D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, no. 2, pp. 129--139, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we need to specify a model to be estimated for the deformation field. To overcome these difficulties (that are classical in computer vision when minimizing a cost function involving a large number of variables) multigrid approaches have been designed and used in the field of computer vision [5, 15, 19]. Multigrid minimization consists in performing the estimation through a set of nested subspaces. As the algorithm goes further, the dimension of these subspaces increases, and the estimation becomes more and more accurate. In practice, the multigrid minimization usually consists in choosing a set ....

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE PAMI, 8(2):129-- 139, 1986.

....moving between different resolutions, thereby propagating information between coarse and fine scales. Multigrid methods have been primarily used for solving partial differential equations [21] but more recently they have been applied to a variety of imaging problems such as image analysis [22], 23] and anisotropic diffusion [24] Perhaps surprisingly, multigrid algorithms have not been widely applied in tomography problems. In earlier work, Bouman and Sauer [25] used multigrid algorithms to solve the nonquadratic optimization problems resulting from projection tomography applications ....

....expressions for optimization of the cost functional in (17) Our approach is unique because it is formulated directly in an optimization framework. This is in contrast to conventional multigrid algorithms which are formulated to solve differential or integro differential equations [19] 21] [22], 25] 26] To derive (a) b) Fig. 3. Multigrid inversion algorithms. Each iteration alternates a Born approximation step with a single iteration of a nonlinear multigrid algorithm. a) V cycle inversion algorithm, and (b) full multigrid inversion algorithm. our method, we start with the ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 129--139, Mar. 1986.

....the computation of PDE solvers by effectively removing smooth error components which are not damped in some fixed grid relaxation schemes. This advantage of the multigrid methods has been used to expedite convergence in various image processing problems, for example, lightness computation [31], shape from shading [31] optical flow estimation [31] 32] 33] 34] adaptive smoothing of signals [35] multispectral MRI image analysis [36] image matching [37] image restoration [38] and anisotropic diffusion [39] More recently, multigrid algorithms have been used to solve image ....

....of PDE solvers by effectively removing smooth error components which are not damped in some fixed grid relaxation schemes. This advantage of the multigrid methods has been used to expedite convergence in various image processing problems, for example, lightness computation [31] shape from shading [31], optical flow estimation [31] 32] 33] 34] adaptive smoothing of signals [35] multispectral MRI image analysis [36] image matching [37] image restoration [38] and anisotropic diffusion [39] More recently, multigrid algorithms have been used to solve image reconstruction problems. ....

[Article contains additional citation context not shown here]

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. PAMI-8, no. 2, pp. 129--139, March 1986.

....contour flow, edge preserving image smoothing, image registration via deformable models, and image segmentation. The advantages of applying PDE methods to image analysis have been summarized in [7] In particular, some of these problems, such as shape from shading [17] surface reconstruction [35] and active surfaces [12] can be formulated in the framework of energy minimization. Variational principles can be applied to find the energy minimizing surface and lead to solving partial differential equation (PDE) of elliptic type for the minimizing surface f f # f # g; 1) where g is ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, pp. 129--139, March 1986.

.... E, v E t (E is image intensity) and the departure from smoothness 8 = IX7ul 2 Ivvl The smoothed flow was that which minimized the total error 82 = jf(8,2 a28, dx dy where a is a blending constant. More recently, this approach has been formalized using the theory of regularization [31] and extended to use two dimensional confidence measures equivalent to local covariance estimates [1, 22] For our application, smoothing is done on the disparity field, using the inverse variance of the disparity Kalman Filter based Algorithms for Estimating Depth from hnage Sequences 217 ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. PAM1 8:129-139, 1986.

....a mathematically ill posed problem. In order to alleviate this problem, additional assumptions on the unknowns are required. The most commonly used assumption is that the spatially smooth parts of S originate from the illumination image, whereas edges in S are due to the reflectance in the image. [13, 15, 8, 9, 14, 5, 22, 2, 7, 21, 18]. In a previous paper [10] a new variational based Retinex formulation to the Retinex problem was introduced. This formulation took into account the illumination smoothness assumption. In addition, it exploited the known limited range of the reflectance image, and the fact that this image, being ....

D. Terzopoulos, "Image Analysis Using Multigrid Relaxation Methods", IEEE Trans. on PAMI, Vol. 8, 129--139, 1986.

....of iterations) but this may be prohibitive in time critical applications. Within the mathematical community, there has been widespread recent interest in multigrid methods [6] 7] 11] 12] Multigrid techniques have already been used to expedite relaxation problems in image processing [15] [16]. The multigrid methods can be used to provide numerical solutions to the anisotropic diffusion problem. With the multigrid approach, high and low frequency error are eliminated rapidly through the use of a multiresolution representation. The original input image provides the initial estimate ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 129--139, 1986.

....a deterministic relaxation algorithm called Iterative Conditional Modes (ICM) Either algorithm is adequate for synthesising texture. A. Multiscale Relaxation A problem with the single scale relaxation process is that global image characteristics evolve indirectly in the relaxation process [8] [16]. Global image characteristics are typically only propagated across the image lattice by local interactions and therefore evolve slowly, requiring long relaxation times to obtain equilibrium, as defined by equation (9) With multiscale relaxation (MR) we attempt to overcome this problem by ....

....is used to constrain the SR at the next highest resolution. By this method, global image characteristics that have been resolved at a low resolution are infused into the relaxation process at the higher resolutions. This helps reduce the number of iterations required to obtain equilibrium [16]. MR also helps the ICM algorithm converge to an image closer to the global maximum of the joint distribution # [3] 5] The multiscale model may be best described by a multigrid representation of the image, as shown in Fig. 3. The grid at level l = 0 represents the increasing image ....

Demetri Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, no. 2, pp. 129--139, 1986.

....the synthesis tends to a texture more conditional on the starting image x(0) Both algorithms are adequate for synthesising texture. A. Multiscale Relaxation A problem with the single scale relaxation process is that global image characteristics evolve indirectly in the relaxation process [24] [44]. Global image characteristics are typically only propagated across the image lattice by local interactions and therefore evolve slowly, requiring long relaxation times to obtain equilibrium, as defined by equation (26) With multiscale relaxation (MR) we attempt to overcome this problem by ....

....is used to constrain the SR at the next highest resolution. By this method, global image characteristics that have been resolved at a low resolution are infused into the relaxation process at the higher resolutions. This helps reduce the number of iterations required to obtain equilibrium [44]. Multiscale relaxation also helps the ICM algorithm converge to an image that is closer to the global maximum of the joint distribution # [6] 16] The multiscale model may be described by a multigrid representation of the image, as shown in Fig. 2. The grid at level l = 0 represents the image ....

Demetri Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, no. 2, pp. 129--139, 1986.

....the synthesis tends to a texture more conditional on the starting image a(0) Both algorithms are adequate for synthesising texture. A. Multiscale Relaxation A problem with the single scale relaxation process is that global image characteristics evolve indirectly in the relaxation process [24,46]. Global image characteristics are typ ically only propagated across the image lattice by local interactions and therefore evolve slowly, requiring long relaxation times to obtain equilibrium, as defined by equation (26) With multiscale relaxation (MR) we attempt to overcome this problem by ....

....is used to constrain the SR at the next highest resolution. By this method, global image characteristics that have been resolved at a low resolution are infused into the relaxation process at the higher resolutions. This helps reduce the number of iterations required to obtain equilibrium [46]. Multiscale relaxation also helps the ICM algorithm converge to an image that is closer to the global maximum of the joint distribution II [6, 15] The multiscale model may be described by a multigrid representation of the image, as shown in Fig. 2. The grid at level I = 0 represents the image ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Mac1ine Intelligence, 8, no. 2, pp. 129-139, 1986.

....developing algorithms at multiple resolutions. In particular, MR algorithms o#er the promise of computational e#ciency. This can be seen in a variety of methods for the solution of large systems of equations (e.g. representing discretizations of partial di#erential equations) Multigrid methods [44, 45, 109, 190, 319] represent one class of examples, in which coarser (and hence computationally simpler) versions of a problem are used to guide (and thus accelerate) the solution of finer versions, with finer versions used in turn to correct for coarsening or aliasing errors in the coarser versions. Multipole ....

....120] have been developed to allow e#cient zooming in on features of interest. There is also an extensive literature on the use of MRF models together with either full multigrid computational algorithms or purely coarse to fine algorithmic structures. Examples of the former can be found in [319, 70, 109, 356], where the treatment in [319] represents what to the author s knowledge is the first thorough examination of the application of multigrid methods to image processing computer vision problems. Full multigrid methods, such as those used in [319] and discussed in much more depth in references ....

[Article contains additional citation context not shown here]

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Trans. on Pattern Analysis and Machine Intelligence, PAMI-8(2):129--139, March 1986.

....[14] relaxation algorithms. Stochastic algorithms may be drastically time consuming while deterministic schemes often get stuck in local minima of the energy function. Besides, it is known that multigrid methods can improve significantly the convergence rate of iterative relaxation schemes [40]. They are useful when the energy to be minimized presents many local minima. It has indeed been conjectured that multiresolution analysis may, to a certain extent, smooth the energy landscape. Deterministic relaxation schemes can then be used at coarse scales to get a good initial guess, which ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. Pattern Anal. Machine Intell., vol. 8, no. 2, pp. 129-139, 1986.

....elsewhere) For example, in the case of optical ow, the Euler equations result in a pair of elliptic second order partial di erential equations. A number of methods have been proposed to speed up the convergence of the resulting numerical problems, including (for example) multigrid techniques [40]. To apply these algorithms to actual imagery, of course, requires discretization. Discrete labels Iterated conditional modes(ICM) is a greedy technique introduced by Besag in [5] This is an iterative algorithm and it works as follows: the sites are processed sequentially, and for each site the ....

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8:129-139, 1986.

....of linear partial di erential equations (PDE) The solution of the discretized PDE s is obtained using multigrid methods which have optimal complexity in contrast to Jacobi iterations, for example. Multigrid methods in digital image processing have also been used in [5] for image reconstruction in [7] for the computation of optical ow and in [8] for image restoration. Finally, in section 3, we present some experimental results for synthetic and real images. 2 Description of the mathematical model. Throughout this text the notations jj jj and h ; i are used for the norms and inner products ....

D. Terzopoulos, Image analysis using multigrid relaxation methods, IEEE Transactions pattern analysis and maschine intelligence, Vol. Pami-8, 2, (1986).

....motion curves have a much higher objective cost than necessary. If too many control points are selected, then the computational complexity is increased unnecessarily due to the larger number of unknowns as well as the resulting ill conditioning of the linear subproblems that arise in the solution [32]. This complexity issue is addressed by reformulating the DOF functions in a hierarchical basis, in particular, in a B spline wavelet (B wavelet) basis. Wavelets provide a natural and elegant means to include the proper amount of local detail in regions of spacetime that require the extra ....

....about changes in one coefficient travels very slowly (in O#n# iterations) to other parts of the trajectory. In contrast, the hierarchical wavelet basis provides a shorter (O#log#n##) communication distance between any two basis functions. This is the basic insight leading to multigrid methods [32], and the related hierarchical methods discussed here. The wavelet representation also allows the user to easily lock in the coarser level solution and only work on details simply by removing the coarser level basis functions from the optimization. This provides the means to create small systems ....

TERZOPOULOS, D. Image analysis using multigrid relaxation methods. IEEE PAMI 8, 2 (March 1986), 129--139.

.... Similar techniques are employed by multi grid algorithms, which have rst been developed for the solution of partial di erential equations [26] Multi grid methods have been adopted to a broad range of optimization problems with locally interacting variables including image processing tasks [27, 28, 29]. They rely on incremental coarse grid corrections and, therefore, on continuous optimization variables. Similar in spirit but technically di erent are Monte Carlo multi grid methods [30] For discrete problems multiscale optimization techniques are better suited than multi grid methods [17] We ....

D. Terzopoulos, \Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, no. 2, pp. 129-139, 1986.

....segmentation. A number of examples and simulation results are presented to illustrate the algorithm. 1. INTRODUCTION The need to solve elliptic equations of the general form r 2 u u = f (1) arises in several computer vision problem, such as shape from shading [1] surface reconstruction [2] and active contours [3] These problems can be formulated in the framework of variational principles and lead to solving Euler Lagrange equations of elliptic type as the necessary condition for a minimum. Although there exist direct analytical methods for solving these equations on 2D rectangular ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, pp. 129--139, March 1986.

....[f6B (11) where and are the vector of the coefficients of the linear combination. In our experience this is a reasonable approximation for low energy residual textures. A multi scale approach using Gaussian pyramids [28] is used so that the system can handle higher energy residual textures [33]. 5 Registration and Tracking During initialization, the model is automatically positioned and scaled to fit the head in the image plane as described in Sec. 4.3. The reference texture h is then obtained by projecting the initial frame of the sequence j onto the visible part of the ....

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Recognition and Machine Intelligence, 8(2):129--139, 1986.

....moving between di#erent resolutions thereby propagating information between coarse and fine scales. Multigrid methods have been primarily used for solving partial di#erential equations 3 [21] but more recently they have been applied to a variety of imaging problems such as image analysis [22, 23] and anisotropic di#usion [24] Perhaps surprisingly, multigrid algorithms have not been widely applied in tomography problems. In earlier work, Bouman and Sauer [25] used multigrid algorithms to solve the non quadratic optimization problems resulting from projection tomography applications such ....

....the specific expressions for optimization of the cost functional in (18) Our approach is unique because it is formulated directly in an optimization framework. This is in contrast to conventional multigrid algorithms which are formulated to solve di#erential or integro di#erential equations [19, 21, 22, 25, 26]. To derive our method, we start with the two grid case, and then generalize this solution using standard recursions for the V cycle and full multigrid cases [41] 3.1 Two Grid Algorithm For the two grid algorithm, we first consider optimization without the positivity constraint. We then discuss ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. PAMI-8, no. 2, pp. 129--139, March 1986.

....or the amount of change allowed in the parameters do not move the current estimate closer to the global minima. As a result the solution gradually drifts into a local trough and eventually gets trapped inside there. Such problems can be handled reliably by using a multigrid relaxation approach[41]. These methods work by taking advantage of multiple discretizations and smoothing of a continuous problem over a range of resolution levels. Solution to a minimization problem requires computations proportional to the spatial distance between the current estimate and the actual solution. This ....

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Recognition and Machine Intelligence, 8(2):129--139, 1986.

....and offered good convergence to acceptable solutions. The implementation can be enhanced by upgrading the geometric model to a volumetric representation that is certainly more realistic. Additionally, multiresolution techniques can improve the convergence by introducing long range interactions [37]. Moreover, stochastic modeling combined with an aspect graph [14] can be used to account for the imprecision in the pose estimate and the geometric fluctuations of the object. However, this expands the state space dimension and makes stochastic search methods almost unavoidable. Acknowledgements ....

D. Terzopoulos, "Image Analysis Using Multigrid Relaxation Methods," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 4, pp. 413-424, 1986.

....that can be extremely large for real images. 2 Gauss Seidel relaxation methods has a time complexity of O(n 2 ) which is not at all conducive for real time processing. Consequently, multigrid methods have been used for solving computer vision problems with O(n) time complexity [7] 17] [18], 19] The rate of convergence of standard multigrid algorithms is h independent for second order problems; however, it depends adversely on the coefficients. We employ an algebraic multigrid algorithm, reported in [13] that can handle second order elliptic problems with general coefficients, ....

.... S l (x l ; b l ) For more details about basic algebraic multigrid components, see [14] Generation of the family of fP l g is key to the performance of the algebraic multigrid algorithm, since, coarse level matrices are obtained using fP l g. Standard multigrid algorithms used in [17] [18], 19] build the 9 prolongators based on uniform averaging without obeying the coefficients of the stiffness matrix, which is appropriate only for special cases, e.g. Laplacian regularization. In more general cases, as for our proposed technique (also for oriented smoothness [15] 16] and ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. Pattern Anal. Machine Intell., vol. 8, no. 2, pp. 129--138, 1986.

....they give rise to false matches [24] which cannot be avoided simply by increasing the likelihood of large deformations in the prior model. A strategy is needed to search for the solution, and one widely used approach involves solving the matching problem at progressively finer spatial scales [25]. Larger scale motions are inferred by matching the lower spatial frequencies of the images, and then used to remove their effect in motion estimation of the higher frequency content. Priors can be imposed that are scale specific [21] and the uncertainty in the estimates propagated from one scale ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. Pattern Anal. Machine Intell., 8, pp. 129--139, 1986.

.... P k (F k 1 (m) Gamma F k 2 (f 1 (m; Z(m) F 2 Z Gamma 2 ( Phi 0 (jrmZj) jrmZj Z Phi 00 (j rmZ j)Z jj ) j n i;j Phi Boundary conditions on the depth (6) Finally, we apply a Gauss Seidel relaxation method for moving iteratively towards the solution of this problem [27]. 5.1 Discretization Scheme In the following we present the consistent way to discretize the divergence term that appears in the equation (1) Denoting by the angle that the unit gradient rZ= j rZ j) makes with the x axis, we have the well known expressions: Z = sin( 2 Z xx Gamma ....

D. Terzopoulos. Image Analysis Using Multigrid Relaxation Methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(2):129--139, March 1986.

....estimate E(x, y, s) The first contribution shows spatio temporal gradients, while the second one represents low edges characterizing the texture of moving objects. 4. Pyramidal relaxation Multi grid or pyramidal relaxation methods have been successfully applied to a variety of vision problems [8][9] A relaxation method with original incrementing functions is proposed in order to obtain compact and significant shapes of moving objects. Pyramidal relaxation performs an integration of images belonging to different pyramid levels combining high resolution estimates (useful to constraint the ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods", IEEE Trans. Patt. Anal. Mach. Intell, Vol. 8, No. 2, March 1986.

....is available. Deterministic approaches converge to configurations corresponding to local minima of the global energy function. On the other hand, 2 E. Memin, F. Heitz and F. Charot it is known that multigrid methods can significantly improve the convergence rate of iterative relaxation schemes [6, 22, 37]. The major drawback of relaxation algorithms is the amount of computation required to update the image. For real world applications the computation time quickly becomes prohibitive on workstations. On the other hand, in low level vision, the global energy functions usually adopted decompose into ....

....This algorithm is described in Fig. 3. It is used in this paper as a standard example of non linear deterministic relaxation. 2.2. 3 Multigrid relaxation It is well known that multigrid methods can significantly improve the convergence rate of linear and non linear iterative relaxation schemes [6, 19, 22, 27, 37]. Multigrid methods may also be useful when the energy to be minimized has many local minima, as is often the case with non linear models. It has indeed been conjectured that multigrid analysis may, to a certain extent, smooth the energy landscape. Fast deterministic relaxation schemes can then be ....

[Article contains additional citation context not shown here]

D. TERZOPOULOS. -- Image analysis using multigrid relaxation methods. -- IEEE Trans. Pattern Anal. Machine Intell., Vol. 8, No 2: pages 129--139, March 1986. Parallel Non-linear Multigrid Relaxation 39

....3, 4, 12, 13] These formulations result in partial differential equations which when discretized lead to large sparse linear systems. Numerical iterative methods such as the Gauss Seidal, Jacobi and the conjugate gradient technique [8] have been popular until the inception of multi grid methods [24] and multiresolution methods [22, 19, 32] More recently, the capacitance matrix technique [5] has been generalized to solve the linear system arising from the early vision problems very efficiently [28, 17] However, this technique can prove to be inefficient for dense data problems with ....

....conjugate gradient algorithm in section 4. Algorithm implementation on synthetic and real data are presented in section 5. In section 6, we end with a discussion and conclusion. 5 2 Variational Formulations Variational formulations for various early vision problems have been reported in [24] and references therein. These formulations make use of the popular theory of regularization. In a regularization framework, generic smoothness assumptions are imposed on the solution space prior to attempting any functional minimization. The smoothness constraints are well characterized by a ....

[Article contains additional citation context not shown here]

D. Terzopoulos. "Image analysis using multigrid relaxation methods". IEEE Trans. Pattern Anal. Machine Intell., 8:129--139, 1986.

....algorithms remains the amount of computation required to update the image. For real world applications the computation time quickly becomes prohibitive on workstations. Several efficient approaches have been proposed to alleviate this computational burden. Among them, multigrid techniques [7, 23, 43]. have shown to significantly improve the convergence rate of linear and non linear relaxation schemes. It is also well known that the computations involved by these algorithms are regular and local, and lead naturally to massive data parallelism, which is well suited for parallel processing on ....

....[6] can often be used instead, when a good initial guess is available. Deterministic approaches converge to configurations corresponding to local minima of the global energy function. They may be combined with multigrid methods to improve the convergence rate of iterative relaxation schemes [7, 23, 43]. The major drawback of relaxation algorithms is the amount of computation required to update the image. For real world applications the computation time quickly becomes prohibitive on workstations. On the other hand, in low level vision, RR n2184 4 E. M emin, F. Heitz and F. Charot the global ....

[Article contains additional citation context not shown here]

D. TERZOPOULOS. -- Image analysis using multigrid relaxation methods. -- IEEE Trans. Pattern Anal. Machine Intell., Vol. 8, No 2: pages 129--139, March 1986.

....Bq Uc (11) where c and q are the vector of the coefficients of the linear combination. In our experience this is a reasonable approximation for low energy residual textures. A multi scale approach using Gaussian pyramids [28] is used so that the system can handle higher energy residual textures [33]. 5 Registration and Tracking During initialization, the model is automatically positioned and scaled to fit the head in the image plane as described in Sec. 4.3. The reference texture T 0 is then obtained by projecting the initial frame of the sequence I 0 onto the visible part of the ....

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Recognition and Machine Intelligence, 8(2):129--139, 1986.

....support the fusion of different data sources and integrate uncertainty and errorness of observed data. Especially Bayesian networks supply a close connection to pyramidal image interpretation as will be shown below. Hierarchical approaches for segmentation tasks reduce the complexity of search (Terzopoulos, 1986). The algorithm we use differs from complex approaches (e.g. Bouman and Liu (1991) because we use a simpler, more straight forward propagation to reduce the computational effort. 3.1 General strategy The focusing procedure consists of two steps: the generation of a feature pyramid and the ....

Terzopoulos, D. (1986). Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-8 (2), 129--139.

....or the amount of change allowed in the parameters do not move the current estimate closer to the global minima. As a result the solution gradually drifts into a local trough and eventually gets trapped inside there. Such problems can be handled reliably by using a multigrid relaxation approach[41]. These methods work by taking advantage of multiple discretizations and smoothing of a continuous problem over a range of resolution levels. Solution to a minimization problem requires computations proportional to the spatial distance between the current estimate and the actual solution. This ....

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Recognition and Machine Intelligence, 8(2):129--139, 1986.

....models offer a reasonable approach to solving these problems, due to their stability, controllability, and their property of regularizing data gathered over regions of the image. Regularization techniques, or penalized optimization, are used for many applications in vision (see for example [21, 28, 27, 30] and references there) In our application, we recover surfaces in 3D medical data, locating surface boundaries of organs and structures, and providing an approximating differentiable description (see Section 6) The differential description may be used for measurements, recognition, ....

Demetri Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8(2):129--139, March 1986.

.... Horn [12] Variational techniques use the Euler equations, which are guaranteed to hold at a local minimum (although they may also hold elsewhere) A number of methods have been proposed to speed up the convergence of the resulting numerical problems, including (for example) multigrid techniques [18]. To apply these algorithms to actual imagery, of course, requires discretization. An alternative is to use discrete relaxation methods; this has been done by many authors, including [7, 16, 17] It is important to note that a local minimum is defined relative to a set of allowed moves. Most ....

Demetri Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(2):129--139, 1986.

....cost function usually prohibit many local minima, and a very common result is that the final restoration corresponds to one of these minima. On the other hand it has been shown that the multiscale techniques reduce in a significant ratio the required computational cost of the restoration operation [29], and perform a smooth operation in the observed cost functions, which eliminates a large percentage of local minima. These techniques have been widely used in image analysis problems with a positive influence in the restoration process, as well as in the computation complexity. The main idea is ....

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Trans. on Pattern Analysis Machine Inteligence, pages 129--139, 1986.

....optimization. Technically similar are multi grid methods, which have first been developed for the solution of partial differential equations [17] and have since then been adopted to a broad range of optimization problems with locally interacting variables including image processing tasks [18, 19, 20]. Multi grid methods rely on incremental coarse grid corrections and therefore on continuous optimization variables. Similar in spirit but technically different are renormalization group approaches [21] For discrete problems multiscale optimization techniques are better suited. As texture ....

D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, no. 2, pp. 129--139, 1986.

....one equipotential contour can be built for a certain potential value, and hence, will affect the results. The problem is not inherent to the EFT but iss a result of spatial quantization due to the presence of the bottlenecks. A possible solution to this problem is to use the multigrid approach [28] to refine the potential distribution in regions near the bottlenecks, the development of which is beyond the scope of this paper. A natural extension to the EFT based approach is to use it in representing threedimensional objects because the Poisson equation is general and is not restricted to ....

D. Terzopoulos, Image analysis using multigrid relaxation methods, IEEE Trans. Patt. Anal. Machine Intell., vol. 8, pp. 129-139, 1986.

....that is a function graph from sparse data [1] Such techniques allow the generation of an implicit function f : R n R (i.e. a hypersurface in R n 1 ) that is constrained to have as one of its isocontours the input surface S. There are several numerical algorithms to implement these methods [2], 3] In this paper we take a different approach, and propose to exploit the multiscale edges corresponding to the boundary of a solid shape. Our method employs a wavelet al..gorithm [4] to generate a smooth implicit function. It essentially synthesizes a tubular neighborhood corresponding to the ....

D. Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8(2):129--139, March 1986.

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D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 8, no. 2, pp. 129--139, 1986.

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D. Terzopoulos, Image analysis using multigrid relaxation methods, IEEE Trans. Pattern Anal. Mach. Intell. (1986) 129---139.

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D. Terzopoulos, "Image analysis using multigrid relaxation methods," IEEE Trans. Pattern Anal. Machine Intell., vol. 8, pp. 129--139, Mar. 1986.

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Terzopoulos, D. 1986. Image analysis using multigrid relaxation methods. IEEE Trans. on PAMI, 8:129--139.

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Demetri Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(2):129--139, 1986.

No context found.

Demetri Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Trans. PAMI, 8(2):129--139, 1986.

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Terzopoulos, D. Image analysis using multigrid relaxation methods. IEEE PAMI 8, 2 (March 1986), 129--139.

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D. Terzopoulos, "Image Analysis Using Multigrid Relaxation Methods", IEEE Trans. on PAMI, Vol. 8, 129--139, 1986.

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D. Terzopoulos, "Image Analysis Using Multigrid Relaxation Methods", IEEE Trans. on PAMI, Vol. 8, 129-139, 1986.

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D. Terzopoulos, "Image analysis using multigrid relaxation methods", IEEE Trans. on PAMI, vol. 8, no. 2, pp. 129-139, June 1986.

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Demetri Terzopoulos. Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8(2):129--139, March 1986.

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TERZOPOULOS, D. Image analysis using multigrid relaxation methods. IEEE PAMI 8, 2 (March 1986), 129--139.

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Terzopoulos D., `Image analysis using multigrid relaxation methods', IEEE Trans. on Pattern Anal. and Machine Intell., vol. 8, no. 2, (1986).

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