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Reconstructing sharply folding surfaces: A convex formulation

Reconstructing sharply folding surfaces: A convex formulation,10.1109/CVPRW.2009.5206759,Mathieu Salzmann,Pascal Fua

Reconstructing sharply folding surfaces: A convex formulation   (Citations: 14)
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In recent years, 3D deformable surface reconstruction from single images has attracted renewed interest. It has been shown that preventing the surface from either shrink- ing or stretching is an effective way to resolve the ambigu- ities inherent to this problem. However, while the geodesic distances on the surface may not change, the Euclidean ones decrease when folds appear. Therefore, when applied to discrete surface representations, such constant-distance constraints are only effective for smoothly deforming sur- faces, and become inaccurate for more e xible ones that can exhibit sharp folds. In such cases, surface points must be allowed to come closer to each other. In this paper, we show that replacing the equality con- straints of earlier approaches by inequality constraints that let the mesh representation of the surface shrink but not ex- pand yields not only a more faithful representation, but also a convex formulation of the reconstruction problem. As a result, we can accurately reconstruct surfaces undergoing complex deformations that include sharp folds from indi- vidual images.
Conference: Computer Vision and Pattern Recognition - CVPR , pp. 1054-1061, 2009
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    • ...which attempt to account for clothing variation [18], and methods for estimating folds in deformable surfaces [22]...

    Stephen Milleret al. Parametrized shape models for clothing

    • ...In section 3.2, we explain in more detail the algorithm from [13, 14], which introduces the SOCP with local linear deformation models...
    • ...In [13], Salzmann and Fua propose a convex formulation to non-rigid surface registration that is also described in [14]...
    • ...Next we present the algorithm from [13], on which our work is based...
    • ...Other layouts are possible, although they complicate the use of local deformation models (see [13])...
    • ...(together with the corresponding face) are called barycentric coordinates of p. Following the notation in [13], we will denote the set of vertices – stacked on top of each other –b yX...
    • ...These correspondences can be created if the 3D shape is known for one reference image, as feature matching then produces the necessary information. As in [13], we assume w.l.o.g...
    • ...where the notation again follows [13] So given the image to surface correspondences, we seek to find the shape configuration X that minimizes one of the above projection errors...
    • ...Our approach builds on the algorithm presented in [13], which we will therefore describe here in some detail...
    • ...We refer the reader to [13] for an in-depth treatment...
    • ...In order to fit an SOCP formulation, Salzmann and Fua in [13] use the algebraic error (4) above...
    • ...In [13], temporal motion models are also used to regularize the problem...
    • ...However, the corresponding ideas in [13] apply directly...
    • ...To conclude this section, we state the final single-frame optimization problem as in [13]...
    • ...where the weights according to [13] can be chosen as wd =...
    • ...In [13], the authors therefore suggest iteratively polishing the dataset...
    • ...Here, the scale factor is chosen to correspond to a 1D error of 3σ. For most of our experiments, we have also not implemented the error-based feature weighting described in [13], as in our opinion, this scheme overemphasizes a few observations with small errors...
    • ...Our experiments presented in figure 1 illustrate the sensitivity of the approach from [13] to rigid deformations...
    • ...This iteration scheme might seem like a drawback with respect to [13] because although we iteratively solve sub-problems that are convex, the global problem is not...
    • ...Figure 2. Rotation of the model yields a planar shape with our method (left), but a twisted shape with the algorithm from [13] (right)...
    • ...The red solid line are the errors using [13] while the dashed line are ours...
    • ...The dotted graph was obtained by running [13] on the original data set without rotation...
    • ...They clearly demonstrate that the algorithm from [13] deviates from the simple planar shape under rotation...
    • ...The same holds when adding noise to the observations, which is depicted in figure 3. To visualize the effect that the rotation has on the estimation, figure 2 shows a frontal view of the reconstructions made with our algorithm and with [13]...
    • ...The reconstruction errors with respect to ground truth are plotted in figures 5 and 6. To compare the results, we have also plotted the errors obtained from applying [13 ]t o observations of the original, non-rotated mesh...
    • ...It is clearly visible that applying [13] to the rotated data introduces a relatively large error...
    • ...We have demonstrated that the deformation model used by Salzmann and Fua in [13] suffers from not being invariant to rigid transformations...
    • ...To this end, we have shown how to incorporate rotation estimation into the SOCP framework suggested by Salzmann and Fua in [13]...

    Markus Mollet al. Separating rigid motion from linear local deformation models

    • ...The last decade has seen two new principal approaches to the problem: methods that rely on an initial reference frame in which the 3D shape of the object is known [21, 22] and those that require 2D tracking data throughout the sequence as input, and make use of spatio-temporal smoothness priors to resolve ambiguous cues...

    Chris Russellet al. Energy Based Multiple Model Fitting for Non-Rigid Structure from Motio...

    • ...We make two basic assumptions, shared with many state-of-the-art approaches [16, 19, 21, 22]...
    • ...Several recent methods have been proposed to recover non-rigid shape from single images, by using deformation modes in conjunction with local rigidity constraints to reconstruct inextensible surfaces [9, 19, 21, 22], and in conjunction with shading constraints to reconstruct stretchable surfaces [16]...
    • ...In addition we implemented a procedure similar to what was proposed in [21] to detect and remove 3D-to-2D correspondences with very large errors...

    Francesc Moreno-Nogueret al. Probabilistic simultaneous pose and non-rigid shape recovery

    • ...To reconstruct surfaces in 3-D, many template-based [1], [2], [21]–[24] and machine learning-based [25]–[28] methods are proposed...

    Shuhan Shenet al. Monocular 3-D Tracking of Inextensible Deformable Surfaces Under L2 No...

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