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

design of steerable wavelets to detect multifold junctions we propose a framework for the detection of junctions in images although the detection of edges and key points is a well examined and described area the multiscale detection of junction centers especially for odd orders poses a challenge in pattern analysis the goal of this paper is to build optimal junction detectors based on 2d steerable wavelets that are polar separable in the fourier domain the approaches we develop are general and can be used for the detection of arbitrary symmetric and asymmetric junctions the backbone of our construction is a multiscale pyramid with a radial wavelet function where the directional components are represented by circular harmonics and encoded in a shaping matrix we are able to detect m fold junctions in different scales and orientations we provide experimental results on both simulated and real data to demonstrate the effectiveness of the algorithm

multiresolution monogenic signal analysis using the riesz laplace wavelet transform the monogenic signal is the natural 2d counterpart of the 1d analytic signal we propose to transpose the concept to the wavelet domain by considering a complexified version of the riesz transform which has the remarkable property of mapping a real valued primary wavelet basis of l2 r2 into a complex one the riesz operator is also steerable in the sense that it give access to the hilbert transform of the signal along any orientation having set those foundations we specify a primary polyharmonic spline wavelet basis of l2 r2 that involves a single mexican hat like mother wavelet laplacian of a b spline the important point is that our primary wavelets are quasi isotropic they behave like multiscale versions of the fractional laplace operator from which they are derived which ensures steerability we propose to pair these real valued basis functions with their complex riesz counterparts to specify a multiresolution monogenic signal analysis this yields a representation where each wavelet index is associated with a local orientation an amplitude and a phase we give a corresponding wavelet domain method for estimating the underlying instantaneous frequency we also provide a mechanism for improving the shift and rotation invariance of the wavelet decomposition and show how to implement the transform efficiently using perfect reconstruction filterbanks we illustrate the specific feature extraction capabilities of the representation and present novel examples of wavelet domain processing in particular a robust tensor based analysis of directional image patterns the demodulation of interferograms and the reconstruction of digital holograms

the monogenic signal of potential field data a python implementation abstractwe have developed codes to calculate the local amplitude the local phase and the local orientation of the nonscale and the poisson s scale space monogenic signals of potential field data in version 1 0 of the open source program monogenic the monogenic vector of a generic function is calculated in the wavenumber domain and then transformed back into the space domain to find the monogenic signal attributes we compare the use of the nonscale monogenic signal with the poisson s scale space monogenic signal in magnetic data this comparison shows that the latter can produce better results as an edge detection filter the implementation of the monogenic signal can be used to enhance other geophysical data such as seismic ground penetrating radar gravity multiple component gravity gradiometry and magnetic gradient data

on miura transformations among nonlinear partial differential equationsa in this paper we study miura transformations u v from partial differential equations uxxx f u ux ut to nonlinear partial differential equations g v vx vt xlv t1v 0 defined using integrable systems on v we classify all such miura transformations under some restrictions and hence generalize the classical miura transformation to a large class of nonlinear partial differential equations for some examples by applying miura transformations found in this paper we derive exact solutions v from known solutions u in particular kink and soliton kink solutions of vt 32vxsin2v 12vx3 vxxx are obtained from constant solutions and soliton solutions of the mkdv equation as another application of miura transformations of this paper we deduce a new backlund transformation for each of vt 32vxsin2v 12vx3 vxxx and vt 32vxsinh2v 12vx3 vxxx from the known backlund transformations for the mkdv equations

multiple multidimensional morse wavelets this paper defines a set of operators that localize a radial image in space and radial frequency simultaneously the eigenfunctions of the operator are determined and a nonseparable orthogonal set of radial wavelet functions are found the eigenfunctions are optimally concentrated over a given region of radial space and scale space defined via a triplet of parameters analytic forms for the energy concentration of the functions over the region are given the radial function localization operator can be generalised to an operator localizing any l2 ropf2 function it is demonstrated that the latter operator given an appropriate choice of localization region approximately has the same radial eigenfunctions as the radial operator based on a given radial wavelet function a quaternionic wavelet is defined that can extract the local orientation of discontinuous signals as well as amplitude orientation and phase structure of locally oscillatory signals the full set of quaternionic wavelet functions are component by component orthogonal their statistical properties are tractable and forms for the variability of the estimators of the local phase and orientation are given as well as the local energy of the image by averaging estimators across wavelets a substantial reduction in the variance is achieved

two view line matching algorithm based on context and appearance in low textured images a novel approach for line detection and matching is proposed aimed at achieving good performance with low textured scenes under uncontrolled illumination conditions line detection is performed by means of phase based edge detector over gaussian scale space followed by a multi scale fusion stage which has been proven to be profitable in minimizing the number of fragmented and overlapped segments line matching is performed by an iterative process that uses structural information collected through the use of different line neighborhoods making the set of matched lines grow robustly at each iteration results show that this approach is suitable to deal with low textured scenes and also robust under a wide variety of image transformations highlightsa novel approach for line detection and matching is proposed lines are detected through a fusion process over the gaussian scale space lines are matched based on their appearance geometric properties and neighborhoods our proposal minimizes the number of fragmented and overlapped segments our proposal is suitable for low textured scenes under uncontrolled illumination

toward a three dimensional automatic interpretation of potential field data via generalized hilbert transforms fundamental relations the paper extends to three dimensions 3 d the two dimensional 2 d hilbert transform relations between potential field components for the 3 d case it is shown that the hilbert transform is composed of two parts with one part acting on the x component and one part on the y component as for the previously developed 2 d case it is shown that in 3 d the vertical and horizontal derivatives are the hilbert transforms of each other the 2 d cauchy riemann relations between a potential function and its hilbert transform are generalized for the 3 d case finally the previously developed concept of analytic signal in 2 d can be extended to 3 d as a first step toward the development of an automatic interpretation technique for potential field data

on the determination of functions from their integral values along certain manifolds when one integrates a function of two variables x y a point function f p in the plane subject to suitable regularity conditions along an arbitrary straight line g then one obtains in the integral values f g a line function in part a of the present paper the problem which is solved is the inversion of this linear functional transformation that is the following questions are answered can every line function satisfying suitable regularity conditions be regarded as constructed in this way if so is f uniquely known from f and how can f be calculated in part b a solution of the dual problem of calculating a line function f g from its point mean values f p is solved in a certain sense finally in part c certain generalizations are discussed prompted by consideration of non euclidean manifolds as well as higher dimensional spaces the treatment of these problems themselves of interest gains enhanced importance through the numerous relationships that exist between this topic and the theory of logarithmic and newtonian potentials these are mentioned at appropriate places in the text

hypercomplex correlation techniques for vector images correlation techniques have been applied to almost every area of signal processing over the past century yet their use has in general been limited to scalar signals while there have been implementations in multichannel applications these can be characterized as a combination of single channel processes true vector correlation techniques with global and interchannel measures have only recently been demonstrated and are still in their infancy by comparison this paper describes our work on vector correlation based on the use of hypercomplex fourier transforms and presents for the first time a unified theory behind the information contained in the peak of a vector correlation response by using example applications for color images we also demonstrate some of the practical implications together with our latest results

myocardial motion estimation from medical images using the monogenic signal we present a method for the analysis of heart motion from medical images the algorithm exploits monogenic signal theory recently introduced as an n dimensional generalization of the analytic signal the displacement is computed locally by assuming the conservation of the monogenic phase over time a local affine displacement model is considered to account for typical heart motions as contraction expansion and shear a coarse to fine b spline scheme allows a robust and effective computation of the model s parameters and a pyramidal refinement scheme helps to handle large motions robustness against noise is increased by replacing the standard point wise computation of the monogenic orientation with a robust least squares orientation estimate given its general formulation the algorithm is well suited for images from different modalities in particular for those cases where time variant changes of local intensity invalidate the standard brightness constancy assumption this paper evaluates the method s feasibility on two emblematic cases cardiac tagged magnetic resonance and cardiac ultrasound in order to quantify the performance of the proposed method we made use of realistic synthetic sequences from both modalities for which the benchmark motion is known a comparison is presented with state of the art methods for cardiac motion analysis on the data considered these conventional approaches are outperformed by the proposed algorithm a recent global optical flow estimation algorithm based on the monogenic curvature tensor is also considered in the comparison with respect to the latter the proposed framework provides along with higher accuracy superior robustness to noise and a considerably shorter computation time

classification via sparse representation of steerable wavelet frames on grassmann manifold application to target recognition in sar image automatic target recognition has been widely studied over the years yet it is still an open problem the main obstacle consists in extended operating conditions e g depression angle change configuration variation articulation and occlusion to deal with them this paper proposes a new classification strategy we develop a new representation model via the steerable wavelet frames the proposed representation model is entirely viewed as an element on grassmann manifolds to achieve target classification we embed grassmann manifolds into an implicit reproducing kernel hilbert space rkhs where the kernel sparse learning can be applied specifically the mappings of training sample in rkhs are concatenated to form an overcomplete dictionary it is then used to encode the counterpart of query as a linear combination of its atoms by designed grassmann kernel function it is capable to obtain the sparse representation from which the inference can be reached the novelty of this paper comes from 1 the development of representation model by the set of directional components of riesz transform 2 the quantitative measure of similarity for proposed representation model by grassmann metric and 3 the generation of global kernel function by grassmann kernel extensive comparative studies are performed to demonstrate the advantage of proposed strategy

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

coherent multiscale image processing using quaternion wavelets coherent multiscale image processing using quaternion wavelets by wai lam chan this thesis develops a quaternion wavelet transform qwt as a new multiscale analysis tool for geometric image features the qwt is a near shift invariant tight frame representation whose coefficients sport a magnitude and three phases two phases encode local image shifts while the third contains textural information the qwt can be computed using a dual tree filter bank with linear computational complexity we develop the qwt by applying an alternative theory of 2 d hilbert transforms and analytic signals to the 1 d complex wavelet transform to demonstrate the properties of the qwt s coherent magnitude phase representation we develop a efficient and accurate algorithm for estimating the local geometrical structure of images and a multiscale algorithm for estimating the disparity between a pair of images the latter algorithm has potentials for various applications in image registration and flow estimation it uses an interesting multiscale phase unwrapping procedure and features linear complexity and sub pixel accuracy

3d steerable wavelets in practice we introduce a systematic and practical design for steerable wavelet frames in 3d our steerable wavelets are obtained by applying a 3d version of the generalized riesz transform to a primary isotropic wavelet frame the novel transform is self reversible tight frame and its elementary constituents riesz wavelets can be efficiently rotated in any 3d direction by forming appropriate linear combinations moreover the basis functions at a given location can be linearly combined to design custom and adaptive steerable wavelets the features of the proposed method are illustrated with the processing and analysis of 3d biomedical data in particular we show how those wavelets can be used to characterize directional patterns and to detect edges by means of a 3d monogenic analysis we also propose a new inverse problem formalism along with an optimization algorithm for reconstructing 3d images from a sparse set of wavelet domain edges the scheme results in high quality image reconstructions which demonstrate the feature reduction ability of the steerable wavelets as well as their potential for solving inverse problems

non parametric mixture model based evolution of level sets and application to medical images we present a novel region based curve evolution algorithm which has three primary contributions i non parametric estimation of probability distributions using the recently developed np windows method ii an inequality constrained least squares method to model the image histogram with a mixture of nonparametric probability distributions and iii accommodation of the partial volume effect which is primarily due to low resolution images and which often poses a significant challenge in medical image analysis our primary application area we first approximate the image intensity histogram as a mixture of non parametric probability density functions pdfs justifying its use with respect to medical image analysis the individual densities in the mixture are estimated using the recent np windows pdf estimation method which builds a continuous representation of discrete signals a bayesian framework is then formulated in which likelihood probabilities are given by the non parametric pdfs and prior probabilities are calculated using an inequality constrained least squares method the non parametric pdfs are then learnt and the segmentation solution is spatially regularised using a level sets framework the log ratio of the posterior probabilities is used to drive the level set evolution as background to our approach we recall related developments in level set methods results are presented for a set of synthetic and natural images as well as simulated and real medical images of various anatomical organs results on a range of images show the effectiveness of the proposed algorithm

generalized likelihood ratios for testing the properness of quaternion gaussian vectors in a recent paper the second order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing which are associated with the two main types of quaternion properness in this paper we consider the problem of determining from a finite number of independent vector observations whether a quaternion gaussian vector is proper or not specifically we derive three generalized likelihood ratio tests glrts for testing the two main kinds of quaternion properness and show that the glrts reduce to the estimation of three previously proposed quaternion improperness measures interestingly the three glrt statistics improperness measures can be interpreted as an estimate of the entropy loss due to the quaternion improperness additionally we analyze the case in which the orthogonal basis for the representation of the quaternion vector is unknown which results in the problem of estimating the principal bbc properness direction i e the pure unit quaternion minimizing the bbc improperness measure although this estimation problem is not convex we propose a technique based on successive convex approximations which can be solved in closed form finally some simulation examples illustrate the performance and practical application of the proposed tests

a differential model of the complex cell the receptive fields of simple cells in the visual cortex can be understood as linear filters these filters can be modeled by gabor functions or gaussian derivatives gabor functions can also be combined in an energy model of the complex cell response this letter proposes an alternative model of the complex cell based on gaussian derivatives it is most important to account for the insensitivity of the complex response to small shifts of the image the new model uses a linear combination of the first few derivative filters at a single position to approximate the first derivative filter at a series of adjacent positions the maximum response over all positions gives a signal that is insensitive to small shifts of the image this model unlike previous approaches is based on the scale space theory of visual processing in particular the complex cell is built from filters that respond to the 2d differential structure of the image the computational aspects of the new model are studied in one and two dimensions using the steerability of the gaussian derivatives the response of the model to basic images such as edges and gratings is derived formally the response to natural images is also evaluated using statistical measures of shift insensitivity the neural implementation and predictions of the model are discussed

nonlinear image operators for the evaluation of local intrinsic dimensionality local intrinsic dimensionality is shown to be an elementary structural property of multidimensional signals that cannot be evaluated using linear filters we derive a class of polynomial operators for the detection of intrinsically 2 d image features like curved edges and lines junctions line ends etc although it is a deterministic concept intrinsic dimensionality is closely related to signal redundancy since it measures how many of the degrees of freedom provided by a signal domain are in fact used by an actual signal furthermore there is an intimate connection to multidimensional surface geometry and to the concept of gaussian curvature nonlinear operators are inevitably required for the processing of intrinsic dimensionality since linear operators are by the superposition principle restricted to or combinations of their intrinsically 1 d eigenfunctions the essential new feature provided by polynomial operators is their potential to act on multiplicative relations between frequency components therefore such operators can provide the and combination of complex exponentials which is required for the exploitation of intrinsic dimensionality using frequency design methods we obtain a generalized class of quadratic volterra operators that are selective to intrinsically 2 d signals these operators can be adapted to the requirements of the signal processing task for example one can control the curvature tuning by adjusting the width of the stopband for intrinsically 1 d signals or the operators can be provided in isotropic and in orientation selective versions we first derive the quadratic volterra kernel involved in the computation of gaussian curvature and then present examples of operators with other arrangements of stop and passbands some of the resulting operators show a close relationship to the end stopped and dot responsive neurons of the mammalian visual cortex

dynamics of the bilinear mindlin plate element plate and shell finite elements based upon mindlin plate theory and evaluated by one point quadrature are attractive because of their effectiveness for both thick and thin shells at minimal computing cost however these elements are rank deficient and must be stabilized to obtain reliable behaviour methods for achieving full rank of the stiffness and for stabilizing element behaviour in static analysis and in explicit dynamic calculations exist and are quite effective this paper addresses the remaining issue of controlling spurious modes of response in vibration analysis and implicit dynamic solutions several alternatives for the element mass formulation are examined in detail we show that non physical dynamic modes present a potential problem with most mass matrix formulations and that spurious modes other than the familiar hourglassing motion are possible a combination of projection methods and reduced quadrature is suggested which eliminates these deficiencies and produces accurate numerical results the remaining techniques investigated give rise to anomalous behaviour which make them unsuitable for general use

a novel image representation via local frequency analysis for illumination invariant stereo matching in this paper we propose a novel image representation approach to tackle illumination variations in stereo matching problems images are mapped using their fourier transforms which are convolved with a set of monogenic filters frequency analysis is carried out at different scales to account for most image content the phase congruency and the local weighted mean phase angle are then computed over all the scales the original image is transformed into a new representation using these two mappings this representation is invariant to illumination and contrast variations more importantly it is generic and can be used with most sparse as well as dense stereo matching algorithms in addition sequential feature matching or tracking can also benefit from our approach in varying radiometric conditions we demonstrate the improvements introduced with our image mappings on well established data sets in the literature as well as on our own experimental scenarios that include high dynamic range imagery the experiments include both dense and sparse stereo and sequential matching algorithms where the latter is considered in the very challenging visual odometry framework