b rep object description from multiple range views we present a system which computes an integrated description of an object from multiple range images the object description is in the form of b rep boundary representation which has not been achieved by the computer vision community to do so we emphasize the inherent difficulties and ambiguities in the low to mid level vision and present novel techniques of resolving them in this system each view of the object is represented as an attributed graph where nodes correspond to the surfaces vertices and links represent the relationship between surfaces the main issue in surface extraction is contour closure which is formulated as a dynamic network the underlying principle for this network is weak smoothness and geometric cohesion and is modeled as the interaction between long and short term variables long term variables represent the initial boundary grouping computed from the low level surface features and short term variables represent the competing hypotheses that cooperate with the long term variables the matching problem involves matching visible surfaces and vertices and provides the necessary basis for volumetric reconstruction from multiple views the matching strategy is a two step process where in each step uses the hopfield network at each step we specify a set of local adjacency and global constraints and define an appropriate energy function to be minimized at the first level of this hierarchy surface patches are matched and the rigidity transformation is computed at the second level the mapping is refined by matching the corresponding vertices and the transformation is verified the multiple view reconstruction consists of two steps first we build a composite graph that contains the bounding surfaces and their corresponding attributes and then intersect these surfaces so that the edges and vertices corresponding to the b rep description are identified we present results on objects with planar as well as quadratically curved surfaces

singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

α continuity properties of one dimensional quasicrystals we apply the jitomirskaya last extension of the gilbert pearson theory to discrete one dimensional schrodinger operators with potentials arising from generalized fibonacci sequences we prove for certain rotation numbers that for every value of the coupling constant there exists an 0 such that the corresponding operator has purely continuous spectrum this result follows from uniform upper and lower bounds for the l norm of the solutions corresponding to energies from the spectrum of the operator

noise reduction in surface reconstruction from a given gradient field we present a gradient space technique for noise reduction in surfaces reconstructed from a noisy gradient field we first analyze the error sources in the recovered gradient field of a surface using a three image photometric stereo method based on this analysis we propose an additive noise model to describe the errors in the surface gradient estimates we then use a vector space formulation and construct a multiscale orthonormal expansion for gradient fields using the sparse representation properties of this expansion we develop techniques for reducing the gradient field noise by coefficient selection with thresholding the simulation results indicate that the proposed technique provides significant improvement on the noise levels of both the estimated gradient fields and the reconstructed surfaces under heavy noise levels furthermore the experiments using noisy photometric stereo image triplets of real range data suggest that the additive model remains viable after the nonlinear photometric stereo operation to provide accurate noise removal

perceptual grouping for generic recognition we address the problem of recognition of generic objects from a single intensity image this precludes the use of purely geometric methods which assume that models are geometrically and precisely designed instead we propose to use descriptions in terms of features and their qualitative geometric relationships to succeed it is clear that these features need to be high level rather than points or lines we propose to detect groups using perceptual organization criteria such as proximity symmetry parallelism and closure the detection of these features is performed in an efficient way using proximity indexing since many groups are created we also perform selection of relevant groups by organizing them into sets of similar perceptual content finally we present an implementation of a recognition system using these sets as primitives it is an efficient colored graph matching algorithm using the adjacency matrix representation of a graph using indexing we retrieve matching hypotheses which are verified against each other with respect to topological constraints groups of consistent hypotheses represent detected model instances in a scene the complete system is illustrated on real images we also discuss further extensions

substitution hamiltonians with bounded trace map orbits we investigate discrete one dimensional schrodinger operators with aperiodic potentials generated by primitive invertible substitutions on a two letter alphabet we prove that the spectrum coincides with the set of energies having a bounded trace map orbit and show that it is a cantor set of zero lebesgue measure this result confirms a suggestion arising from a study of roberts and complements results obtained by bovier and ghez as an application we present a class of models exhibiting a purely singular continuous spectrum with probability one

substitution dynamical systems characterization of linear repetitivity and applications we consider dynamical systems arising from substitutions over a finite alphabet we prove that such a system is linearly repetitive if and only if it is minimal based on this characterization we extend various results from primitive substitutions to minimal substitutions this includes applications to random schrodinger operators and to number theory

spectra and transport in almost periodic dimers we study spectral properties of discrete schrodinger operators with potentials obtained via dimerization of a class of aperiodic sequences it is shown that both the nature of the autocorrelation measure of a regular sequence and the presence of generic full probability singular continuous spectrum in the hull of primitive and palindromic four block substitution potentials are robust under dimerization generic results also hold for circle potentials we illustrate these results with numerical studies of the quantum mean square displacement as a function of time the numerical techniques provide a very fast algorithm for the time evolution of wave packets

some remarks on discrete aperiodic schrodinger operators we consider schrodinger operators onl2 with deterministic aperiodic potential and schrodinger operators on the l2 space of the set of vertices of penrose tilings and other aperiodic self similar tilings the operators onl2 fit into the formalism of ergodic random schrodinger operators hence their lyapunov exponent integrated density of states and spectrum are almost surely constant we show that they are actually constant the lyapunov exponent for one dimensional schrodinger operators with potential defined by a primitive substitution the integrated density of states and the spectrum in arbitrary dimension if the system is strictly ergodic we give examples of strictly ergodic schrodinger operators that include several kinds of almost periodic operators that have been studied in the literature for schrodinger operators on penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular penrose tiling under consideration

the spectrum of a quasiperiodic schrodinger operator the spectrum h of the tight binding fibonacci hamiltonian hmn m n 1 m 1 n m n v n v n chi omega 3 omega 2 n 1 1 is the golden number is shown to coincide with the dynamical spectrum the set on which an infinite subsequence of traces of transfer matrices is bounded the point spectrum is absent for any and h is a cantor set for 4 combining this with casdagli s earlier result one finds that the spectrum is singular continuous for 16

3d free form surface registration and object recognition a new technique to recognise 3d free form objects via registration is proposed this technique attempts to register a free form surface represented by a set of mathtype mtef 2 1 feaafeart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepekkfr0xfr x fr xb9adbaqaaegacigaaiaabeqaamaabaabaagcbagaagomamaala aabagaagymaaqaaiaaikdaaagaamiraaaa 38f8 2 frac 1 2 d sensed data points to the model surface represented by another set of mathtype mtef 2 1 feaafeart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepekkfr0xfr x fr xb9adbaqaaegacigaaiaabeqaamaabaabaagcbagaagomamaala aabagaagymaaqaaiaaikdaaagaamiraaaa 38f8 2 frac 1 2 d model data points without prior knowledge of correspondence or view points between the two point sets with an initial assumption that the sensed surface be part of a more complete model surface the algorithm begins by selecting three dispersed reliable points on the sensed surface to find the three corresponding model points the method uses the principal curvatures and the darboux frames to restrict the search over the model space invariably many possible model 3 typles will be found for each hypothesized model 3 tuple the transformation to match the sensed 3 tuple to the model 3 tuple can be determined a heuristic search is proposed to single out the optimal transformation in low order time for realistic object recognition or registration where the two range images are often extracted from different view points of the model the earlier assumption that the sensed surface be part of a more complete model surface cannot be relied on with this the sensed 3 tuple must be chosen such that the three sensed points lie on the common region visible to both the sensed and model views we propose an algorithm to select a minimal non redundant set of 3 tuples such that at least one of the 3 tuples will lie on the overlap applying the previous algorithm to each 3 tuple within this set the optimal transformation can be determined experiments using data obtained from a range finder have indicated fast registration for relatively complex test cases if the optimal registrations between the sensed data candidate and each of a set of model data are found then for 3d object recognition purposes the minimal best fit error can be used as the decision rule

surface signatures an orientation independent free form surface representation scheme for the purpose of objects registration and matching this paper introduces anew free form surface representation scheme for the purpose of fast and accurate registration and matching accurate registration of surfaces is a common task in computer vision the proposed representation scheme captures the surface curvature information seen from certain points and produces images called surface signatures at these points matching signatures of different surfaces enables the recovery of the transformation parameters between these surfaces we propose using template matching to compare the signature images to enable partial matching another criterion the overlap ratio is used this representation scheme can be used as a global representation of the surface as well as a local one and performs near real time registration we show that the signature representation can be used to recover scaling transformation as well as matching objects in 3d scenes in the presence of clutter and occlusion applications presented include free form object matching multimodal medical volumes registration and dental teeth reconstruction from intraoral images

quantum return probability for substitution potentials we propose an effective exponent ruling the algebraic decay of the average quantum return probability for discrete schrodinger operators we compute it for some non periodic substitution potentials with different degrees of randomness and do not find a complete qualitative agreement with the spectral type of the substitution sequences themselves i e the more random the sequence the smaller such an exponent

symbolic dynamics for the renormalization map of a quasiperiodic schrodinger equation a rigorous analysis is given of the dynamics of the renormalization map associated to a discrete schrodinger operatorh onl 2 defined byh n n 1 n 1 vf n n wherev is a real parameter f is a certain discontinuous period 1 function and sigma left 1 sqrt 5 right mathord left vphantom left 1 sqrt 5 right 2 right kern nulldelimiterspace 2 is the golden mean the renormalization map forh is a diffeomorphism t of 3 preserving a cubic surfaces v forv 8 we prove that the non wandering set of the restriction oft tos v is a hyperbolic set on whicht is conjugate to a subshift on six symbols it follows from results in dynamical systems theory that the optimally approximating periodic operators toh have spectra which obey a global scaling law we also define a set which we call the pseudospectrum of the operatorh we prove it to be a cantor set of measure zero and obtain bounds on its hausdorff dimension it is an open question whether the pseudospectrum coincides with the spectrum ofh

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

on the dimensions of the spectral measure of symmetric binary substitutions substitution rules defined on the binary alphabet and invariant under the interchange of both participating symbols are considered typically the resulting symbolic sequences are weakly disordered and their fourier spectra are singular continuous exact relations are derived which express the correlation dimension of the multifractal spectral measure in terms of the entries in the substitution pattern recurrence relations are obtained for the finite length spectral sums and for the autocorrelation function

polyhedral object recognition by indexing in computer vision the indexing problem is the problem of recognizing a few objects in a large database of objects while avoiding the help of the classical image feature to object feature matching paradigm in this paper we address the problem of recognizing three dimensional 3 d polyhedral objects from 2 d images by indexing both the objects to be recognized and the images are represented by weighted graphs the indexing problem is therefore the problem of determining whether a graph extracted from the image is present or absent in a database of model graphs we introduce a novel method for performing this graph indexing process which is based both on polynomial characterization of binary and weighted graphs and on hashing we describe in detail this polynomial characterization and then we show how it can be used in the context of polyhedral object recognition next we describe a practical recognition by indexing system that includes the organization of the database the representation of polyhedral objects in terms of 2 d characteristic views the representation of this views in terms of weighted graphs and the associated image processing finally some experimental results allow the evaluation of the system performance

snapshots a novel local surface descriptor and matching algorithm for robust 3d surface alignment in this paper a novel local surface descriptor is proposed and applied to the problem of aligning partial views of a 3d object the descriptor is based on taking snapshots of the surface over each point using a virtual camera oriented perpendicularly to the surface this representation has the advantage of imposing minimal loss of information be robust to self occlusions and also be very efficient to compute then we describe an efficient search technique to deal with the rotation ambiguity of our representation and experimentally demonstrate the benefits of our approaches which are pronounced especially when we align views with small overlap

function described graphs for modelling objects represented by sets of attributed graphs we present in this article the model function described graph fdg which is a type of compact representation of a set of attributed graphs ags that borrow from random graphs the capability of probabilistic modelling of structural and attribute information we define the fdgs their features and two distance measures between ags unclassified patterns and fdgs models or classes and we also explain an efficient matching algorithm two applications of fdgs are presented in the former fdgs are used for modelling and matching 3d objects described by multiple views whereas in the latter they are used for representing and recognising human faces described also by several views

remarks on the spectral properties of tight binding and kronig penney models with substitution sequences we comment on some recent investigations on the electronic properties of models associated with the thue morse chain point out that their conclusions contradict rigorously proven theorems and indicate some of the sources of these misinterpretations we briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems

towards effective planar shape representation with multiscale digital curvature analysis based on signal processing techniques abstract this paper presents and discusses a new approach to multiscale curvature analysis of digital contours that is based on digital signal processing techniques the shape contour is expressed in terms of two one dimensional signals linearized x and y coordinates and the derivative theorem is applied as a means of obtaining an interesting expression relating contour curvature and the spectra of its parametrized x and y signals multiscale curvature analysis is achieved through gaussian lowpass filtering and the shrinkage of the original signals implied by such a process is effectively circumvented by an energy based compensation scheme which has allowed accurate quantitative identification of the curvature value the concept of curvegram is introduced and exemplified and the overall performance of the proposed technique for curvature estimation is formally assessed with respect to a contour defined in terms of b splines application examples to synthetic and real images have been included

robust image registration by increment sign correlation abstract a novel and robust statistic as a similarity measure for robust image registration is proposed the statistic is named as increment sign correlation because it is based on the average evaluation of incremental tendency of brightness in adjacent pixels it is formalized to be a binary distribution or a gaussian distribution for a large image size through statistical analysis and modeling by utilizing the proposed statistical model for example we can theoretically determine a reasonable value of threshold for verification of matching this sign correlation can also be proved to expectedly have the constant value 0 5 for any uncorrelated images to a template image and then the property of the constancy can be utilized to analyze the high robustness for occlusion the good performance for the case of saturation or highlight can also be proved through theoretical analysis and fundamental experiments a basic algorithm for image scanning search and registration over a large scene is represented with a technique for a fast version by the branch and bound approach many experimental evidences with real images are provided and discussed

point signatures a new representation for 3d object recognition few systems capable of recognizing complex objects with free form sculptured surfaces have been developed the apparent lack of success is mainly due to the lack of a competent modelling scheme for representing such complex objects in this paper a new form of point representation for describing 3d free form surfaces is proposed this representation which we call the point signature serves to describe the structural neighbourhood of a point in a more complete manner than just using the 3d coordinates of the point being invariant to rotation and translation the point signature can be used directly to hypothesize the correspondence to model points with similar signatures recognition is achieved by matching the signatures of data points representing the sensed surface to the signatures of data points representing the model surface the use of point signatures is not restricted to the recognition of a single object scene to a small library of models instead it can be extended naturally to the recognition of scenes containing multiple partially overlapping objects which may also be juxtaposed with each other against a large model library no preliminary phase of segmenting the scene into the component objects is required in searching for the appropriate candidate model recognition need not proceed in a linear order which can become prohibitive for a large model library for a given scene signatures are extracted at arbitrarily spaced seed points each of these signatures is used to vote for models that contain points having similar signatures inappropriate models with low votes can be rejected while the remaining candidate models are ordered according to the votes they received in this way efficient verification of the hypothesized candidates can proceed by testing the most likely model first experiments using real data obtained from a range finder have shown fast recognition from a library of fifteen models whose complexities vary from that of simple piecewise quadric shapes to complicated face masks results from the recognition of both single object and multiple object scenes are presented

uniform cantor singular continuous spectrum for nonprimitive schrodinger operators it is shown that some schrodinger operators with nonprimitive substitution potentials have pure singular continuous cantor spectrum with null lebesgue measure for all elements in the respective hulls

lie transformations similarity reduction and solutions for the nonlinear madelung fluid equations with external potential the application of lie group methods to a system of coupled nonlinear partial differential equations representing what is usually called a madelung fluid is shown the generating operators of the transformation group that depends on five arbitrary group constants will be constructed and all subclasses of systems of ordinary differential equations derived by similarity reduction will be presented in tabular form two subclasses of physical interest are investigated in detail and the similarity solutions are compared with solutions found earlier by the application of inverse scattering transform techniques to the cubic nonlinear schrodinger equation similarity solutions for the madelung equations with linear external potential x f0x are presented

on the hausdorff dimension of the spectrum of the thue morse hamiltonian we show that the hausdorff dimension of the spectrum of the thue morse hamiltonian has a common positive lower bound for all coupling

representation and shape matching of 3 d objects a three dimensional scene analysis system for the shape matching of real world 3 d objects is presented various issues related to representation and modeling of 3 d objects are addressed a new method for the approximation of 3 d objects by a set of planar faces is discussed the major advantage of this method is that it is applicable to a complete object and not restricted to single range view which was the limitation of the previous work in 3 d scene analysis the method is a sequential region growing algorithm it is not applied to range images but rather to a set of 3 d points the 3 d model of an object is obtained by combining the object points from a sequence of range data images corresponding to various views of the object applying the necessary transformations and then approximating the surface by polygons a stochastic labeling technique is used to do the shape matching of 3 d objects the technique matches the faces of an unknown view against the faces of the model it explicitly maximizes a criterion function based on the ambiguity and inconsistency of classification it is hierarchical and uses results obtained at low levels to speed up and improve the accuracy of results at higher levels the objective here is to match the individual views of the object taken from any vantage point details of the algorithm are presented and the results are shown on several unknown views of a complicated automobile casting

free electron gas in the apollonian network multifractal energy spectrum and its thermodynamic fingerprints we study the free electron gas in an apollonian network within the tight binding framework the scale free and small world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities gaps and minibands after an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum the fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat the absence of chiral symmetry of the apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations

remarks on the relation between quantum dynamics and fractal spectra abstract we investigate the large time behaviour of various solutions of the schrodinger equation in terms of some local and global dimensions of the spectral measure we emphasize in particular the role of the hausdorff and correlation dimensions for the growth exponents of position moments we also discuss the stability of such exponents under local perturbations

on kicked systems modulated along the thue morse sequence we give necessary conditions for the fourier transform of the time evolution operator of kicked systems modulated along the thue morse sequence to present dirac s delta functions the case of two level systems is analysed in detail and it is found based on the trace map and numerical investigations that generically its time evolution has no quasi periodic component

singular continuous spectrum on a cantor set of zero lebesgue measure for the fibonacci hamiltonian it is rigorously proven that the spectrum of the tight binding fibonacci hamiltonian h mn m n 1 m n 1 m n n 1 n where 5 1 2 and means integer part is a cantor set of zero lebesgue measure for all real nonzero and the spectral measures are purely singular continuous this follows from a recent result by kotani coupled with the vanishing of the lyapunov exponent in the spectrum

planar object recognition using projective shape representation we describe a model based recognition system called lewis for the identification of planar objects based on a projectively invariant representation of shape the advantages of this shape description include simple model acquisition direct from images no need for camera calibration or object pose computation and the use of index functions we describe the feature construction and recognition algorithms in detail and provide an analysis of the combinatorial advantages of using index functions index functions are used to select models from a model base and are constructed from projective invariants based on algebraic curves and a canonical projective coordinate frame examples are given of object recognition from images of real scenes with extensive object libraries successful recognition is demonstrated despite partial occlusion by unmodelled objects and realistic lighting conditions

hybrid quasicrystals transport and localization in products of minimal sets we consider convex combinations of finite valued almost periodic sequences mainly substitution sequences and put them as potentials of one dimensional tight binding models we prove that these sequences are almost periodic we call such combinations hybrid quasicrystals and these studies are related to the minimality under the shift on both coordinates of the product space of the respective minimal hulls we observe a rich variety of behaviors on the quantum dynamical transport ranging from localization to transport

using spin images for efficient object recognition in cluttered 3d scenes we present a 3d shape based object recognition system for simultaneous recognition of multiple objects in scenes containing clutter and occlusion recognition is based on matching surfaces by matching points using the spin image representation the spin image is a data level shape descriptor that is used to match surfaces represented as surface meshes we present a compression scheme for spin images that results in efficient multiple object recognition which we verify with results showing the simultaneous recognition of multiple objects from a library of 20 models furthermore we demonstrate the robust performance of recognition in the presence of clutter and occlusion through analysis of recognition trials on 100 scenes

shapeme histogram projection and matching for partial object recognition histograms of shape signature or prototypical shapes called shapemes have been used effectively in previous work for 2d 3d shape matching and recognition we extend the idea of shapeme histogram to recognize partially observed query objects from a database of complete model objects we propose representing each model object as a collection of shapeme histograms and match the query histogram to this representation in two steps 1 compute a constrained projection of the query histogram onto the subspace spanned by all the shapeme histograms of the model and 2 compute a match measure between the query histogram and the projection the first step is formulated as a constrained optimization problem that is solved by a sampling algorithm the second step is formulated under a bayesian framework where an implicit feature selection process is conducted to improve the discrimination capability of shapeme histograms results of matching partially viewed range objects with a 243 model database demonstrate better performance than the original shapeme histogram matching algorithm and other approaches

lie backlund symmetries of certain nonlinear evolution equations under perturbation around their solutions a generalization of the lie backlund lb theory for coupled evolution equations is discussed as a consequence the direct connection between the lb symmetries and constants of motion of these systems is also established furthermore as an application of this theory we investigate the existence of infinitely many commuting lb symmetries of certain nonlinear evolution equations under perturbation around their solutions then the corresponding constants of motion are derived which are in agreement with the known results

matching 3 d models to 2 d images we consider the problem of analytically characterizing the set of all 2 d images that a group of 3 d features may produce and demonstrate that this is a useful thing to do our results apply for simple point features and point features with associated orientation vectors when we model projection as a 3 d to 2 d affine transformation we show how to represent the set of images that a group of 3 d points produces with two lines 1 d subspaces one in each of two orthogonal high dimensional spaces where a single image group corresponds to one point in each space the images of groups of oriented point features can be represented by a 2 d hyperbolic surface in a single high dimensional space the problem of matching an image to models is essentially reduced to the problem of matching a point to simple geometric structures moreover we show that these are the simplest and lowest dimensional representations possible for these cases we demonstrate the value of this way of approaching matching by applying our results to a variety of vision problems in particular we use this result to build a space efficient indexing system that performs 3 d to 2 d matching by table lookup this system is analytically built and accessed accounts for the effects of sensing error and is tested on real images we also derive new results concerning the existence of invariants and non accidental properties in this domain finally we show that oriented points present unexpected difficulties indexing requires fundamentally more space with oriented than with simple points we must use more images in a motion sequence to determine the affine structure of oriented points and the linear combinations result does not hold for oriented points

registration of 3d points using geometric algebra and tensor voting we address the problem of finding the correspondences of two point sets in 3d undergoing a rigid transformation using these correspondences the motion between the two sets can be computed to perform registration our approach is based on the analysis of the rigid motion equations as expressed in the geometric algebra framework through this analysis it was apparent that this problem could be cast into a problem of finding a certain 3d plane in a different space that satisfies certain geometric constraints in order to find this plane in a robust way the tensor voting methodology was used unlike other common algorithms for point registration like the iterated closest points algorithm ours does not require an initialization works equally well with small and large transformations it cannot be trapped in local minima and works even in the presence of large amounts of outliers we also show that this algorithm is easily extended to account for multiple motions and certain non rigid or elastic transformations

line pattern retrieval using relational histograms this paper presents a new compact shape representation for retrieving line patterns from large databases the basic idea is to exploit both geometric attributes and structural information to construct a shape histogram we realize this goal by computing the n nearest neighbor graph for the lines segments for each pattern the edges of the neighborhood graphs are used to gate contributions to a two dimensional pairwise geometric histogram shapes are indexed by searching for the line pattern that maximizes the cross correlation of the normalized histogram bin contents we evaluate the new method on a database containing over 2 500 line patterns each composed of hundreds of lines

robust matching of 3d contours using iterative closest point algorithm improved by m estimation abstract an extension of the iterative closest point matching by m estimation is proposed for realization of robustness to non overlapping data or outlying data in two sets of contour data or depth images for rigid bodies an objective function which includes independent residual components for each of x y and z coordinates is originally defined and proposed to evaluate the fitness simultaneously dealing with a distribution of outlying gross noise the proposed procedure is based on modified m estimation iterations with bi weighting coefficients for selecting corresponding points for optimization of estimating the transforms for matching the transforms can be represented by quaternions in the procedure to eliminate redundancy in representation of rotational degree of freedom by linear matrices optimization steps are performed by the simplex method because it does not need computation of differentiation some fundamental experiments utilizing real data of 2d and 3d measurement show effectiveness of the proposed method when reasonable initial positions are given the unique solution of position could be provided in spite of surplus point data in the objects and then the outlying data could be filtered out from the normal ones by the proposed method

spectrum and extended states in a harmonic chain with controlled disorder effects of the thue morse symmetry along the lines of previous work we give the general framework together with a detailed and rigorous study of the spectrum and born von karman eigenstates of a 1d harmonic chain with controlled disorder determined by the thue morse sequence the spectrum is a cantor like set we prove numerically that its measure is zero and calculate its bouligand minkowski dimension box dimension we prove that the value of the ids on each of the gaps is 2k 1 3 2 p withk andp integers finally we also prove that points in a dense subset of the spectrum give rise to extended states an exceptional property due to the symmetry of the thue morse substitution which can have important applications to multilayered structures and we illustrate this situation

uniform spectral properties of one dimensional quasicrystals i absence of eigenvalues we consider discrete one dimensional schrodinger operators with sturmian potentials for a full measure set of rotation numbers including the fibonacci case we prove absence of eigenvalues for all elements in the hull

real time object recognition in sparse range images using error surface embedding a novel object recognition algorithm is introduced to identify objects and recover their pose from sparse range data the method is based upon comparing the 7 d error surfaces of objects in various poses which result from the registration error function between two convolved surfaces the objects and their pose values are encoded by a small set of feature vectors extracted from the minima of the error surfaces the problem of object recognition is thus reduced to comparing these feature vectors to find the corresponding error surfaces between the runtime data and a preprocessed database the algorithm called potential well space embedding pwse has been implemented and tested on both simulated and real data the experimental results show the technique to be both effective and efficient executing at 122 frames per second on standard hardware and with recognition rates exceeding 97 for a database of 60 objects the performance of pwse on the large size database was also evaluated on the princeton shape benchmark containing 1 814 objects in addition it functions well with very sparse data possibly comprising only hundreds of points per image and is shown to be robust to measurement error and outliers with some small modifications we applied pwse to the problem of object class recognition in experiments with the princeton shape benchmark pwse is able to provides better classification rates than the previous methods in terms of nearest neighbour classification

polynomial constants for the quantized nls equation the classical nonlinear schrodinger equation nls is known to have an infinite number of polynomial constants while recursion relations to compute these are available no general expressions in terms of the fields have been found however general expressions have been obtained in terms of the reflection coefficients when we turn to the quantum case where the fields become operators with conventional commutation relations the polynomials with suitable ordering are still constants the classical expression for the constants in terms of the reflection coefficients strongly suggests what the quantum form should be this conjecture is proved for the repulsive case the expression is significantly simpler than the classical one it is in 1 2 k nr k r k dk

on the leading term and the degree of the polynomial trace mapping associated with a substitution using the trace mapping and the reduced trace mapping associated with a substitution one obtaines the spectral properties of one dimensional schrodinger operators of the formh v onl2 where is the discrete laplacian andv is a diagonal operator with elements derived from a substitution rule in particular the reduced trace mapping is closely related to the leading term of the original trace mapping in this paper the explicit expression of the leading term is given and its properties are discussed

measure zero spectrum of a class of schrodinger operators we study the measure of the spectrum of a class of one dimensional discrete schrodinger operators hv with potential v generated by any primitive substitutions it is well known that the spectrum of hv is singular continuous 1 we will give a more exact result that the spectrum of hv is a set of lebesgue measure zero by removing two hypotheses the semi primitive of a certain induced substitution and the existence of square word from a theorem due to bovier and ghez 2

adaptively merging large scale range data with reflectance properties in this paper we tackle the problem of geometric and photometric modeling of large intricately shaped objects typical target objects we consider are cultural heritage objects when constructing models of such objects we are faced with several important issues that have not been addressed in the past issues that mainly arise due to the large amount of data that has to be handled we propose two novel approaches to efficiently handle such large amounts of data a highly adaptive algorithm for merging range images and an adaptive nearest neighbor search to be used with the algorithm we construct an integrated mesh model of the target object in adaptive resolution taking into account the geometric and or photometric attributes associated with the range images we use surface curvature for the geometric attributes and laser reflectance values for the photometric attributes this adaptive merging framework leads to a significant reduction in the necessary amount of computational resources furthermore the resulting adaptive mesh models can be of great use for applications such as texture mapping as we will briefly demonstrate additionally we propose an additional test for the k d tree nearest neighbor search algorithm our approach successfully omits back tracking which is controlled adaptively depending on the distance to the nearest neighbor since the main consumption of computational cost lies in the nearest neighbor search the proposed algorithm leads to a significant speed up of the whole merging process in this paper we present the theories and algorithms of our approaches with pseudo code and apply them to several real objects including large scale cultural assets

a fast and robust local descriptor for 3d point cloud registration this paper proposes a novel local feature descriptor called a local feature statistics histogram lfsh for efficient 3d point cloud registration an lfsh forms a comprehensive description of local shape geometries by encoding their statistical properties on local depth point density and angles between normals the sub features in the lfsh descriptor are low dimensional and quite efficient to compute in addition an optimized sample consensus osac algorithm is developed to iteratively estimate the optimum transformation from point correspondences osac can handle the challenging cases of matching highly self similar models based on the proposed lfsh and osac a coarse to fine algorithm can be formed for 3d point cloud registration experiments and comparisons with the state of the art descriptors demonstrate that lfsh is highly discriminative robust and significantly faster than other descriptors meanwhile the proposed coarse to fine registration algorithm is demonstrated to be robust to common nuisances including noise and varying point cloud resolutions and can achieve high accuracy on both model data and scene data

localizing overlapping parts by searching the interpretation tree this paper discusses how local measurements of positions and surface normals may be used to identify and locate overlapping objects the objects are modeled as polyhedra or polygons having up to six degrees of positional freedom relative to the sensors the approach operates by examining all hypotheses about pairings between sensed data and object surfaces and efficiently discarding inconsistent ones by using local constraints on distances between faces angles between face normals and angles relative to the surface normals of vectors between sensed points the method described here is an extension of a method for recognition and localization of nonoverlapping parts previously described in 18 and 15

spectral properties of one dimensional quasi crystals in this paper we prove that the one dimensional schrodinger operator onl 2 with potential given by upsilon n lambda chi 1 alpha 1 x n alpha alpha notin mathbb q has a cantor spectrum of zero lebesgue measure for any irrational and any 0 we can thus extend the kotani result on the absence of absolutely continuous spectrum for this model to all mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepge9fr xfr x frpewzqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiaadihaiiaacq wfiiiztuudjxwak1uy0hmmaehbfv3yslgzg0uy0hgiud3bagqbbiab nj8ubaa 4628 x in mathbb t

escaping orbits in trace maps abstract we study the finitely generated group a of invertible polynomial mappings from c 3 to itself or r 3 to itself which preserve the fricke vogt invariant i x y z x2 y2 z2 2xyz 1 using properties of suitably chosen generators we give a necessary condition and sufficient conditions for infinite order elements of a to have an unbounded orbit escaping to infinity in forward or backward time our main motivation for this study is that a includes the so called trace maps derived from transfer matrix approaches to various physical processes displaying non periodicity in space or time as shown previously characterising escaping orbits leads to various conclusions for the physical model and vice versa e g electronic properties of id quasicrystals our results generalize in a simple constructive way those previously proved for fibonacci type trace maps

recognition of occluded polyhedra from range images abstract occlusion remains a major hindrance for automatic recognition of 3 d objects in this paper we address the occlusion problem in the context of polyhedral object recognition from range data a novel approach is presented for object recognition based on sound occlusion guided reasoning for feature distortion analysis and perceptual organization this type of reasoning enables us to maximize the amount of information extracted from the scene data thus leading to robust and efficient recognition the proposed approach is based on a multi stage matching process which attempts to recognize scene objects according to their order in the occlusion hierarchy i e an object is recognized before those that are occluded by it such a strategy helps in resolving some occlusion induced ambiguities in feature distortion analysis furthermore it leads to verification of object pose hypotheses with greater confidence matching is based on a hypothesize cluster and verify approach hypotheses are generated using an occlusion tolerant composite feature a fork which is a pair of non parallel edges that belong to the same surface generated hypotheses are then clustered and verified using a robust pixel based technique indexing is performed using distortion adaptive bounds on a rich set of viewpoint invariant fork attributes for high selectivity even in the presence of heavy occlusion performance of the system is demonstrated using complex multi object scenes

autonomous construction of three dimensional models from range data a substantial body of literature exists in the field of model based vision and object recognition a model based vision system assumes that some kind of model exists for each object the system is to recognize unfortunately in the past each new recognition scheme has frequently required its own specialized models this in turn has required that a time consuming and often manual model construction process be completed first development of an automated method for constructing reasonably detailed general purpose models would both simplify the model construction process and provide models that could be used by many different recognition schemes this dissertation presents a method for autonomously constructing a complete general purpose surface description model of an object the models are constructed based on a large number of noisy range images from an essentially arbitrary set of viewpoints around the object the resulting models consist of a set of parametric surface patches results of testing the model building algorithm on six synthetic test objects are also presented in addition an object recognition scheme is presented which uses models constructed by the model builder the object recognition results clearly demonstrate that the models constructed by the system are of sufficient accuracy and detail for use in model based vision tasks

a proof of the dynamical version of the bombieri taylor conjecture an adaptation of the bombieri taylor conjecture on diffraction has been applied to some time dependent quantum systems here the problem is mathematically stated and a proof is presented simple examples give an idea as to what extent the conjecture is valid

correlation dimension and spectrum for the almost periodic bouncing ball abstract we analyze the dependence of the autocorrelation functions and correlation dimensions on the degree of almost periodicity of the classical bouncing ball model

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

spectral properties of one dimensional schrodinger operators with potentials generated by substitutions we investigate one dimensional discrete schrodinger operators whose potentials are invariant under a substitution rule the spectral properties of these operators can be obtained from the analysis of a dynamical system called the trace map we give a careful derivation of these maps in the general case and exhibit some specific properties under an additional easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying schrodinger operator is singular and supported on a set of zero lebesgue measure a condition allowing to exclude point spectrum is also given the application of our theorems is explained on a series of examples

unbounded trace orbits of thue morse hamiltonian it is well known that an energy is in the spectrum of fibonacci hamiltonian if and only if the corresponding trace orbit is bounded however it is not known whether the same result holds for the thue morse hamiltonian in this paper we give a negative answer to this question more precisely we construct two subsets sigma ii and sigma iii of the spectrum of the thue morse hamiltonian both of which are dense and uncountable such that each energy in sigma ii cup sigma iii corresponds to an unbounded trace orbit exact estimates on the norm of the transfer matrices are also obtained for these energies for e in sigma ii cup sigma iii the norms of the transfer matrices behave like begin aligned e c 1 gamma sqrt n le vert t n e vert le e c 2 gamma sqrt n end aligned however two types of energies are quite different in the sense that each energy in sigma ii is associated with a two sided pseudo localized state while each energy in sigma iii is associated with a one sided pseudo localized state the difference is also reflected by the local dimensions of the spectral measure the local dimension is 0 for energies in sigma ii and is larger than 1 for energies in sigma iii as a comparison we mention another known countable dense subset sigma i each energy in sigma i corresponds to an eventually constant trace map and the associated eigenvector is an extended state in summary the thue morse hamiltonian exhibits mixed spectral nature

a spherical representation for recognition of free form surfaces introduces a new surface representation for recognizing curved objects the authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object the mesh is computed from the data by deforming a standard shaped mesh for example an ellipsoid until it fits the surface of the object the authors define local regularity constraints that the mesh must satisfy the authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh a surface curvature index that is pose invariant is stored at every node of the mesh the authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene the authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation the authors present results on real range images which show that this approach to modelling and recognizing 3d objects has three main advantages 1 it is applicable to complex curved surfaces that cannot be handled by conventional techniques 2 it reduces the recognition problem to the computation of similarity between spherical distributions in particular the recognition algorithm does not require any combinatorial search and 3 even though it is based on a spherical mapping the approach can handle occlusions and partial views

structural matching in computer vision using probabilistic relaxation in this paper we develop the theory of probabilistic relaxation for matching features extracted from 2d images derive as limiting cases the various heuristic formulae used by researchers in matching problems and state the conditions under which they apply we successfully apply our theory to the problem of matching and recognizing aerial road network images based on road network models and to the problem of edge matching in a stereo pair for this purpose each line network is represented by an attributed relational graph where each node is a straight line segment characterized by certain attributes and related with every other node via a set of binary relations

schrodinger operators with rudin shapiro potentials are not palindromic we prove a conjecture of a hof o knill and b simon commun math phys 174 149 159 1995 by showing that the rudin shapiro sequence is not palindromic i e does not contain arbitrarily long palindromes we prove actually this property for all paperfolding sequences and all rudin shapiro sequences deduced from paperfolding sequences as a consequence and as guessed by the above authors their method cannot be used for establishing that discrete schrodinger operators with rudin shapiro potentials have a purely singular continuous spectrum

spectral properties of a tight binding hamiltonian with period doubling potential we study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence we prove that its spectrum is purely singular continuous and supported on a cantor set of zero lebesgue measure for all nonzero values of the potential strength moreover we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength

structural indexing efficient 3 d object recognition the authors present an approach for the recognition of multiple 3 d object models from three 3 d scene data the approach uses two different types of primitives for matching small surface patches where differential properties can be reliably computed and lines corresponding to depth or orientation discontinuities these are represented by splashes and 3 d curves respectively it is shown how both of these primitives can be encoded by a set of super segments consisting of connected linear segments these super segments are entered into a table and provide the essential mechanism for fast retrieval and matching the issues of robustness and stability of the features are addressed in detail the acquisition of the 3 d models is performed automatically by computing splashes in highly structured areas of the objects and by using boundary and surface edges for the generation of 3 d curves the authors present results with the current system 3 d object recognition based on super segments and discuss further extensions

3d geometric scale variability in range images features and descriptors despite their ubiquitous presence little has been investigated about the scale variability the relative variations in the spatial extents of local structures of 3d geometric data in this paper we present a comprehensive framework for exploiting this 3d geometric scale variability in range images that provides rich information for characterizing the overall geometry we derive a sound scale space representation which we refer to as the geometric scale space that faithfully encodes the scale variability of the surface geometry and derive novel detectors to extract prominent features and identify their natural scales the result is a hierarchical set of features of different scales which we refer to as scale dependent geometric features we then derive novel local shape descriptors that represent the surface structures that give rise to those features by carving out and encoding the local surface that fall within the support regions of the features this leads to scale dependent or scale invariant local shape descriptors that convey significant discriminative information of the object geometry we demonstrate the effectiveness of geometric scale analysis on range images and show that it enables novel applications in particular fully automatic registration of multiple objects from a mixed set of range images and 3d object recognition in highly cluttered range image scenes

cosmos a representation scheme for 3d free form objects we address the problem of representing and recognizing 3d free form objects when 1 the object viewpoint is arbitrary 2 the objects may vary in shape and complexity and 3 no restrictive assumptions are made about the types of surfaces on the object we assume that a range image of a scene is available containing a view of a rigid 3d object without occlusion we propose a new and general surface representation scheme for recognizing objects with free form sculpted surfaces in this scheme an object is described concisely in terms of maximal surface patches of constant shape index the maximal patches that represent the object are mapped onto the unit sphere via their orientations and aggregated via shape spectral functions properties such as surface area curvedness and connectivity which are required to capture local and global information are also built into the representation the scheme yields a meaningful and rich description useful for object recognition a novel concept the shape spectrum of an object is also introduced within the framework of cosmos for object view grouping and matching we demonstrate the generality and the effectiveness of our scheme using real range images of complex objects

on a new hierarchy of hamiltonian soliton equations a method is suggested for studying the hamiltonian structure of the nonlinear partial differential equations that can be solved by the use of the spectral transform soliton equations the method is applied to a new hierarchy of n 1 coupled partial differential equations related to a schrodinger like spectral problem it is shown that these soliton equations are integrable hamiltonian equations with commuting flows for n 1 and n 2 a miura like transformation is computed and the corresponding modified equations are explicitly given

power law bounds on transfer matrices and quantum dynamics in one dimension ii abstract we establish quantum dynamical lower bounds for a number of discrete one dimensional schrodinger operators these dynamical bounds are derived from power law upper bounds on the norms of transfer matrices we develop further the approach from part i and study many examples particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power law bounds on transfer matrices the models discussed in this paper include substitution models sturmian models a hierarchical model the prime model and a class of moderately sparse potentials

a condition of boshernitzan and uniform convergence in the multiplicative ergodic theorem this paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts if such a subshift satisfies a certain condition originally introduced by boshernitzan every locally constant sl 2 r valued cocycle is uniform as a consequence the corresponding schrodinger operators exhibit cantor spectrum of lebesgue measure zero an investigation of boshernitzan s condition then shows that these results cover all earlier results of this type and moreover provide various new ones in particular boshernitzan s condition is shown to hold for almost all circle maps and almost all arnoux rauzy subshifts

sine gordon and modified korteweg de vries charges the sine gordon and the integrated modified korteweg de vries equations are shown to conserve the same infinite set of charges the charges are determined by a recursion relation as a consequence the solutions of all the equations generated by the charges have in common all time independent properties

singular continuous electron spectrum for a class of circle sequences we derive substitution rules for a class of binary quasiperiodic sequences generated by circle maps whose rotation numbers are obtained from the precious means the nature of the electron spectra for the corresponding diatomic chains is studied in the nearest neighbour on site tight binding approximation using the transfer matrix technique by studying properties of the trace maps we find that the spectrum is purely singular continuous in most of the studied cases

three dimensional model based object recognition and segmentation in cluttered scenes viewpoint independent recognition of free form objects and their segmentation in the presence of clutter and occlusions is a challenging task we present a novel 3d model based algorithm which performs this task automatically and efficiently a 3d model of an object is automatically constructed offline from its multiple unordered range images views these views are converted into multidimensional table representations which we refer to as tensors correspondences are automatically established between these views by simultaneously matching the tensors of a view with those of the remaining views using a hash table based voting scheme this results in a graph of relative transformations used to register the views before they are integrated into a seamless 3d model these models and their tensor representations constitute the model library during online recognition a tensor from the scene is simultaneously matched with those in the library by casting votes similarity measures are calculated for the model tensors which receive the most votes the model with the highest similarity is transformed to the scene and if it aligns accurately with an object in the scene that object is declared as recognized and is segmented this process is repeated until the scene is completely segmented experiments were performed on real and synthetic data comprised of 55 models and 610 scenes and an overall recognition rate of 95 percent was achieved comparison with the spin images revealed that our algorithm is superior in terms of recognition rate and efficiency

how easy is matching 2d line models using local search local search is a well established and highly effective method for solving complex combinatorial optimization problems here local search is adapted to solve difficult geometric matching problems matching is posed as the problem of finding the optimal many to many correspondence mapping between a line segment model and image line segments image data is assumed to be fragmented noisy and cluttered the algorithms presented have been used for robot navigation photo interpretation and scene understanding this paper explores how local search performs as model complexity increases image clutter increases and additional model instances are added to the image data expected run times to find optimal matches with 95 percent confidence are determined for 48 distinct problems involving six models nonlinear regression is used to estimate run time growth as a function of problem size both polynomial and exponential growth models are fit to the run time data for problems with random clutter the polynomial model fits better and growth is comparable to that for tree search for problems involving symmetric models and multiple model instances where tree search is exponential the polynomial growth model is superior to the exponential growth model for one search algorithm and comparable for another

structure factor of 1d systems superlattices based on two letter substitution rules i delta bragg peaks the recent generalization to the case of arbitrary tile lengths and arbitrary scattering factors of the calculation of the structure factor of 1d substitutional systems is studied in detail this method makes a easy to find all the peaks in the diffraction spectrum of a system the well known periodic and quasiperiodic spectra with delta peaks at integer multiples of a single number and integer linear combinations of two incommensurate frequencies respectively were found to be the l 0 subsets of two more general types of spectra infinite periodic or limit periodic and infinite quasiperiodic or limit quasiperiodic characterized by rational numbers of the type m nl l 0 infinity in place of the above integers substitution rules that produce quasicrystalline quasiperiodic and infinite quasiperiodic spectra give the same type of spectrum for all values of the ratio rho rho a rho b of the two tile lengths rho a and rho b this is not the case for the other rules thus the same substitution rule such as the copper mean rule can give an infinite periodic spectrum for a single rational ratio rho rho a rho b of the two tile lengths rho a and rho b a periodic like spectrum for other rational rho and a spectrum in some aspects similar to that of a random system when rho is an irrational number on the other hand a thue morse system diffracts as a periodic crystal when p not 1 but has no non trivial delta peaks when rho 1 other thue morse like systems can have infinite periodic spectra for all rho

singular continuous spectrum for the period doubling hamiltonian on a set of full measure we consider the discrete one dimensional schrodinger operator with potential generated by the period doubling substitution we show that for almost every element in the hull with respect to the unique ergodic measure on there are no eigenvalues combining this with a result proven by kotani we establish purely singular continuous spectrum on a set of full measure

gradient theorem for completely integrable hamiltonian systems for evolution equations which can be written in hamiltonian form two ways there exists a relation between two functions q 1 and q 2 both of which are gradients of conserved functionals the relation can be extended to define recursively functions q n it is shown that the q n corresponding to the general evolution equation associated with the zakharov shabat eigenvalue problem are all gradients of conserved functionals this in turn implies all these functionals are in involution