Konrad J. and Dubois E. (1992) Bayesian Estimation of Motion Vector Fields, IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 910- 927.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the position, the gray level and the local motion information. The proposed method is based on the unsupcrviscd classification of the feature vectors by considering the displaced frame difference as well. The classification is done according to a decision criterion derived from the Bayesian theory [2] and representing a metric in the parameter space. A radial basis functions (RBF s) decomposition is known to bca good functional approximator and has bccn used in many applications. The first layer units implement Gaussian functions and the output units arc assigned to the moving objects. The ....

.... a posteror distributions in (5) cn be fctored s follows: P(flf l) P( I lI , j,j )P( j lBj,I )P( j lI ) m ) 9) where P( lf 0 psnts the a priori probability of the segmentation and P(j Ij , ft )is the probability of the optic flow estimation depending on the segmen tation and image [2]. Each of the above conditional probabilities can be expressed as an energy function: e(x) xp 00) where Z is a normalizing constant and is a constant controlling the properties of E(X) The probability estimation problem (5) is converted into the minimization of an energy function: E. ....

J. Konrad, E. Dubois, "Bayesian estimation of motion vec- tor fields" IEEE 2'rans. on Pattern Analtsis and Machine Intelligence, vol. 14 no. 9 pp. 910-927 Sep. 1992.

....motion in the image sequence [1] Block matching algorithms, widely used in video coding, do not model the optical flow explicitly and in many situations fail to provide the optimal estimation. Different approaches were investigated for simultaneous optical flow estima tion and segmentation. In [2] the theoretical framework was derived from the maximum a posteriori probability. Simulated annealing algorithm was employed for optical flow and line field estimation. This algorithm however, does not always provide closed contours and usually requires many iterations until the conver gence is ....

....further expressed as: where P(j If l) represents the a priori probability of the segmentation and P(jlj, f l) is the optical flow probability depending on the segmentation and image. Each of the above probabilities can be expressed as an energy functional depending on the image feature vector X [2]: P(X) 2 exp (4) where E is an energy function, Z is a normalizing constant and fl is a constant controlling the properties of the E. The problem of maximizing the probability from (1) is converted into the minimization of an energy function: z(flf l, glf l) The image is partitioned in ....

J. Konrad, E. Dubois, "Bayesian estimation of motion vector fields" IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. PAMI-14, No. 9, pp. 910-927, Sep. 1992.

.... can be further described as: P(J4j,jlf(2) f(1) P(f(2)lJ4j,j,f(1) P(A4JlJ f(1) P(Jlf(1) 5) P(f(2) lf(1) where P(jlf(1) represents the a priori probability of the segmentation and P(A4jlj,f(1) is the probability of the optical flow estimation depending on the segmentation map and image [7]. After expressing each probability as an energy function, we model them with Gaussian functions. The Gaussian function associated with a moving region and implemented by a hidden unit of the MRBF network is given by: qbj(uj) exp [ ua j)T4(ua [ j) WDFD(J4j) 6) where j and j are the center ....

J. Konrad, E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. on Pat. Anal. Mach. Intel., vol. 14, No. 9, pp. 910-927, Sep. 1992.

.... each observation point and arranges the labels in all points according to the expectation (prior knowledge) 2 Key concepts and novel features # MAP motion estimation The motion estimation problem in this MAP framework is shown in equation (1) It is an extension of what Dubois and Konrad [4] [6] have developed. The u in equation (1) is a novel variable for blocksize that will be used on the observation model. The other variables, q and y , are respectively the displacement field and line field, t g and t g are respectively the first and previous image sequences. max ....

....It is indeterminate if after the largest possible block size has been tried, the occurrence is more than once. It will be the best proposed motion vector if the best probability is achieved and the occurrence is only once. # Displacement and line model The model here is the same as that in [6]. The equation of displacement model is: # # # # r a t u C y G q 9 Q # # 0 . # = 8) where d d = is a variance for energy function and d Z is the partition function. These functions are designated as an interaction between motion vectors. The line field indicates discontinuity ....

J. Konrad and E. Dubois, " Bayesian Estimation of Motion Vector Fields," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 910-927, Sept. 1992.

.... (MRF) models have been successfully introduced in many fundamental issues of image analysis and computer vision such as image restoration [5] 14] edge detection [13] image segmentation [9] 13] computed tomography [11] surface reconstruction [9] 30] stereovision [2] motion analysis [18] [24], 37] or scene interpretation [33] The mathematical framework is a statistical one: entities of interest in a given task are described by statistical models (Markov random fields) and Bayesian estimation theory is used to extract the relevant information from the observed images. By defining ....

....of MRF models with multiresolution decompositions of the images to process. Gaussian pyramids, wavelet decompositions have for instance been used, but generally the same model was considered at each resolution (same parameters, same neighborhood structure and same potential functions) 2] [24]. Yet, in multigrid implementations of statistical models such as MRF s, the key problem remains the derivation of the model parameters at different scales. When global mathematical consistency is not guaranteed, the parameters and the neighborhood structure associated with the model can only be ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, no. 9, pp. 910-927, 1992.

....in the lattice, and DeltaU is the maximum difference in energy function for two configurations which differ at only one site. Clearly this is a huge number, too large to be used in practice. Also the logarithmic schedule in equation 3. 7 is very slow, and T = C ; 0 a 1 is often used [61]. First Order Mean Field Approximation Whilst SSA finds the MAP configuration, it is slow and computationally intensive. The mean field approximation (MFA) provides a deterministic optimisation technique which retains many of the features of SSA. Consider Var d = d Gamma d) p(d) ....

....be explicitly specified in the values the potential function takes for each line configuration. Figure 4.4 illustrates the various configurations possible (up to a rotation) together with the associated potential function values. These configurations and costs (or similar costs) are commonly used [33, 61, 105]. Also included in the cliques and costs illustrated in figure 4.4 is a cost which serves to exclude isolated pixels, which rarely occur in real images. We now have a model which specifies the joint probability of two fields, the pixel field, h h h h h h h h h h h h h h h h h h h h h . ....

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J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Trans PAMI, 14(9), September 1992.

....of the formulation, the resulting estimator must be able to manage the unavoidable presence of discontinuities in the DVF in order to obtain a reasonable approximation of image flow. Managing the discontinuities in the motion field is an active area of research. For example, Konrad and Dubois [1] and Brailean and Katsaggelos [2] utilized Boolean line processes in a MAP formulation to account for motion boundaries in their estimates. motion boundaries in their estimates. To perform improved estimation in the presence of noise, Geman and Reynolds [3] introduced a family of robust objective ....

J. Konrad and E. Dubois. "Bayesian estimation of motion vector fields". IEEE Trans. on PAMI, 14(9):910-- 927, September 1992.

....across motion discontinuities. However, motion discontinuities are usually not known prior to motion computation the notable chicken and egg problem [17, ch. 12] Many approaches have been attempted to avoid the erroneous smoothing across motion discontinuities, e.g. 4] 6] 15] 16] [20], 23] 25] 28, p. 77] 30] 33] Among the most notable and successful ones seem to be the Markov random field (MRF) based approaches [15] 20] 30] 32] No closed form solution is available with the MRF based approaches, instead an iterative algorithm is used to optimize a highly ....

....[17, ch. 12] Many approaches have been attempted to avoid the erroneous smoothing across motion discontinuities, e.g. 4] 6] 15] 16] 20] 23] 25] 28, p. 77] 30] 33] Among the most notable and successful ones seem to be the Markov random field (MRF) based approaches [15] [20], 30] 32] No closed form solution is available with the MRF based approaches, instead an iterative algorithm is used to optimize a highly nonlinear and nonconvex function. Approaches for motion estimation can be categorized into two groups: i) techniques which usually exploit feature ....

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J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 910--927, Sept. 1992.

....be quadratic. As shown in Section 5. unfortunately (unweighted) Tikhonov regularized mo tion estimators tend to over smooth important discontinu ous features of the DVF. For this reason, the management of discontinuities is an active area of research in motion estimation. In Konrad and Dubois [1] and Brailean and Katsaggelos [2] a hne process was used to adapt their motion models in the presence of abrupt motion boundaries in a Markov random field (MRF) formulation. Black and Ariandan in [3] first introduced the use of robust measures in Eq. 2) They proposed robust non convex measures ....

....Eq. 2) They proposed robust non convex measures for ;b and b and demonstrated the relationship between their robust approach and MRF formulations with a hne process . Although significantly improved results are obtained in both formulations, the criterion to be minimized in the MRF formulation [1, 2] and in the robust formulation of [3] is non convex, and have many local minima requiring costly minimization techniques that only approximate a globally optimum solution. In [4] the authors considered robust non convex d ( b quadratic) in an iterative half quadratic regularization algorithm. The ....

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J. Konrad and E. Dubois. "Bayesian estimation of motion vector fields". 1EEE Trans. on PAMI, 14(9):910- 927, September 1992.

....model, like, for instance, edges to guide the line extraction process [10] or confidence data to guide the the reconstruction of underwater acoustic images [19] More specifically, MRFs are also frequently used in computer vision applications in the context of stereo or motion. For example, in [14] the motion vector is computed adopting a stochastic relaxation criterion, or, in [16] MRFs are used to detect occlusions in image sequences. Stereo disparity estimation methods using MRFs have been proposed in several papers. In [21] a stereo matching algorithm is presented as a regularized ....

J.Konrad and E.Dubois. Bayesian estimation of motion vector field. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(9):910--927, September 1992.

....4 Image Sequence Moving Blocks Temporal Clustering Object Identification Object Memory Object Memory Moving Objects Moving Objects Moving Regions Target Object Tracking Block Based Motion Estimation Spatial Clustering Fig. 1. Overview of a gaseous object tracking system. region based method [18], or the intensity based methods [19] are not well suited for gaseous object segmentation. In this work, we propose a novel temporal clustering method based on tracking object paths. The proposed method records the previous object regions to track the paths of spreading objects and clusters the ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields", IEEE Trans. on Pattern Anal. Machine Intell., vol. 14, pp. 910--927, Sept. 1992.

....in terms of the motion field itself while Nagel [3] 4] introduces an oriented smoothness constraint which suppresses the smoothing in the direction of the local intensity gradient. In other works a line process [5] is used in order to explicitly model the motion discontinuities. Konrad and Dubois [6] model the motion and the discontinuity fields as a pair of coupled MRFs and minimize the resulting energy function by means of stochastic relaxation. Identifying the need to exploit intensity discontinuities in order to detect motion discontinuities, they propose a potential function for the line ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 9, pp. 910--927, Sep. 1992.

....of relations are the spatial and temporal derivatives. A smooth transition will be characterized by having small derivatives, as the velocity levels in neighboring locations will be fairly similar. One way of formulating the smoothness constraint is by using a Markov random field (MRF) model [4] [5], 6] The Markov property states, that the true velocity in a given location (x;z; t) depends on the velocities in a finite neighborhood of (x;z; t) Implementing the derivatives as finite differences obeys the Markov property : v dx (x;z; t) v c (x Dx;z; t 1 ) v c (x;z; t 1 ) v dz ....

J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Trans. Pattern Anal. Mach. Intell., 14:910--927, 1992.

.... 1 Introduction Gibbs random fields (GRF) and their equivalent Markov random fields, have recently been applied to image segmentation [1] 2] 3] 4] 5] 6] 7] edge detection [8] restoration [9] 10] 11] 12] 13] reconstruction [14] 15] coding [16] 17] and motion estimation [18]. Underlying these applications is the notion of representing an image as a random field of primarily local interactions, i.e. using the GRF as an image model. However it has proven difficult to control scale and patterning within the GRF model. Meanwhile, alternate models that provide ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE T. Patt. Analy. and Mach. Intell., vol. PAMI-14, pp. 910--927, Sept. 1992.

....generic knowledge about the features to be estimated. For instance, global energy functions have been successfully introduced in image restoration [4, 9, 16] edge detection [14] image segmentation [14] stereovision [2] computed tomography, surface reconstruction [11] visual motion analysis [7, 8, 20, 26, 31] and scene interpretation [30] However, minimizing a global energy function is often an intricate problem: the number of possible label configurations is generally very large and the global energy function may exhibit many local minima. Computationally demanding stochastic relaxation algorithms ....

....landscape. Fast deterministic relaxation schemes can then be used at coarse scales to obtain a good initial guess that may be refined at the finer scales. This procedure often yields fast convergence towards good estimates, even if the initial configuration of the system is far from the optimum [6, 22, 23, 26]. Multigrid relaxation techniques have been considered for image analysis models based on partial differential equations [13, 37] as well as on MRF Parallel Non linear Multigrid Relaxation 7 function ICM( e; U(o; e) result e 2 Omega ; e : initial configuration ; e(k) current ....

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J. KONRAD and E. DUBOIS. -- Bayesian estimation of motion vector fields. -- IEEE Trans. Pattern Anal. Machine Intell., Vol. 14, No 9: pages 910--927, 1992. 38 E. Memin, F. Heitz and F. Charot

....each of its components are positive, negative, or zero. This is shown in Figure 7.3 where v x denotes the horizontal component of a motion vector and v y its vertical component. The idea is to implicitly model the discontinuity in the motion field and is similar to the approaches considered in [82, 83], wherein the discontinuities of motion fields were modeled by means of binary MRFs. After all the neighboring motion vectors have been classified we then determine the class to which the missing motion vector belongs. This is done by assigning a cost to each class and then choosing the class ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 9, pp. 910--926, September 1992.

....an 8 nearest neighbour support, this effectively allows for implicit motion field interpolation in the missing region. To reduce the complexity of the final solution the motion field is block based, with one motion vector being employed for each specified block in the image. In the manner of e.g. [9, 10] the prior for d n;n Gamma1 (x) the motion vector mapping the pixel at x in frame n into frame n Gamma 1, is as follows. p(d n;n Gamma1 (x)jS n (x) exp Gamma 0 X v2Sn (x) v) d n;n Gamma1 (x) Gamma v] 2 1 A (6) where v is each vector in the neighborhood represented by Sn (x) and ....

J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Trans PAMI, 14(9), September 1992.

....an 8 nearest neighbour support. This effectively allows for implicit motion field interpolation in the missing region. To reduce the complexity of the final solution the motion field is block based, with one motion vector being employed for each specified block in the image. In the manner of e.g. [7, 8] the prior for d n;n Gamma1 (x) the motion vector mapping the pixel at x in frame n into frame n Gamma 1, is as follows. p(d n;n Gamma1 (x)jS n (x) exp Gamma 0 X v2Sn (x) v) d n;n Gamma1 (x) Gamma v] 2 1 A (5) where v is each vector in the neighborhood represented by Sn (x) ....

J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Trans PAMI, 14(9), September 1992.

....generic knowledge about the features to be estimated. For instance, global energy functions have been successfully introduced in image restoration [4, 10, 18] edge detection [16] image segmentation [16] stereovision [2] computed tomography, surface reconstruction [12] visual motion analysis [8, 9, 21, 29] and scene interpretation [35] However, minimizing a global energy function is often an intricate problem: the number of possible label configurations is generally very large and the global energy function may exhibit many local minima. Computationally demanding stochastic relaxation algorithms ....

....relaxation converges to a local minimum of the energy function, but this loss of optimality may be compensated for by an appropriate initial guess. When a relevant initial guess is not available, the solution at convergence may however be far from the optimum and lead to poor performances [23, 29]. A very popular non linear deterministic relaxation scheme, known as the Iterated Conditional Mode (ICM) algorithm, has been proposed by Besag in 1986 [4] ICM basically corresponds to non linear Gauss Seidel relaxation. It typically converges within about ten iterations (i.e. full scans of ....

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J. KONRAD and E. DUBOIS. -- Bayesian estimation of motion vector fields. -- IEEE Trans. Pattern Anal. Machine Intell., Vol. 14, No 9: pages 910--927, 1992.

....problem. It is also well suited to specify non linear interactions between features of different nature, that is for combining information. This approach has already proven fruitful in many computer vision applications [16, 6] image restoration [15] segmentation [4, 43, 23] motion analysis [1, 9, 12, 30, 35, 41], 3D reconstruction [32] tomography [24] In such a formulation, two sets of elements must be defined: observations, i.e. input data to be considered; labels, i.e. output primitives to be determined. We have now to formalize these two sets of variables. Intermediate decision maps Motion ....

J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. Ieee Transactions on Pattern Analysis and Machine Intelligence, 14(9):910--927, September 1992.

....one frame and a block from the reference frame. 1 Scene segmentation formulation from optical flow, based on a posteriori criterion, was introduced in [4] A more general framework for motion and line field estimation based on Markov Random Fields (MRF) and Gibbs distributions [5] was used in [6]. Simulated annealing was employed in these approaches for estimating the optical flow and line field. Iterated Conditional Modes (ICM) was proposed for the restoration of color images in [8] and was applied to motion field smoothing in [9, 10] ICM is a deterministic local optimization approach ....

....Highest Confidence First and provides improvements over the segmentation on the basis of a separate estimation. A gradient based algorithm derived from the Mean Field Theory using global optimization was introduced in [20] Some of these algorithms have been embedded in multiresolution structures [6, 12, 13, 20]. In this study, a cost function associated with the minimization of a global criterion is proposed for simultaneous estimation of the optical flow and segmentation of moving objects. The image is partitioned in block sites situated on a rectangular lattice. Each block site is associated with a ....

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J. Konrad, E. Dubois, "Bayesian estimation of motion vector fields" IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 14, No. 9, pp. 910-927, Sep. 1992.

....the Hadamard transform for real time image coding, in Proc. Soc. Photo Optical Instrumentation Engineers, 1975, pp. 207 211. 10] P. C. Ching and C. C. Goodyear, Walsh transform coding of the speech residual in RELP coders, Proc. Inst. Elect. Eng. G, vol. 131, no. 1, pp. 29 34, Feb. 1984. [11] Y. Tadokoro and T. Higuchi, Conversion factors from Walsh coefficients to Fourier coefficients, IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP 31, pp. 231 232, Feb. 1983. Estimation of Depth Fields Suitable for Video Compression Based on 3 D Structure and Motion of Objects A. Aydn ....

....coupled with each other [12] segmentation and finding correspondences are achieved simultaneously before 3 D motion and depth estimation. A. 2 D Motion Estimation and Object Segmentation Two dimensional motion analysis using Gibbs formulation has been shown to be successful for both estimation [11] and segmentation [12] The Gibbs energy function U , which is the negative exponent of the exponential joint probability density function (pdf) can be formulated in terms of 2 D motion field D, segmentation field R and temporally unpredictable (TU) regions S, as follows: U(D; R; SjI t ;I t01 ....

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J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 910--927, Sept. 1992.

....and the type of smoothness assumed. In optical ow computations [3] smoothness is established by requiring video data to satisfy the augmented optical ow equation as much as possible. Pel recursive algorithms [2] constrain the movement of neighboring pixels to be similar, stochastic frameworks [6] incorporate smoothness by requiring smooth elds to be more probable, etc. Finally, the block matching algorithm, 1 By smoothness we mean dense motion elds that vary slowly except for possible discontinuities in the eld which is perhaps the most widely used motion estimation method [7] ....

J. Konrad and E. Dubois, "Bayesian Estimation of Motion Vector Fields," IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 14, pp. 910-927, September 1992.

....as a system that integrates visual sensing and action. There are two common tasks to be solved in active vision systems: one is the correspondence problem in stereo imaging, the other is motion estimation to dynamically track the objects in the scene. Many researchers have proposed algorithms [2, 5, 19, 22, 23, 26, 32, 51] for these tasks. Our objective here is to provide researchers a simulation environment to simulate image sensing process in motion and stereo systems. In this Chapter, a computer simulation system called Active Vision Simulator (AVS) is presented. AVS is an extension of the Image Defocus 86 87 ....

J. Konrad and E. Dubois, "Bayesian Estimation of Motion Vector Fields," IEEE Transaction on Pattern Analysis and Machine Intelligence, Vol. 14, No. 9, pp. 910-927, Sep. 1992.

....as a system that integrates visual sensing and action. There are two common tasks to be solved in active vision systems: one is the correspondence problem in stereo imaging, the other is motion estimation to dynamically track the objects in the scene. Many researchers have proposed algorithms [1, 2, 3, 4, 5, 6, 8, 11] for these tasks. Our objective here is to provide researchers a simulation environment to simulate image sensing process in motion and stereo systems. In our previous work[7, 10] we proposed a computational model on the image sensing and formation process of a CCD camera system. This model 2 ....

J. Konrad and E. Dubois, "Bayesian Estimation of Motion Vector Fields," IEEE Transaction on Pattern Analysis and Machine Intelligence, Vol. 14, No. 9, pp. 910-927, Sep. 1992.

....SSA repeatedly samples from the distribution as the temperature is reduced. A logarithmic cooling schedule will cause convergence to the MAP configuration [4] Practically the logarithmic cooling schedule is too slow, and the sub optimal exponential schedule, T = Ca k ; a 1, is often used [5]. 2 2.3 First Order Mean Field Approximation (MFA) SSA finds the MAP configuration but is slow and computationally intensive. MFA is a deterministic optimisation technique which retains many of the features of SSA. The mean value of the field is d = X d dp(d) which is the minimum variance ....

J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Trans PAMI, 14(9), September 1992.

....be estimated. Energy minimization techniques have been used in a wide range of applications including image restoration [3, 21] edge detection [20] luminance and texture segmentation [15, 17, 20, 23, 36] stereovision [1] computed tomography, surface reconstruction [11] visual motion analysis [5, 6, 27, 34] and scene interpretation [37] Energy functions involve generally two components, one of which expresses the interaction between the hidden labels and the observations and the other which encodes constraints on the desired solution [20] The choice of these energy functions is either heuristic or ....

J. KONRAD and E. DUBOIS. -- Bayesian estimation of motion vector fields. -- IEEE Trans. Pattern Anal. Machine Intell., Vol. 14, No 9: pages 910--927, 1992.

....and optical flow computation problems verify the superior modeling provided by the new model. 1. INTRODUCTION In the past decade, Bayesian methods have gained popularity in image processing applications such as computerized tomography [1] image restoration [2] and the computation of optical flow [3] to name a few. Consequently image modeling has become increasingly important. Since natural images often have regions with different local characteristics, it is imperative for a good image model to be non homogeneous to adapt to the local behavior of the image. Past approaches to solve this ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 14, no. 9, pp. 910--927, September 1992.

....are susceptible to getting stuck to local minima. For global methods, the global minima can be attained, however, often with extremely high computational cost. For the purpose of estimating motion vectors in the framework of MAP MRF, many methods have been developed with various degrees of success [20, 14, 30, 1, 33]. In these existing methods, either global methods or local methods are used to search for the MAP configuration of the MRF based on their respective prior beliefs. Thereby the strong and weak points of the two different schools of MAP search schemes are all inherited. In order to strike a ....

....of MAP MRF can be stated as the process of maximizing the conditional pdf P ( d t jf t 1 ; f t ) i.e. d t = argmax d t P ( d t jf t 1 ; f t ) 18) which reads: in the presence of f t 1 and f t locate the d t that maximizes the conditional pdf. As described in [36, 20], since P (f t 1 jf t ) is not a function of d t , it can be ignored in maximizing P ( d t jf t 1 ; f t ) with respect to d t . Therefore, after the application of Bayesian theorem, eq. 18) turns out to be: d t = argmax d t [P (f t j d t ; f t 1 ) Delta P ( ....

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J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14:910--927, 1992.

....energy functions. There are many optimization methods that can be used for the calculation, for instance, simulated annealing (SA) iterated conditional modes (ICM) mean field annealing (MFA) and the highest confidence first (HCF) A survey of these schemes can be found in [28] and [29] In [16], J.Konrad and E.Dubois presented stochastic approach based on a Bayesian maximum a posteriori (MAP) estimation for estimating optical flow field. They proposed two motion models (i.e. a globally smooth model and a piecewise smooth model) Together with these models, two estimation criteria i.e. ....

J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE transactions on pattern analysis and machine intelligence, 14(9):910--927, September 1992.

....of motion estimation algorithms extends from machine vision devices to video coding systems. In particular optic flow computation can help in recovering the 3D motion of the objects relative to the viewer [1] in estimating the shapes of the moving objects [2] in segmenting the dynamic scene [3] or in predictively coding video sequences [4] Optic flow represents the apparent motion of the viewed objects onto the image plane: discontinuities of the optic flow vector field correspond to boundaries between objects moving in different ways. In optic flow computation a particular attention ....

.... have been made in order to relax this smoothness constraint, either by imposing it only along objects contours [6] or by weighting it accordingly to the gradient direction [7] More recently a bayesian approach has been proposed which allows for simultaneous motion estimation and segmentation [3]. In this paper the original Horn and Schunck algorithm is modified in such a way to weaken the smoothness effect. This is achieved by introducing nonlinear filtering techniques capable of smoothing signals without deteriorating discontinuities. 2. THE SMOOTHNESS CONSTRAINT In this section an ....

J. Konrad and E. Dubois, Bayesian estimation of motion vector fields. IEEE Trans. on Pattern Analysis and Machine Intelligence 14, 1992, pp. 910--927.

....the position, the gray level and the local motion information. The proposed method is based on the unsupervised classification of the feature vectors by considering the displaced frame difference as well. The classification is done according to a decision criterion derived from the Bayesian theory [2] and representing a metric in the parameter space. A radial basis functions (RBF s) decomposition is known to be a good functional approximator and has been used in many applications. The first layer units implement Gaussian functions and the output units are assigned to the moving objects. The ....

.... S j )P ( M j j S j ;f t Gamma1 )P ( S j jf t Gamma1 ) P (f t jf t Gamma1 ) 9) where P ( S j jf t Gamma1 ) represents the a priori probability of the segmentation and P ( M j j S j ; f t Gamma1 ) is the probability of the optical flow estimation depending on the segmentation and image [2]. Each of the above conditional probabilities can be expressed as an energy functional : P (X) 1 Z exp Gamma E(X) fi (10) where Z is a normalizing constant and fi is a constant controlling the properties of E(X) The probability estimation problem (5) is converted into the minimization ....

J. Konrad, E. Dubois, "Bayesian estimation of motion vector fields" IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 14, no. 9, pp. 910-927, Sep. 1992.

....equivalent to forcing the parameter d and the image I 1 to be statistically independent. Yet, such an assumption is usually false; if we consider stereo or motion pairs, for example, there often exists a high spatial correlation between disparity or motion boundaries and image intensity boundaries [2]. Discarding this correlation yields estimates which are often imprecise at object boundaries, suffering from excessive smoothness. The problem is serious for applications that require estimates very close to the original (true) parameters. For example, high fidelity disparity and motion ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 910--927, Sept. 1992.

.... has been dedicated to this specific problem of discontinuity preserving regularization in computing optical flow (and in computer vision in general) within Markovian framework, binary edge variables similar to Geman and Geman s line processes [19] have thus been introduced (see for instance [22] [28], 41] within anisotropic diffusion framework, non linear Euler Lagrange PDEs have been devised along the same philosophy [11] 14] 15] 32] 35] Adopting a more global viewpoint, Black has pointed out in [3] that the different problems we have just evoked can all be seen as deviations ....

....subset of (R Theta R) S . Assuming that the luminance of a given physical point does not change much between times t and t 1, and that the velocity field is reasonably smooth, one often addresses the optical flow recovery problem by minimizing an objective function of the following type [28]: U(w; f) 4 = X s2S [f(s w s ; t 1) Gamma f(s; t) 2 ff X hs;ri2C kw s Gamma w r k 2 (1) where C is the set of neighboring site pairs (with respect to the first or second order neighborhood system ) and ff 0 is a parameter controlling the balance between the smoothness ....

J. Konrad and E. Dubois, Bayesian estimation of motion vector fields, IEEE Trans. Pattern Anal. Machine Intell., 14(9):910--927, 1992.

....the minimization of a high dimensional (but loosely coupled) objective function such as that presented in the previous section without being trapped in local minima. We have studied this method extensively and shown it to give very good results, although the computational complexity is very high [7]. Alternative methods to avoid local optima and reduce complexity are based on hierarchical approaches. Both the image data and the estimated motion field can be represented at different levels of resolution or detail, in a pyramidal fashion. The solution obtained at one level can be used as a ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-14, pp. 910--927, Sept. 1992.

.... Such a model is possible (and natural) in the framework of Markov random fields (MRFs) via the inclusion of a prior probability distribution (prior knowledge) Early MRF based methods have considered a planar scene composition only [8] 9] while more sophisticated models were proposed later [10], 11] These methods have modeled the image partitions implicitly via a coupled binary MRF (line process) resulting in segmentation boundaries that need not be connected nor closed. To deal with this, an explicit MRF modeling of the field of segmentation labels has been proposed [12] In the same ....

J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, no. 9, pp. 910--927, Sept. 1992.

....camera pan zoom, substantial rate savings have been achieved compared to standard methods based solely on local block motion estimation. B.2 Motion of individual image points At the other extreme of the spectrum, region of support may consist of a single image point (Fig. 3. b) 43] 68] 2] [59]: R x = fxg; x 2 : Then, motion of each image point can be described by a set of parameters such as displacement in the case of linear motion, or velocity acceleration in the case of quadratic trajectories (equation (5) This pixel based or dense motion representation is the least constrained ....

....the constant intensity assumption against motion smoothness is termed a regularization constant. For practical reasons, the equation (17) is often expressed in discrete form where the first term is replaced by J 1 (d) 13) and the second term becomes a discrete version of the Laplacian operator [59]. This formulation is often referred to as regularized, although formally it is not since the first term is no longer quadratic but highly irregular in d. This irregularity is due to the dependence of J 1 on d through the image data g (x d(x) Hence, the overall criterion may have multiple ....

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J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 910--927, Sept. 1992.

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Konrad J. and Dubois E. (1992) Bayesian Estimation of Motion Vector Fields, IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 910- 927.

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J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(9), September 1992.

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E. Dubois and J. Konrad. Bayesian estimation of motion vector fields. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14:910--927, September 1992.

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J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, no. 9, pp. 910-927, 1992.

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J. Konrad and E. Dubois, "Bayesian estimation of motion vector fields," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 9, September 1992. 5

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E. Dubois J. Konrad. Bayesian estimation of motion vector fields. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14(9):910--927, 1992.

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J. Konrad and E. Dubois. Bayesian estimation of motion vector field. IEEE Tr. On PAMI, 14:910-927, September 1992.

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J. Konrad and E. Dubois, "Bayesian Estimation of Motion Vector Fields," IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 14, pp. 910-927, September 1992.

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E. Dubois and J. Konrad, "Bayesian estimation of motion vector fields," IEEE Trans. on Pattern Analysis and Machine Intelligent , vol. 14, pp. 910-927, September 1992.

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J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(9):910--927, 1992.

No context found.

J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(9):910--927, 1992.

No context found.

J. Konrad and E. Dubois. Bayesian estimation of motion vector fields. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(9), September 1992.

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J. Konard and E. Dubois, "Bayesian Estimation of Motion Vector Fields", IEEE Trans. on . Pattern analysis and Machine Intelligence, Vol. 14, pp. 910-927, September 1992.

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