We present an implicit surface reconstruction algorithm for point clouds. We view the implicit surface reconstruction as a three dimensional binary image segmentation problem that segments the entire space \(\mathbb R ^3\) or the computational domain into an interior region and an exterior region while the boundary between these two regions fits the data points properly. The key points with using an image segmentation formulation are: (1) an edge indicator function that gives a sharp indicator of the surface location, and (2) an initial image function that provides a good initial guess of the interior and exterior regions. In this work we propose novel ways to build both functions directly from the point cloud data. We then adopt recent convexified image segmentation models and fast computational algorithms to achieve efficient and robust implicit surface reconstruction for point clouds. We test our methods on various data sets that are noisy, non-uniform, and with holes or with open boundaries. Moreover, comparisons are also made to current state of the art point cloud surface reconstruction techniques.

We propose a novel fourth order dual method for the minimization of the non-smooth semi-norm ∥Δ·∥1 when in amalgamation with a new staircase reducing texture decomposition model of image processing. The proposed model incorporating this high order energy is a variant of the Chambolle Lions denoising model that additionally utilizes a negative Sobolev norm. We claim that the dual method is faster and more stable than the current gradient descent time marching algorithms often used to minimize such energies. Moreover, a proof of convergence of the proposed method, in conjunction with the new model, will be provided. Lastly, we provide guidelines on how the new energy and proposed framework can be naturally incorporated into many popular texture extraction and restoration models of image processing.

This paper proposes a natural and efficient way to achieve staircase reduction in texture extraction models of image processing. Moreover, we propose a precise framework for this amalgamation. In a sense, we utilize the best of both worlds: (I) the use of higher order derivatives through a variant of the Chambolle–Lions inf convolution energy (an image decomposition model in itself) along with (II) approximations to Meyer’s G and E norms including the H−1 negative norm for ameliorating staircasing in image decomposition and restoration problems.

We propose a total variation based model for simultaneous image inpainting and blind deconvolution. We demonstrate that the tasks are inherently coupled together and that solving them individually will lead to poor results. The main advantages of our model are that (i) boundary conditions for deconvolution required near the interface between observed and occluded regions are naturally generated through inpainting; (ii) inpainting results are enhanced through deconvolution (as opposed to inpainting blurry images). As a result, ringing effects due to imposing improper boundary conditions and errors due to imperfection of inpainting blurry images are reduced. Moreover, our model can also be used to generate boundary conditions for regular deconvolution problems that yields better results than previous methods.© 2005 Wiley Periodicals, Inc. © 2005 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 15, 92–102, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.20041

Since their introduction in a classic paper by Rudin, Osher and Fatemi [26], total variation minimizing models have become one of the most popular and successful methodology for image restoration. More recently, there has been a resurgence of interest and exciting new developments, some extending the applicabilities to inpainting, blind deconvolution and vector-valued images, while others offer improvements in better preservation of contrast, geometry and textures, in ameliorating the staircasing effect, and in exploiting the multiscale nature of the models. In addition, new computational methods have been proposed with improved computational speed and robustness. We shall review some of these recent developments.