L.Robert, R.Deriche, "Dense Depth Map Reconstruction: A Minimization and Regularization Approach which Preserves Discontinuities," Fourth European Conference on Computer Vision, Cambridge, UK, 1996. (a) Disparity map ZNCC (b) Correlated pixels ZNCC (c) Disparity map CT (d) Correlated pixels CT (e) ZNCC (f) CT  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... matrix: P = p ij = P tr(i, Z t 1 ) j i, j # # recurrence equation X t 1 = tr(X t , Z t 1 ) as HMC # ergodicity and stationarity # componentwise HMC Gibbs Sampler Goodness of an Estimator: # Systematic Errors stationary chain and correlations between samples # Random Error 3 11 Coupling From the Past Nonconstructive Exact Sampling through Monitoring Stationarity [2] 1. Set starting value for the time to go back, t # 1. 2. Generate new random pair M t # 1 = U t , v t ) 3. Start a chain in each labelling fm m = 1, K at time t and run ....

....Errors stationary chain and correlations between samples # Random Error 3 11 Coupling From the Past Nonconstructive Exact Sampling through Monitoring Stationarity [2] 1. Set starting value for the time to go back, t # 1. 2. Generate new random pair M t # 1 = U t , v t ) 3. Start a chain in each labelling fm m = 1, K at time t and run the chains X t 1 (f m ) tr X t (f m ) M t 1 to time 0 i.e. t # t # , t 1, 1 reusing old random pairs of former rounds. 4. Check for coalescence at time 0, that is check if X 0 (f m ) occupy ....

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L. Robert and R Deriche. Dense Depth Map Reconstruction: A Minimization and Regularization Approach which Preserves Discontinuities. In Bernard Buxton, editor, Proceedings of the 4th European Conference on Computer Vision, Cambridge, UK, 1996.

....this path and tried to pose the problem in the framework of statistical estimation theory with a probabilistic continuity prior. The ML estimators of early 90 s, based on dynamic programming, were later put on a more sound basis by [11,12] assuming continuity along epipolar lines) and by others [13,14,15] (assuming isotropic continuity prior) Recent disparity component matching [16] and network flow [17,18,19] formulations of the matching task also include isotropic prior model but their computational complexity is lower. Since the most important cause of ambiguity is poor SNR, one has to be ....

....erroneous, especially in scenes of deep range and relatively thin objects. Various attempts to switch the regularizer o# at discontinuities rely on either detecting image edges under the assumption that they coincide with object boundaries [20,21] or by weighting the regularizer by image gradient [13]. This coincidence does not always occur. Alternatively, they rely on identifying hightension regions in regularized solution [22] releasing the continuity prior there, and re computing the solution. Yet another alternative is to use robust regularizers [15] None of those methods proved fully ....

Robert, L., Deriche, R.: Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In: Proc ICIP. (1992) 123--127

....(France) for his support. nalizes large variations of the function to recover and tends to isotropically smooth the solution, no matter what the true variations of the function ought to be. This problem has been especially addressed in the case of image restoration [13] stereo reconstruction [12], optical flow computation [3] and motion estimation [1] These approaches, however, are image based whereas, given the task of reconstructing a surface from multiple images whose vantage points may be very different, we need a surface representation that can be used to generate images of the ....

....principal curvatures (details can be found in [10] 3.2. Non quadratic regularization A non quadratic regularizer over a bidimensional domain # can be formally expressed as E reg = # #( #z )dxdy where (x, y) is a parametrization of# and z is the function we want to recover. It is shown in [12] that such a regularization can be decomposed in 2 terms respectively expressing the regularization amount in the direction of #z and the direction orthogonal to #z. This leads to specific conditions on the # function and to different kinds of such functions. One of the most popular ones is the ....

L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In European Conference on Computer Vision, Cambridge, U.K., volume 1, pages 439--451, 1996.

....by x Delta y. The cross product of two 3 Theta 1 vectors x and y is noted x Theta y. Partial derivatives will be indicated either using the symbol, e.g. f x , or as a lower index, e.g. fx . Our approach is an extension of previous work by Robert et al. and Robert and Deriche, 22] [21], where the idea of using a variational approach for solving the stereo problem was proposed rst in the classical Tikhonov regularization framework and then by using regularization functions more proper to preserve discontinuities. Our work can be seen as a 3D extension of the approach proposed in ....

....term (f) It is known that if care is not taken, for example by adding a regularizing term to (1) the solution f is likely not to be smooth and therefore any noise in the images may cause the solution to dioeer widely from the real objects. This is more or less the approach taken in [22] [21]. We will postpone the solution of this problem until Sect. IV and in fact solve it dioeerently from the usual way which consists in adding a regularization term to C1(f) Another strategy is to apply the level set idea [20] 24] Consider the family of surfaces S dened by S(x; y; t) x; y; ....

L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In Bernard Buxton, editor, Proceedings of the 4th European Conference on Computer Vision, Cambridge, UK, April 1996.

....pixel m k . The pixels in the images are considered as functions of the 3D geometry of the scene, i.e. of some 3D point M on the surface of an object in the scene, and of the unit normal vector N to this surface. Our approach is an extension of previous work by Robert et al. Robert and Deriche, [19,18], and Deriche et al. 4] For a comparison with the work of those authors see [7] M z y x x 1 y 1 x 2 y 2 xn yn C 1 C 2 Cn m 2 mn m 1 z x 1 x y 1 1 y C F R c M m x 1 = x=z y 1 = y=z z y x (a) b) Fig. 1. a) The multicamera stereo vision problem is, given a pixel ....

L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In Bernard Buxton, editor, Proceedings of the 4th European Conference on Computer Vision, Cambridge, UK, April 1996.

....of two 3 Theta 1 vectors x and y is noted x Theta y. Partial derivatives will be indicated either using the symbol, e.g. f x , or as a lower index, e.g. f x . 2 Comparison with previous work Our approach is an extension of previous work by Robert et al. and Robert and Deriche, [25, 24], where the idea of using a variational approach for solving the stereo problem was proposed rst in the classical Tikhonov regularization framework and then by using regularization functions more proper to preserve discontinuities. Our work can be seen as a 3D extension of the approach proposed in ....

....term (f) It is known that if care is not taken, for example by adding a regularizing term to (1) the solution f is likely not to be smooth and therefore any noise in the images may cause the solution to dioeer widely from the real objects. This is more or less the approach taken in [25, 24]. We will postpone the solution of this problem until Sect. 5 and in fact solve it dioeerently from the usual way which consists in adding a regularization term to C 1 (f) Another strategy is to apply the level set idea [23, 27] Consider the family of surfaces S dened by S(x; y; t) x; y; ....

L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In Bernard Buxton, editor, Proceedings of the 4th European Conference on Computer Vision, Cambridge, UK, April 1996.

....[3] for a comprehensive survey. The common goal of existing stereo algorithm is to assign a single disparity value to each point in the image, producing a 2 1 2 D sketch [10] of the scene. In this framework, stereo matching is usually cast as a constrained functional optimization problem [11]. Optimization techniques such as relaxation, dynamic programming, and stochastic methods are widely used in stereo algorithms. This formulation results in iterative, initialization and parameter dependent solutions, which often fail to handle surface discontinuities and half occlusion (the ....

L. Robert and R. Deriche, "Dense Depth map Reconstruction: A Minimization and Regularization Approach which Preserve Discontinuities", Proc. 4th ECCV, 1996

....order to facilitate a subsequent planar facet segmentation step, we present a geometrically constrained disparity field estimation. This technique is derived from a robust optical flow estimation approach. Unlike classical correlation methods, it provides a reliable piecewise smooth motion field [2, 12]. Moreover the disparity estimation is constrained by the associated epipolar geometry so that the estimated field is explicitly forced to be geometrically consistent with a perspective projection model and with the fixed scene assumption. This constraint also yields a substantial computational ....

L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In Proceedings of the 4th European Conf. on Computer Vision, volume 1, pages 439--451, 1996.

....costs is seemingly an arbitrary choice. We call both types the functional based matching. The popular implementations include the dynamic programming methods [5, 6, 12] and the network optimization methods [11, 15] Other modern implementations include general MAP estimators and related approaches [7, 4, 16, 14, 2, 1, 17, 3]. From the practical point of view, the maximization type of functional based matching tends to work better when ordering constraint is imposed while the continuity term is omitted. This is due to the fact that the trivial solution is the matching table diagonal (i.e. the zero disparity 1 ....

L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In Proc. Int. Conf. Image Processing, pp. 123--127, 1992.

....be able to avoid using tensor calculus altogether. Partial derivatives will be indicated either using the symbol, e.g. f x , or as a lower index, e.g. f x . INRIA PDE s, level set methods, and Stereo 7 Our approach is an extension of previous work by Robert et al. and Robert and Deriche, [18, 17], where the idea of using a variational approach for solving the stereo problem was proposed rst in the classical Tikhonov regularization framework and then by using regularization functions more proper to preserve discontinuities. We dioeer from this work because we do not assume that the depth ....

....term (f) It is known that if care is not taken, for example by adding a regularizing term to (1) the solution f is likely not to be smooth and therefore any noise in the images may cause the solution to dioeer widely from the real objects. This is more or less the approach taken in [18, 17]. We will postpone the solution of this problem until section 4 and in fact solve it dioeerently from the usual way which consists of adding a regularization term to C 1 (f) Another strategy is to apply the level set idea [16, 20] Consider the family of surfaces S dened by S(x; y; t) x; y; ....

L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In Bernard Buxton, editor, Proceedings of the 4th European Conference on Computer Vision, Cambridge, UK, April 1996.

....term. These deviations occur especially at spatial discontinuity locations. The archetype objective function we focus on is representative of most of the energy functions developed in reconstruction problems (such as image restoration [2, 10] optical flow estimation [2] stereovision [26], computed tomography [11] In image restoration, f denotes the degraded observed intensity image and is the point spread function of the imaging system [10] In computed tomography is the Radon transform, the data being photons counted over an array of detectors. In dense matching problem ....

....intensity image and is the point spread function of the imaging system [10] In computed tomography is the Radon transform, the data being photons counted over an array of detectors. In dense matching problem such as binocular stereo disparities estimation, dense depth map reconstruction [26] or optical flow estimation, the corresponding energy function may be written in this general form after a linearization of a non linear data term arising from a constancy assumption [19] In optical flow estimation this linearization leads to the well known optical flow constraint equation [15] ....

L. Robert and R. Deriche. Dense depth map reconstruction: a minimization and regularization approach which preserves discontinuities. In B. Buxton and R. Cipolla, editors, Proc. Europ. Conf. Computer Vision, number 1064 in LNCS, pages 439--451. Springer-Verlag, April 1996. PI n1220 24 ' Etienne M'emin and Tanguy Risset

....must have an explicit method for handling discontinuity, otherwise the matching (and hence the reconstructed depth) will be inappropriately smoothed over the discontinuities. If regularisation is used within each level, it must be sensitive to the possibility of discontinuity. Robert and Deriche [11] followed this strategy, bypassing disparity computation altogether to obtain depth directly from pixel intensity values. We hope to gain a greater robustness by working in the CDWT feature space. This paper reports on the further development of the CDWT based stereo matching algorithm described ....

L. Robert and R. Deriche. Dense depth map reconstruction: a minimization and regularization approach which preserves discontinuities. In Proc. Fourth European Conference on Computer Vision (ECCV '96), Lecture Notes on Computer Science, pages 439-451. Springer-Verlag, 1996.

....29, 58] Hierarchical methods are also used here in order not to get trapped in some local minima. # Energy based: A last kind of approach which does not suoeer any of the shortcomings presented above, consists of solving the correspondence problem in a minimization and regularization formulation [6, 8, 24, 45, 46, 51, 59]. An iterative solution of the discrete version of the associated EulerLagrange equation is then used in order to estimate depth. For instance, in [46] the authors propose a method to directly compute the depth map Z(x; y) as the minimum of the following energy: S(Z) ZZ (I l (x; y) Gamma I ....

....shortcomings presented above, consists of solving the correspondence problem in a minimization and regularization formulation [6, 8, 24, 45, 46, 51, 59] An iterative solution of the discrete version of the associated EulerLagrange equation is then used in order to estimate depth. For instance, in [46], the authors propose a method to directly compute the depth map Z(x; y) as the minimum of the following energy: S(Z) ZZ (I l (x; y) Gamma I r (f 1 (x; y; Z(x; y) 2 dx dy ZZ Phi(jrZj) dx dy RR n# 3874 4 Alvarez Deriche S#nchez Weickert where f 1 ( depends on the ....

[Article contains additional citation context not shown here]

L. Robert, R. Deriche, Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities, B. Buxton, R. Cipolla (Eds.), Computer vision ECCV '96, Volume I, Lecture Notes in Computer Science, Vol. 1064, Springer, Berlin, 439451, 1996.

....the iterations, we use a focusing strategy based on a linear scalespace. Experimental results on both synthetic and real images are presented to illustrate the capabilities of this PDE and scale space based method. 1 Introduction Energy based methods have been extensively used in the last years [4, 5, 6, 8, 9, 11, 12] for estimating the disparity map between images. The goal Address: Edicio de Inform#tica y Sistemas, Campus Universitario de Tara. SP 35017 Las Palmas, Spain. E mail: lalvarez dis.ulpgc.es y Address: 2004 Route des Lucioles. F06902 Sophia Antipolis, France. E mail: ....

L. Robert and R. Deriche, Dense depth map reconstruction: a minimization and regularization approach which preserves discontinuities, in Computer Vision ECCV '96 (B. Buxton, R. Cipolla, Eds.), Vol. I, pp. 439451, Lecture Notes in Computer Science, Vol. 1064, Springer, Berlin, 1996.

....Hierarchical methods are also used here in order not to get trapped in some local minimum. Energy based: A last kind of approach which does not su er from any of the shortcomings presented above, consists of solving the correspondence problem in a minimization and regularization formulation [7, 9, 11, 12, 17, 23, 40, 41, 42, 46, 53]. An iterative solution of the discrete version of the associated Euler Lagrange equation is then used in order to estimate disparity or depth. For instance, in [42] a variational approach for solving the stereo problem was proposed using the classical quadratic Tikhonov regularization term in ....

....Euler Lagrange equation is then used in order to estimate disparity or depth. For instance, in [42] a variational approach for solving the stereo problem was proposed using the classical quadratic Tikhonov regularization term in order to obtain a smooth depth map. This work has been extended in [41] where the authors propose a method to directly compute the depth map Z(x; y) between two images I l and I r as the minimum of the following energy: S(Z) ZZ (I l (x; y) I r (f(x; y; Z(x; y) 2 dx dy ZZ (jrZj) dx dy (1) DENSE DISPARITY MAP ESTIMATION 3 where f( depends on ....

[Article contains additional citation context not shown here]

L. Robert and R. Deriche, Dense depth map reconstruction: a minimization and regularization approach which preserves discontinuities, in Computer Vision { ECCV '96 (B. Buxton, R. Cipolla, Eds.), Vol. I, pp. 439-451, Lecture Notes in Computer Science, Vol. 1064, Springer, Berlin, 1996.

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L.Robert, R.Deriche, "Dense Depth Map Reconstruction: A Minimization and Regularization Approach which Preserves Discontinuities," Fourth European Conference on Computer Vision, Cambridge, UK, 1996. (a) Disparity map ZNCC (b) Correlated pixels ZNCC (c) Disparity map CT (d) Correlated pixels CT (e) ZNCC (f) CT

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L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In Bernard Buxton, editor, Proc. of the 4th European Conf. on Computer Vision, Cambridge, UK, April 1996. 30

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L. Robert and R. Deriche. Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities. In Proceedings of the 4th European Conference on Computer Vision, Cambridge, England, pages 439--451, April 1996.

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Robert L. and Deriche R., 1996, "Dense depth map reconstruction: A minimization and regularization approach which preserves discontinuities". Proc. 4th ECCV. AKNOWLEDGEMENTS This work was partially supported by grants GV97TI -05-27 (Generalitat Valenciana) and TIC98-0677C02 -01 (CICYT, Ministerio de Educaci'on y Ciencia)

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