rate distortion efficient piecewise planar 3 d scene representation from 2 d images in any practical application of the 2 d to 3 d conversion that involves storage and transmission representation efficiency has an undisputable importance that is not reflected in the attention the topic received in order to address this problem a novel algorithm which yields efficient 3 d representations in the rate distortion sense is proposed the algorithm utilizes two views of a scene to build a mesh based representation incrementally via adding new vertices while minimizing a distortion measure the experimental results indicate that in scenes that can be approximated by planes the proposed algorithm is superior to the dense depth map and in some practical situations to the block motion vector based representations in the rate distortion sense

regional bit allocation and rate distortion optimization for multiview depth video coding with view synthesis distortion model in this paper we propose a view synthesis distortion model vsdm that establishes the relationship between depth distortion and view synthesis distortion for the regions with different characteristics color texture area corresponding depth ctad region and color smooth area corresponding depth csad respectively with this vsdm we propose regional bit allocation rba and rate distortion optimization rdo algorithms for multiview depth video coding mdvc by allocating more bits on ctad for rendering quality and fewer bits on csad for compression efficiency experimental results show that the proposed vsdm based rba and rdo can improve the coding efficiency significantly for the test sequences in addition for the proposed overall mdvc algorithm that integrates vsdm based rba and rdo it achieves 9 99 and 14 51 bit rate reduction on average for the high and low bit rate respectively it can improve virtual view image quality 0 22 and 0 24 db on average at the high and low bit rate respectively when compared with the original joint multiview video coding model the rd performance comparisons using five different metrics also validate the effectiveness of the proposed overall algorithm in addition the proposed algorithms can be applied to both intra and inter frames

a bit allocation method for sparse source coding in this paper we develop an efficient bit allocation strategy for subband based image coding systems more specifically our objective is to design a new optimization algorithm based on a rate distortion optimality criterion to this end we consider the uniform scalar quantization of a class of mixed distributed sources following a bernoulli generalized gaussian distribution this model appears to be particularly well adapted for image data which have a sparse representation in a wavelet basis in this paper we propose new approximations of the entropy and the distortion functions using piecewise affine and exponential forms respectively because of these approximations bit allocation is reformulated as a convex optimization problem solving the resulting problem allows us to derive the optimal quantization step for each subband experimental results show the benefits that can be drawn from the proposed bit allocation method in a typical transform based coding application

invertible temporal subband wavelet filter banks with half pixel accurate motion compensation three dimensional 3 d subband wavelet coding with motion compensation has been demonstrated to be an efficient technique for video coding applications in some recent research works when motion compensation is performed with half pixel accuracy images need to be interpolated in both temporal subband analysis and synthesis stages the resulting subband filter banks developed in these former algorithms were not invertible due to image interpolation in this paper an invertible temporal analysis synthesis system with half pixel accurate motion compensation is presented we look at temporal decomposition of image sequences as a kind of down conversion of the sampling lattices the earlier motion compensated mc interlaced progressive scan conversion scheme is extended for temporal subband analysis synthesis the proposed subband wavelet filter banks allow perfect reconstruction of the decomposed video signal while retaining high energy compaction of subband transforms the invertible filter banks are then utilized in our 3 d subband video coder this video coding system does not contain the temporal dpcm loop employed in the conventional hybrid coder and the earlier mc 3 d subband coders the experimental results show a significant psnr improvement by the proposed method the generalization of our algorithm for mc temporal filtering at arbitrary subpixel accuracy is also discussed

quantization of multiple sources using nonnegative integer bit allocation asymptotically optimal real valued bit allocation among a set of quantizers for a finite collection of sources was derived in 1963 by huang and schultheiss and an algorithm for obtaining an optimal nonnegative integer valued bit allocation was given by fox in 1966 we prove that for a given bit budget the set of optimal nonnegative integer valued bit allocations is equal to the set of nonnegative integer valued bit allocation vectors which minimize the euclidean distance to the optimal real valued bit allocation vector of huang and schultheiss we also give an algorithm for finding optimal nonnegative integer valued bit allocations the algorithm has lower computational complexity than fox s algorithm as the bit budget grows finally we compare the performance of the huang schultheiss solution to that of an optimal integer valued bit allocation specifically we derive upper and lower bounds on the deviation of the mean squared error mse using optimal integer valued bit allocation from the mse using optimal real valued bit allocation it is shown that for asymptotically large transmission rates optimal integer valued bit allocations do not necessarily achieve the same performance as that predicted by huang schultheiss for optimal real valued bit allocations

scalable coding of depth maps with r d optimized embedding recent work on depth map compression has revealed the importance of incorporating a description of discontinuity boundary geometry into the compression scheme we propose a novel compression strategy for depth maps that incorporates geometry information while achieving the goals of scalability and embedded representation our scheme involves two separate image pyramid structures one for breakpoints and the other for sub band samples produced by a breakpoint adaptive transform breakpoints capture geometric attributes and are amenable to scalable coding we develop a rate distortion optimization framework for determining the presence and precision of breakpoints in the pyramid representation we employ a variation of the ebcot scheme to produce embedded bit streams for both the breakpoint and sub band data compared to jpeg 2000 our proposed scheme enables the same the scalability features while achieving substantially improved rate distortion performance at the higher bit rate range and comparable performance at the lower rates

singularities of first kind in the harmonic map and yang mills heat flows by constructing examples which are explicit up to solving an ode we prove that singularities of first kind exist for the harmonic map and yang mills heat flows as a by product we also get a simplified proof of ratto s ding s theorem about the existence of harmonic hopf constructions

context tree based image contour coding using a geometric prior efficient encoding of object contours in images can facilitate advanced image video compression techniques such as shape adaptive transform coding or motion prediction of arbitrarily shaped pixel blocks we study the problem of lossless and lossy compression of detected contours in images specifically we first convert a detected object contour into a sequence of directional symbols drawn from a small alphabet to encode the symbol sequence using arithmetic coding we compute an optimal variable length context tree vct mathcal t via a maximum a posterior map formulation to estimate symbols conditional probabilities map can avoid overfitting given a small training set mathcal x of past symbol sequences by identifying a vct mathcal t with high likelihood p mathcal x mathcal t of observing mathcal x given mathcal t using a geometric prior p mathcal t stating that image contours are more often straight than curvy for the lossy case we design fast dynamic programming dp algorithms that optimally trade off coding rate of an approximate contour hat text x given a vct mathcal t with two notions of distortion of hat text x with respect to the original contour x to reduce the size of the dp tables a total suffix tree is derived from a given vct mathcal t for compact table entry indexing reducing complexity experimental results show that for lossless contour coding our proposed algorithm outperforms state of the art context based schemes consistently for both small and large training datasets for lossy contour coding our algorithms outperform comparable schemes in the literature in rate distortion performance

elliptic equations strongly degenerate at a point abstract in this paper we will mainly propose some problems for a class of degenerate elliptic equations either linear or nonlinear we will study some special cases of these problems and reveal some phenomena which may not have been noticed previously our problems originated from the self similar solutions of the heat flow of harmonic maps we will prove that the self similar solutions or the so called quasi harmonic spheres are discontinuous at infinity for the equivariant case in other words the equivariant quasi harmonic spheres are not continuous images of topological spheres

efficient quadtree coding of images and video the quadtree data structure is commonly used in image coding to decompose an image into separate spatial regions to adaptively identify the type of quantizer used in various regions of an image the authors describe the theory needed to construct quadtree data structures that optimally allocate rate given a set of quantizers a lagrange multiplier method finds these optimal rate allocations with no monotonicity restrictions they use the theory to derive a new quadtree construction method that uses a stepwise search to find the overall optimal quadtree structure the search can be driven with either actual measured quantizer performance or ensemble average predicted performance they apply this theory to the design of a motion compensated interframe video coding system using a quadtree with vector quantization

quantization of time like energy for wave maps into spheres in this article we consider large energy wave maps in dimension 2 1 as in the resolution of the threshold conjecture by sterbenz and tataru commun math phys 298 1 139 230 2010 commun math phys 298 1 231 264 2010 but more specifically into the unit euclidean sphere mathbb s n 1 subset mathbb r n with n geq2 and study further the dynamics of the sequence of wave maps that are obtained in sterbenz and tataru commun math phys 298 1 231 264 2010 at the final rescaling for a first finite or infinite time singularity we prove that on a suitably chosen sequence of time slices at this scaling there is a decomposition of the map up to an error with asymptotically vanishing energy into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm concentrating on the null boundary and converging to a constant locally in the interior of the cone in the energy space similar and stronger results have been recently obtained in the equivariant setting by several authors cote commun pure appl math 68 11 1946 2004 2015 cote commun pure appl math 69 4 609 612 2016 cote am j math 137 1 139 207 2015 cote et al am j math 137 1 209 250 2015 krieger commun math phys 250 3 507 580 2004 where better control on the dispersive term concentrating on the null boundary of the cone is provided and in some cases the asymptotic decomposition is shown to hold for all time here however we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to lin and riviere ann math 149 2 785 829 1999 duke math j 111 177 193 2002 which we make work here in the hyperbolic context of sterbenz and tataru commun math phys 298 1 231 264 2010

nonuniqueness for the heat flow of harmonic maps on the disk we prove that for suitable initial data the heat flow of harmonic maps from the disk to the sphere admits infinitely many solutions characterised by backward bubbling at some arbitrarily large time all having uniformly bounded energy

homotopy classes of harmonic maps of the stratified 2 spheres and applications to geometric flows abstract we show that the set of harmonic maps from the 2 dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes we apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2 sphere in the connected sum of two closed 3 dimensional manifolds m 1 s 3 and m 2 s 3 r p 3 which must develop finite time singularity

estimation of depth fields suitable for video compression based on 3 d structure and motion of objects intensity prediction along motion trajectories removes temporal redundancy considerably in video compression algorithms in three dimensional 3 d object based video coding both 3 d motion and depth values are required for temporal prediction the required 3 d motion parameters for each object are found by the correspondence based e matrix method the estimation of the correspondences two dimensional 2 d motion field between the frames and segmentation of the scene into objects are achieved simultaneously by minimizing a gibbs energy the depth field is estimated by jointly minimizing a defined distortion and bit rate criterion using the 3 d motion parameters the resulting depth field is efficient in the rate distortion sense bit rate values corresponding to the lossless encoding of the resultant depth fields are obtained using predictive coding prediction errors are encoded by a lempel ziv algorithm the results are satisfactory for real life video scenes

distortion policy of buffer constrained rate control for real time vbr coders with a rate policy we can achieve buffer constrained rate control for variable bit rate coder since the rate policy emphasizes the stability of the state of the buffer we can loose the advantage of variable bit rate coders i e a nearly constant quality distortion to overcome the difficulty the distortion policy is considered in this paper based on both the state of the buffer and the distortion we can formulate the distortion policy by choosing a proper weighting factor of the cost function in the distortion policy the state of the buffer can be stable and or the variation of the distortion sequence can be kept small in addition the properties of the distortion policy are obtained we also consider switching and adaptive methods for the weighting factor to adjust time varying characteristics of blocks of images using a jpeg like image coder the distortion policy is applied to buffer constrained rate control

rate distortion optimized graph based representation for multiview images with complex camera configurations graph based representation gbr has recently been proposed for describing color and geometry of multiview video content the graph vertices represent the color information while the edges represent the geometry information i e the disparity by connecting corresponding pixels in two camera views in this paper we generalize the gbr to multiview images with complex camera configurations compared with the existing gbr the proposed representation can handle not only horizontal displacements of the cameras but also forward backward translations rotations etc however contrary to the usual disparity that is a 2 d vector denoting horizontal and vertical displacements each edge in gbr is represented by a 1 d disparity this quantity can be seen as the disparity along an epipolar segment in order to have a sparse i e easy to code graph structure we propose a rate distortion model to select the most meaningful edges hence the graph is constructed with just enough information for rendering the given predicted view the experiments show that the proposed gbr allows high reconstruction quality with lower or equivalent coding rate than traditional depth based representations

blowup dynamics for smooth data equivariant solutions to the critical schrodinger map problem we consider the energy critical schrodinger map problem with the 2 sphere target for equivariant initial data of homotopy index k 1 we show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm which generates finite time blowup solutions we give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy

subband coding systems incorporating quantizer models a new method for dealing with the effects of quantization in a subband system is proposed it uses the gain plus additive noise linear model for the lloyd max quantizer based on this it is demonstrated how by an appropriate choice of synthesis filters one can cancel all signal dependent errors at the output of the system the only remaining error is random in nature and not correlated with the input signal we therefore have a tradeoff between the error being only random or having signal dependent components as well since the error variances in both cases are comparable as a result of having only a random error it is possible to reduce this error using for example a noise removal technique the result is then extended to the case where the input is a multidimensional signal and arbitrary sampling lattices are used as well as to the qmf alias cancellation case to demonstrate the validity of the proposed approach two types of experiments on images are carried out in a toy example it is shown that using noise removal could be beneficial for a more realistic coding scheme however it is demonstrated that even in the case when the model is no longer valid when some of the subbands are discarded the output error is still much less correlated with the input signal as opposed to the commonly used subband system while visually the reconstructed images look very similar

singularity formation for the two dimensional harmonic map flow into s 2 we construct finite time blow up solutions to the 2 dimensional harmonic map flow into the sphere s 2 begin aligned u t delta u nabla u 2 u quad text in omega times 0 t u varphi quad text on partial omega times 0 t u cdot 0 u 0 quad text in omega end aligned where omega is a bounded smooth domain in mathbb r 2 u omega times 0 t rightarrow s 2 u 0 bar omega rightarrow s 2 is smooth and varphi u 0 big partial omega given any k points q 1 ldots q k in the domain we find initial and boundary data so that the solution blows up precisely at those points the profile around each point is close to an asymptotically singular scaling of a 1 corotational harmonic map we build a continuation after blow up as a h 1 weak solution with a finite number of discontinuities in space time by reverse bubbling which preserves the homotopy class of the solution after blow up furthermore we prove the codimension one stability of the one point blow up phenomenon

context coding of depth map images under the piecewise constant image model representation this paper introduces an efficient method for lossless compression of depth map images using the representation of a depth image in terms of three entities 1 the crack edges 2 the constant depth regions enclosed by them and 3 the depth value over each region the starting representation is identical with that used in a very efficient coder for palette images the piecewise constant image model coding but the techniques used for coding the elements of the representation are more advanced and especially suitable for the type of redundancy present in depth images initially the vertical and horizontal crack edges separating the constant depth regions are transmitted by 2d context coding using optimally pruned context trees both the encoder and decoder can reconstruct the regions of constant depth from the transmitted crack edge image the depth value in a given region is encoded using the depth values of the neighboring regions already encoded exploiting the natural smoothness of the depth variation and the mutual exclusiveness of the values in neighboring regions the encoding method is suitable for lossless compression of depth images obtaining compression of about 10 65 times and additionally can be used as the entropy coding stage for lossy depth compression

3d planar representation of stereo depth images for 3dtv applications the depth modality of the multiview video plus depth mvd format is an active research area whose main objective is to develop depth image based rendering friendly efficient compression methods as a part of this research a novel 3d planar based depth representation is proposed the planar approximation of multiple depth images are formulated as an energy based co segmentation problem by a markov random field model the energy terms of this problem are designed to mimic the rate distortion tradeoff for a depth compression application a novel algorithm is developed for practical utilization of the proposed planar approximations in stereo depth compression the co segmented regions are also represented as layered planar structures forming a novel single reference mvd format the ability of the proposed layered planar mvd representation in decoupling the texture and geometric distortions make it a promising approach proposed 3d planar depth compression approaches are compared against the state of the art image video coding standards by objective and visual evaluation and yielded competitive performance

energy based geometric multi model fitting geometric model fitting is a typical chicken egg problem data points should be clustered based on geometric proximity to models whose unknown parameters must be estimated at the same time most existing methods including generalizations of ransac greedily search for models with most inliers within a threshold ignoring overall classification of points we formulate geometric multi model fitting as an optimal labeling problem with a global energy function balancing geometric errors and regularity of inlier clusters regularization based on spatial coherence on some near neighbor graph and or label costs is np hard standard combinatorial algorithms with guaranteed approximation bounds e g expansion can minimize such regularization energies over a finite set of labels but they are not directly applicable to a continuum of labels e g mathcal r 2 in line fitting our proposed approach pearl combines model sampling from data points as in ransac with iterative re estimation of inliers and models parameters based on a global regularization functional this technique efficiently explores the continuum of labels in the context of energy minimization in practice pearl converges to a good quality local minimum of the energy automatically selecting a small number of models that best explain the whole data set our tests demonstrate that our energy based approach significantly improves the current state of the art in geometric model fitting currently dominated by various greedy generalizations of ransac

energy identity for approximate harmonic maps from surfaces to general targets abstract let u n be a sequence of mappings from a closed riemannian surface m to a general riemannian manifold n if u n satisfies sup n u n l 2 m u n l p m for some p 1 where u n is the tension field of u n then the so called energy identity and neckless property hold during blowing up this result is sharp by parker s example where the tension fields of the mappings from riemannian surface are bounded in l 1 m but the energy identity fails

solution of a quadratic nonlinear oscillator by the method of harmonic balance abstract this paper deals with the quadratic nonlinear oscillator qno x x e x 2 0 this oscillator is compared with the nonlinear oscillators x x e x x 0 which are solved by the method of harmonic balance therefore we obtain the first approximate solution to the qno this solution is more accurate than the second approximate solution obtained by the lindstedt poincare method for qnos it has been pointed out that the angular frequency for x 0 should be different from the angular frequency for x 0 if solutions of higher degree of accuracy are required

an embedded still image coder with rate distortion optimization it is well known that the fixed rate coder achieves optimality when all coefficients are coded with the same rate distortion r d slope in this paper we show that the performance of the embedded coder can be optimized in a rate distortion sense by coding the coefficients with decreasing r d slope we denote such a coding strategy as rate distortion optimized embedding rde rde allocates the available coding bits first to the coefficient with the steepest r d slope i e the largest distortion decrease per coding bit the resultant coding bitstream can be truncated at any point and still maintain an optimal r d performance to avoid the overhead of coding order transmission we use the expected r d slope which can be calculated from the coded bits and is available in both the encoder and the decoder with the probability estimation table of the qm coder the calculation of the r d slope can be just a lookup table operation experimental results show that the rate distortion optimization significantly improves the coding efficiency in a wide bit rate range

exact solution of the quadratic mixed parity helmholtz duffing oscillator abstract in this paper the exact solution of the quadratic mixed parity helmholtz duffing oscillator is derived by using jacobi elliptic functions it is also shown that the exact period of oscillation is given as a function of the complete elliptic integral of the first kind at the end of the paper we examine the stability of the system and determine the regions for which periodic and unbounded motions take place

multiview image coding using depth layers and an optimized bit allocation in this paper we present a novel wavelet based compression algorithm for multiview images this method uses a layer based representation where the 3 d scene is approximated by a set of depth planes with their associated constant disparities the layers are extracted from a collection of images captured at multiple viewpoints and transformed using the 3 d discrete wavelet transform dwt the dwt consists of the 1 d disparity compensated dwt across the viewpoints and the 2 d shape adaptive dwt across the spatial dimensions finally the wavelet coefficients are quantized and entropy coded along with the layer contours to improve the rate distortion performance of the entire coding method we develop a bit allocation strategy for the distribution of the available bit budget between encoding the layer contours and the wavelet coefficients the achieved performance of our proposed scheme outperforms the state of the art codecs for several data sets of varying complexity

random walk graph laplacian based smoothness prior for soft decoding of jpeg images given the prevalence of joint photographic experts group jpeg compressed images optimizing image reconstruction from the compressed format remains an important problem instead of simply reconstructing a pixel block from the centers of indexed discrete cosine transform dct coefficient quantization bins hard decoding soft decoding reconstructs a block by selecting appropriate coefficient values within the indexed bins with the help of signal priors the challenge thus lies in how to define suitable priors and apply them effectively in this paper we combine three image priors laplacian prior for dct coefficients sparsity prior and graph signal smoothness prior for image patches to construct an efficient jpeg soft decoding algorithm specifically we first use the laplacian prior to compute a minimum mean square error initial solution for each code block next we show that while the sparsity prior can reduce block artifacts limiting the size of the overcomplete dictionary to lower computation would lead to poor recovery of high dct frequencies to alleviate this problem we design a new graph signal smoothness prior desired signal has mainly low graph frequencies based on the left eigenvectors of the random walk graph laplacian matrix lerag compared with the previous graph signal smoothness priors lerag has desirable image filtering properties with low computation overhead we demonstrate how lerag can facilitate recovery of high dct frequencies of a piecewise smooth signal via an interpretation of low graph frequency components as relaxed solutions to normalized cut in spectral clustering finally we construct a soft decoding algorithm using the three signal priors with appropriate prior weights experimental results show that our proposal outperforms the state of the art soft decoding algorithms in both objective and subjective evaluations noticeably

an optimal quadtree based motion estimation and motion compensated interpolation scheme for video compression we propose an optimal quadtree qt based motion estimator for video compression it is optimal in the sense that for a given bit budget for encoding the displacement vector field dvf and the qt segmentation the scheme finds a dvf and a qt segmentation which minimizes the energy of the resulting displaced frame difference dfd we find the optimal qt decomposition and the optimal dvf jointly using the lagrangian multiplier method and a multilevel dynamic program we introduce a new very fast convex search for the optimal lagrangian multiplier spl lambda which results in a very fast convergence of the lagrangian multiplier method the resulting dvf is spatially inhomogeneous since large blocks are used in areas with simple motion and small blocks in areas with complex motion we also propose a novel motion compensated interpolation scheme which uses the same mathematical tools developed for the qt based motion estimator one of the advantages of this scheme is the globally optimal control of the tradeoff between the interpolation error energy and the dvf smoothness another advantage is that no interpolation of the dvf is required since we directly estimate the dvf and the qt segmentation for the frame which needs to be interpolated we present results with the proposed qt based motion estimator which show that for the same dfd energy the proposed estimator uses about 25 fewer bits than the commonly used block matching algorithm we also experimentally compare the interpolated frames using the proposed motion compensated interpolation scheme with the reconstructed original frames

graph based representation for multiview image geometry in this paper we propose a new geometry representation method for multiview image sets our approach relies on graphs to describe the multiview geometry information in a compact and controllable way the links of the graph connect pixels in different images and describe the proximity between pixels in 3d space these connections are dependent on the geometry of the scene and provide the right amount of information that is necessary for coding and reconstructing multiple views our multiview image representation is very compact and adapts the transmitted geometry information as a function of the complexity of the prediction performed at the decoder side to achieve this our graph based representation gbr carefully selects the amount of geometry information needed before coding this is in contrast with depth coding which directly compresses with losses the original geometry signal thus making it difficult to quantify the impact of coding errors on geometry based interpolation we present the principles of this gbr and we build an efficient coding algorithm to represent it we compare our gbr approach to classical depth compression methods and compare their respective view synthesis qualities as a function of the compactness of the geometry description we show that gbr can achieve significant gains in geometry coding rate over depth based schemes operating at similar quality experimental results demonstrate the potential of this new representation

multiresolution graph fourier transform for compression of piecewise smooth images piecewise smooth pws images e g depth maps or animation images contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces leveraging on recent advances in graph signal processing in this paper we propose to compress the pws images using suitable graph fourier transforms gfts to minimize the total signal representation cost of each pixel block considering both the sparsity of the signal s transform coefficients and the compactness of transform description unlike fixed transforms such as the discrete cosine transform we can adapt gft to a particular class of pixel blocks in particular we select one among a defined search space of gfts to minimize total representation cost via our proposed algorithms leveraging on graph optimization techniques such as spectral clustering and minimum graph cuts furthermore for practical implementation of gft we introduce two techniques to reduce computation complexity first at the encoder we low pass filter and downsample a high resolution hr pixel block to obtain a low resolution lr one so that a lr gft can be employed at the decoder upsampling and interpolation are performed adaptively along hr boundaries coded using arithmetic edge coding so that sharp object boundaries can be well preserved second instead of computing gft from a graph in real time via eigen decomposition the most popular lr gfts are pre computed and stored in a table for lookup during encoding and decoding using depth maps and computer graphics images as examples of the pws images experimental results show that our proposed multiresolution gft scheme outperforms h 264 intra by 6 8 db on average in peak signal to noise ratio at the same bit rate

harmonic maps from degenerating riemann surfaces we study harmonic maps from degenerating riemann surfaces with uniformly bounded energy and show the so called generalized energy identity we find conditions that are both necessary and sufficient for the compactness in w 1 2 and c 0 modulo bubbles of sequences of such maps

lossless image compression by quantization feedback in a content driven enhanced laplacian pyramid in this paper the effects of quantization noise feedback on the entropy of laplacian pyramids are investigated this technique makes it possible for the maximum absolute reconstruction error to be easily and strongly upper bounded near lossless coding and therefore allows reversible compression the entropy minimizing optimum quantizer is obtained by modeling the first order distributions of the differential signals as laplacian densities and by deriving a model for the equivalent memoryless entropy a novel approach based on an enhanced laplacian pyramid is proposed for the compression either lossless or lossy of gray scale images major details are prioritized through a content driven decision rule embedded in a uniform threshold quantizer with noise feedback lossless coding shows improvements over reversible joint photographers expert group jpeg and the reduced difference pyramid schemes while lossy coding outperforms jpeg with a significant peak signal to noise ratio psnr gain also subjective quality is higher even at very low bit rates due to the absence of the annoying impairments typical of jpeg moreover image versions having resolution and snr that are both progressively increasing are made available at the receiving end from the earliest retrieval stage on as intermediate steps of the decoding procedure without any additional cost

brownian motion and the formation of singularities in the heat flow for harmonic maps we develop a general framework for a stochastic interpretation of certain nonlinear pdes on manifolds the linear operation of takin expectations is replaced by the concept of martingale means namely the notion of deterministic starting points of martingales with respect to the levi civita connection ending up at a prescribed state we formulate a monotonicity condition for the riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation such estimates lead to simple criteria for blow ups in the nonlinear heat flow for harmonic maps with small initial energy

framing pyramids burt and adelson 1983 introduced the laplacian pyramid lp as a multiresolution representation for images we study the lp using the frame theory and this reveals that the usual reconstruction is suboptimal we show that the lp with orthogonal filters is a tight frame and thus the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform in more general cases we propose an efficient filterbank fb for the reconstruction of the lp using projection that leads to a proved improvement over the usual method in the presence of noise setting up the lp as an oversampled fb we offer a complete parameterization of all synthesis fbs that provide perfect reconstruction for the lp finally we consider the situation where the lp scheme is iterated and derive the continuous domain frames associated with the lp

a weak energy identity and the length of necks for a sequence of sacks uhlenbeck α harmonic maps abstract in this paper we discuss the convergence behavior of a sequence of harmonic maps u with e u c from a compact surface m g into a compact riemannian manifold n h without boundary generally such a sequence converges weakly to a harmonic map in w 1 2 m n and strongly in c away from a finite set of points in m if energy concentration phenomena appears we show a generalized energy identity and discover a direct convergence relation between the blow up radius and the parameter which ensures the energy identity and no neck property we show that the necks converge to some geodesics moreover in the case there is only one bubble a length formula for the neck is given in addition we also give an example which shows that the necks contain at least a geodesic of infinite length

entropy based distortion measure and bit allocation for wavelet image compression quality criteria for image coding are often based on mean square error however this is not always a relevant measure of visual quality at low bit rates here we investigate the properties of a distortion measure based on the conditional differential entropy of the input signal given its quantized value the proposed measure appears to be a correct representation of the amount of information lost by quantization an adaptive bit allocation algorithm is proposed in order to take advantage of this criterion experimental results illustrate the behavior of the proposed distortion measure and exhibit interesting visual properties for low bit rate subband image coding

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

solutions of a quadratic nonlinear oscillator iteration procedure a modified iteration procedure is applied to a quadratic nonlinear oscillator qno when the solutions of the two auxiliary equations are available we obtain the first and second analytical approximate solutions to the qno our second approximation result significantly improves on the accuracy of the approximate solution obtained by using the first order harmonic balance method

optimized update prediction assignment for lifting transforms on graphs transformations on graphs can provide compact representations of signals with many applications in denoising feature extraction or compression in particular lifting transforms have the advantage of being critically sampled and invertible by construction but the efficiency of the transform depends on the choice of a good bipartition of the graph into update mathcal u and prediction mathcal p nodes this is the update prediction mathcal u mathcal p assignment problem which is the focus of this paper we analyze this problem theoretically and derive an optimal mathcal u mathcal p assignment under assumptions about signal model and filters furthermore we prove that the best mathcal u mathcal p partition is related to the correlation between nodes on the graph and is not the one that minimizes the number of conflicts connections between nodes of same label or maximizes the weight of the cut we also provide experimental results in randomly generated graph signals and real data from image and video signals that validate our theoretical conclusions demonstrating improved performance over state of the art solutions for this problem

efficient multiview depth coding optimization based on allowable depth distortion in view synthesis depth video is used as the geometrical information of 3d world scenes in 3d view synthesis due to the mismatch between the number of depth levels and disparity levels in the view synthesis the relationship between depth distortion and rendering position error can be modeled as a many to one mapping function in which different depth distortion values might be projected to the same geometrical distortion in the synthesized virtual view image based on this property we present an allowable depth distortion add model for 3d depth map coding then an add based rate distortion model is proposed for mode decision and motion disparity estimation modules aiming at minimizing view synthesis distortion at a given bit rate constraint in addition an add based depth bit reduction algorithm is proposed to further reduce the depth bit rate while maintaining the qualities of the synthesized images experimental results in intra depth coding show that the proposed overall algorithm achieves bjontegaard delta peak signal to noise ratio gains of 1 58 and 2 68 db on average for half and integer pixel rendering precisions respectively in addition the proposed algorithms are also highly efficient for inter depth coding when evaluated with different metrics

iterative rate distortion optimization of h 264 with constant bit rate constraint in this paper we apply the primal dual decomposition and subgradient projection methods to solve the rate distortion optimization problem with the constant bit rate constraint the primal decomposition method enables spatial or temporal prediction dependency within a group of picture gop to be processed in the master primal problem as a result we can apply the dual decomposition to minimize independently the lagrangian cost of all the mbs using the reference software model of h 264 furthermore the optimal lagrange multiplier is iteratively derived from the solution of the dual problem as an example we derive the optimal bit allocation condition with the consideration of temporal prediction dependency among the pictures experimental results show that the proposed method achieves better performance than the reference software model of h 264 with rate control

an optimal polygonal boundary encoding scheme in the rate distortion sense in this paper we present fast and efficient methods for the lossy encoding of object boundaries that are given as eight connect chain codes we approximate the boundary by a polygon and consider the problem of finding the polygon which leads to the smallest distortion for a given number of bits we also address the dual problem of finding the polygon which leads to the smallest bit rate for a given distortion we consider two different classes of distortion measures the first class is based on the maximum operator and the second class is based on the summation operator for the first class we derive a fast and optimal scheme that is based on a shortest path algorithm for a weighted directed acyclic graph for the second class we propose a solution approach that is based on the lagrange multiplier method which uses the above mentioned shortest path algorithm since the lagrange multiplier method can only find solutions on the convex hull of the operational rate distortion function we also propose a tree pruning based algorithm that can find all the optimal solutions finally we present results of the proposed schemes using objects from the miss america sequence

bubbling of the heat flows for harmonic maps from surfaces in this article we prove that any palais smale sequence of the energy functional on surfaces with uniformly l2 bounded tension fields converges pointwise by taking a subsequence if necessary to a map from connected possibly singular surfaces which consist of the original surfaces and finitely many bubble trees we therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time 1997 john wiley sons inc

image coding using adaptive recursive interpolative dpcm a predictive image coder having minimal decoder complexity is presented the image coder utilizes recursive interpolative dpcm in conjunction with adaptive classification entropy constrained trellis coded quantization and optimal rate allocation to obtain signal to noise ratios snrs in the range of those provided by the most advanced transform coders

motion compensated 3 d subband coding of video this paper describes a video coding system based on motion compensated three dimensional 3 d subband wavelet coding mc 3dsbc which can overcome the limits of both 3 d sbc and mc prediction based coding in this new system spatio temporal subbands are generated by mc temporal analysis and a spatial wavelet transform and then encoded by 3 d subband finite state scalar quantization 3dsb fssq the rate allocation from the gop level to each class of subbands is optimized by utilizing the structural property of mc 3dsbc that additive superposition approximately holds for both rate and distortion the proposed video coding system is applied to several test video clips its performance exceeds that of both a known mpeg 1 implementation and a similar subband mc predictive coder while maintaining modest computational complexity and memory size

asymptotic stability of harmonic maps under the schrodinger flow for schr odinger maps from r 2 r to the 2 sphere s 2 it is not known if finite energy solutions can form singularities blow up in finite time we consider equivariant solutions with energy near the energy of the two parameter family of equivariant harmonic maps we prove that if the topological degree of the map is at least four blowup does not occur and global solutions converge in a dispersive sense i e scatter to a fixed harmonic map as time tends to infinity the proof uses among other things a time dependent splitting of the solution the generalized hasimoto transform and strichartz dispersive estimates for a certain two space dimensional linear schr odinger equation whose potential has critical power spatial singularity and decay along the way we establish an energy space local well posedness result for which the existence time is determined by the length scale of a nearby harmonic map

approximations for motion of the oscillators with a non negative real power restoring force abstract oscillators with a non negative real power restoring force are considered in this paper this type of restoring force is related to systems with a quasi zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature examples of these types of restoring force are grounded in real physical and engineering systems periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and hamilton s variational principle then the approach is extended to non conservative oscillators by adjusting the elliptic krylov bogoliubov method the methods proposed for the conservative and non conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force several examples the majority of which are so far unsolved are given to illustrate the methods proposed and to demonstrate their generality which permits unforeseen solutions for motion containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force the results obtained are compared with numerical results and have excellent accuracy

wavelet based joint estimation and encoding of depth image based representations for free viewpoint rendering we propose a wavelet based codec for the static depth image based representation which allows viewers to freely choose the viewpoint the proposed codec jointly estimates and encodes the unknown depth map from multiple views using a novel rate distortion rd optimization scheme the rate constraint reduces the ambiguity of depth estimation by favoring piece wise smooth depth maps the optimization is efficiently solved by a novel dynamic programming along trees of integer wavelet coefficients the codec encodes the image and the depth map jointly to decrease their redundancy and to provide a rd optimized bitrate allocation between the two the codec also offers scalability both in resolution and in quality experiments on real data show the effectiveness of the proposed codec

arbitrarily shaped motion prediction for depth video compression using arithmetic edge coding depth image compression is important for compact representation of 3d visual data in texture plus depth format where texture and depth maps from one or more viewpoints are encoded and transmitted a decoder can then synthesize a freely chosen virtual view via depth image based rendering using nearby coded texture and depth maps as reference further depth information can be used in other image processing applications beyond view synthesis such as object identification segmentation and so on in this paper we leverage on the observation that neighboring pixels of similar depth have similar motion to efficiently encode depth video specifically we divide a depth block containing two zones of distinct values e g foreground and background into two arbitrarily shaped regions sub blocks along the dividing boundary before performing separate motion prediction mp while such arbitrarily shaped sub block mp can lead to very small prediction residuals resulting in few bits required for residual coding it incurs an overhead to transmit the dividing boundaries for sub block identification at decoder to minimize this overhead we first devise a scheme called arithmetic edge coding aec to efficiently code boundaries that divide blocks into sub blocks specifically we propose to incorporate the boundary geometrical correlation in an adaptive arithmetic coder in the form of a statistical model then we propose two optimization procedures to further improve the edge coding performance of aec for a given depth image the first procedure operates within a code block and allows lossy compression of the detected block boundary to lower the cost of aec with an option to augment boundary depth pixel values matching the new boundary given the augmented pixels do not adversely affect synthesized view distortion the second procedure operates across code blocks and systematically identifies blocks along an object contour that should be coded using sub block mp via a rate distortion optimized trellis experimental results show an average overall bitrate reduction of up to 33 over classical h 264 avc

bit allocation for dependent quantization with applications to multiresolution and mpeg video coders we address the problem of efficient bit allocation in a dependent coding environment while optimal bit allocation for independently coded signal blocks has been studied in the literature we extend these techniques to the more general temporally and spatially dependent coding scenarios of particular interest are the topical mpeg video coder and multiresolution coders our approach uses an operational rate distortion r d framework for arbitrary quantizer sets we show how a certain monotonicity property of the dependent r d curves can be exploited in formulating fast ways to obtain optimal and near optimal solutions we illustrate the application of this property in specifying intelligent pruning conditions to eliminate suboptimal operating points for the mpeg allocation problem for which we also point out fast nearly optimal heuristics additionally we formulate an efficient allocation strategy for multiresolution coders using the spatial pyramid coder as an example we then extend this analysis to a spatio temporal 3 d pyramidal coding scheme we tackle the compatibility problem of optimizing full resolution quality while simultaneously catering to subresolution bit rate or quality constraints we show how to obtain fast solutions that provide nearly optimal typically within 0 3 db full resolution quality while providing much better performance for the subresolution layer typically 2 3 db better than the full resolution optimal solution

motion vector optimization of control grid interpolation and overlapped block motion compensation using iterated dynamic programming the application of advanced motion compensation techniques control grid interpolation cgi and overlapped block motion compensation obmc to video coding systems provides significant performance advantages terms of compression ratio and visual quality over traditional block matching motion compensation however the two dimensional 2 d interdependence among motion vectors introduced by these compensation frameworks makes the problem of finding rate distortion optimal motion vectors computationally prohibitive thus iterative optimization techniques are often used to achieve good compensation performance while most reported optimization algorithms adopt an approach that uses a block matching algorithm to obtain an initial estimate and then successively optimize each motion vector the over relaxed motion vector dependency relations often result in considerable performance degradation in view of this problem we present a new optimization scheme for dependent motion vector optimization problems one based on dynamic programming our approach efficiently decomposes 2 d dependency problems into a series of one dimensional 1 d dependency problems we show that a reliable initial estimate of motion vectors can be obtained efficiently by only considering the dependency in the rate term we also show that at the iterative optimization stage an effective logarithmic search strategy can be used with dynamic programming to reduce the necessary complexity involved in distortion computation compared to conventional iterative approaches our experimental results demonstrate that our algorithm provides superior rate and distortion performance while maintaining reasonable complexity

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

long time existence for yang mills flow we establish that finite time singularities do not occur in four dimensional yang mills flow confirming the conjecture of schlatter et al am j math 120 117 128 1998 the proof relies on a weighted energy identity and sharp decay estimates in the neck region

optimal buffered compression and coding mode selection for mpeg 4 shape coding we propose an optimal buffered compression algorithm for shape coding as defined in the forthcoming mpeg 4 international standard the mpeg 4 shape coding scheme consists of two steps first distortion is introduced by down and up scaling then context based arithmetic encoding is applied since arithmetic coding is lossless the down up scaling step is considered as a virtual quantizer we first formulate the buffer constrained adaptive quantization problem for shape coding and then propose an algorithm for the optimal solution under buffer constraints previously the fact that a conversion ratio cr of 1 4 makes a coded image irritating to human observers for qcif size was reported for mpeg 4 shape coding therefore careful consideration for small size images such as qcif should be given to prevent coded images from being unacceptable to this end a low bit rate tuned algorithm is proposed in this paper as well experimental results are given using an mpeg 4 shape codec

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

stable blowup dynamics for the 1 corotational energy critical harmonic heat flow we exhibit a stable finite time blowup regime for the 1 corotational energy critical harmonic heat flow from 2 into a smooth compact revolution surface of 3 that reduces to the semilinear parabolic problem for a suitable class of functions f the corresponding initial data can be chosen smooth well localized and arbitrarily close to the ground state harmonic map in the energy critical topology we give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den berg hulshof and king our approach lies in the continuation of the study of the 1 equivariant energy critical wave map and schrodinger map with 2 target by merle raphael and rodnianski 2012 wiley periodicals inc

remarks on the regularity criterion for the nematic liquid crystal flows in r 3 in this paper we investigate the regularity criteria for the 3d nematic liquid crystal flows in terms of the components of the velocity u or the gradient of the velocity field u more precisely the anisotropic integrability regularity criterion to the 3d nematic liquid crystal flows involving i u i 1 2 3 is obtained firstly and then it is proved that the solution is regular if u3 3 or 3u3 3 satisfies the prodi serrin type conditions

asymptotic stability concentration and oscillation in harmonic map heat flow landau lifshitz and schrodinger maps on mathbb r 2 we consider the landau lifshitz equations of ferromagnetism including the harmonic map heat flow and schrodinger flow as special cases for degree m equivariant maps from mathbb r 2 to mathbb s 2 if m 3 we prove that near minimal energy solutions converge to a harmonic map as t asymptotic stability extending previous work gustafson et al duke math j 145 3 537 583 2008 down to degree m 3 due to slow spatial decay of the harmonic map components a new approach is needed for m 3 involving among other tools a normal form for the parameter dynamics and the 2d radial double endpoint strichartz estimate for schrodinger operators with sufficiently repulsive potentials which may be of some independent interest when m 2 this asymptotic stability may fail in the case of heat flow with a further symmetry restriction we show that more exotic asymptotics are possible including infinite time concentration blow up and even eternal oscillation

on dependent bit allocation for multiview image coding with depth image based rendering the encoding of both texture and depth maps of multiview images captured by a set of spatially correlated cameras is important for any 3 d visual communication system based on depth image based rendering dibr in this paper we address the problem of efficient bit allocation among texture and depth maps of multiview images more specifically suppose we are given a coding tool to encode texture and depth maps at the encoder and a view synthesis tool to construct intermediate views at the decoder using neighboring encoded texture and depth maps our goal is to determine how to best select captured views for encoding and distribute available bits among texture and depth maps of selected coded views such that the visual distortion of desired constructed views is minimized first in order to obtain at the encoder a low complexity estimate of the visual quality of a large number of desired synthesized views we derive a cubic distortion model based on basic dibr properties whose parameters are obtained using only a small number of viewpoint samples then we demonstrate that the optimal selection of coded views and quantization levels for corresponding texture and depth maps is equivalent to the shortest path in a specially constructed 3 d trellis finally we show that using the assumptions of monotonicity in the predictor s quantization level and distance suboptimal solutions can be efficiently pruned from the feasible space during solution search experiments show that our proposed efficient selection of coded views and quantization levels for corresponding texture and depth maps outperforms an alternative scheme using constant quantization levels for all maps commonly used in video standard implementations by up to 1 5 db moreover the complexity of our scheme can be reduced by at least 80 over the full solution search

winding behaviour of finite time singularities of the harmonic map heat flow we settle a number of questions about the possible behaviour of the harmonic map heat flow at finite time singularities in particular we show that a type of nonuniqueness of bubbles can occur at finite time we show that the weak limit of the flow at the singular time can be discontinuous we determine exactly the polynomial rate of blow up in one particular example and we show that winding behaviour of the flow can lead to an unexpected failure of convergence when the flow is locally lifted to the universal cover of the target manifold

optimization of mpeg 2 snr scaleable codecs it is shown how the signal to noise snr scaleable coder can benefit from optimizing quantified discrete cosine transform dct coefficients in a rate distortion context in order to reduce bit rate overheads the technique is based on adjusting the quantized coefficients rather than dropping them since the former gives finer control over rate distortion trade offs the widely used lagrangian optimization technique is then applied to arrive at the optimally adjusted coefficients block we show that such an optimization is very efficient for the second layer but has little effect on the base layer

compression of laplacian pyramids through orthogonal transforms and improved prediction scalable representation of visual signals such as image and video signals has become a subject of active research since early 1980s scalability allows the adaptation of the bit rate and or the resolution of the transmitted data to the network bandwidth and or the rendering capability of the receiving device for many years spatial scalability has been achieved through wavelets but recently the laplacian pyramid lp has become an alternative choice because of reduced aliasing in the lower resolutions in this paper we focus on the coding efficiency of the lp with a view to transmitting it over a communication channel in particular we aim to improve the compression efficiency of the lp detail layers through improved interlayer prediction and orthogonal spatial transforms first we consider an lp in the open loop configuration and propose to improve its rate distortion performance by compressing it to a critically sampled representation we derive four different orthogonal spatial transforms from the upsampling and downsampling filters that can achieve this representation and apply them on the detail layers the application of these transforms to the detail layers renders a fixed number of transform coefficients either zero or redundant thus making their transmission unnecessary then we consider the compression of an lp in the closed loop configuration through similar spatial transforms because of the introduction of quantization in the prediction loop these spatial transforms applied on the detail layers do not produce the same number of zero or redundant transform coefficients as in the open loop case nevertheless the insight obtained from the open loop coding leads us to enhance the interlayer prediction and the subsequent application of the spatial transforms to the new detail layers aims to achieve better energy compaction

an operational rate distortion optimal single pass snr scalable video coder we introduce a new methodology for signal to noise ratio snr video scalability based on the partitioning of the dct coefficients the dct coefficients of the displaced frame difference dfd for inter blocks or the intensity for intra blocks are partitioned into a base layer and one or more enhancement layers thus producing an embedded bitstream subsets of this bitstream can be transmitted with increasing video quality as measured by the snr given a bit budget for the base and enhancement layers the partitioning of the dct coefficients is done in a way that is optimal in the operational rate distortion sense the optimization is performed using lagrangian relaxation and dynamic programming dp experimental results are presented and conclusions are drawn

consistent picture quality control strategy for dependent video coding typically a video rate control algorithm minimizes the average distortion denoted as minave at the cost of large temporal quality variation especially for videos with high motion and frequent scene changes to alleviate the negative effect on subjective video quality another criterion that restricts a small amount of quality variation among adjacent frames is preferred for practical applications as pointed out by although some existing proposals can produce consistent quality videos they often fail to fully utilize the available bits to minimize the global total distortion in this paper we would like to achieve the triple goal of consistent quality video minimizing the total distortion and meeting the bit budget strictly all at the same time on the interframe dependent coding structure two approaches are taken to accomplish this goal in the first algorithm a trellis based framework is proposed one of our contributions is to derive an equivalent condition between the distortion minimization problem and the budget minimization problem second our trellis state tree node is defined in terms of distortion which facilitates the consistent quality control third by adjusting one key parameter in our algorithm a solution in between the minave and the constant quality criteria can be obtained the second approach is to combine the lagrange multipliers method together with the consistent quality control the psnr performance is degraded slightly but the computational complexity is significantly reduced simulation results show that both our approaches produce a much smaller psnr variation at a slight average psnr loss as compared to the mpeg jm rate control when they are compared to the other consistent quality proposals only the proposed algorithms can strictly meet the target bit budget requirement no more no less and produce the largest average psnr at a small psnr variation

heat flow of harmonic maps whose gradients belong to l n _ x l infty _ t for any compact n dimensional riemannian manifold m g without boundary a compact riemannian manifold n subset mathbb r k without boundary and 0 t we prove that for n 4 if u m 0 t n is a weak solution to the heat flow of harmonic maps such that nabla u in l n x l infty t m times 0 t then u c m 0 t n as a consequence we show that for n 3 if 0 t is the maximal time interval for the unique smooth solution u c m 0 t n of 1 1 then nabla u t l n m blows up as t t

rate distortion optimization for h 264 interframe coding a general framework and algorithms rate distortion rd optimization for h 264 interframe coding with complete baseline decoding compatibility is investigated on a frame basis using soft decision quantization sdq rather than the standard hard decision quantization we first establish a general framework in which motion estimation quantization and entropy coding in h 264 for the current frame can be jointly designed to minimize a true rd cost given previously coded reference frames we then propose three rd optimization algorithms a graph based algorithm for near optimal sdq in h 264 baseline encoding given motion estimation and quantization step sizes an algorithm for near optimal residual coding in h 264 baseline encoding given motion estimation and an iterative overall algorithm to optimize h 264 baseline encoding for each individual frame given previously coded reference frames with them embedded in the indicated order the graph based algorithm for near optimal sdq is the core given motion estimation and quantization step sizes it is guaranteed to perform optimal sdq if the weak adjacent block dependency utilized in the context adaptive variable length coding of h 264 is ignored for optimization the proposed algorithms have been implemented based on the reference encoder jm82 of h 264 with complete compatibility to the baseline profile experiments show that for a set of typical video testing sequences the graph based algorithm for near optimal sdq the algorithm for near optimal residual coding and the overall algorithm achieve on average 6 8 and 12 respectively rate reduction at the same psnr ranging from 30 to 38 db when compared with the rd optimization method implemented in the h 264 reference software

depth map coding optimization using rendered view distortion for 3d video coding in order to improve 3d video coding efficiency we propose methods to estimate rendered view distortion in synthesized views as a function of the depth map quantization error our approach starts by calculating the geometric error caused by the depth map error based on the camera parameters then we estimate the rendered view distortion based on the local video characteristics the estimated rendered view distortion is used in the rate distortion optimized mode selection for depth map coding a lagrange multiplier is derived using the proposed distortion metric which is estimated based on an autoregressive model experimental results show the efficiency of the proposed methods with average savings of 43 in depth map bitrate as compared with encoding the depth maps using the same coding tools but with the rate distortion optimization based on the conventional distortion metric

a regularity criterion for the solution of nematic liquid crystal flows in terms of the b 1 norm abstract in this paper we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions n 3 and in two dimensions n 2 we prove that the solution u d is smooth up to time t provided that there exists a positive constant e 0 0 such that i for n 3 u d l 0 t b 1 e 0 and ii for n 2 d l 0 t b 1 e 0

exact solution of a quadratic nonlinear oscillator abstract the exact solution of a quadratic nonlinear oscillator qno is presented by using the jacobian elliptic sine function the constants in the solution are expressed in terms of the initial conditions the formulas for the period of oscillation t and amplitude b for x 0 are also given this exact solution can serve as a benchmark solution for various approximate solutions it also can be utilized for educational purpose

a video compression scheme with optimal bit allocation among segmentation motion and residual error we present a theory for the optimal bit allocation among quadtree qt segmentation displacement vector field dvf and displaced frame difference dfd the theory is applicable to variable block size motion compensated video coders vbsmcvc where the variable block sizes are encoded using the qt structure the dvf is encoded by first order differential pulse code modulation dpcm the dfd is encoded by a block based scheme and an additive distortion measure is employed we derive an optimal scanning path for a qt that is based on a hilbert curve we consider the case of a lossless vbsmcvc first for which we develop the optimal bit allocation algorithm using dynamic programming dp we then consider a lossy vbsmcvc for which we use lagrangian relaxation and show how an iterative scheme which employs the dp based solution can be used to find the optimal solution we finally present a vbsmcvc which is based on the proposed theory which employs a dct based dfd encoding scheme we compare the proposed coder with h 263 the results show that it outperforms h 263 significantly in the rate distortion sense as well as in the subjective sense

a fast continuation scheme for accurate tracing of nonlinear oscillator frequency response functions a new algorithm is proposed to combine the split frequency harmonic balance method sf hbm with arc length continuation alc for accurate tracing of the frequency response of oscillators with non expansible nonlinearities alc is incorporated into the sf hbm in a two stage procedure stage i involves finding a reasonably accurate response frequency and solution using a relatively large number of low frequency harmonics this step is achieved using the sf hbm in conjunction with alc stage ii uses the sf hbm to obtain a very accurate solution at the frequency obtained in stage i to guarantee rapid path tracing the frequency axis is appropriately subdivided this gives high chance of success in finding a globally optimum set of harmonic coefficients when approaching a turning point however arc lengths are adaptively reduced to obtain a very accurate solution the combined procedure is tested on three hardening stiffness examples a duffing model an oscillator with non expansible stiffness and single harmonic forcing and an oscillator with non expansible stiffness and multiple harmonic forcing the results show that for non expansible nonlinearities and multiple harmonic forcing the proposed algorithm is capable of tracing out frequency response functions with high accuracy and efficiency

repulsion and quantization in almost harmonic maps and asymptotics of the harmonic map flow we present an analysis of bounded energy low tension maps between 2 spheres by deriving sharp estimates for the ratio of length scales on which bubbles of opposite orientation develop we show that we can establish a quantization estimate which constrains the energy of the map to lie near to a discrete energy spectrum one application is to the asymptotics of the harmonic map flow we find uniform exponential convergence in time in the case under consideration

a novel method of minimizing view synthesis distortion based on its non monotonicity in 3d video in depth based 3d video the view synthesis distortion vsd is generally measured by modeling the effect of texture and depth errors separately with such a development it has been referred that the vsd changes monotonically with respect to to both the texture and depth distortions in this paper we find that the vsd does not always change monotonically with them by both theoretical analysis and experimental test when the effect of the texture and depth errors is considered together specifically first we prove that the vsd is non monotonic with the texture distortion that is the vsd increases with the increasing texture distortion at higher distortion range but conversely decreases with it at lower range it is different from the general scenario that only considering the effect of the texture errors we also analytically depict their relationship with low computational cost and identify the turning point at which the change of the vsd is converted second we confirm that the vsd is always monotonic with the depth distortion which is consistent with the general scenario that only considering the effect of the depth errors the non monotonicity property of the vsd can be utilized to improve the viewing performance of 3d video in relevant applications since a minimal value of the vsd exists at the turning point we conduct two applications for this purpose first it is used to generate the synthesis view of minimal distortion which achieves 0 51 db gain of psnr on average for the tested scenarios second it is used for lossy compression of texture videos in 3d video which reduces the coding rate by 24 on average for the tested scenarios meanwhile keeps the vsd not increased simultaneously