hamiltonian gravity and noncommutative geometry a version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry the adm version of the gravitational action is expressed in terms of these variables it is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action

multiscattering inversion for low model wavenumbers abstracta successful full waveform inversion implementation updates the low wavenumber model components first for a proper description of the wavefield propagation and slowly adds the high wavenumber potentially scattering parts of the model the low wavenumber components can be extracted from the transmission parts of the recorded wavefield emanating directly from the source or the transmission parts from the single or double scattered wavefield computed from a predicted scatter field acting as secondary sources we use a combined inversion of data modeled from the source and those corresponding to single and double scattering to update the velocity model and the component of the velocity perturbation responsible for the single and double scattering the combined inversion helps us access most of the potential model wavenumber information that may be embedded in the data a scattering angle filter is used to divide the gradient of the combined inversion so initially the high wavenumber low scatterin

spectral triples for the sierpinski gasket abstract we construct a family of spectral triples for the sierpinski gasket k for suitable values of the parameters we determine the dimensional spectrum and recover the hausdorff measure of k in terms of the residue of the volume functional a tr a d s at its abscissa of convergence d d which coincides with the hausdorff dimension d h of the fractal we determine the associated connes distance showing that it is bi lipschitz equivalent to the distance on k induced by the euclidean metric of the plane and show that the pairing of the associated fredholm module with odd k theory is non trivial when the parameters belong to a suitable range the abscissa of convergence d of the energy functional a tr d s 2 d a 2 d s 2 takes the value d e log 12 5 log 2 which we call energy dimension and the corresponding residue gives the standard dirichlet form on k

application of the finite difference contrast source inversion algorithm to seismic full waveform data we have applied the finite difference contrast source inversion fdcsi method to seismic full waveform inversion problems the fdcsi method is an iterative nonlinear inversion algorithm however unlike the nonlinear conjugate gradient method and the gauss newton method fdcsi does not solve any full forward problem explicitly in each iterative step of the inversion process this feature makes the method very efficient in solving large scale computational problems it is shown that fdcsi with a significant lower computation cost can produce inversion results comparable in quality to those produced by the gauss newton method and better than those produced by the nonlinear conjugate gradient method another attractive feature of the fdcsi method is that it is capable of employing an inhomogeneous background medium without any extra or special effort this feature is useful when dealing with time lapse inversion problems where the objective is to reconstruct changes between the baseline and the monitor model by using the baseline model as the background medium in crosswell seismic monitoring problems high quality time lapse inversion results are obtained

dixmier traces and extrapolation description of noncommutative lorentz spaces we study the connections between dixmier traces functions and traces of heat semigroups beyond the dual of the macaev ideal and in the general context of semifinite von neumann algebras we show that the correct framework for this investigation is that of operator lorentz spaces possessing an extrapolation description we demonstrate the applicability of our results to hormander weyl pseudo differential calculus on rn in this context we prove that the dixmier trace of a pseudo differential operator coincides with the dixmier integral of its symbol

the earth mover s distance as a metric for image retrieval we investigate the properties of a metric between two distributions the earth mover s distance emd for content based image retrieval the emd is based on the minimal cost that must be paid to transform one distribution into the other in a precise sense and was first proposed for certain vision problems by peleg werman and rom for image retrieval we combine this idea with a representation scheme for distributions that is based on vector quantization this combination leads to an image comparison framework that often accounts for perceptual similarity better than other previously proposed methods the emd is based on a solution to the transportation problem from linear optimization for which efficient algorithms are available and also allows naturally for partial matching it is more robust than histogram matching techniques in that it can operate on variable length representations of the distributions that avoid quantization and other binning problems typical of histograms when used to compare distributions with the same overall mass the emd is a true metric in this paper we focus on applications to color and texture and we compare the retrieval performance of the emd with that of other distances

trace class operators and commutators abstract we identify the linear span of commutators ab ba where a is a trace class operator and b is any bounded operator on a separable hilbert space or where a b are both hilbert schmidt operators thus answering questions raised by work of anderson vaserstein and weiss

a statistical approach to texture classification from single images we investigate texture classification from single images obtained under unknown viewpoint and illumination a statistical approach is developed where textures are modelled by the joint probability distribution of filter responses this distribution is represented by the frequency histogram of filter response cluster centres textons recognition proceeds from single uncalibrated images and the novelty here is that rotationally invariant filters are used and the filter response space is low dimensional classification performance is compared with the filter banks and methods of leung and malik ijcv 2001 schmid cvpr 2001 and cula and dana ijcv 2004 and it is demonstrated that superior performance is achieved here classification results are presented for all 61 materials in the columbia utrecht texture database we also discuss the effects of various parameters on our classification algorithm such as the choice of filter bank and rotational invariance the size of the texton dictionary as well as the number of training images used finally we present a method of reliably measuring relative orientation co occurrence statistics in a rotationally invariant manner and discuss whether incorporating such information can enhance the classifier s performance

joint migration inversion simultaneous determination of velocity fields and depth images using all orders of scattering the future in seismic exploration and seismic monitoring is the inclusion of all orders of scattering in the imaging and inversion algorithms using the full wavefield in an inversion process allows us to surpass imprints provided by incomplete acquisition and provide more accurate information on the subsurface as noise becomes signal we describe an inversion process that does not use the standard parameters from finite difference type modeling being local velocity and density but that describes the seismic data in terms of elastic reflectivity operators and propagation operators with these operators our full wavefield modeling process builds the complete two way seismic response a major advantage of our alternative to the traditional full waveform inversion methodology is that reflectivity operators are traveltime free and propagation operators are scattering free making the inversion problem significantly more linear a major advantage of our algorithm is 1 that multiples are utilized to impro

on the local behavior of spaces of natural images in this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images to this end we use a space of 3 by 3 high contrast patches we develop a theoretical model for the high density 2 dimensional submanifold of showing that it has the topology of the klein bottle using our topological software package plex we experimentally verify our theoretical conclusions we use polynomial representation to give coordinatization to various subspaces of we find the best fitting embedding of the klein bottle into the ambient space of our results are currently being used in developing a compression algorithm based on a klein bottle dictionary

measures from dixmier traces and zeta functions for l functions on a closed compact riemannian manifold the noncommutative residue and the dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the lebesgue integral the identifications are shown to continue to and be sharp at l2 functions for functions strictly in lp 1 p 2 symmetrised noncommutative residue and dixmier trace formulas must be introduced for which the identification is shown to continue for the noncommutative residue however a failure is shown for the dixmier trace formulation at l1 functions it is shown the noncommutative residue remains finite and recovers the lebesgue integral for any integrable function while the dixmier trace expression can diverge the results show that a claim in the monograph j m gracia bondia j c varilly h figueroa elements of noncommutative geometry birkhauser adv texts birkhauser boston 2001 that the equality on c functions between the lebesgue integral and an operator theoretic expression involving a dixmier trace obtained from connes trace theorem can be extended to any integrable function is false the results of this paper include a general presentation for finitely generated von neumann algebras of commuting bounded operators including a bounded borel or l functional calculus version of c results in iv 2 of a connes noncommutative geometry academic press new york 1994

spectral flow and dixmier traces abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi finite von neumann algebra our results have several applications we deduce a formula for the chern character of an odd l 1 summable breuer fredholm module in terms of a hochschild 1 cycle we explain how to derive a wodzicki residue for pseudo differential operators along the orbits of an ergodic r n action on a compact space x finally we give a short proof of an index theorem of lesch for generalised toeplitz operators

which traces are spectral abstract among ideals of compact operators on a hilbert space we identify a subclass of those closed with respect to the logarithmic submajorization within this subclass we answer the questions asked by pietsch 22 and by dykema figiel weiss and wodzicki 7 in the first case we show that lidskii type formulae hold for every trace on such ideal in the second case we provide the description of the commutator subspace associated with a given ideal finally we prove that a positive trace on an arbitrary ideal is spectral if and only if it is monotone with respect to the logarithmic submajorization

the dixmier trace and asymptotics of zeta functions abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup we prove our results in a general semi finite von neumann algebra we find for p 1 that the asymptotics of the zeta function determines an ideal strictly larger than l p on which the dixmier trace may be defined we also establish stronger versions of other results on dixmier traces and zeta functions

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

fubini theorem in noncommutative geometry abstract we discuss the fubini formula in alain connes noncommutative geometry we present a sufficient condition on spectral triples for which a fubini formula holds true the condition is natural and related to heat semigroup asymptotics we provide examples of spectral triples for which the fubini formula fails

traces of compact operators and the noncommutative residue abstract we extend the noncommutative residue of m wodzicki on compactly supported classical pseudo differential operators of order d and generalise a connes trace theorem which states that the residue can be calculated using a singular trace on compact operators contrary to the role of the noncommutative residue for the classical pseudo differential operators a corollary is that the pseudo differential operators of order d do not have a unique trace pseudo differential operators can be non measurable in connes sense other corollaries are given clarifying the role of dixmier traces in noncommutative geometry including the definitive statement of connes original theorem

combining full wavefield migration and full waveform inversion a glance into the future of seismic imaging the next generation seismic migration and inversion technology considers multiple scattering as vital information allowing the industry to derive significantly better reservoir models with more detail and less uncertainty while requiring a minimum of user intervention three new insights have been uncovered with respect to this fundamental transition unblended or blended multiple scattering can be included in the seismic migration process and it has been proposed to formulate the imaging principle as a minimization problem the resulting process yields angle dependent reflectivity and is referred to as recursive full wavefield migration wfm the full waveform inversion process for velocity estimation can be extended to a recursive optionally blended anisotropic multiple scattering algorithm the resulting process yields angle dependent velocity and is referred to as recursive full waveform inversion wfi the mathematical equations of wfm and wfi have an identical structure but the physical meaning behind the expressions is fundamentally different in wfm the reflection process is central and the aim is to estimate reflection operators of the subsurface using the up and downgoing incident wavefields including the codas in each gridpoint in wfi however the propagation process is central and the aim is to estimate velocity operators of the subsurface using the total incident wavefield sum of up and downgoing in each gridpoint angle dependent reflectivity in wfm corresponds with angle dependent velocity anisotropy in wfi the algorithms of wfm and wfi could be joined into one automated joint migration inversion process in the resulting hybrid algorithm being referred to as recursive joint migration inversion jmi the elaborate volume integral solution was replaced by an efficient alternative wfm and wfi are alternately applied at each depth level where wfm extrapolates the incident wavefields and wfi updates the velocities without any user interaction the output of the jmi process offers an integrated picture of the subsurface in terms of angle dependent reflectivity as well as anisotropic velocity this two fold output reflectivity image and velocity model offers new opportunities to extract accurate rock and pore properties at a fine reservoir scale

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

the nonlinear statistics of high contrast patches in natural images recently there has been a great deal of interest in modeling the non gaussian structures of natural images however despite the many advances in the direction of sparse coding and multi resolution analysis the full probability distribution of pixels values in a neighborhood has not yet been described in this study we explore the space of data points representing the values of 3 3 high contrast patches from optical and 3d range images we find that the distribution of data is extremely sparse with the majority of the data points concentrated in clusters and non linear low dimensional manifolds furthermore a detailed study of probability densities allows us to systematically distinguish between images of different modalities optical versus range which otherwise display similar marginal distributions our work indicates the importance of studying the full probability distribution of natural images not just marginals and the need to understand the intrinsic dimensionality and nature of the data we believe that object like structures in the world and the sensor properties of the probing device generate observations that are concentrated along predictable shapes in state space our study of natural image statistics accounts for local geometries such as edges in natural scenes but does not impose such strong assumptions on the data as independent components or sparse coding by linear change of bases

dixmier traces on noncompact isospectral deformations abstract we extend the isospectral deformations of connes landi and dubois violette to the case of riemannian spin manifolds carrying a proper action of the noncompact abelian group r l under deformation by a torus action a standard formula relates dixmier traces of measurable operators to integrals of functions on the manifold we show that this relation persists for actions of r l under mild restrictions on the geometry of the manifold which guarantee the dixmier traceability of those operators

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

type ii non commutative geometry i dixmier trace in von neumann algebras abstract we define the notion of connes von neumann spectral triple and consider the associated index problem we compute the analytic chern connes character of such a generalized spectral triple and prove the corresponding local formula for its hochschild class this formula involves the dixmier trace for ii von neumann algebras in the case of foliations we identify this dixmier trace with the corresponding measured wodzicki residue

the second order local image structure solid characterization of second order local image structure by a 6d vector or jet of gaussian derivative measurements is considered we consider the affect on jets of a group of transformations affine intensity scaling image rotation and reflection and their compositions that preserve intrinsic image structure we show how this group stratifies the jet space into a system of orbits considering individual orbits as points a 3d orbifold is defined we propose a norm on jet space which we use to induce a metric on the orbifold the metric tensor shows that the orbifold is intrinsically curved to allow visualization of the orbifold and numerical computation with it we present a mildly distorting but volume preserving embedding of it into euclidean 3 space we call the resulting shape which is like a flattened lemon the second order local image structure solid as an example use of the solid we compute the distribution of local structures in noise and natural images for noise images analytical results are possible and they agree with the empirical results for natural images an excess of locally 1d structure is found

integration on locally compact noncommutative spaces abstract we present an ab initio approach to integration theory for nonunital spectral triples this is done without reference to local units and in the full generality of semifinite noncommutative geometry the main result is an equality between the dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators we also examine definitions for integrable bounded elements of a spectral triple based on zeta function heat kernel and dixmier trace techniques we show that zeta functions and heat kernels yield equivalent notions of integrability which imply dixmier traceability

elastic reflection based waveform inversion with a nonlinear approach abstractfull waveform inversion fwi is a highly nonlinear problem due to the complex reflectivity of the earth and this nonlinearity only increases under the more expensive elastic assumption in elastic media we need a good initial p wave velocity and even better initial s wave velocity models with accurate representation of the low model wavenumbers for fwi to converge however inverting for the low wavenumber components of p and s wave velocities using reflection waveform inversion rwi with an objective to fit the reflection shape rather than produce reflections may mitigate the limitations of fwi because fwi performing as a migration operator is preferred of the high wavenumber updates along reflectors we have developed an elastic rwi that inverts for the low wavenumber and perturbation components of the p and s wave velocities to generate the full elastic reflection wavefields we derive an equivalent stress source made up by the inverted model perturbations and incident wavefields w

dixmier traces as singular symmetric functionals and applications to measurable operators abstract we unify various constructions and contribute to the theory of singular symmetric functionals on marcinkiewicz function operator spaces this affords a new approach to the non normal dixmier and connes dixmier traces introduced by dixmier and adapted to non commutative geometry by connes living on a general marcinkiewicz space associated with an arbitrary semifinite von neumann algebra the corollaries to our approach stated in terms of the operator ideal l 1 which is a special example of an operator marcinkiewicz space are i a new characterization of the set of all positive measurable operators from l 1 i e those on which an arbitrary connes dixmier trace yields the same value in the special case when the operator ideal l 1 is considered on a type i infinite factor a bounded operator x belongs to l 1 if and only if the sequence of singular numbers s n x n 1 in the descending order and counting the multiplicities satisfies x 1 sup n 1 1 log 1 n n 1 n s n x in this case our characterization amounts to saying that a positive element x l 1 is measurable if and only if lim n 1 log n n 1 n s n x exists ii the set of dixmier traces and the set of connes dixmier traces are norming sets up to equivalence for the space l 1 l 0 1 where the space l 0 1 is the closure of all finite rank operators in l 1 in the norm 1

distance metric learning for large margin nearest neighbor classification the accuracy of k nearest neighbor knn classification depends significantly on the metric used to compute distances between different examples in this paper we show how to learn a mahalanobis distance metric for knn classification from labeled examples the mahalanobis metric can equivalently be viewed as a global linear transformation of the input space that precedes knn classification using euclidean distances in our approach the metric is trained with the goal that the k nearest neighbors always belong to the same class while examples from different classes are separated by a large margin as in support vector machines svms the margin criterion leads to a convex optimization based on the hinge loss unlike learning in svms however our approach requires no modification or extension for problems in multiway as opposed to binary classification in our framework the mahalanobis distance metric is obtained as the solution to a semidefinite program on several data sets of varying size and difficulty we find that metrics trained in this way lead to significant improvements in knn classification sometimes these results can be further improved by clustering the training examples and learning an individual metric within each cluster we show how to learn and combine these local metrics in a globally integrated manner

using basic image features for texture classification representing texture images statistically as histograms over a discrete vocabulary of local features has proven widely effective for texture classification tasks images are described locally by vectors of for example responses to some filter bank and a visual vocabulary is defined as a partition of this descriptor response space typically based on clustering in this paper we investigate the performance of an approach which represents textures as histograms over a visual vocabulary which is defined geometrically based on the basic image features of griffin and lillholm proc spie 6492 09 1 11 2007 rather than by clustering bifs provide a natural mathematical quantisation of a filter response space into qualitatively distinct types of local image structure we also extend our approach to deal with intra class variations in scale our algorithm is simple there is no need for a pre training step to learn a visual dictionary as in methods based on clustering and no tuning of parameters is required to deal with different datasets we have tested our implementation on three popular and challenging texture datasets and find that it produces consistently good classification results on each including what we believe to be the best reported for the kth tips and equal best reported for the uiuctex databases

multidimensional linearized inversion and seismic migration this paper discusses the close relationship between seismic migration and multidimensional inversion according to the linearized inverse scattering theory the linearized inverse scattering approach represents a mixed modeling inversion procedure unlike seismic migration the actual inversion process is carried out on the difference between a modeled reference response and the actually measured data the output is generally presented in terms of the elastic parameters of the medium seismic migration represents a direct inversion method the downward extrapolation process is carried out directly on the measured data output is presented in terms of reflectivity if the reference medium has been chosen in such a way that 1 the total wave field in the reference medium can be split into a downward traveling source wave field and an upward traveling response the propagation of both wave fields being defined by the one way wave equation and that 2 the upward traveling response in the reference medium can be neglected with respect to the upward traveling response in the actual medium then seismic migration and linearized inversion define identical inversion processes typically the above conditions are fulfilled in a homogeneous reference medium in iterative multidimensional inversion the full inverse scattering problem is approached by a number of linearized inversion steps i show that each linear step consists of a prestack migration process and a prestack modeling process the modeling output being used to remove the contribution of multiple scattering finally i argue that for a proper inversion process information on the elastic parameters outside the seismic frequency bandwidth temporally and spatially should be accounted for in the reference medium

sums of commutators in non commutative banach function spaces let m be a ii factor and denote by its normal faithful semi finite trace for any rearrangement invariant kothe function space x on 0 let x m be the associated non commutative banach function space this paper is concerned with ideals in m of the form ix m m x m that are contained in lp m for some p 0 it is proved that an element t in ix m is a finite sum of commutators of the form a b with a ix m and b m if and only if the function t 1t t t d t belongs to x where t is the brown spectral measure of t and t t t is the non increasing rearrangement of the function with respect to t this extends to general banach function spaces a result obtained by kalton for quasi banach ideals of compact operators and implies that the dixmier s trace of a quasi nilpotent element in l1 m is always zero

connes s trace theorem for curved noncommutative tori application to scalar curvature in this paper we prove a version of connes s trace theorem for noncommutative tori of any dimension n 2 this allows us to recover and improve earlier versions of this result in dimensions n 2 and n 4 by fathizadeh and khalkhali we also recover connes s integration formula for flat noncommutative tori of mcdonald sukochev and zanin as a further application we prove a curved version of this integration formula in terms of the laplace beltrami operator defined by an arbitrary riemannian metric for the class of the so called self compatible riemannian metrics including the conformally flat metrics of connes and tretkoff this shows that connes s noncommutative integral allows us to recover the riemannian density this exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras as an application of these results we set up a natural notion of scalar curvature for curved noncommutative tori

subsurface reflectivity estimation from imaging of primaries and multiples using amplitude normalized separated wavefields abstractseveral studies have shown the benefits of including multiple reflections together with primaries in the structural imaging of subsurface reflectors however to characterize the reflector properties there is a need to compensate for propagation effects due to multiple scattering and to properly combine the information from primaries and all orders of multiples from this perspective and based on the wave equation and rayleigh s reciprocity theorem recent works have suggested computing the subsurface image from the green s function reflection response or reflectivity by inverting a fredholm integral equation in the frequency space domain by following claerbout s imaging principle and assuming locally reacting media the integral equation may be reduced to a trace by trace deconvolution imaging condition for a complex overburden and considering that the structure of the subsurface is angle dependent this trace by trace deconvolution does not properly solve the fredholm integral equation we

commutator structure of operator ideals commutator structure of operator ideals ken dykema 1 tadeusz figiel gary weiss and mariusz wodzicki department of mathematics texas a m university college station tx 77843 3368 usa b instytut matematyczny polskiej akademii nauk 81 825 sopot poland department of mathematical sciences university of cincinnati cincinnati oh 45221 0025 usa department of mathematics university of california berkeley ca 94720 3840 usa

estimating probability densities from short samples a parametric maximum likelihood approach a parametric method similar to autoregressive spectral estimators is proposed to determine the probability density function pdf of a random set the method proceeds by maximizing the likelihood of the pdf yielding estimates that perform equally well in the tails as in the bulk of the distribution it is therefore well suited for the analysis short sets drawn from smooth pdfs and stands out by the simplicity of its computational scheme its advantages and limitations are discussed

connes integration formula for the noncommutative plane our aim is to prove the integration formula on the noncommutative moyal plane in terms of singular traces a la connes

banach limits and traces on l1 abstract we introduce a new approach to traces on the principal ideal l 1 generated by any positive compact operator whose singular value sequence is the harmonic sequence distinct from the well known construction of j dixmier the new approach provides the explicit construction of every trace of every operator in l 1 in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues the approach is based on a remarkable bijection between the set of all traces on l 1 and the set of all translation invariant functionals on l this bijection allows us to identify all known and commonly used subsets of traces dixmier traces connes dixmier traces etc in terms of invariance properties of linear functionals on l and definitively classify the measurability of operators in l 1 in terms of qualified convergence of sums of eigenvalues this classification has led us to a resolution of several open problems for the class l 1 from 7 as an application we extend connes classical trace theorem to positive normalised traces

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes

a klein bottle based dictionary for texture representation a natural object of study in texture representation and material classification is the probability density function in pixel value space underlying the set of small patches from the given image inspired by the fact that small n times n n n high contrast patches from natural images in gray scale accumulate with high density around a surface fancyscript k subset mathbb r n 2 k r n 2 with the topology of a klein bottle carlsson et al international journal of computer vision 76 1 1 12 2008 we present in this paper a novel framework for the estimation and representation of distributions around fancyscript k k of patches from texture images more specifically we show that most n times n n n patches from a given image can be projected onto fancyscript k k yielding a finite sample s subset fancyscript k s k whose underlying probability density function can be represented in terms of fourier like coefficients which in turn can be estimated from s s we show that image rotation acts as a linear transformation at the level of the estimated coefficients and use this to define a multi scale rotation invariant descriptor we test it by classifying the materials in three popular data sets the curet uiuctex and kth tips texture databases