Filling-in by joint interpolation of vector fields and gray levels

Ballester, Coloma; Bertalmío, Marcelo; Caselles, Vicent; Sapiro, Guillermo; Verdera, Jоan

doi:10.1109/83.935036

Cited by 889 publications

(511 citation statements)

References 26 publications

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“…Another splitting strategy can be found in [37], where the authors penalize the smoothness of a vector field while simultaneously forcing the gradient directions of a gray-scale image to closely match the vector field. The penalty function on the normal field is proportional to the divergence of the normal vectors.…”

Section: A Splitting Strategy For Higher-order Priorsmentioning

confidence: 99%

“…This is found to be computationally more efficient than simultaneously solving the two equations and coupling them with a penalty term as in Ballester et al [37]. Fig.…”

Section: A Splitting Strategy For Higher-order Priorsmentioning

confidence: 99%

“…The proposed method is a second-order flow which has a fundamental relationship to the second-order penalty function described in Section 4. As noted by Ambrosio and Masnou [36], the relationship between such second-order systems (see also [37]) and the corresponding fourth-order variational problem is not fully understood.…”

Section: Appendix a Mathematical Foundationmentioning

confidence: 99%

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Higher-order nonlinear priors for surface reconstruction

Taşdizen

Whitaker

2004

IEEE Trans. Pattern Anal. Machine Intell.

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Abstract-For surface reconstruction problems with noisy and incomplete range data, a Bayesian estimation approach can improve the overall quality of the surfaces. The Bayesian approach to surface estimation relies on a likelihood term, which ties the surface estimate to the input data, and the prior, which ensures surface smoothness or continuity. This paper introduces a new high-order, nonlinear prior for surface reconstruction. The proposed prior can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion. An exact solution would require solving a fourth-order partial differential equation (PDE), which can be difficult with conventional numerical techniques. Our approach is to solve a cascade system of two second-order PDEs, which resembles the original fourth-order system. This strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals. We solve one PDE for processing the normals and one for refitting the surface to the normals. Furthermore, we implement the associated surface deformations using level sets. Hence, the algorithm can accommodate very complex shapes with arbitrary and changing topologies. This paper gives the mathematical formulation and describes the numerical algorithms. We also show results using range and medical data.

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Section: A Splitting Strategy For Higher-order Priorsmentioning

confidence: 99%

“…This is found to be computationally more efficient than simultaneously solving the two equations and coupling them with a penalty term as in Ballester et al [37]. Fig.…”

Section: A Splitting Strategy For Higher-order Priorsmentioning

confidence: 99%

Section: Appendix a Mathematical Foundationmentioning

confidence: 99%

See 1 more Smart Citation

Higher-order nonlinear priors for surface reconstruction

Taşdizen

Whitaker

2004

IEEE Trans. Pattern Anal. Machine Intell.

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“…These manipulations are beneficial when the analyzed data consists of directional information in addition to positional information on the manifold. For example, the goal in [2] is to recover missing data in images utilizing interpolation of the appropriate vector field. Another example is the utilization of tangential vector fields interpolation on S 2 for modeling atmospheric air flow and oceanic water velocity [8].…”

Section: Introductionmentioning

confidence: 99%

Dictionary Construction for Patch-to-Tensor Embedding

Salhov

Wolf

Bermanis

et al. 2012

Advances in Intelligent Data Analysis XI

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The incorporation of a matrix relations, which encompass multidimensional similarities between local neighborhoods of data points in the underlying manifold of a data, improves the utilization of kernel based data analysis methodologies. However, the utilization of multidimensional similarities results in a larger kernel and hence the computational complexity of the corresponding spectral decomposition increases dramatically. In this paper, we propose an efficient approximation to a spectral decomposition of a multidimensional similarity based kernel. Furthermore, we propose a dictionary construction that approximates the oversized kernel in this case and its associated embedding. The performance of the proposed dictionary construction is demonstrated on an example of a super-kernel that utilizes the Diffusion Maps methodology together with linear-projection operators between tangent spaces in the manifold. I. INTRODUCTIONRecent methods for advanced massive high dimensional data analysis utilize a manifold structure on which data points are assumed to lie. This manifold is immersed (or submersed) in an ambient space that is defined by observable parameters. Kernel methods such as k-PCA and Diffusion Maps (DM) [4] have provided good results in analyzing such massive high dimensional data. The defined kernel can be thought of as an adjacency matrix of a graph whose vertices are the data points in the dataset. The analysis of the eigenvalues and the corresponding eigenvectors of this matrix reveals many properties and connections in the graph. These methods are based on the spectral decomposition of a kernel that was designed to incorporate a scalar similarity measure between data points. The resulting embedding of the data points into an Euclidean space preserves the qualities represented by the designed kernel. This approach extends the core of the classical Multi-Dimensional Scaling (MDS) method [6], [9] by considering non-linear relations instead of just a linear one in its original Gram matrix.Recently, DM was extended in several different ways to handle the orientation in local tangent spaces [10]- [13]. The relation between two patches is described by a matrix instead of a scalar value. The resulting kernel captures enriched similarities between local structures in the underlying manifold. These enriched similarities can be used to analyze local areas around data points instead of analyzing their specific locations. For example, this analysis can be beneficial in image processing (analyzing regions instead of individual pixels) and when the data points are perturbed so that their surrounding area is more important than their specific position. Since the constructions of these similarities are based on local tangent spaces, they provide methods to manipulate tangential vector fields (e.g., perform out-ofsample extensions). These manipulations are beneficial when the analyzed data consists of directional information in addition to positional information on the manifold. For example, the goal in [2] is ...

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“…Many methods for achieving accurate reconstruction of important image features, such as edges and textures, have been proposed for these applications. Most algorithms reported in the literature are based on structure inpainting techniques for accurate reconstruction of missing edges [1]- [3]. These techniques are effective for pure structure images.…”

Section: Introductionmentioning

confidence: 99%

POCS-Based Texture Reconstruction Method Using Clustering Scheme by Kernel PCA

Ogawa¹,

Haseyama²

2007

IEICE Transactions on Fundamentals of Electronics, Communicatio

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SUMMARYA new framework for reconstruction of missing textures in digital images is introduced in this paper. The framework is based on a projection onto convex sets (POCS) algorithm including a novel constraint. In the proposed method, a nonlinear eigenspace of each cluster obtained by classification of known textures within the target image is applied to the constraint. The main advantage of this approach is that the eigenspace can approximate the textures classified into the same cluster in the leastsquares sense. Furthermore, by monitoring the errors converged by the POCS algorithm, a selection of the optimal cluster to reconstruct the target texture including missing intensities can be achieved. This POCS-based approach provides a solution to the problem in traditional methods of not being able to perform the selection of the optimal cluster due to the missing intensities within the target texture. Consequently, all of the missing textures are successfully reconstructed by the selected cluster's eigenspaces which correctly approximate the same kinds of textures. Experimental results show subjective and quantitative improvement of the proposed reconstruction technique over previously reported reconstruction techniques.

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Filling-in by joint interpolation of vector fields and gray levels

Cited by 889 publications

References 26 publications

Higher-order nonlinear priors for surface reconstruction

Higher-order nonlinear priors for surface reconstruction

Dictionary Construction for Patch-to-Tensor Embedding

POCS-Based Texture Reconstruction Method Using Clustering Scheme by Kernel PCA

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