Given a bounded domain, we deal with the problem of estimating the distance function from the internal points of the domain to the boundary of the domain. Two simple extensions of the heat method for distance computation are introduced and evaluated. The extensions are based on first- and second-order Taylor series extrapolations. Numerical experiments demonstrate that the extensions deliver more accurate and robust estimates of the distance function.

Abstract We consider a variational approach to the problem of structure + texture decomposition (also known as cartoon + texture decomposition). As usual for many variational problems in image analysis and processing, the energy we minimize consists of two terms: a data-fitting term and a regularization term. The main feature of our approach consists of choosing parameters in the regularization term adaptively. Namely, the regularization term is given by a weighted $$p(\cdot )$$ p(·) -Dirichlet-based energy $$\int \!a({\varvec{x}}){\,\!|\,\!}\nabla u {\,\!|\,\!}^{p({\varvec{x}})}$$ ∫a(x)|∇u|p(x) , where the weight and exponent functions are determined from an analysis of the spectral content of the image curvature. Our numerical experiments, both qualitative and quantitative, suggest that the proposed approach delivers better results than state-of-the-art methods for extracting the structure from textured and mosaic images, as well as competitive results on image enhancement problems.

Given an image filter, defiltering refers to the problem of recovering an original image from its filtered version, assuming that the internal structure of the filter is not known. In this paper, we propose five iterative image defiltering schemes and use them for a semi-blind image deblurring problem. Namely, given an image resulting from applying a blurring image filter corrupted by noise to a clean image, we use the proposed iterative schemes to achieve a restoration of the clean image. In particular, for the motion deblurring problem, we show that our defiltering schemes are competitive with modern non-blind image deconvolution methods while using less information. The schemes are inspired by classical methods solving inverse problems and consist of properly modified and extended versions of the Van Cittert iterations, Levenberg–Marquardt method, Wiener filter, Landweber iterations, and Richardson–Lucy algorithm. In addition to dealing with image deblurring problems, we show that the proposed schemes can be used for inverting non-linear filters, and show that they are competitive with state-of-the-art black-box defiltering methods for these problems.

The extraction of 3D models represented by Constructive Solid Geometry (CSG) trees from point clouds is a common problem in reverse engineering pipelines as used by Computer Aided Design (CAD) tools. We propose three independent enhancements on state-of-the-art Genetic Algorithms (GAs) for CSG tree extraction: (1) A deterministic point cloud filtering mechanism that significantly reduces the computational effort of objective function evaluations without loss of geometric precision, (2) a graph-based partitioning scheme that divides the problem domain in smaller parts that can be solved separately and thus in parallel and (3) a 2-level improvement procedure that combines a recursive CSG tree redundancy removal technique with a local search heuristic, which significantly improves GA running times. We show in an extensive evaluation that our optimized GA-based approach provides faster running times and scales better with problem size compared to state-of-the-art GA-based approaches.

A preliminary structure + texture image decomposition is very useful for a number of digital image processing tasks, as different strategies are supposed to be employed for processing the structure and texture image components. In this paper, a new variational structure + texture image decomposition method is developed. The main ingredients of the proposed approach are: (1) using a low-pass filtered level-set curvature of the input image as a guidance image; (2) texture suppressing by minimizing a variable exponent energy, where the variable exponent is learned from the result of the curvature-guided image filtering. Numerical experiments demonstrate that the method is competitive with the current state of the art in structure + texture image decomposition. Several applications are considered.

This paper presents a system for user-assisted reverse modeling: from digitized point-cloud to solid models ready to be used in a CAD modeling system. Our approach consists in the following steps: segmentation, fitting, and constructive model discovery. Each of these steps are based on evolutionary algorithms. The obtained objects can then be further edited or parameterized by users and fitted to adapt their shape to different point-clouds.

Abstract In this paper, we deal with the problem of computing the distance to a surface (a curve in two dimensional) and consider several distance function approximation methods which are based on solving partial differential equations (PDEs) and finding solutions to variational problems. In particular, we deal with distance function estimation methods related to the Poisson‐like equations and generalized double‐layer potentials. Our numerical experiments are backed by novel theoretical results and demonstrate efficiency of the considered PDE‐based distance function approximations.

Abstract Signed distance fields obtained from polygonal meshes are commonly used in various applications. However, they can haveC1discontinuities causing creases to appear when applying operations such as blending or metamorphosis. The focus of this work is to efficiently evaluate the signed distance function and to apply a smoothing filter to it while preserving the shape of the initial mesh. The resulting function is smooth almost everywhere, while preserving the exact shape of the polygonal mesh. Due to its low complexity, the proposed filtering technique remains fast compared to its main alternatives providingC1‐continuous distance field approximation. Several applications are presented such as blending, metamorphosis and heterogeneous modelling with polygonal meshes.

We propose a method which helps to fit existing parameterized function representation (FRep) models to a given dataset of 3D surface points. Best fitted parameters of the model are obtained by using a hybrid algorithm combining simulated annealing and Levenberg-Marquardt methods. The efficiency of the approach is shown for recovery of two test items. We show through the CAD model processing an application of the proposed approach to the shape recovery followed by finite element mesh generation and adaptation.

Abstract Issues of digital preservation of shapes and internal structures of historical cultural objects are discussed. An overview of existing approaches to digital preservation related to shape modeling is presented and corresponding problems are considered. We propose a new digital preservation paradigm based on both constructive modeling reflecting the logical structure of the objects and open standards and procedures. Constructive Solid Geometry (CSG) and Function Representation (FRep) are examined and practically applied as mathematical representations producing compressed yet precise data structures, thus providing inter‐operability between current and future computer platforms crucial to archiving. Examples of CSG reconstruction of historical temples and FRep modeling of traditional lacquer ware are given. We examine the application of fitting of a parameterized FRep model to a cloud of data points as a step towards automation of the modeling process. Virtual venues for public access to cultural heritage objects including real time interactive simulation of cultural heritage sites over the Web are discussed and illustrated.