multimodality image registration by maximization of mutual information a new approach to the problem of multimodality medical image registration is proposed using a basic concept from information theory mutual information mi or relative entropy as a new matching criterion the method presented in this paper applies mi to measure the statistical dependence or information redundancy between the image intensities of corresponding voxels in both images which is assumed to be maximal if the images are geometrically aligned maximization of mi is a very general and powerful criterion because no assumptions are made regarding the nature of this dependence and no limiting constraints are imposed on the image content of the modalities involved the accuracy of the mi criterion is validated for rigid body registration of computed tomography ct magnetic resonance mr and photon emission tomography pet images by comparison with the stereotactic registration solution while robustness is evaluated with respect to implementation issues such as interpolation and optimization and image content including partial overlap and image degradation our results demonstrate that subvoxel accuracy with respect to the stereotactic reference solution can be achieved completely automatically and without any prior segmentation feature extraction or other preprocessing steps which makes this method very well suited for clinical applications

nonlinear wave interactions in a complex hamiltonian formalism a hamiltonian formalism is used to study the set of equations governing the changes in the amplitudes of some coherent waves due to their nonlinear interactions the main point of the present description is that it uses these complex wave amplitudes as canonical variables for a real hamiltonian thus greatly simplifying the algebra three examples are then dealt with first the hamiltonian for the four wave interactions is derived in accordance with previous studies such systems are then known to be integrable as a second example one looks at a very special case of three interacting waves where a fundamental interacts simultaneously with its second and third harmonic even though this system can be hamiltonian it seems to be nonintegrable finally attention is focused on the elliptically polarised third harmonic generation both in anisotropic and in isotropic media this set of equations can be hamiltonian under certain restrictions on the coupling coefficients but it also appears to be nonintegrable because not enough independent invariants are found

on tracial approximation let c be a class of unital c algebras the class tac of c algebras which can be tracially approximated in the egorov like sense first considered by lin by the c algebras in c is studied lin considered the case that c consists of finite dimensional c algebras or the tensor products of such with c 0 1 in particular the question is considered whether for any simple separable a tac there is a c algebra b which is a simple inductive limit of certain basic homogeneous c algebras together with c algebras in c such that the elliott invariant of a is isomorphic to the elliott invariant of b an interesting case of this question is answered in the final part of the paper the question is also considered which properties of c algebras are inherited by tracial approximation results of this kind are obtained which are used in the proof of the main theorem of the paper and also in the proof of the classification theorem of the second author given in z niu a classification of tracially approximately splitting tree algebra in preparation and z niu a classification of certain tracially approximately subhomogeneous c algebras phd thesis university of toronto 2005 which also uses the main result of the present paper

third order equation for harmonic generation complex canonical transformation and jwkb solution the quantum description of third harmonic generation can be formulated as an eigenvalue problem for a third order linear differential equation we perform a semiclassical study of this third order equation generalizing the familiar jwkb theory for the second order schrodinger equation and deriving explicit albeit approximate formulas for the eigenvalues within this semiclassical context a central role in this analysis is played by a nonlinear complex canonical transformation which permits a complete description of the classical motion generated by a complex polynomial hamiltonian function in the complexified position and momentum planes

simple c algebras with perforation abstract simple c algebras are constructed for which the ordered k 0 group is not weakly unperforated the algebras are approximately homogeneous and do not have slow dimension growth

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

drift estimation in sparse sequential dynamic imaging with application to nanoscale fluorescence microscopy summary a major challenge in many modern superresolution fluorescence microscopy techniques at the nanoscale lies in the correct alignment of long sequences of sparse but spatially and temporally highly resolved images this is caused by the temporal drift of the protein structure e g due to temporal thermal inhomogeneity of the object of interest or its supporting area during the observation process we develop a simple semiparametric model for drift correction in single marker switching microscopy then we propose an m estimator for the drift and show its asymptotic normality this is used to correct the final image and it is shown that this purely statistical method is competitive with state of the art calibration techniques which require the incorporation of fiducial markers in the specimen moreover a simple bootstrap algorithm allows us to quantify the precision of the drift estimate and its effect on the final image estimation we argue that purely statistical drift correction is even more robust than fiducial tracking rendering the latter superfluous in many applications the practicability of our method is demonstrated by a simulation study and by a single marker switching application this serves as a prototype for many other typical imaging techniques where sparse observations with high temporal resolution are blurred by motion of the object to be reconstructed

a new pt symmetric complex hamiltonian with a real spectrum we construct an isospectrum system in terms of a real and a complex potential to show that the underlying pt symmetric complex hamiltonian possesses a real spectrum which is shared by its real partner

a low dimensional fluid motion estimator in this paper we propose a new motion estimator for image sequences depicting fluid flows the proposed estimator is based on the helmholtz decomposition of vector fields this decomposition consists in representing the velocity field as a sum of a divergence free component and a vorticity free component the objective is to provide a low dimensional parametric representation of optical flows by depicting them as deformations generated by a reduced number of vortex and source particles both components are approximated using a discretization of the vorticity and divergence maps through regularized dirac measures the resulting so called irrotational and solenoidal fields consist of linear combinations of basis functions obtained through a convolution product of the green kernel gradient and the vorticity map or the divergence map respectively the coefficient values and the basis function parameters are obtained by minimization of a functional relying on an integrated version of mass conservation principle of fluid mechanics results are provided on synthetic examples and real world sequences

computation of 3 d velocity fields from 3 d cine ct images of a human heart a method of computing the three dimensional 3 d velocity field from 3 d cine computer tomographs cts of a beating heart is proposed using continuum theory the authors develop two constraints on the 3 d velocity field generated by a beating heart with these constraints the computation of the 3 d velocity field is formulated as an optimization problem and a solution to the optimization problem is developed using the euler lagrange method the solution is then discretized for computer implementation the authors present the results for both simulated images and clinical cine ct images of a beating heart

a stochastic filtering technique for fluid flow velocity fields tracking in this paper we present a method for the temporal tracking of fluid flow velocity fields the technique we propose is formalized within a sequential bayesian filtering framework the filtering model combines an ito diffusion process coming from a stochastic formulation of the vorticity velocity form of the navier stokes equation and discrete measurements extracted from the image sequence in order to handle a state space of reasonable dimension the motion field is represented as a combination of adapted basis functions derived from a discretization of the vorticity map of the fluid flow velocity field the resulting nonlinear filtering problem is solved with the particle filter algorithm in continuous time an adaptive dimensional reduction method is applied to the filtering technique relying on dynamical systems theory the efficiency of the tracking method is demonstrated on synthetic and real world sequences

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

reliable estimation of dense optical flow fields with large displacements in this paper we show that a classic optical flow technique by nagel and enkelmann 1986 ieee trans pattern anal mach intell vol 8 pp 565 593 can be regarded as an early anisotropic diffusion method with a diffusion tensor we introduce three improvements into the model formulation that i avoid inconsistencies caused by centering the brightness term and the smoothness term in different images ii use a linear scale space focusing strategy from coarse to fine scales for avoiding convergence to physically irrelevant local minima and iii create an energy functional that is invariant under linear brightness changes applying a gradient descent method to the resulting energy functional leads to a system of diffusion reaction equations we prove that this system has a unique solution under realistic assumptions on the initial data and we present an efficient linear implicit numerical scheme in detail our method creates flow fields with 100 density over the entire image domain it is robust under a large range of parameter variations and it can recover displacement fields that are far beyond the typical one pixel limits which are characteristic for many differential methods for determining optical flow we show that it performs better than the optical flow methods with 100 density that are evaluated by barron et al 1994 int j comput vision vol 12 pp 43 47 our software is available from the internet

global convergence properties of conjugate gradient methods for optimization this paper explores the convergence of nonlinear conjugate gradient methods without restarts and with practical line searches the analysis covers two classes of methods that are globally convergent on smooth nonconvex functions some properties of the fletcher reeves method play an important role in the first family whereas the second family shares an important property with the polak ribiere method numerical experiments are presented

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

closed form solutions of the schrodinger equation for a coupled harmonic potential in three dimensions using the ansatz method we obtain the exact closed form solutions of a time independent schrodinger equation for a coupled harmonic potential and its variant in three dimensions some authors r s kaushal quantum mechanics of noncentral harmonic and anharmonic potentials in two dimensions ann phys n y 206 1991 90 105 have raised the difficulty to solve the schrodinger equation for coupled potentials but we found that by imposing some restrictions on ansatz parameters and at the cost of certain constraints on the potential parameters ground state as well as excited state solutions of the schrodinger equation are obtained the number of constraints increases with increase in anharmonicity of the potential

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

dense estimation and object based segmentation of the optical flow with robust techniques we address the issue of recovering and segmenting the apparent velocity field in sequences of images as for motion estimation we minimize an objective function involving two robust terms the first one cautiously captures the optical flow constraint while the second a priori term incorporates a discontinuity preserving smoothness constraint to cope with the nonconvex minimization problem thus defined we design an efficient deterministic multigrid procedure it converges fast toward estimates of good quality while revealing the large discontinuity structures of flow fields we then propose an extension of the model by attaching to it a flexible object based segmentation device based on deformable closed curves different families of curve equipped with different kinds of prior can be easily supported experimental results on synthetic and natural sequences are presented including an analysis of sensitivity to parameter tuning

dense estimation of fluid flows in this paper we address the problem of estimating and analyzing the motion of fluids in image sequences due to the great deal of spatial and temporal distortions that intensity patterns exhibit in images of fluids the standard techniques from computer vision originally designed for quasi rigid motions with stable salient features are not well adapted in this context we thus investigate a dedicated minimization based motion estimator the cost function to be minimized includes a novel data term relying on an integrated version of the continuity equation of fluid mechanics which is compatible with large displacements this term is associated with an original second order div curl regularization which prevents the washing out of the salient vorticity and divergence structures the performance of the resulting fluid flow estimator is demonstrated on meteorological satellite images in addition we show how the sequences of dense motion fields we estimate can be reliably used to reconstruct trajectories and to extract the regions of high vorticity and divergence

on the structure of simple c algebras tensored with a uhf algebra ii let d be a simple unital c algebra let b be a uhf algebra and put a b d it is proved that if p q a are projections and p q for all quasitraces on a then p q in the sense of murray and von neumann a more general result involving positive operators in a is also proved if d has finitely many extremal quasi traces and the projections in d k separate these then it is proved that a has real rank zero finally it is proved that provided d is stably finite then each positive state on k0 d lifts to a quasi trace

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

quantum mechanics of complex hamiltonian systems in one dimension with a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one dimensional schrodinger equation corresponding to a non hermitian hamiltonian h x p we investigate the ground state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x x1 ip2 p p1 ix2 where x1 p1 and x2 p2 are real and considered as canonical pairs the analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems it is noted that the imaginary part of the eigenvalue ei turns out to be zero for all potentials v x with real couplings whereas it turns out to be nonzero for the case when the couplings are complex the prescription is also extended to study the excited states the problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed

robust processing of optical flow of fluids this paper proposes a new approach coupling physical models and image estimation techniques for modelling the movement of fluids the fluid flow is characterized by turbulent movement and dynamically changing patterns which poses challenges to existing optical flow estimation methods the proposed methodology which relies on navier stokes equations is used for processing fluid optical flow by using a succession of stages such as advection diffusion and mass conservation a robust diffusion step jointly considering the local data geometry and its statistics is embedded in the proposed framework the diffusion kernel is gaussian with the covariance matrix defined by the local second derivatives such an anisotropic kernel is able to implicitly detect changes in the vector field orientation and to diffuse accordingly a new approach is developed for detecting fluid flow structures such as vortices the proposed methodology is applied on artificially generated vector fields as well as on various image sequences

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

an investigation of smoothness constraints for the estimation of displacement vector fields from image sequences a mapping between one frame from an image sequence and the preceding or following frame can be represented as a displacement vector field in most situations the mere gray value variations do not provide sufficient information in order to estimate such a displacement vector field supplementary constraints are necessary for example the postulate that a displacement vector field varies smoothly as a function of the image position taken as a general requirement this creates difficulties at gray value transitions which correspond to occluding contours nagel therefore introduced the oriented smoothness requirement which restricts variations of the displacement vector field only in directions with small or no variation of gray values this contribution reports results of an investigation about how such an oriented smoothness constraint may be formulated and evaluated

on the stable rank of simple c algebras in 16 m a rieffel introduced the notion of stable rank for c algebras for a a unital c algebra the stable rank denoted by sr a is the least integer n such that the set of n tuples over a which generate a as a left ideal is dense in an if no such n exists we set sr a oo for unital c algebras the stable rank was shown to coincide with the bass stable rank which is defined for rings see 12 the stable rank can also be defined for nonunital c algebras but we shall only consider the unital case stable rank is related to nonstable k theory in 16 theorem 10 12 and 3 corollary 7 14 it was shown that if a is a simple unital c algebra of stable rank one then the natural map u a u a o 4 ki a is an isomorphism see also 17 2 and 18 it is a natural question which values occur as the stable rank of simple c algebras suppose that a is simple and infinite then by 7 proposition 1 5 a contains two isometries with orthogonal ranges and so by 16 proposition 6 5 sr a oo in the case of finite simple c algebras the following is known whenever a is simple and stably finite and b is a uhf algebra the tensor product a0 b has stable rank one see 19 this is also the case for every simple approximately homogeneous ah algebra with slow dimension growth see 4 and 8 so in short sr oo for every infinite simple c algebra and sr 1 for various classes of finite simple c algebras in this paper we answer the question above by showing that every n c n u oo is the stable rank of some simple finite c algebra more precisely it is shown that for every n 2 3 4 oo there exists a simple separable and unital ah algebra a of stable rank n in particular the group of invertible elements in a is not dense and like the examples in 20 a does not have slow dimension growth and is not approximately divisible see 5 i thank marius dadarlat for suggesting that i study ah algebras with connecting homomorphisms as described in the following section and juliana erlijman for drawing my attention to some of the powerful tools of differential topology and i thank george elliott for my stay at the fields institute where this paper was written

the solution of the schrodinger equation for complex hamiltonian systems in two dimensions we investigate the ground state solutions of the schrodinger equation for complex non hermitian hamiltonian systems in two dimensions within the framework of an extended complex phase space approach the eigenvalues and eigenfunctions of some two dimensional complex potentials are found

bayesian inference of models and hyperparameters for robust optical flow estimation selecting optimal models and hyperparameters is crucial for accurate optical flow estimation this paper provides a solution to the problem in a generic bayesian framework the method is based on a conditional model linking the image intensity function the unknown velocity field hyperparameters and the prior and likelihood motion models inference is performed on each of the three levels of this so defined hierarchical model by maximization of marginalized a posteriori probability distribution functions in particular the first level is used to achieve motion estimation in a classical a posteriori scheme by marginalizing out the motion variable the second level enables to infer regularization coefficients and hyperparameters of non gaussian m estimators commonly used in robust statistics the last level of the hierarchy is used for selection of the likelihood and prior motion models conditioned to the image data the method is evaluated on image sequences of fluid flows and from the middlebury database experiments prove that applying the proposed inference strategy yields better results than manually tuning smoothing parameters or discontinuity preserving cost functions of the state of the art methods

harmonic oscillator well with a screened coulombic core is quasi exactly solvable in the quantization scheme which weakens the hermiticity of a hamiltonian to its mere t invariance the superposition v x x2 ze2 x of the harmonic and coulomb potentials is defined at the purely imaginary effective charges ze2 if and regularized by a purely imaginary shift of x this model is quasi exactly solvable we show that at each excited n 1 th harmonic oscillator energy e 2n 3 there exists not only the well known harmonic oscillator bound state at the vanishing charge f 0 but also a normalizable n 1 plet of the further elementary sturmian eigenstates n x at eigencharges f f n 0 n 0 1 n beyond the smallest multiplicities n we recommend perturbative methods for their construction

pseudo reality and pseudo adjointness of hamiltonians we define pseudo reality and pseudo adjointness of a hamiltonian h as h 1 h and h 1 h respectively we prove that the former yields the necessary condition for a spectrum to be real whereas the latter helps in fixing a definition for the inner product of the eigenstates here we separate out the adjointness of an operator from its hermitian adjointness it turns out that a hamiltonian possessing a real spectrum is first pseudo real further it could be hermitian pt symmetric or pseudo hermitian

non hermitian susy hydrogen like hamiltonians with real spectra it is shown that the radial part of the hydrogen hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant t he 1 susy approach is used to construct non hermitian operators with hydrogen spectra other non hermitian hamiltonians are shown to admit an extra complex energy at n ew self adjoint hydrogen like hamiltonians are also derived by using a 2 susy transformation with complex conjugate pairs

the cancellation theorem for projective modules over irrational rotation c algebras on montre que deux modules projectifs sur a qui sont isomorphes de facon stable sont en fait isomorphes

ilogdemons a demons based registration algorithm for tracking incompressible elastic biological tissues tracking soft tissues in medical images using non linear image registration algorithms requires methods that are fast and provide spatial transformations consistent with the biological characteristics of the tissues logdemons algorithm is a fast non linear registration method that computes diffeomorphic transformations parameterised by stationary velocity fields although computationally efficient its use for tissue tracking has been limited because of its ad hoc gaussian regularisation which hampers the implementation of more biologically motivated regularisations in this work we improve the logdemons by integrating elasticity and incompressibility for soft tissue tracking to that end a mathematical justification of demons gaussian regularisation is proposed building on this result we replace the gaussian smoothing by an efficient elastic like regulariser based on isotropic differential quadratic forms of vector fields the registration energy functional is finally minimised under the divergence free constraint to get incompressible deformations as the elastic regulariser and the constraint are linear the method remains computationally tractable and easy to implement tests on synthetic incompressible deformations showed that our approach outperforms the original logdemons in terms of elastic incompressible deformation recovery without reducing the image matching accuracy as an application we applied the proposed algorithm to estimate 3d myocardium strain on clinical cine mri of two adult patients results showed that incompressibility constraint improves the cardiac motion recovery when compared to the ground truth provided by 3d tagged mri

mean shift mode seeking and clustering mean shift a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper this generalization makes some k means like clustering algorithms its special cases it is shown that mean shift is a mode seeking process on the surface constructed with a shadow kernal for gaussian kernels mean shift is a gradient mapping convergence is studied for mean shift iterations cluster analysis if treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data applications in clustering and hough transform are demonstrated mean shift is also considered as an evolutionary strategy that performs multistart global optimization

the diffusion reaction d r hamiltonian and the solutions of certain types of linear and nonlinear d r equations in one dimension with a view to having further insight into the mathematical content of the non hermitian hamiltonian associated with the diffusion reaction d r equation in one dimension we investigate a the solitary wave solutions of certain types of its nonlinear versions and b the problem of real eigenvalue spectrum associated with its linear version or with this class of non hermitian hamiltonians for case a we use the standard techniques to handle the quadratic and cubic nonlinearities in the d r equation whereas for case b a newly proposed method based on an extended complex phase space is employed for a particular class of solutions an ermakov system of equations is also found for the linear case further corresponding to the classical version of the above one dimensional complex hamiltonian an equivalent integrable system of two two dimensional real hamiltonians is suggested

eigenspectra of a complex coupled harmonic potential in three dimensions within the framework of extended complex phase space approach characterized by position and momentum coordinates we investigate the quasi exact solutions of the schrodinger equation for a coupled harmonic potential and its variants in three dimensions for this purpose ansatz method is employed and nature of the eigenvalues and eigenfunctions is determined by the analyticity property of the eigenfunctions alone the energy eigenvalue is real for the real coupling parameters and becomes complex if the coupling parameters are complex however in case of complex coupling parameters the imaginary component of energy eigenvalue reduces to zero if the p t symmetric condition is satisfied thus a non hermitian hamiltonian possesses real eigenvalue if it is p t symmetric