantly exploit all the available pieces of information on these peaks results of the tests on synthetic and real images are reported which show that this method achieves satisfactory results making the detection task highly reliable in the presence of both noise and clutter

the generalization of the r transform for invariant pattern representation the beneficial properties of the radon transform make it a useful intermediate representation for the extraction of invariant features from pattern images for the purpose of indexing matching this paper revisits the problem of radon image utilization with a generic view on a popular radon transform based transform and pattern descriptor the r transform and r signature bringing in a class of transforms and descriptors spatially describing patterns at all directions and at different levels while maintaining the beneficial properties of the conventional r transform and r signature the domain of this class which is delimited due to the existence of singularities and the effect of sampling quantization and additive noise is examined moreover the ability of the generic r transform to encode the dominant directions of patterns is also discussed adding to the robustness to additive noise of the generic r signature the stability of dominant direction encoding by the generic r transform and the superiority of the generic r signature over existing invariant pattern descriptors on grayscale and binary noisy datasets have been confirmed by experiments

a forward regridding method with minimal oversampling for accurate and efficient iterative tomographic algorithms reconstruction of underconstrained tomographic data sets remains a major challenge standard analytical techniques frequently lead to unsatisfactory results due to insufficient information several iterative algorithms which can easily integrate a priori knowledge have been developed to tackle this problem during the last few decades most of these iterative algorithms are based on an implementation of the radon transform that acts as forward projector this operator and its adjoint the backprojector are typically called few times per iteration and represent the computational bottleneck of the reconstruction process here we present a fourier based forward projector founded on the regridding method with minimal oversampling we show that this implementation of the radon transform significantly outperforms in efficiency other state of the art operators with o n2log2n complexity despite its reduced computational cost this regridding method provides comparable accuracy to more sophisticated projectors and can therefore be exploited in iterative algorithms to substantially decrease the time required for the reconstruction of underconstrained tomographic data sets without loss in the quality of the results

natural representations for straight lines and the hough transform on discrete arrays the grid of discrete sampled data points in a digital image supports a limited set of lines at angles and displacements natural to that grid the effect of this implicit line quantization on the parametrization of the hough transform is presented the function describing the discrete hough transforms line detection sensitivity is derived expressions for the orientation frequency and popularity of lines in the natural set are given the results obtained are of importance to data arrays of small size the distribution of lines in the natural set is also important as it determines the precision and reliability with which straight lines can be measured on a discrete imaging array from the natural line set concept a general a d slope offset straight line parametrization is developed for which the hough transform is compact and fast to compute and which is as easy to interpret as the class p theta parametrization

existence and geometric properties of solutions of a free boundary problem in potential theory let 0 g h l rn n 2 be two given density functions at least one of them bounded away from zero outside a compact set and g holder continuous we prove that for any compactly supported positive measure which is sufficiently concentrated e g has sufficiently high n 1 dimensional density there exists a bounded open set rn such that the newtonian potential of the measure hln ghn 1 agrees with that of outside some regularity of is obtained as well as several results on the geometry of example if h and g are constant then for any x the inward normal ray of at x if it exists intersects the closed convex hull of typeset by ams tex 1 2 bjorn gustafsson and henrik shahgholian

the adaptive hough transform we introduce the adaptive hough transform aht as an efficient way of implementing the hough transform ht method for the detection of 2 d shapes the aht uses a small accumulator array and the idea of a flexible iterative coarse to fine accumulation and search strategy to identify significant peaks in the hough parameter spaces the method is substantially superior to the standard ht implementation in both storage and computational requirements in this correspondence we illustrate the ideas of the aht by tackling the problem of identifying linear and circular segments in images by searching for clusters of evidence in 2 d parameter spaces we show that the method is robust to the addition of extraneous noise and can be used to analyze complex images containing more than one shape

a multilevel domain decomposition algorithm for fast o n sup 2 logn reprojection of tomographic images a novel algorithm for fast computation of tomographic image projections is presented the method comprises a decomposition of an image into subimages followed by an aggregation of projections computed for the subimages the multilevel domain decomposition algorithm is formulated as a recursive procedure the computational cost of the proposed algorithm is comparable to that of fft based techniques while it appears to be more flexible than the latter numerical results demonstrate the effectiveness of the method

discretization errors in the hough transform abstract straight line segments in a two dimensional image can be detected with the hough transform by searching peaks in a parameter space the influence on the hough transform of the quantization of the parameter space the quantization of the image and the width of the line segment is investigated in this paper the hough transform was improved by o gorman and clowes by taking into account the gradient direction the resulting scatter of the peaks can be reduced by using a weighting function in the transform examples of asbestos preparations are given

multiple confidence sets based on stagewise tests abstract recent years have seen suggested constructions of multiple confidence sets related to stagewise multiple tests by some authors these methods are a type of mixture between test and confidence interval methods because confidence interval statements are made only for some parameters whereas test statements for fixed parameter values are made for the other parameters in this article i define a concept confidence directional set giving a confidence bound for one parameter which may depend on other parameters using this concept one can construct multiple confidence sets which are always confidence set statements and not test statements for fixed parameter values the confidence sets correspond exactly to stagewise tests which is theoretically appealing special examples of the general technique are given for the independent test statistic case and for comparison of a number of treatments to a control in the case of normally distributed observations with the same variance

topology of quadrature domains we address the problem of topology of quadrature domains namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity

hierarchical generalized hough transforms and line segment based generalized hough transforms abstract generalized hough transforms are fast point pattern matching algorithms which have applications in image processing two extensions to generalized hough transforms are described the first allows for matching hierarchically organized point patterns an example concerning image registration using hierarchical point patterns is described the second extension concerns matching patterns of simple geometric objects such as line segments rather than patterns of points

constrained simultaneous confidence intervals for multiple comparisons with the best for comparing k treatment effects 01 02 ok often the parameters of primary interest are 0i maxeioi0 i 1 k in this article we develop constrained 100p two sided simultaneous confidence intervals for 0i maxj i0j which we refer to as constrained mcb intervals it turns out that the lower bounds of the intervals imply indifference zone selection inference and the upper bounds of these intervals imply subset selection inference each given at the same confidence level 100p as mcb we also extend our

fast inversion of the radon transform using log polar coordinates and partial back projections in this paper a novel filtered back projection algorithm for inversion of a discretized radon transform is presented it makes use of invariance properties possessed by both the radon transform and its dual by switching to log polar coordinates both operators can be expressed in a displacement invariant manner explicit expressions for the corresponding transfer functions are calculated furthermore by dividing the back projection into several partial back projections inversion can be performed by means of finite convolutions and hence implemented by an fft algorithm in this way a fast and accurate reconstruction method is obtained

a fast digital radon transform an efficient means for evaluating the hough transform a fast digital radon transform based on recursively defined digital straight lines is described which has the sequential complexity of n 2 log n additions for an n x n image this transform can be used to evaluate the hough transform to detect straight lines in a digital image whilst a parallel implementation of the hough transform algorithm is difficult because of global memory access requirements the fast digital radon transform is vectorizable and therefore well suited for parallel computation the structure of the fast algorithm is shown to be quite similar to the fft algorithm for decimation in frequency it is demonstrated that even for sequential computation the fast radon transform is an attractive alternative to the classical hough transform algorithm

a framework for discrete integral transformations ii the 2d discrete radon transform although naturally at the heart of many fundamental physical computations and potentially useful in many important image processing tasks the radon transform lacks a coherent discrete definition for two dimensional 2d discrete images which is algebraically exact invertible and rapidly computable we define a notion of 2d discrete radon transforms for 2d discrete images which is based on summation along lines of absolute slope less than 1 values at nongrid locations are defined using trigonometric interpolation on a zero padded grid our definition is shown to be geometrically faithful the summation avoids wrap around effects our proposal uses a special collection of lines in mathbb r 2 for which the transform is rapidly computable and invertible we describe a fast algorithm using o n log n operations where n n 2 is the number of pixels in the image the fast algorithm exploits a discrete projection slice theorem which associates the discrete radon transform with the pseudopolar fourier transform a averbuch et al siam j sci comput 30 2008 pp 764 784 our definition for discrete images converges to a natural continuous counterpart with increasing refinement

some properties of two phase quadrature domains abstract in this paper we investigate general properties of the two phase quadrature domain which was recently introduced by emamizadeh prajapat and shahgholian the concept which is a generalization of the well known one phase domain introduces substantial difficulties with interesting features even richer than those of the one phase counterpart for given positive constants and two bounded and compactly supported measures we investigate the uniqueness of the solution of the following free boundary problem 1 u in r n n 2 u 0 in r n where it is further required that the supports of should be inside this in general may fail and give rise to non existence of solutions along the paths to various properties that we state and prove here we also present several conjectures and open problems that we believe should be true

similarity transformation parameters recovery based on radon transform application in image registration and object recognition the radon transform since its introduction in the beginning of the last century has been studied deeply and applied by researchers in a great number of applications especially in the biomedical imaging fields by using the radon transform properties the issue is to recover the transformation parameters regarding the rotation scaling and translation by handling only the image projections assuming no access to the spatial domain of the image this paper proposes an algorithm using an extended version of the radon transform to recover such parameters relating to two unknown images directly from their projection data especially our approach deals with the problem of the estimation accuracy of the rotation angle and its finding in one step instead of two steps as it is reported in the literature this method may be applied in image registration as well in object recognition the results are for the first time exploited in object recognition where comparison with powerful descriptors shows the outstanding performance of the proposed paradigm moreover the influence of additive noise on registration and recognition experiments is discussed and shows the efficiency of the method to reduce the effect of the noise highlightswe define 2 based radon transform rt and motivate its use we propose an algorithm to recover rst transforms using only rt the algorithm is tested on images in real world application e g ct mri etc this rst parameter recovery method is newly applied in object recognition a new confusion matrix is designed to identify also a mislabeled class element

a fast and accurate multilevel inversion of the radon transform a number of imaging technologies reconstruct an image function from its radon projection using the convolution backprojection method the convolution is an o n2 log n algorithm where the image consists of n x n pixels while the backprojection is an o n3 algorithm thus constituting the major computational burden of the convolution backprojection method an o n2 log n multilevel backprojection method is presented here when implemented with a fourier domain postprocessing technique also presented here the resulting image quality is similar or superior to the image quality of the classical backprojection technique

a step up multiple test procedure abstract we consider the problem of simultaneously testing k 2 hypotheses on parameters 1 k in a typical application the s may be a set of contrasts for instance a set of orthogonal contrasts among population means or a set of differences between k treatment means and a standard treatment mean it is assumed that least squares estimators 1 k are available that are jointly normally distributed with a common variance known up to a scalar namely the error variance 2 and a common known correlation an independent 2 distributed unbiased estimator of 2 is also assumed to be available we propose a step up multiple test procedure for this problem which tests the t statistics for the k hypotheses in order starting with the least significant one and continues as long as an acceptance occurs by contrast the step down approach which is usually used starts with the most significant and continues as long as a rejection occurs critical constants required by this step up procedure to co

global solutions to bubble growth in porous media abstract we study a moving boundary problem modeling an injected fluid into another viscous fluid the viscous fluid is withdrawn at infinity and governed by darcy s law we present solutions to the free boundary problem in terms of time derivative of a generalized newtonian potentials of the characteristic function of the bubble this enables us to show that the bubble occupies the entire space as the time tends to infinity if and only if the internal generalized newtonian potential of the initial bubble is a quadratic polynomial howison 1985 7 and dibenedetto and friedman 1986 2 studied such behavior but for bounded bubbles we extend their results to unbounded bubbles

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

block decomposition and segmentation for fast hough transform evaluation the decomposition of binary images using rectangular blocks of foreground pixels as primitives is considered based on this type of decomposition a fast method for evaluating the hough transform is introduced a complexity analysis of the proposed block hough transform algorithm sets constraints on the complexity of algorithms used for block decomposition so that the total decomposition and hough transform application time is much less than the time consumed by the usual point hough transform using this analysis we propose two algorithms for the decomposition and segmentation of binary images into rectangular blocks a combination of these methods leads to significant acceleration in the identification of linear features which is demonstrated in various image processing experiments 1999 pattern recognition society published by elsevier science ltd all rights reserved

step down multiple tests for comparing treatments with a control in unbalanced one way layouts we show how a well known multiple step down significance testing procedure for comparing treatments with a control in balanced one way layouts can be applied in unbalanced layouts unequal sample sizes for the treatments the method we describe has the advantage that it provides p values for each treatment versus control comparison that take account of the multiple step down testing nature of the procedure these joint p values can be used with any value of alpha the fixed type i family wise error rate bound that may be specified by the investigator to determine the p values it is necessary to compute a multivariate student t integral for which a computer program is available this procedure is more powerful than the step down bonferroni procedure of holm and the single step procedure of dunnett an example from the pharmaceutical literature is used to illustrate the procedure

discrete radon transform has an exact fast inverse and generalizes to operations other than sums along lines gotz druckmuller and independently brady have defined a discrete radon transform drt that sums an image9s pixel values along a set of aptly chosen discrete lines complete in slope and intercept the transform is fast o n 2 log n for an n n image it uses only addition not multiplication or interpolation and it admits a fast exact algorithm for the adjoint operation namely backprojection this paper shows that the transform additionally has a fast exact although iterative inverse the inverse reproduces to machine accuracy the pixel by pixel values of the original image from its drt without artifacts or a finite point spread function fourier or fast fourier transform methods are not used the inverse can also be calculated from sampled sinograms and is well conditioned in the presence of noise also introduced are generalizations of the drt that combine pixel values along lines by operations other than addition for example there is a fast transform that calculates median values along all discrete lines and is able to detect linear features at low signal to noise ratios in the presence of pointlike clutter features of arbitrarily large amplitude

on quadrature domains and the schwarz potential abstract let t be a distribution in r n whose support is compact a domain in r n is said to be a quadrature domain with respect to t if supp t and h h t for all functions h which are harmonic and integrable over we denote this by q t hl 1 this is equivalent to saying that there is a solution u called the modified schwarz potential to the following overdetermined cauchy problem u 1 t in and u u 0 on in this paper we investigate different properties of quadrature domains we show that if t is a fixed distribution then bounded domains in q t hl1 are uniformly bounded later we prove that if the modified schwarz potential of a domain is a polynomial then is a half space we also study the uniqueness problem namely we show that if d and are in q t hl1 v and u are the msps of d and respectively and sup d u 0 then meets x v x consequently if v 0 then d 0

on the relationship between stepwise decision procedures and confidence sets abstract this article considers the relationship between two standard methods of inference on a set of a unknown parameters namely stepwise decision procedures and the construction of simultaneous confidence intervals illustrative examples with k 2 are discussed for one sided step down and step up decision procedures and for a two sided step down decision procedure for these examples with normal distributional assumptions a set of confidence intervals associated with the decision procedure is constructed that provides additional information at no increase in the error rate the implication of this work is that commonly used stepwise decision procedures are per se not providing as complete an inference as is available the construction of new decision procedures and confidence sets designed specifically to meet an experimenter s particular requirements is also discussed

invariant pattern recognition using the rfm descriptor a pattern descriptor invariant to rotation scaling translation rst and robust to additive noise is proposed by using the radon fourier and mellin transforms the radon transform converts the rst transformations applied on a pattern image into transformations in the radial and angular coordinates of the pattern s radon image these beneficial properties of the radon transform make it an useful intermediate representation for the extraction of invariant features from pattern images for the purpose of indexing matching in this paper invariance to rst is obtained by applying the 1d fourier mellin and discrete fourier transforms on the radial and angular coordinates of the pattern s radon image respectively the implementation of the proposed descriptor is reasonably fast and correct based mainly on the fusion of the radon and fourier transforms and on a modification of the mellin transform theoretical arguments validate the robustness of the proposed descriptor to additive noise and empirical evidence on both occlusion deformation and noisy datasets shows its effectiveness

incomplete block designs for comparing treatments wth a control general theory in this paper we develop a theory of optimal incomplete block designs for comparing several treatments with a control this class of designs is appropriate for comparing simultaneously p 2 test treatments with a control treatment the so called multiple comparisons with a control mcc problem when the observations are taken in incomplete blocks of common size k p 1 for this problem we propose a new general class of incomplete block designs that are balanced with respect to wrt test treatments we shall use the abbreviation btib to refer to such designs we study their structure and give some methods of construction a procedure for making exact joint confidence statements for this multiple comparisons problem is described by using a new concept of admissibility of designs it is shown how inferior designs can be eliminated from consideration and attention limited to a small class of btib designs that can be constructed from so called generator designs in the minimal complete class of such d

p value adjustments for multiple tests in multivariate binomial models abstract data from rodent carcinogenicity preclinical and clinical studies involving new drugs may be modeled as having come from multivariate binomial distributions in two year rodent carcinogenicity studies there are typically 20 50 tissues examined for occurrence of any of several possible lesions for a particular treatment group the number of occurrences of a particular lesion at a particular tissue may be modeled as binomial and the vector of such frequencies may be considered multivariate binomial with unspecified dependence structure the same model may also apply to clinical side effects data in this case the marginal frequencies may represent occurrences of events ranging from headaches to ingrown toenails frequently the goal of such studies is to isolate site specific significant differences between treatment and control groups for example in rodent carcinogenicity analyses it is generally not sufficient to claim that a new compound causes an increase in tumors at some unspecified si

rotation invariant textural feature extraction for image retrieval using eigen value analysis of intensity gradients and multi resolution analysis in this paper a novel rotation invariant multi resolution based texture retrieval technique is proposed the rotation invariance is achieved by aligning the direction of maximum variation of intensity gradient defined as principal texture direction along the reference axis the principal direction is determined using eigen value analysis of gradient image wavelet transform based techniques are applied on the rotated image the independent representation of textural energies along various directions enhances the retrieval performance over the existing rotation invariant wavelet based techniques which achieve rotation invariance by averaging the direction sensitive components extensive experiments on brodatz database support this postulate