multiresolution moment filters theory and applications we introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales when the window function satisfies a two scale relation we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution wavelet like algorithm we show that b splines are well suited window functions because in addition to being refinable they are positive symmetric separable and very nearly isotropic gaussian shape we present three applications of these multiscale local moments the first is a feature extraction method for detecting and characterizing elongated structures in images the second is a noise reduction method which can be viewed as a multiscale extension of savitzky golay filtering the third is a multiscale optical flow algorithm that uses a local affine model for the motion field extending the lucas kanade optical flow method the results obtained in all cases are promising

finding the piecewise linear frontier production function in data envelopment analysis data envelopment analysis dea is a mathematical programming technique for identifying efficiency scores of decision making units dmus the ability of identifying efficient frontier prior to the dea calculation is of extreme importance to an effective and efficient dea computation in order to do this in this paper we introduce a method for identifying efficient frontier and efficient dmus

ultrasonic texture motion analysis theory and simulation a theoretical model was previously developed to evaluate the relationship between the dynamics of ultrasonic speckle and its underlying tissue the model is divided into an instrumental part represented by the point spread function in the far field of the ultrasonic apparatus and a moving tissue component described by a collection of scatterers by computing the convolution of these terms and then the envelope one obtains a simulated ultrasonic speckle pattern sequence which shows speckle motions closely linked to the tissue dynamics when small motion amplitudes are involved here a theoretical study of the correlation between various linear transformations of the tissue and the corresponding ultrasonic speckle motions is performed based on a 2d extension of the envelope cross correlation analysis of a narrow band gaussian noise in the linear scan case obviously tissue translation generates an identical speckle translation however tissue speckle motion correlation decreases with increasing rotation and or biaxial deformation lateral deformation perpendicular to the beam propagation axis being much less sensitive with respect to the transducer frequency the rotation and the axial deformation of the tissue show a better relationship with their respective speckle motion at lower frequencies while lateral deformation correlation is independent of the pulse frequency with respect to beam pulse size parameters tissue speckle correlation decreases with rotation when a wide ultrasonic beam is used while the axial deformation correlation decreases with the axial duration of the pulse this study sets the ground for the development of an ultrasonic strain gauge particularly useful for the assessment of biomechanical soft tissue and fluid flow properties based on speckle tracking

three dimensional cardiac strain estimation using spatio temporal elastic registration of ultrasound images a feasibility study current ultrasound methods for measuring myocardial strain are often limited to measurements in one or two dimensions cardiac motion and deformation however are truly 3 d with the introduction of matrix transducer technology 3 d ultrasound imaging of the heart has become feasible but suffers from low temporal and spatial resolution making 3 d strain estimation challenging in this paper it is shown that automatic intensity based spatio temporal elastic registration of currently available 3 d volumetric ultrasound data sets can be used to measure the full 3 d strain tensor the method was validated using simulated 3 d ultrasound data sets of the left ventricle lv three types of data sets were simulated a normal and symmetric lv with different heart rates a more realistic asymmetric normal lv and an infarcted lv the absolute error in the estimated displacement was between 0 47 plusmn0 23 and 1 00 plusmn0 59 mm depending on heart rate and amount of background noise the absolute error on the estimated strain was 9 21 for the radial strain and 1 4 for the longitudinal and circumferential strains no large differences were found between the different types of data sets the shape of the strain curves was estimated properly and the position of the infarcts could be identified correctly preliminary results on clinical data taken in vivo from three healthy volunteers and one patient with an apical aneurism confirmed these findings in a qualitative manner as the strain curves obtained with the proposed method have an amplitude and shape similar to what could be expected

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

irreducible pattern k potent ray pattern matrices abstract a ray pattern is the class of all complex matrices with a specified zero nonzero pattern such that the argument of each nonzero entry is specified the ray pattern a is called pattern k potent if k is the smallest positive integer for which ak 1 a as ray patterns we characterize the irreducible ray patterns that are pattern k potent and we provide a canonical form for patterns with respect to permutation similarity and signature similarity by diagonal unitary matrices

moms maximal order interpolation of minimal support we consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly supported basis function spl phi z we first give the expression for the cases of spl phi that have minimal support for a given accuracy also known as approximation order this class of functions which we call maximal order minimal support functions moms is made of linear combinations of the b spline of the same order and of its derivatives we provide an explicit form of the moms that maximizes the approximation accuracy when the step size is small enough we compute the sampling gain obtained by using these optimal basis functions over the splines of the same order we show that it is already substantial for small orders and that it further increases with the approximation order l when l is large this sampling gain becomes linear more specifically its exact asymptotic expression is 2 spl pi e l since the optimal functions are continuous but not differentiable for even orders and even only piecewise continuous for odd orders our result implies that regularity has little to do with approximating performance these theoretical findings are corroborated by experimental evidence that involves compounded rotations of images

variational image reconstruction from arbitrarily spaced samples a fast multiresolution spline solution we propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations we adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms 1 the sum of squared errors at the specified points and 2 a quadratic functional that penalizes the lack of smoothness we search for a solution that is a uniform spline and show how it can be determined by solving a large sparse system of linear equations we interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach using the two scale relation for b splines we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution we use this relation to develop a fast multigrid algorithm we demonstrate the effectiveness of our approach on some image reconstruction examples

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

parameterized strain estimation for vascular ultrasound elastography with sparse representation ultrasound vascular strain imaging has shown its potential to interrogate the motion of the vessel wall induced by the cardiac pulsation for predicting plaque instability in this study a sparse model strain estimator smse is proposed to reconstruct a dense strain field at a high resolution with no spatial derivatives and a high computation efficiency this sparse model utilizes the highly compacted property of discrete cosine transform dct coefficients thereby allowing to parameterize displacement and strain fields with truncated dct coefficients the derivation of affine strain components axial and lateral strains and shears was reformulated into solving truncated dct coefficients and then reconstructed with them moreover an analytical solution was derived to reduce estimation time with simulations the smse reduced estimation errors by up to 50 compared with the state of the art window based lagrangian speckle model estimator lsme the smse was also proven to be more robust than the lsme against global and local noise for in vitro and in vivo tests residual strains assessing cumulated errors with the smse were 2 to 3 times lower than with the lsme regarding computation efficiency the processing time of the smse was reduced by 4 to 25 times compared with the lsme according to simulations in vitro and in vivo results finally phantom studies demonstrated the enhanced spatial resolution of the proposed smse algorithm against lsme

full motion and flow field recovery from echo doppler data we present a new computational method for reconstructing a vector velocity field from scattered pulsed wave ultrasound doppler data the main difficulty is that the doppler measurements are incomplete for they do only capture the velocity component along the beam direction we thus propose to combine measurements from different beam directions however this is not yet sufficient to make the problem well posed because 1 the angle between the directions is typically small and 2 the data is noisy and nonuniformly sampled we propose to solve this reconstruction problem in the continuous domain using regularization the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms 1 the sum of squared difference between the doppler data and the projected velocities 2 a quadratic regularization functional that imposes some smoothness on the velocity field we express our solution for this minimization problem in a b spline basis obtaining a sparse system of equations that can be solved efficiently using synthetic phantom data we demonstrate the significance of tuning the regularization according to the a priori knowledge about the physical property of the motion next we validate our method using real phantom data for which the ground truth is known we then present reconstruction results obtained from clinical data that originate from 1 blood flow in carotid bifurcation and 2 cardiac wall motion

comparison of kasai autocorrelation and maximum likelihood estimators for doppler optical coherence tomography in optical coherence tomography oct and ultrasound unbiased doppler frequency estimators with low variance are desirable for blood velocity estimation hardware improvements in oct mean that ever higher acquisition rates are possible which should also in principle improve estimation performance paradoxically however the widely used kasai autocorrelation estimator s performance worsens with increasing acquisition rate we propose that parametric estimators based on accurate models of noise statistics can offer better performance we derive a maximum likelihood estimator mle based on a simple additive white gaussian noise model and show that it can outperform the kasai autocorrelation estimator in addition we also derive the cramer rao lower bound crlb and show that the variance of the mle approaches the crlb for moderate data lengths and noise levels we note that the mle performance improves with longer acquisition time and remains constant or improves with higher acquisition rates these qualities may make it a preferred technique as oct imaging speed continues to improve finally our work motivates the development of more general parametric estimators based on statistical models of decorrelation noise

diagonally scaled permutations and circulant matrices abstract if r is an n n matrix over the complex field which is the product of a diagonal matrix d and a permutation matrix p then r is called a diagonally scaled permutation matrix we present the eigenstructure of r by observing that r is permutation similar to the direct sum of diagonally scaled permutation matrices of the form dc where d is a diagonal matrix and c is the circulant permutation 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 the matrix dc is called a scaled circulant permutation matrix we consider two cases for r dc when the scaling matrix d is nonsingular and when d is singular in the singular case r is nilpotent and we are able to obtain upper and lower bounds on the index of nilpotency of r we conclude with information about matrices that commute with a scaled permutation matrix we are also able to represent an arbitrary n n toeplitz matrix as a sum of matrices of the form d k ck for k 1 n where d k is a diagonal matrix

speckle reduction and contrast enhancement of echocardiograms via multiscale nonlinear processing this paper presents an algorithm for speckle reduction and contrast enhancement of echocardiographic images within a framework of multiscale wavelet analysis the authors apply wavelet shrinkage techniques to eliminate noise while preserving the sharpness of salient features in addition nonlinear processing of feature energy is carried out to enhance contrast within local structures and along object boundaries the authors show that the algorithm is capable of not only reducing speckle but also enhancing features of diagnostic importance such as myocardial walls in two dimensional echocardiograms obtained from the parasternal short axis view shrinkage of wavelet coefficients via soft thresholding within finer levels of scale is carried out on coefficients of logarithmically transformed echocardiograms enhancement of echocardiographic features is accomplished via nonlinear stretching followed by hard thresholding of wavelet coefficients within selected midrange spatial frequency levels of analysis the authors formulate the denoising and enhancement problem introduce a class of dyadic wavelets and describe their implementation of a dyadic wavelet transform their approach for speckle reduction and contrast enhancement was shown to be less affected by pseudo gibbs phenomena the authors show experimentally that this technique produced superior results both qualitatively and quantitatively when compared to results obtained from existing denoising methods alone a study using a database of clinical echocardiographic images suggests that such denoising and enhancement may improve the overall consistency of expert observers to manually defined borders

sign k potent sign patterns and ray k potent ray patterns that admit k potence abstract we prove that for each positive integer k all irreducible k potent ray pattern classes allow k potence we construct all irreducible real complex k potent matrices in an irreducible sign ray k potent sign ray pattern class for a given reducible sign ray pattern that is sign ray k potent for some positive integer k we determine necessary conditions for that pattern to allow k potence this generalizes recent work by lee and park 2015 2 who determined necessary conditions for a sign idempotent sign pattern to allow idempotence and who showed that every irreducible sign k potent sign pattern must allow k potence

relationships between pareto optimality in multi objective 0 1 linear programming and dea efficiency in a paper that appeared in ejor f h f liu c c huang and y l yen analyzed the relationships between data envelopment analysis dea and multi objective linear programming with binary variables moblp in particular they stated that a dea efficient solution in the sense of banker charnes and cooper bcc is a pareto optimal solution of moblp in this note we show that this property is false we establish the correct relations between these two notions and we illustrate these statements by some examples

finding the efficiency score and rts characteristic of dmus by means of identifying the efficient frontier in dea data envelopment analysis dea is basically a linear programming based technique used for measuring the relative performance of organizational units referred to as decision making units dmus where the presence of multiple inputs and outputs makes comparisons difficult the ability of identifying frontier dmus prior to the dea calculation is of extreme importance to an effective and efficient dea computation in this paper a method for identifying the efficient frontier is introduced then the efficiency score and returns to scale rts characteristic of dmus will be produced by means of the equation of efficient frontier

left ventricular motion reconstruction based on elastic vector splines in medical imaging it is common to reconstruct dense motion estimates from sparse measurements of that motion using some form of elastic spline thin plate spline snakes and other deformable models etc usually the elastic spline uses only bending energy second order smoothness constraint or stretching energy first order smoothness constraint or a combination of the two these elastic splines belong to a family of elastic vector splines called the laplacian splines this spline family is derived from an energy minimization functional which is composed of multiple order smoothness constraints these splines can be explicitly tuned to vary the smoothness of the solution according to the deformation in the modeled material tissue in this context it is natural to question which members of the family will reconstruct the motion more accurately the authors compare different members of this spline family to assess how well these splines reconstruct human cardiac motion they find that the commonly used splines containing first order and or second order smoothness terms only are not the most accurate for modeling human cardiac motion

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

on the period and base of a sign pattern matrix abstract a square sign pattern matrix a whose entries are or 0 is said to be powerful if all the powers a 1 a 2 a 3 are unambiguously defined for a powerful pattern a if a l a l p with l and p minimal then l is called the base of a and p is called the period of a we characterize irreducible powerful sign pattern matrices and investigate the period and base of a powerful sign pattern matrix we also consider some connections with real matrices and give some significant classes of powerful patterns

irreducible powerful ray pattern matrices a ray pattern is a matrix each of whose entries is either 0 or a ray in the complex plane originating from 0 but not including 0 a ray pattern is a natural generalization of the concept of a sign pattern whose entries are from the set 0 powers of sign patterns and ray patterns especially patterns whose powers are periodic have been studied in several recent papers a ray pattern a is said to be powerful if ak is unambiguously defined for all positive integers k irreducible powerful ray patterns have been characterized recently in this paper reducible powerful ray patterns are investigated in particular for a powerful ray pattern in frobenius normal form it is shown that the existence of a nonzero entry in an off diagonal block implies that the corresponding irreducible components are related in a certain way further the structure of each of the off diagonal blocks is characterized

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

periodic irreducible powerful ray pattern matrices abstract a ray pattern is a complex matrix each of whose entries is either 0 or a ray e i where is a real number for a ray pattern a a st we define the ray pattern a a st of a where a st 1 if a st 0 and a st 0 if a st 0 in this paper we first show that an irreducible powerful ray pattern a is ray diagonally similar to a for some ray by using this representation we obtain several results on irreducible powerful ray patterns and irreducible periodic ray patterns then we show that the number of such rays is k a where k a is the index of imprimitivity of a as an application to complex matrices we generalize the perron frobenius theorem to a subclass of complex matrices

an exact method for computing the area moments of wavelet and spline curves we present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis using green s theorem we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product the multidimensional filter coefficients are precomputed exactly as the solution of a two scale relation to demonstrate the performance improvement of the new method we compare it with existing methods such as pixel based approaches and approximation of the region by a polygon we also propose an alternate scheme when the scaling function is sinc x

spatio temporal nonrigid registration for ultrasound cardiac motion estimation we propose a new spatio temporal elastic registration algorithm for motion reconstruction from a series of images the specific application is to estimate displacement fields from two dimensional ultrasound sequences of the heart the basic idea is to find a spatio temporal deformation field that effectively compensates for the motion by minimizing a difference with respect to a reference frame the key feature of our method is the use of a semi local spatio temporal parametric model for the deformation using splines and the reformulation of the registration task as a global optimization problem the scale of the spline model controls the smoothness of the displacement field our algorithm uses a multiresolution optimization strategy to obtain a higher speed and robustness we evaluated the accuracy of our algorithm using a synthetic sequence generated with an ultrasound simulation package together with a realistic cardiac motion model we compared our new global multiframe approach with a previous method based on pairwise registration of consecutive frames to demonstrate the benefits of introducing temporal consistency finally we applied the algorithm to the regional analysis of the left ventricle displacement and strain parameters were evaluated showing significant differences between the normal and pathological segments thereby illustrating the clinical applicability of our method

restoration of the velocity field of the heart from two dimensional echocardiograms a method to quantify the motion of the heart from digitized sequences of two dimensional echocardiograms 2 d echos was recently proposed this method computes on every point of the 2 d echoes the 2 d apparent velocity vector or optical flow which characterizes its interframe motion however further analysis is required to determine what part of this motion is due to translation rotation contraction and deformation of the myocardium a method to locally obtain this information is presented the proposed method assumes that the interframe velocity field u xy v x y can be locally described by linear equations in the form u x y a ax by v x y b cx dy the additional constraint was introduced in the computation of the local velocity field by the method of projections onto convex sets since this constraint is only valid locally the myocardium must be first divided into sectors and the velocity fields computed independently for each sector

myocardial motion analysis from b mode echocardiograms the quantitative assessment of cardiac motion is a fundamental concept to evaluate ventricular malfunction we present a new optical flow based method for estimating heart motion from two dimensional echocardiographic sequences to account for typical heart motions such as contraction expansion and shear we analyze the images locally by using a local affine model for the velocity in space and a linear model in time the regional motion parameters are estimated in the least squares sense inside a sliding spatiotemporal b spline window robustness and spatial adaptability is achieved by estimating the model parameters at multiple scales within a coarse to fine multiresolution framework we use a wavelet like algorithm for computing b spline weighted inner products and moments at dyadic scales to increase computational efficiency in order to characterize myocardial contractility and to simplify the detection of myocardial dysfunction the radial component of the velocity with respect to a reference point is color coded and visualized inside a time varying region of interest the algorithm was first validated on synthetic data sets that simulate a beating heart with a speckle like appearance of echocardiograms the ability to estimate motion from real ultrasound sequences was demonstrated by a rotating phantom experiment the method was also applied to a set of in vivo echocardiograms from an animal study motion estimation results were in good agreement with the expert echocardiographic reading

on regularized reconstruction of vector fields in this paper we give a general characterization of regularization functionals for vector field reconstruction based on the requirement that the said functionals satisfy certain geometric invariance properties with respect to transformations of the coordinate system in preparation for our general result we also address some commonalities of invariant regularization in scalar and vector settings and give a complete account of invariant regularization for scalar fields before focusing on their main points of difference which lead to a distinct class of regularization operators in the vector case finally as an illustration of potential we formulate and compare quadratic l2 and total variation type l1 regularized denoising of vector fields in the proposed framework

enhancing the velocity data from 4d flow mr images by reducing its divergence velocity measurements from 4d flow mri are prone to be affected by several imperfections of the mr system assuming that blood is incompressible we propose a novel method for enhancing the velocity field by reducing its divergence to enhance the velocity data we added a corrector velocity to each voxel such that the divergence is minimized the method was validated using an analytical womersley flow model for different settings of resolution and noise levels the performance of the proposed method was also assessed in volunteers and patients results demonstrated a significant reduction of the divergence depending on the size of the regularization term obtaining a reduction close to 50 of the mean divergence with negligible modification of flow parameters remarkably we found that the reduction of the divergence in percentage was independent of volunteers resolution or noise

irreducible sign k potent sign pattern matrices abstract the sign pattern matrix a is called sign k potent if k is the smallest positive integer such that a k 1 a the structure of irreducible sign k potent pattern matrices was characterized by stuart et al j stuart c eschenbach s kirkland linear algebra appl 294 1999 85 92 we extend those results to the reducible case providing necessary conditions that characterize the structure of each off diagonal block of the frobenius normal form of a reducible sign k potent matrix

a multiple active contour model for cardiac boundary detection on echocardiographic sequences tracing of left ventricular epicardial and endocardial borders on echocardiographic sequences is essential for quantification of cardiac function the authors designed a method based on an extension of active contour models to detect both epicardial and endocardial borders on short axis cardiac sequences spanning the entire cardiac cycle they validated the results by comparing the computer generated boundaries to the boundaries manually outlined by four expert observers on 44 clinical data sets the mean boundary distance between the computer generated boundaries and the manually outlined boundaries was 2 80 mm spl sigma 1 28 mm for the epicardium and 3 61 spl sigma 1 68 mm for the endocardium these distances were comparable to interobserver distances which had a mean of 3 79 mm spl sigma 1 53 mm for epicardial borders and 2 67 mm spl sigma 0 88 mm for endocardial borders the correlation coefficient between the areas enclosed by the computer generated boundaries and the average manually outlined boundaries was 0 95 for epicardium and 0 91 for endocardium the algorithm is fairly insensitive to the choice of the initial curve thus the authors have developed an effective and robust algorithm to extract left ventricular boundaries from echocardiographic sequences

sign idempotent sign pattern matrices that allow idempotence abstract it is well known that not all sign idempotent sign pattern matrices allow idempotence in this paper we found a necessary and sufficient condition for a sign idempotent sign pattern matrix to allow idempotence in addition we investigated the nonnegativity of sign idempotent sign pattern matrices that allow idempotence finally we show that if a is a sign idempotent sign pattern matrix then a allows idempotence if and only if a a i j also allows idempotence

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed