A regularization approach for surface reconstruction from point clouds

Montegranario, Hebert; Espinosa, Jairo

doi:10.1016/j.amc.2006.10.028

Cited by 7 publications

(4 citation statements)

References 6 publications

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“…In their main result 2, the authors use a particular finite element method. An alternative could be using a Tikhonov regularization scheme with particular adapted functionals, like those shown in Montegranario and Espinosa (2007). This regularization method has the mathematical advantage of using properties of compact operators in Hilbert spaces (Vapnik (l998), chapter 7).…”

Section: Jorge Mateu (University Jaume I Castellón)mentioning

confidence: 99%

An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach

Lindgren

Rue

Lindström

2011

Journal of the Royal Statistical Society Series B: Statistical Methodology

2,242

2,892

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Summary.Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R 2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matérn class, provide an explicit link , for any triangulation of R d , between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.

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Section: Jorge Mateu (University Jaume I Castellón)mentioning

confidence: 99%

An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach

Lindgren

Rue

Lindström

2011

Journal of the Royal Statistical Society Series B: Statistical Methodology

2,242

2,892

View full text Add to dashboard Cite

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“…, that samples the surface Σ ; find a surface ' Σ that approximates Σ by the data set A ; the reconstructed mesh ' Σ must be topologically equivalent to the surface Σ of the original object [12]. The goal of triangulation is to build a 3D triangular mesh from a set of data points, many algorithms are based on the Delaunay triangulation [13].…”

mentioning

confidence: 99%

Bone Trabecula Surface Reconstruction for Cranium Scaffold in Bone Tissue Engineering

Zheng

Gong

2013

AMM

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In bone tissue engineering, the scaffold architecture is very important for cell growth, it is better to make it similar with the native bone trabecula. To imitate the nature bone morphology, this paper presents a 3D reconstruction method of bone trabecula surface for cranium scaffold. Firstly, a native human cranium specimen on forehead was scanned by micro CT equipment and a set of gray level images were obtained. Then through image denoising, image enhancement, contour extraction and triangular surface reconstruction, the 3D structure of the specimen and its internal bone trabecula were reconstructed successfully. Lastly, to evaluate the feasibility of the method, a biomaterial scaffold case was fabrication using lost-foam casting technology. Results shown that the bone trabecula architecture in the scaffold is best retained and the porous structure is highly similar with the native specimen. This reconstructions process is simple and objective, which provide a new way for clinic cranium restoration.

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“…0 f is known as the representer of φ . When Dirac functionals are bounded, H becomes a Reproducing Kernel Hilbert Space (RKHS)[2,6] and the representer is a unique positive definite function ( , ) . Now, suppose we have M fragments or pieces of partial information or data about an unknown function f .…”

mentioning

confidence: 99%

“…Furthermore, if H is a RKHS and ( ) ( ( , )) this is a practical way to generalize the result for many kinds of data. The solution to this problem is known as Smoothing Spline or simply Spline ( ) x S[2,5], 0 λ > is the well known regularization parameter.…”

mentioning

confidence: 99%

A unified framework for 3D reconstruction

Montegranario

2007

Proc Appl Math and Mech

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3D reconstruction is a branch of computer vision with a broad range of applications like computer aided design, animation, medicine and many others. In this talk we use continuous linear functionals for characterizing different kinds of 3D data. These problems can be tackled in the framework of Reproducing Kernel Hilbert Spaces and regularization of inverse problems. It can be said that 3D reconstruction is the general problem of estimating or finding functional dependencies from a three-dimensional data set. The origin of these data covers a wide range that includes tomography, surface reconstruction from point clouds or image and signal processing. Usually the problem of reconstruction has a very close relationship with scientific visualization and computer graphics. Reconstruction and generalized interpolationIn the same way we need data structures for representing data into a computer we also need a mathematical structure to represent different kinds of data. This can be done using the concept of functional on a Hilbert Space and some results from approximation theory [1]. If X is a function space, i.e. a space whose elements are functions then a real function R X → φ :on X is called a functional [3]. Our setting is a real Hilbert space, X =H, whose inner product is written

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A regularization approach for surface reconstruction from point clouds

Cited by 7 publications

References 6 publications

An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach

An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach

Bone Trabecula Surface Reconstruction for Cranium Scaffold in Bone Tissue Engineering

A unified framework for 3D reconstruction

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