dichotomies and asymptotic solutions of nonlinear differential systems the problem of asymptotic equivalence between two systems of ordinary differential equations has been studied by many authors l 5 it was first used by levinson in his theorem on asymptotic integration 6 71 in l 51 this problem is studied when system 1 2 is stable a t is constant and g t y b t is integrable in 3 theorems 5 111 it is assumed that the linear system 1 2 has an ordinary dichotomy and the nonlinear term g t y is lipschitz continuous in 2 the previous results are extended to nonlinear perturbations g t y which are not lipschitz continuous such that g t y l i r t y where r t y is monotone in y we emphasize that in all these works only the asymptotic equivalence between bounded solutions is established nothing is said about a possible correspondence between unbounded solutions of 1 1 and 1 2 in this paper we suppose that the nonlinear term g t y satisfies some integrability conditions that we summarize in hypothesis h2 these conditions permit one to consider systems with large nonlinearities for example equations where g t y is an oscillatory function of the variable y no lipschitz conditions or monotonicity properties are required of g t y however we will use dichotomic properties of system 1 2

riemann and edalat integration on domains the main result of this paper is that the domain theoretic approach to the generalized riemann integral first introduced by edalat extends to a large class of spaces that can be realized as the set of maximal points of domains the approach is based on the theory of a riemann stieltjes type integral on a topological space with respect to a finitely additive measure we develop the theory of this integral for a bounded function f defined on the maximal points of a continuous domain and show that it gives an alternate approach to the edalat integral

a fast approximation of the bilateral filter using a signal processing approach the bilateral filter is a nonlinear filter that smoothes a signal while preserving strong edges it has demonstrated great effectiveness for a variety of problems in computer vision and computer graphics and fast versions have been proposed unfortunately little is known about the accuracy of such accelerations in this paper we propose a new signal processing analysis of the bilateral filter which complements the recent studies that analyzed it as a pde or as a robust statistical estimator the key to our analysis is to express the filter in a higher dimensional space where the signal intensity is added to the original domain dimensions importantly this signal processing perspective allows us to develop a novel bilateral filtering acceleration using downsampling in space and intensity this affords a principled expression of accuracy in terms of bandwidth and sampling the bilateral filter can be expressed as linear convolutions in this augmented space followed by two simple nonlinearities this allows us to derive criteria for downsampling the key operations and achieving important acceleration of the bilateral filter we show that for the same running time our method is more accurate than previous acceleration techniques typically we are able to process a 2 megapixel image using our acceleration technique in less than a second and have the result be visually similar to the exact computation that takes several tens of minutes the acceleration is most effective with large spatial kernels furthermore this approach extends naturally to color images and cross bilateral filtering

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

a computational model for metric spaces for every metric space x we define a continuous poset bx such that x is homeomorphic to the set of maximal elements of bx with the relative scott topology the poset bx is a dcpo iff x is complete and continuous iff x is separable the computational model bx is used to give domain theoretic proofs of banach s fixed point theorem and of two classical results of hutchinson on a complete metric space every hyperbolic iterated function system has a unique non empty compact attractor and every iterated function system with probabilities has a unique invariant measure with bounded support we also show that the probabilistic power domain of bx provides an continuous computational model for measure theory on a separable complete metric space x

dynamical systems measures and fractals via domain theory abstract we introduce domain theory in dynamical systems iterated function systems fractals and measure theory for a discrete dynamical system given by the action of a continuous map f x x on a metric space x we study the extended dynamical systems vx vf ux uf and lx lf where v u and l are respectively the vietoris hyperspace the upper hyperspace and the lower hyperspace functors we show that if x f is chaotic then so is ux uf when x is locally compact ux is a continuous bounded complete dcpo if x is second countable as well then ux will be continuous and can be given an effective structure we show how strange attractors attractors of iterated function systems fractals and julia sets are obtained effectively as fixed points of deterministic functions on ux or fixed points of non deterministic functions on cux where c is the convex plotkin power domain we also show that the set m x of finite borel measures on x can be embedded in pux where p is the probabilistic power domain this provides an effective framework for measure theory we then prove that the invariant measure of an hyperbolic iterated function system with probabilities can he obtained as the unique fixed point of an associated continuous function on pux

asymptotic integration of ordinary different systems an extension and a new proof of hartman and wintner s theorem on asymptotic integration are given

a general framework for low level vision we introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented we consider intensity images as surfaces in the x i space the image is thereby a two dimensional 2 d surface in three dimensional 3 d space for gray level images and 2 d surfaces in five dimensions for color images the new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast and results in new and efficient schemes extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms

the structure of locally orderless images we propose a representation of images in which a global but not a local topology is defined the topology is restricted to resolutions up to the extent of the local region of interest roi although the roi s may contain many pixels there is no spatial order on the pixels within the roi the only information preserved is the histogram of pixel values within the roi s this can be considered as an extreme case of a textel texture element image the histogram is the limit of texture where the spatial order has been completely disregarded we argue that locally orderless images are ubiquitous in perception and the visual arts formally the orderless images are most aptly described by three mutually intertwined scale spaces the scale parameters correspond to the pixellation inner scale the extent of the roi s outer scale and the resolution in the histogram tonal scale we describe how to construct locally orderless images how to render them and how to use them in a variety of local and global image processing operations

channel smoothing efficient robust smoothing of low level signal features in this paper we present a new and efficient method to implement robust smoothing of low level signal features b spline channel smoothing this method consists of three steps encoding of the signal features into channels averaging of the channels and decoding of the channels we show that linear smoothing of channels is equivalent to robust smoothing of the signal features if we make use of quadratic b splines to generate the channels the linear decoding from b spline channels allows the derivation of a robust error norm which is very similar to tukey s biweight error norm we compare channel smoothing with three other robust smoothing techniques nonlinear diffusion bilateral filtering and mean shift filtering both theoretically and on a 2d orientation data smoothing task channel smoothing is found to be superior in four respects it has a lower computational complexity it is easy to implement it chooses the global minimum error instead of the nearest local minimum and it can also be used on nonlinear spaces such as orientation space

susan a new approach to low level image processing this paper describes a new approach to low level image processing in particular edge and corner detection and structure preserving noise reduction non linear filtering is used to define which parts of the image are closely related to each individual pixel each pixel has associated with it a local image region which is of similar brightness to that pixel the new feature detectors are based on the minimization of this local image region and the noise reduction method uses this region as the smoothing neighbourhood the resulting methods are accurate noise resistant and fast details of the new feature detectors and of the new noise reduction method are described along with test results

stereo matching using tree filtering matching cost aggregation is one of the oldest and still popular methods for stereo correspondence while effective and efficient cost aggregation methods typically aggregate the matching cost by summing averaging over a user specified local support region this is obviously only locally optimal and the computational complexity of the full kernel implementation usually depends on the region size in this paper the cost aggregation problem is re examined and a non local solution is proposed the matching cost values are aggregated adaptively based on pixel similarity on a tree structure derived from the stereo image pair to preserve depth edges the nodes of this tree are all the image pixels and the edges are all the edges between the nearest neighboring pixels the similarity between any two pixels is decided by their shortest distance on the tree the proposed method is non local as every node receives supports from all other nodes on the tree the proposed method can be naturally extended to the time domain for enforcing temporal coherence unlike previous methods the non local property guarantees that the depth edges will be preserved when the temporal coherency between all the video frames are considered a non local weighted median filter is also proposed based on the non local cost aggregation algorithm it has been demonstrated to outperform all local weighted median filters on disparity depth upsampling and refinement

speeding up the bilateral filter a joint acceleration way computational complexity of the brute force implementation of the bilateral filter bf depends on its filter kernel size to achieve the constant time bf whose complexity is irrelevant to the kernel size many techniques have been proposed such as 2d box filtering dimension promotion and shiftability property although each of the above techniques suffers from accuracy and efficiency problems previous algorithm designers were used to take only one of them to assemble fast implementations due to the hardness of combining them together hence no joint exploitation of these techniques has been proposed to construct a new cutting edge implementation that solves these problems jointly employing five techniques kernel truncation best n term approximation as well as previous 2d box filtering dimension promotion and shiftability property we propose a unified framework to transform bf with arbitrary spatial and range kernels into a set of 3d box filters that can be computed in linear time to the best of our knowledge our algorithm is the first method that can integrate all these acceleration techniques and therefore can draw upon one another s strong point to overcome deficiencies the strength of our method has been corroborated by several carefully designed experiments in particular the filtering accuracy is significantly improved without sacrificing the efficiency at running time

diffusion tensors for processing sheared and rotated rectangles image restoration and simplification methods that respect important features such as edges play a fundamental role in digital image processing however known edge preserving methods like common nonlinear diffusion methods tend to round vertices for large diffusion times in this paper we adapt the diffusion tensor for anisotropic diffusion to avoid this effects in images containing rotated and sheared rectangles respectively in this context we propose a new method for estimating rotation angles and shear parameters based on the so called structure tensor further we show how the knowledge of appropriate diffusion tensors can be used in variational models numerical examples including orientation estimation denoising and segmentation demonstrate the good performance of our methods

fast domain decomposition for global image smoothing edge preserving smoothing eps can be formulated as minimizing an objective function that consists of data and regularization terms at the price of high computational cost this global eps approach is more robust and versatile than a local one that typically has a form of weighted averaging in this paper we introduce an efficient decomposition based method for global eps that minimizes the objective function of l 2 data and possibly non smooth and non convex regularization terms in linear time different from previous decomposition based methods which require solving a large linear system our approach solves an equivalent constrained optimization problem resulting in a sequence of 1 d sub problems this enables applying fast linear time solver for weighted least squares and l 1 smoothing problems an alternating direction method of multipliers algorithm is adopted to guarantee fast convergence our method is fully parallelizable and its runtime is even comparable to the state of the art local eps approaches we also propose a family of fast majorization minimization algorithms that minimize an objective with non convex regularization terms experimental results demonstrate the effectiveness and flexibility of our approach in a range of image processing and computational photography applications

cauchy matrix for linear almost periodic systems and some consequences abstract a novel method to construct the fundamental matrix for a linear almost periodic system is proposed provided that the diagonal terms satisfy an average separation condition and the off diagonal coefficients are l small the idea is to transform the system in a set of riccati type equations and use exponential dichotomy and its consequences it is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients finally our result enables us to obtain i explicit almost periodic matrices q t q 1 t and q t which diagonalize the original system and ii sufficient conditions for the stability two illustrative examples are shown

fundamental relationship between bilateral filtering adaptive smoothing and the nonlinear diffusion equation in this paper the relationship between bilateral filtering and anisotropic diffusion is examined the bilateral filtering approach represents a large class of nonlinear digital image filters we first explore the connection between anisotropic diffusion and adaptive smoothing and then the connection between adaptive smoothing and bilateral filtering previously adaptive smoothing was considered to be an inconsistent approximation to the nonlinear diffusion equation we extend adaptive smoothing to make it consistent thus enabling a unified viewpoint that relates nonlinear digital image filters and the nonlinear diffusion equation

constant time median and bilateral filtering this paper formulates both the median filter and bilateral filter as a cost volume aggregation problem whose computational complexity is independent of the filter kernel size unlike most of the previous works the proposed framework results in a general bilateral filter that can have arbitrary spatial 1 1 and arbitrary range filter kernels this bilateral filter takes about 3 5 s to exactly filter a one megapixel 8 bit grayscale image on a 3 2 ghz intel core i7 cpu in practice the intensity range and spatial domain can be downsampled to improve the efficiency this compression can maintain very high accuracy e g 40 db but over 100 times 100 faster

orientation guided geodesic weighting for patchmatch based stereo matching we propose an orientation guided geodesic weighting oggw strategy for local stereo matching we propose a method of cost volume filtering combining a multipoint lpa method with our oggw strategy we propose a patchmatch filter with curved surface fitting pmf cs to obtain a disparity map with sub pixel accuracy recently patchmatch based methods for local stereo matching are experiencing great progress with the use of compact and over segmented regions that have similar intensities or colors using patches as support regions this paper proposes an orientation guided geodesic weighting oggw strategy to search for an approximate shortest path from a support pixel in the patch to a pixel of interest along a guided orientation the oggw is computed by accumulating intensity differences or color dissimilarities between connected pixels along the path after obtaining matching cost updates by model fitting the oggw is used for weighted averaging on the updated costs to obtain a filtered cost volume in addition a new filtering method that combines the patchmatch filter with curved surface fitting pmf cs is presented in this paper curved surface fitting along with outliers removal is carried out to seek for a reliable regression model for estimating the disparities on a patch and to achieve a disparity map with sub pixel accuracy we conduct a number of experiments to evaluate the performances of oggw and pmf cs on cost volume filtering and disparity estimation experimental results show that our algorithm produces accurate stereo matching results and outperforms the current state of the art patchmatch based methods

yet another cost aggregation over models in the recent decades we have witnessed the advent of local or non local filters for cost aggregation in stereo matching pushing the envelope of local methods to the degree of global methods while maintaining the efficiency a specific filter with a specific parameter setting may have a potential to best work for an image pair but may not guarantee equally good performance for other image pairs to address this problem we propose a mixture of experts model which applies a heterogeneous set of filters on the cost volume and adaptively combines the results we employ supervised learning to estimate per pixel mixing coefficients which are used to adaptively control the weight of the filter responses through experiments we show that the mixture model significantly reduces errors in disparity estimation and even outperforms the strategy of selecting the best per pixel filter from the pool of filters in the average sense

the hartman wintner theorem for functional differential equations abstract in this paper we present a generalization of the hartman wintner theorem about the asymptotic behavior of the solutions of linear ordinary differential equations to functional differential equations

asymptotic integration of linear differential equations abstract we establish a new result on the asymptotic integration of linear differential systems of the form y r y where is diagonal and r is small more specifically we allow for different scales of asymptotic behaviour of the individual matrix elements of and r we also discuss an application to a self adjoint fourth order equation

roughness of h k dichotomies abstract in this paper we prove a roughness theorem for a class of dichotomies of linear systems of differential equations the so called h k dichotomies these include a great variety of dichotomies among them the ordinary and exponential dichotomies exponential ordinary dichotomies levinson dichotomies for diagonal systems etc we prove that under general uniform conditions upon the functions h and k these dichotomies are not destroyed by integrable perturbations this result generalizes the property of roughness of ordinary dichotomies under integrable perturbations

revisiting the relationship between adaptive smoothing and anisotropic diffusion with modified filters anisotropic diffusion has been known to be closely related to adaptive smoothing and discretized in a similar manner this paper revisits a fundamental relationship between two approaches it is shown that adaptive smoothing and anisotropic diffusion have different theoretical backgrounds by exploring their characteristics with the perspective of normalization evolution step size and energy flow based on this principle adaptive smoothing is derived from a second order partial differential equation pde not a conventional anisotropic diffusion via the coupling of fick s law with a generalized continuity equation where a source or sink exists which has not been extensively exploited we show that the source or sink is closely related to the asymmetry of energy flow as well as the normalization term of adaptive smoothing it enables us to analyze behaviors of adaptive smoothing such as the maximum principle and stability with a perspective of a pde ultimately this relationship provides new insights into application specific filtering algorithm design by modeling the source or sink in the pde we introduce two specific diffusion filters the robust anisotropic diffusion and the robust coherence enhancing diffusion as novel instantiations which are more robust against the outliers than the conventional filters

fusion of median and bilateral filtering for range image upsampling we present a new upsampling method to enhance the spatial resolution of depth images given a low resolution depth image from an active depth sensor and a potentially high resolution color image from a passive rgb camera we formulate it as an adaptive cost aggregation problem and solve it using the bilateral filter the formulation synergistically combines the median and bilateral filters thus it better preserves the depth edges and is more robust to noise numerical and visual evaluations on a total of 37 middlebury data sets demonstrate the effectiveness of our method a real time high resolution depth capturing system is also developed using commercial active depth sensor based on the proposed upsampling method

fast bilateral filtering for denoising large 3d images a fast implementation of bilateral filtering is presented which is based on an optimal expansion of the filter kernel into a sum of factorized terms these terms are computed by minimizing the expansion error in the mean square error sense this leads to a simple and elegant solution in terms of eigenvectors of a square matrix in this way the bilateral filter is applied through computing a few gaussian convolutions for which very efficient algorithms are readily available moreover the expansion functions are optimized for the histogram of the input image leading to improved accuracy it is shown that this further optimization it made possible by removing the commonly deployed constrain of shiftability of the basis functions experimental validation is carried out in the context of digital rock imaging results on large 3d images of rock samples show the superiority of the proposed method with respect to other fast approximations of bilateral filtering

dichotomies and asymptotic formulas for the solutions of differential equations abstract the new concept of h k dichotomy is studied extensively and is used in asymptotic equivalence problems

automatic estimation and removal of noise from a single image image denoising algorithms often assume an additive white gaussian noise awgn process that is independent of the actual rgb values such approaches cannot effectively remove color noise produced by today s ccd digital camera in this paper we propose a unified framework for two tasks automatic estimation and removal of color noise from a single image using piecewise smooth image models we introduce the noise level function nlf which is a continuous function describing the noise level as a function of image brightness we then estimate an upper bound of the real nlf by fitting a lower envelope to the standard deviations of per segment image variances for denoising the chrominance of color noise is significantly removed by projecting pixel values onto a line fit to the rgb values in each segment then a gaussian conditional random field gcrf is constructed to obtain the underlying clean image from the noisy input extensive experiments are conducted to test the proposed algorithm which is shown to outperform state of the art denoising algorithms

efficient and robust specular highlight removal a robust and effective specular highlight removal method is proposed in this paper it is based on a key observation the maximum fraction of the diffuse colour component in diffuse local patches in colour images changes smoothly the specular pixels can thus be treated as noise in this case this property allows the specular highlights to be removed in an image denoising fashion an edge preserving low pass filter e g the bilateral filter can be used to smooth the maximum fraction of the colour components of the original image to remove the noise contributed by the specular pixels recent developments in fast bilateral filtering techniques enable the proposed method to run over 200 times faster than state of the art techniques on a standard cpu and differentiates it from previous work

an asymptotic theory for nonlinear functional differential equations abstract the hartman wintner theorem on asymptotic integration is established for a certain class of functional differential equations with nonlinear perturbations by looking them as abstract ordinary differential equations results for delay differential equations are included in this study consequences and applications are also shown

a computable approach to measure and integration theory we introduce a computable framework for lebesgue s measure and integration theory in the spirit of domain theory for an effectively given second countable locally compact hausdorff space and an effectively given finite borel measure on the space we define a recursive measurable set which extends the corresponding notion due to s anin for the lebesgue measure on the real line we also introduce the stronger notion of a computable measurable set where a measurable set is approximated from inside and outside by sequences of closed and open subsets respectively the more refined property of computable measurable sets give rise to the idea of partial measurable subsets which naturally form a domain for measurable subsets we then introduce interval valued measurable functions and develop the notion of recursive and computable measurable functions using interval valued simple functions this leads us to the interval versions of the main results in classical measure theory the lebesgue integral is shown to be a continuous operator on the domain of interval valued measurable functions and the interval valued lebesgue integral provides a computable framework for integration

separating reflection components of textured surfaces using a single image in inhomogeneous objects highlights are linear combinations of diffuse and specular reflection components to our knowledge all methods that use a single input image require explicit color segmentation to deal with multicolored surfaces unfortunately for complex textured images current color segmentation algorithms are still problematic to segment correctly consequently a method without explicit color segmentation becomes indispensable and this work presents such a method the method is based solely on colors particularly chromaticity without requiring any geometrical information one of the basic ideas is to iteratively compare the intensity logarithmic differentiation of an input image and its specular free image a specular free image is an image that has exactly the same geometrical profile as the diffuse component of the input image and that can be generated by shifting each pixel s intensity and maximum chromaticity nonlinearly unlike existing methods using a single image all processes in the proposed method are done locally involving a maximum of only two neighboring pixels this local operation is useful for handling textured objects with complex multicolored scenes evaluations by comparison with the results of polarizing filters demonstrate the effectiveness of the proposed method