on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

a representation for shape based on peaks and ridges in the difference of low pass transform this paper defines a multiple resolution representation for the two dimensional gray scale shapes in an image this representation is constructed by detecting peaks and ridges in the difference of lowpass dolp transform descriptions of shapes which are encoded in this representation may be matched efficiently despite changes in size orientation or position motivations for a multiple resolution representation are presented first followed by the definition of the dolp transform techniques are then presented for encoding a symbolic structural description of forms from the dolp transform this process involves detecting local peaks and ridges in each bandpass image and in the entire three dimensional space defined by the dolp transform linking adjacent peaks in different bandpass images gives a multiple resolution tree which describes shape peaks which are local maxima in this tree provide landmarks for aligning manipulating and matching shapes detecting and linking the ridges in each dolp bandpass image provides a graph which links peaks within a shape in a bandpass image and describes the positions of the boundaries of the shape at multiple resolutions detecting and linking the ridges in the dolp three space describes elongated forms and links the largest peaks in the tree the principles for determining the correspondence between symbols in pairs of such descriptions are then described such correspondence matching is shown to be simplified by using the correspondence at lower resolutions to constrain the possible correspondence at higher resolutions

sequential convolution techniques for image filtering sequentially convolving images with small size operators is a promising idea for performing image filtering various classes of two dimensional finite impulse response fir filters that can be implemented with such techniques are studied design procedures are presented for each of the classes they are based upon minimizing l2and l criteria results are assessed in an image processing context

super brownian motion as the unique strong solution to an spde a stochastic partial differential equation spde is derived for super brownian motion regarded as a distribution function valued process the strong uniqueness for the solution to this spde is obtained by an extended yamada watanabe argument similar results are also proved for the fleming viot process

pyramid segmentation algorithms revisited the main goal of this work is to compare pyramidal structures proposed to solve segmentation tasks segmentation algorithms based on regular and irregular pyramids are described together with the data structures and decimation procedures which encode and manage the information in the pyramid in order to compare the different segmentation algorithms we have employed three types of quality measurements the shift variance measure the f function and the q function

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

comparison methods for a class of function valued stochastic partial differential equations a comparison theorem is derived for a class of function valued stochastic partial differential equations spde s with lipschitz coefficients driven by cylindrical and regular hilbert space valued brownian motions moreover we obtain necessary and sufficient conditions for the positivity of the mild solutions of the spde s where the sufficiency follows from the comparison theorem thereby it is e g possible to identify a class of spde s which can serve as stochastic space time models for the density of particles as a consequence we can construct unique mild solutions of spde s on the cone of positive functions with non lipschitz drift parts including the case of arbitrary polynomialsr x withr o o and leading negative coefficient

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

stability in semilinear problems abstract in the paper we derive results concerning a continuous dependence of solutions on the right hand side for a semilinear operator equation lu g u by assuming that l d l h h h a hilbert space is self adjont with a closed range and g h r is continuous convex on h and g teaux differentiable on d l using these results we obtain theorems on the continuous dependence of solutions on functional parameters for a semilinear problem of the second order u au d u f t u t 0 a e with the dirichlet boundary conditions u 0 u 0 where a 1 and is a functional parameter

a cyclically catalytic super brownian motion in generalization of the mutually catalytic super brownian motion in r of dawson and perkins and mytnik a function valued cyclically catalytic model x is constructed as a strong markov solution to a martingale problem starting with a finite population x 0 each pair of neighboring types will globally segregate in the long term limit noncoexistence of neighboring types also finer extinction survival properties depending on x 0 are studied in the spirit of mueller and perkins in fact x 0 can be chosen in such a way that all types survive for all finite times on the other hand sufficient conditions on x 0 are stated for the following situation given a type k and a positive time t the kth subpopulation x k dies by time t with a large probability provided that its initial value x k 0 was sufficiently small

efficient synthesis of gaussian filters by cascaded uniform filters gaussian filtering is an important tool in image processing and computer vision in this paper we discuss the background of gaussian filtering and look at some methods for implementing it consideration of the central limit theorem suggests using a cascade of simple filters as a means of computing gaussian filters among simple filters uniform coefficient finite impulse response digital filters are especially economical to implement the idea of cascaded uniform filters has been around for a while 13 16 we show that this method is economical to implement has good filtering characteristics and is appropriate for hardware implementation we point out an equivalence to one of burt s methods 1 3 under certain circumstances as an extension we describe an approach to implementing a gaussian pyramid which requires approximately two addition operations per pixel per level per dimension we examine tradeoffs in choosing an algorithm for gaussian filtering and finally discuss an implementation

elliptic equations and systems with nonstandard growth conditions existence uniqueness and localization properties of solutions abstract we study the dirichlet problem for the elliptic equations i d i a i x u d i u p i x 2 d i u c x u u x 2 u f x in a bounded domain r n and the class of elliptic systems j d j a i j x u f i x u i 1 n u u 1 u n satisfying the growth condition x s v r n 2 i j a i j x v v i j a 0 i j v i j p i j x a 0 const 0 the exponents p i j x p i x x are known functions these equations are usually referred to as elliptic equations with nonstandard growth conditions we prove first the theorems of existence of bounded weak solutions and establish sufficient conditions of uniqueness of a weak solution our main purpose is the study of the localization properties of weak solutions we show that the weak solution may identically vanish on a set of nonzero measure in a dead core and derive estimates on the size and location of these dead cores in terms of the problem data the study of the localization properties is performed via the method of local energy estimates

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

dynamic occlusion analysis in optical flow fields optical flow can be used to locate dynamic occlusion boundaries in an image sequence we derive an edge detection algorithm sensitive to changes in flow fields likely to be associated with occlusion the algorithm is patterned after the marr hildreth zero crossing detectors currently used to locate boundaries in scalar fields zero crossing detectors are extended to identify changes in direction and or magnitude in a vector valued flow field as a result the detector works for flow boundaries generated due to the relative motion of two overlapping surfaces as well as the simpler case of motion parallax due to a sensor moving through an otherwise stationary environment we then show how the approach can be extended to identify which side of a dynamic occlusion boundary corresponds to the occluding surface the fundamental principal involved is that at an occlusion boundary the image of the surface boundary moves with the image of the occluding surface such information is important in interpreting dynamic scenes results are demonstrated on optical flow fields automatically computed from real image sequences

fast convolution with laplacian of gaussian masks we present a technique for computing the convolution of an image with log laplacian of gaussian masks it is well known that a log of variance a can be decomposed as a gaussian mask and a log of variance 1 we take advantage of the specific spectral characteristics of these filters in our computation the log is a bandpass filter we can therefore fold the spectrum of the image after low pass filtering without loss of information which is equivalent to reducing the resolution we present a complete evaluation of the parameters involved together with a complexity analysis that leads to the paradoxical result that the computation time decreases when increases we illustrate the method on two images

a degenerate stochastic partial differential equation for superprocesses with singular interaction the scaling limit for a class of interacting superprocesses and the associated singular degenerate stochastic partial differential equation sdspde are investigated it is proved that the scaling limit is a coalescing purely atomic measure valued process which is the unique strong solution of a reconstructed associated sdspde

a stochastic log laplace equation we study a nonlinear stochastic partial differential equation whose solution is the conditional log laplace functional of a superprocess in a random environment we establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by kurtz and xiong stochastic process appl 83 1999 103 126 we also derive the wong zakai type approximation for this equation as an application we give a direct proof of the moment formulas of skoulakis and adler ann appl probab 11 2001 488 543

a multiresolution image segmentation technique based on pyramidal segmentation and fuzzy clustering in this paper an unsupervised image segmentation technique is presented which combines pyramidal image segmentation with the fuzzy c means clustering algorithm each layer of the pyramid is split into a number of regions by a root labeling technique and then fuzzy c means is used to merge the regions of the layer with the highest image resolution a cluster validity functional is used to find the optimal number of objects automatically segmentation of a number of synthetic as well as clinical images is illustrated and two fully automatic segmentation approaches are evaluated which determine the left ventricular volume lv in 140 cardiovascular magnetic resonance mr images first fuzzy c means is applied without pyramids in the second approach the regions generated by pyramidal segmentation are merged by fuzzy c means the correlation coefficients of manually and automatically defined lv lumen of all 140 and 20 end diastolic images were equal to 0 86 and 0 79 respectively when images were segmented with fuzzy c means alone these coefficients increased to 0 90 and 0 93 when the pyramidal segmentation was combined with fuzzy c means this method can be applied to any dimensional representation and at any resolution level of an image series the evaluation study shows good performance in detecting lv lumen in mr images

fast computation of the difference of low pass transform this paper defines the difference of low pass dolp transform and describes a fast algorithm for its computation the dolp is a reversible transform which converts an image into a set of bandpass images a dolp transform is shown to require o n2 multiplies and produce o n log n samples from an n sample image when gaussian low pass filters are used the result is a set of images which have been convolved with difference of gaussian dog filters from an exponential set of sizes a fast computation technique based on resampling is described and shown to reduce the dolp transform complexity to o n log n multiplies and o n storage locations a second technique cascaded convolution with expansion is then defined and also shown to reduce the computational cost to o n log n multiplies combining these two techniques yields an algorithm for a dolp transform that requires o n storage cells and requires o n multiplies

a computational framework and an algorithm for the measurement of visual the robust measurement of visual motion from digitized image sequences has been an important but difficult problem in computer vision this paper describes a hierarchical computational framework for the determination of dense displacement fields from a pair of images and an algorithm consist ent with that framework our framework is based on the separation of the image intensity information as well as the process of measuring motion according to scale the large scale intensity information is first used to obtain rough estimates of image motion which are then refined by using intensity information at smaller scales the estimates are in the form of displacement or velocity vectors for pixels and are accompanied by a direction dependent confidence measure a smoothness constraint is employed to propagate the measurements with high confidence to their neigboring areas where the confidences are low at all levels the computations are pixel parallel uniform across the image and based on information from a small neighborhood of a pixel for our algorithm the local displacement vectors are determind by mini mizing the sum of squared differences ssd of intensities the confidence measures are derived from the shape of the ssd surface and the smoothness constraint is cast in the form of energy minimization results of applying our algorithm to pairs of real images are included in addition to our own

image filtering using multiresolution representations it is shown how multiresolution representations can be used for filter design and implementation these representations provide a coarse frequency decomposition of the image which forms the basis for two filtering techniques the first method based on image pyramids is used for approximating the convolution of an image with a given mask in this technique a filter is designed using a least squares procedure based on filters synthesized from the basic pyramid equivalent filters the second method is an adaptive noise reduction algorithm an optimally filtered image is synthesized from the multiresolution levels which in this case are maintained at the original sampling density individual pixels of the image representation are linearly combined under a minimum mean square error criterion this uses a local signal to noise ratio estimate to provide the best compromise between noise removal and resolution loss

the fibering map approach to a nonlocal problem involving p x laplacian abstract we study the existence of positive solutions for a p x kirchhoff problem the main tool used is the fibering map approach for the corresponding nehari manifold

picone identities for half linear elliptic operators with p x laplacians and applications to sturmian comparison theory picone identities are established for a class of half linear elliptic operators with p x laplacians and sturmian comparison theorems are obtained on the basis of the picone identities specializations to half linear partial or ordinary differential inequalities with p x laplacians are shown

on the existence and stability of solutions for dirichlet problem with p x laplacian abstract we show the existence and stability of solutions for a family of dirichlet problems div a x u x p x 2 u x b x u x p x 2 u x f u k x u x u x 0 u w 0 1 p x with nonlinearity satisfying some local growth conditions we construct a new duality theory which differs from the known ones in that it does not require a type of a palais smale condition

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

on a dirichlet problem with p x laplacian abstract we show the existence and stability of solutions for a family of dirichlet problems div v z 1 x u v z n x u l u x u f u k x u u w 0 1 p x in a bounded domain and with nonconvex nonlinearity satisfying some local growth conditions the conditions upon v and l allow for considering the p x laplacian equation we use the relations between critical points and critical values to the primal and a suitable dual action functional to get the existence stability and some properties of the solutions

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved