A Smoothed Particle Image Reconstruction method

Blasi, Gianpiero Di; Francomano, Elisa; Tortorici, A.; Toscano, Elena

doi:10.1007/s10092-010-0028-3

Cited by 9 publications

(14 citation statements)

References 29 publications

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“…These functions are popular for scattered data interpolation, and some of them have also been used for inpainting corrupted images [6,19]. However, such problems usually do not allow to optimise the location and the grey values of the inpainting data set.…”

Section: Introductionmentioning

confidence: 99%

Optimising Spatial and Tonal Data for Homogeneous Diffusion Inpainting

Mainberger

Hoffmann

Weickert

et al. 2012

Lecture Notes in Computer Science

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Abstract. Finding optimal inpainting data plays a key role in the field of image compression with partial differential equations (PDEs). In this paper, we optimise the spatial as well as the tonal data such that an image can be reconstructed with minimised error by means of discrete homogeneous diffusion inpainting. To optimise the spatial distribution of the inpainting data, we apply a probabilistic data sparsification followed by a nonlocal pixel exchange. Afterwards we optimise the grey values in these inpainting points in an exact way using a least squares approach. The resulting method allows almost perfect reconstructions with only 5% of all pixels. This demonstrates that a thorough data optimisation can compensate for most deficiencies of a suboptimal PDE interpolant.

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Section: Introductionmentioning

confidence: 99%

Optimising Spatial and Tonal Data for Homogeneous Diffusion Inpainting

Mainberger

Hoffmann

Weickert

et al. 2012

Lecture Notes in Computer Science

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“…In the following, the same notation will be used every time a discretization is performed. Moreover, the smoothing length h and the number of particles determine the resolution of the approximation so the NNP spotting is a crucial task before performing any computation using the SPH method [28]. Finally, in approximating a function with the particle approximation provided by (3), the integration error generated from the discretization of the integral depends on the particles' arrangement.…”

Section: Fundamental Issues Of Smoothed Particle Hydrodynamics Methodsmentioning

confidence: 99%

Section: Fundamental Issues Of Smoothed Particle Hydrodynamics Methodsmentioning

confidence: 99%

A numerical meshless particle method in solving the magnetoencephalography forward problem

Ala

Blasi

Francomano

2012

Int J Numerical Modelling

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In this paper, a numerical meshless particle method is presented in order to solve the magnetoencephalography\ud forward problem for analyzing the complex activation patterns in the human brain. The forward problem is\ud devoted to compute the scalp potential and magnetic field distribution generated by a set of current sources\ud representing the neural activity, and in this paper, it has been approached by means of the smoothed particle\ud hydrodynamics method suitably handled. The Poisson equation generated by the quasi-stationary Maxwell’s curl\ud equations, by assuming Neumann boundary conditions has been considered, and the current sources have been\ud simulated by current dipoles. The adopted meshless particle model has provided good results in agreement with\ud the analytical ones and by overcoming the drawback of the mesh generation. The numerical model has been validated,\ud at first, in computing the electric potential and the external magnetic field for a dipole plunged near the upper\ud surface of a homogeneous sphere simulating the human brain. Simulation results obtained by simulating two\ud concentric spheres with different conductivities are also reported. Moreover, in order to better assess the validity of\ud the proposed approach, a realistic human brain cortex model has been also simulated and compared with boundary\ud element method results. A satisfactory agreement has been reached

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“…Since our tonal optimisation methods are tailored towards linear least squares formulations, specific challenges arise when we want to extend them to EED-based inpainting. The nonlinearity of the EED inpainting scheme prevents closed form expressions such as (37) and (38). This is caused by the fact that the matrix A is now a function of the inpainted image u k , which itself depends on the grey value data g k in the specified pixels.…”

Section: Optimising Tonal Datamentioning

confidence: 99%

2. Optimizing spatial and tonal data for PDE-based inpainting

Hoeltgen¹,

Mainberger²,

Hoffmann³

et al. 2016

Variational Methods

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Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. Apart from these specific contributions, we also give an extensive literature survey on PDE-based image compression methods. Summary and ConclusionsThe contributions of our work are twofold: On the one hand we gave a comprehensive survey on the state-of-the-art in PDE-based image compression.

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A Smoothed Particle Image Reconstruction method

Cited by 9 publications

References 29 publications

Optimising Spatial and Tonal Data for Homogeneous Diffusion Inpainting

Optimising Spatial and Tonal Data for Homogeneous Diffusion Inpainting

A numerical meshless particle method in solving the magnetoencephalography forward problem

2. Optimizing spatial and tonal data for PDE-based inpainting

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