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 This paper presents a deep learning method for image dehazing and clarification. The main advantages of the method are high computational speed and using unpaired image data for training. The method adapts the Zero-DCE approach (Li et al. in IEEE Trans Pattern Anal Mach Intell 44(8):4225–4238, 2021) for the image dehazing problem and uses high-order curves to adjust the dynamic range of images and achieve dehazing. Training the proposed dehazing neural network does not require paired hazy and clear datasets but instead utilizes a set of loss functions, assessing the quality of dehazed images to drive the training process. Experiments on a large number of real-world hazy images demonstrate that our proposed network effectively removes haze while preserving details and enhancing brightness. Furthermore, on an affordable GPU-equipped laptop, the processing speed can reach 1000 FPS for images with 2K resolution, making it highly suitable for real-time dehazing applications.

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.

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.

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.

We propose a novel bolt-on module capable of boosting the robustness of various single compact 2D gait representations. Gait recognition is negatively influenced by covariate factors including clothing and time which alter the natural gait appearance and motion. Contrary to traditional gait recognition, our bolt-on module remedies this by a dedicated covariate factor detection and removal procedure which we quantitatively and qualitatively evaluate. The fundamental concept of the bolt-on module is founded on exploiting the pixel-wise composition of covariate factors. Results demonstrate how our bolt-on module is a powerful component leading to significant improvements across gait representations and datasets yielding state-of-the-art results.

This paper demonstrates potential advantages of using implicit finite differencing and filtering schemes for fast, accurate, and reliable differentiating and filtering of multidimensional signals defined on regular grids. In particular, applications to image enhancement and Gaussian image deblurring are considered. The theoretical contribution of the paper is threefold. The first adapts the Fourier--Padé--Galerkin approximation approach for constructing compact implicit finite difference schemes with desirable spectral resolution properties. The second establishes a link between implicit and explicit finite differences used for gradient estimation. Finally, the third one consists of introducing new implicit finite difference schemes with good spectral resolution properties.

Abstract In this paper, we present an overview of the achievements accomplished to date in the field of computer‐aided relief generation. We delineate the problem, classify different solutions, analyse similarities, investigate developments and review the approaches according to their particular relative strengths and weaknesses. Moreover, we describe remaining challenges and point out prospective extensions. In consequence, this survey is addressed to both researchers and artists, through providing valuable insights into the theory behind the different concepts in this field and augmenting the options available among the methods presented with regard to practical application.

Fast and reliable estimation of image derivatives is among the most fundamental tasks of low level image processing. Implicit finite differences offer much higher spectral resolving efficiency compared to explicit finite differences while the computational effort increases only slightly. Although implicit finite differences had become known to a general audience of numerical mathematicians and computational physicists after Collatz’ book [2], their heyday began after Lele’s seminal paper [4] where a remarkable performance of implicit finite differences for computational problems with a range of spatial scales was analyzed and demonstrated.

Abstract In spite of high computational complexity, the bilateral filter and its modifications and extensions have recently become very popular image and shape processing tools. In this paper, we propose a fast and accurate approximation of the bilateral filter. Our approach combines a dimension elevation trick with a Fast Gauss Transform. First we represent the bilateral filter as a convolution in a high dimensional space. Then the convolution is efficiently approximated by using space partitioning and Gaussian function expansions. Advantages of our approach include linear computational complexity, user‐specified precision, and an ability to process high dimensional and non‐uniformly sampled data. We demonstrate capabilities of the approach by considering its applications to the image and volume denoising and high‐dynamic‐range tone mapping problems.

Among all forms of sculpture, bas-relief is arguably the closest to painting. Although inherently a two dimensional sculpture, a bas-relief suggests a visual spatial extension of the scene in depth through the combination of composition, perspective, and shading. There have been significant results on digital bas-relief generation but many of the existing techniques may wash out high level surface detail during the compression process. The primary goal of this work is to address the problem of fine features by tailoring a filtering technique that achieves good compression without compromising the quality of surface details. As a secondary application we explore the generation of artistic relief which mimic cubism in painting and we show how it could be used for generating Picasso like portraits.

A new geometry-based finite difference method for a fast and reliable detection of perceptually salient curvature extrema on surfaces approximated by dense triangle meshes is proposed. The foundations of the method are two simple curvature and curvature derivative formulas overlooked in modern differential geometry textbooks and seemingly new observation about inversion-invariant local surface-based differential forms.

We present a new method for noise removal on static and time-varying range data. Our approach predicts the restored position of a perturbed vertex using similar vertices in its neighborhood. It defines the required similarity measure in a new non-local fashion which compares regions of the surface instead of point pairs. This allows our algorithm to obtain a more accurate denoising result than previous state-of-the-art approaches and, at the same time, to better preserve fine features of the surface. Furthermore, our approach is easy to implement, effective, and flexibly applicable to different types of scanned data. We demonstrate this on several static and interesting new time-varying datasets obtained using laser and structured light scanners.

As large shape repositories become more common, the problem of automatically generating good views of shapes has recently gained prominence. However, very few of the proposed methods take into account the orientation of the shape in the resulting view, and none presents a satisfactory solution. In this paper, we present a simple, example based method to correct the orientation of a shape in a query image. Our method depends on the availability of a database of classified images containing correctly oriented shapes. In the first step, a candidate class for the query shape is identified, and in the second, the query shape is aligned with a target shape from the candidate class.

In this paper, we introduce and study a new problem of converting a given 3D shape (or a 2.5D range data) into a bas-relief. The problem can be considered as a geometry counterpart of the HDR image compression problem widely studied in computer graphics. In our approach to the shape bas-reliefing problem, we combine the concepts of mesh saliency, shape exaggerating, and discrete differential coordinates. The final bas-relief has a small width, preserves salient features of the original 3D shape, and, therefore, can be used for shape decorating purposes.

Abstract In this paper, a new free‐form shape deformation approach is proposed. We combine a skeleton‐based mesh deformation technique with discrete differential coordinates in order to create natural‐looking global shape deformations. Given a triangle mesh, we first extract a skeletal mesh, a two‐sided Voronoibased approximation of the medial axis. Next the skeletal mesh is modified by free‐form deformations. Then a desired global shape deformation is obtained by reconstructing the shape corresponding to the deformed skeletal mesh. The reconstruction is based on using discrete differential coordinates. Our method preserves fine geometric details and original shape thickness because of using discrete differential coordinates and skeleton‐based deformations. We also develop a new mesh evolution technique which allow us to eliminate possible global and local self‐intersections of the deformed mesh while preserving fine geometric details. Finally, we present a multi‐resolution version of our approach in order to simplify and accelerate the deformation process. In addition, interesting links between the proposed free‐form shape deformation technique and classical and modern results in the differential geometry of sphere congruences are established and discussed.

We develop spherical barycentric coordinates. Analogous to classical, planar barycentric coordinates that describe the positions of points in a plane with respect to the vertices of a given planar polygon, spherical barycentric coordinates describe the positions of points on a sphere with respect to the vertices of a given spherical polygon. In particular, we introduce spherical mean value coordinates that inherit many good properties of their planar counterparts. Furthermore, we present a construction that gives a simple and intuitive geometric interpretation for classical barycentric coordinates, like Wachspress coordinates, mean value coordinates, and discrete harmonic coordinates. One of the most interesting consequences is the possibility to construct mean value coordinates for arbitrary polygonal meshes. So far, this was only possible for triangular meshes. Furthermore, spherical barycentric coordinates can be used for all applications where only planar barycentric coordinates were available up to now. They include Bézier surfaces, parameterization, free-form deformations, and interpolation of rotations.

An approach to automatically select stable and salient representative views of a given 3D object is proposed. Initially, a set of viewpoints are uniformly sampled along the surface of a bounding sphere. The sampled viewpoints are connected to their closest points to form a spherical graph in which each edge is weighted by a similarity measure between the two views from its incident vertices. Partitions of similar views are obtained using a graph partitioning procedure and their "centroids" are considered to be their representative views. Finally, the views are ranked based on a saliency measure to form the object’s representative views. This leads to a compact, human-oriented 2D description of a 3D object, and as such, is useful both for traditional applications like presentation and analysis of 3D shapes, and for emerging ones like indexing and retrieval in large shape repositories.

In this paper, we propose a new and powerful shape denoising technique for processing surfaces approximated by triangle meshes and soups. Our approach is inspired by recent non-local image denoising schemes and naturally extends bilateral mesh smoothing methods. The main idea behind the approach is very simple. A new position of vertex P of a noisy mesh is obtained as a weighted mean of mesh vertices Q with nonlinear weights reflecting a similarity between local neighborhoods of P and Q. We demonstrate that our technique outperforms recent state-of-the-art smoothing methods. We also suggest a new scheme for comparing different mesh/soup denoising methods.

In this paper, we propose a new method for approximating an unorganized set of points scattered over a piecewise smooth surface by a triangle mesh. The method is based on the Garland-Heckbert local quadric error minimization strategy. First an adaptive spherical cover and auxiliary points corresponding to the cover elements are generated. Then the intersections between the spheres of the cover are analyzed and the auxiliary points are connected. Finally the resulting mesh is cleaned from non-manifold parts. The method allows us to control the approximation accuracy, process noisy data, and reconstruct sharp edges and corners. Further, the vast majority of the triangles of the generated mesh have their aspect ratios close to optimal. Thus our approach integrates the mesh reconstruction, smoothing, decimation, feature restoration, and remeshing stages together.

We propose a fast and robust method for detecting crest lines on surfaces approximated by dense triangle meshes. The crest lines, salient surface features defined via first- and second-order curvature derivatives, are widely used for shape matching and interrogation purposes. Their practical extraction is difficult because it requires good estimation of high-order surface derivatives. Our approach to the crest line detection is based on estimating the curvature tensor and curvature derivatives via local polynomial fitting.Since the crest lines are not defined in the surface regions where the surface focal set (caustic) degenerates, we introduce a new thresholding scheme which exploits interesting relationships between curvature extrema, the so-called MVS functional of Moreton and Sequin, and Dupin cyclides,An application of the crest lines to adaptive mesh simplification is also considered.

Feature sensitive mesh segmentation is important for many computer graphics and geometric modeling applications. In this paper, we develop a mesh segmentation method, which is capable of producing high-quality shape partitioning. It respects fine shape features and works well on various types of shapes, including natural shapes and mechanical parts. The method combines a procedure for clustering mesh normals with a modification of the mesh clarification technique. For clustering of mesh normals, we adopt Mean Shift, a powerful general purpose technique for clustering scattered data. We demonstrate advantages of our method by comparing it with two state-of-the-art mesh segmentation techniques.

We develop an adaptive RBF fitting procedure for a high quality approximation of a set of points scattered over a piecewise smooth surface. We use compactly supported RBFs whose centers are randomly chosen from the points. The randomness is controlled by the point density and surface geometry. For each RBF, its support size is chosen adoptively according to surface geometry at a vicinity of the RBF center. All these lead to a noise-robust high quality approximation of the set. We also adapt our basic technique for shape reconstruction from registered range scans by taking into account measurement confidences. Finally, an interesting link between our RBF fitting procedure and partition of unity approximations is established and discussed.

We propose a fast and simple method for generating a low-stretch mesh parameterization. Given a triangle mesh, we start from the floater shape preserving parameterization and then improve the parameterization gradually. At each improvement step, we optimize the parameterization generated at the previous step by minimizing a weighted quadratic energy where the weights are chosen in order to minimize the parameterization stretch. This optimization procedure does not generate triangle flips if the boundary of the parameter domain is a convex polygon. Moreover already the first optimization step produces a high-quality mesh parameterization. We compare our parameterization procedure with several state-of-the-art mesh parameterization methods and demonstrate its speed and high efficiency in parameterizing large and geometrically complex models.

We propose a simple and effective method for detecting view-and scale-independent ridge-valley lines defined via first- and second-order curvature derivatives on shapes approximated by dense triangle meshes. A high-quality estimation of high-order surface derivatives is achieved by combining multi-level implicit surface fitting and finite difference approximations. We demonstrate that the ridges and valleys are geometrically and perceptually salient surface features, and, therefore, can be potentially used for shape recognition, coding, and quality evaluation purposes.

We propose a new interpolatory subdivision scheme for generating nice-looking curvature-continuous curves of round shapes. The scheme is based on a diffusion of normals. Given a subdivided polyline, the new polyline vertices inserted at the splitting step are updated in order to fit diffused (averaged with appropriate weights) normals. Although the resulting interpolatory subdivision scheme is nonstationary, nonlinear, and nonuniform from the traditional point of view, the scheme is easy to implement because the same simple geometric procedure for generating new vertices is used at each subdivision step. According to our experiments, the scheme is robust and demonstrates very good convergence properties.

We present two simple yet effective mesh denoising methods. The methods combine a diffusion (repeated local averaging) of mesh normals with histogram-based filtering procedures. According to our experimental results, the methods outperform conventional smoothing techniques in denoising meshes with sharp features.

We propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed along a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate the sets starting from the coarsest level. We interpolate a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. An original point set and its coarse-to-fine hierarchy of interpolated sets is presented. According to our numerical experiments, the method is essentially faster than the state-of-the-art scattered data approximation with globally supported RBFs (Carr et al., 2001) and much simpler to implement.

We present a new shape representation, the multi-level partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape.Our approach gives us considerable flexibility in the choice of local shape functions, and in particular we can accurately represent sharp features such as edges and corners by selecting appropriate shape functions. An error-controlled subdivision leads to an adaptive approximation whose time and memory consumption depends on the required accuracy. Due to the separation of local approximation and local blending, the representation is not global and can be created and evaluated rapidly. Because our surfaces are described using implicit functions, operations such as shape blending, offsets, deformations and CSG are simple to perform.

In this paper we introduce a new mesh filtering method: a mesh median filter. This is an application of the median filter to smoothen 3-D noisy shapes given by triangle meshes. An algorithm of the mesh median filter is realized by applying the median filter to face normals on triangle meshes and updating mesh vertex positions to make them fit to the filtered normals. As an advanced modification of the mesh median filter we further introduce a weighted mesh median filter. The weighted mesh median filter has a reinforced feature preservation effect. The weighted mesh median filter with positive weighting has the smoothing effect, and the one with negative weighting has the enhancing effect. The two kinds of mesh median filters are compared with two conventional mesh filtering methods: the Laplacian smoothing flow and the mean curvature flow. Experimental results demonstrate that the mesh median filter does not induce oversmoothing.

This paper presents frameworks to extend the mean and median filtering schemes in image processing to smoothing noisy 3D shapes given by triangle meshes. The frameworks consist of the application of the mean and median filters to face normals on triangle meshes and the editing of mesh vertex positions to make them fit the modified normals. We also give a quantitative evaluation of the proposed mesh filtering schemes and compare them with conventional mesh smoothing methods such as Laplacian smoothing flow and mean curvature flow. The quantitative evaluation is performed in error metrics on mesh vertices and normals. Experimental results demonstrate that our mesh mean and median filtering methods are more stable than conventional Laplacian and mean curvature flows. We propose thee new mesh smoothing methods as one possible solution of the oversmoothing problem.

Surface fairing, generating free-form surfaces satisfying aesthetic requirements, is important for many computer graphics and geometric modeling applications. A common approach for fair surface design consists of minimization of fairness measures penalizing large curvature values and curvature oscillations. The paper develops a numerical approach for fair surface modeling via curvature-driven evolutions of triangle meshes. Consider a smooth surface each point of which moves in the normal direction with speed equal to a function of curvature and curvature derivatives. Chosen the speed function properly, the evolving surface converges to a desired shape minimizing a given fairness measure. Smooth surface evolutions are approximated by evolutions of triangle meshes. A tangent speed component is used to improve the quality of the evolving mesh and to increase computational stability. Contributions of the paper include also art improved method for estimating the mean curvature.

In this paper we propose a set of free-form shape deformation techniques. The basic approach can be described as follows. Given a surface represented by a mesh and a control point, for every mesh vertex let us consider the difference between the control point and the vertex. The vertex is shifted by a displacement equal to the difference, times a scale factor where the scale factor is given by a function depending nonlinearly on the difference. The function is bump-shaped and depends on a number of parameters. Varying the parameters leads to a rich palette of shape deformations. The proposed approach includes also shape deformations with multiple (real, auxiliary, and virtual) control points and constrained, directional, and anisotropic deformations. It allows a user to edit a given shape interactively and intuitively. No mesh connectivity information is used and, therefore, the proposed techniques can be applied directly to a shape given as a cloud of points.

Abstract The paper presents a skeleton‐based approach for robust detection of perceptually salient shape features. Given ashape approximated by a polygonal surface, its skeleton is extracted using a three‐dimensional Voronoi diagramtechnique proposed recently by Amenta et al. [ 3 ]. Shape creases, ridges and ravines, are detected as curvescorresponding to skeletal edges. Salient shape regions are extracted via skeleton decomposition into patches.The approach explores the singularity theory for ridge and ravine detection, combines several filtering methodsfor skeleton denoising and for selecting perceptually important ridges and ravines, and uses a topological analysisof the skeleton for detection of salient shape regions. ACM CSS: I.3.5 Computational Geometry and Object Modeling

The paper proposes an approach for stable extraction of the skeleton of a polygonal surface and detection of surface features, ridges and ravines, corresponding to skeletal edges. The approach adapts the three-dimensional Voronoi diagram technique for skeleton extraction, explores the singularity theory for ridge and ravine detection, and combines several filtering methods for skeleton denoising and for selecting perceptually salient ridges and ravines. Rather than extract the skeleton as a CW-complex, we approximate it with a two-sided surface. It allows us to use standard mesh editing tools for skeleton denoising and achieve stable extraction of the ridges and ravines.

The paper develops a simple and effective method for adaptive smoothing tangential direction fields defined on piecewise smooth surfaces approximated by triangle meshes. The method consists of simultaneous and iterative updating each vertex direction by a weighted sum of the directions at the neighboring vertices. The weights themselves depend on directions: if the direction at a given vertex differs strongly from the direction at a neighboring vertex, then the weight associated with that neighboring direction is chosen small. Choosing such nonlinear weights allows us to preserve important direction field discontinuities during the smoothing process. Applications of the developed smoothing technique to curvature extrema detection and non-photorealistic rendering with curvature lines are given.

Consider a 2D smooth closed curve evolving in time, the skeleton (medial axis) of the figure bounded by the curve, and the evolute of the curve. A new branch of the skeleton can appear/disappear when an evolute cusp intersects the skeleton. In this paper, we describe exact conditions of the skeleton bifurcations corresponding to such intersections. Similar results are also obtained for 3D surfaces evolving in time.

The paper presents a novel approach for accurate polygonization of implicit surfaces with sharp features. The approach is based on mesh evolution towards a given implicit surface with simultaneous control of the mesh vertex positions and mesh normals.

In this paper, we propose a method for improvement of isosurface polygonizations. Given an initial polygonization of an isosurface, we introduce a mesh evolution process initialized by the polygonization. The evolving mesh converges quickly to its limit mesh which provides with a high quality approximation of the isosurface even if the isosurface has sharp features, boundary, complex topology. To analyze how close the evolving mesh approaches its destined isosurface, we introduce error estimators measuring the deviations of the mesh vertices from the isosurface and mesh normals from the isosurface normals. A new technique for mesh editing with isosurfaces is also proposed. In particular, it can be used for creating carving effects.

We develop an approach for stable detection of perceptually salient curvature features on surfaces approximated by dense triangle meshes. The approach explores an “area degenerating” effect of the focal surface near its singularities and combines together a new approximations of the mean and Gaussian curvatures, nonlinear averaging of curvature maps, histogram-based curvature extrema filtering, and an image processing skeletonization procedure adapted for triangular meshes. Finally we use perceptually significant curvature extrema triangles to enhance the Garland-Heckbert mesh decimation method.

A computer graphics object reconstructed from real-world data often contains undesirable noise and small-scale oscillations. An important problem is how to remove the noise and oscillations while preserving desirable geometric features of the object. We develops methods for polyhedral surface smoothing and denoising with simultaneous increasing mesh regularity. We also propose an adaptive smoothing method allowing to reduce possible oversmoothing. Roughly speaking, our smoothing schemes consist of moving every vertex in the direction defined by the Laplacian flow with speed equal to a properly chosen function of the mean curvature at the vertex.

The paper develops a method to design nonlinear splines on a plane via curve evolutions driven by curvature. We consider a curve passing through two given end points and satisfying prescribed boundary conditions at them (for example, curvature values or tangent directions are specified at the end points). Each point of the curve moves in the normal direction with speed equal to a function of the curvature and curvature derivatives at the point. Choosing the speed function properly, the evolving curve converges to a desired nonlinear spline. We also consider evolutions of closed curves for purposes of multiscale shape analysis. Smooth curve evolutions are approximated by evolutions of polygonal curves. Discrete analogs of the curvature and its derivatives are considered.

Surface creases provide us with important information about the shapes of objects and can be intuitively defined as curves on a surface along which the surface bends sharply. Our mathematical description of such surface creases is based on a study of extrema of the principal curvatures along their curvature lines. On a smooth generic surface we define ridges to be the local positive maxima of the maximal principal curvature along its associated curvature line and ravines to be the local negative minima of the minimal principal curvature along its associated curvature line. The ridges and ravines are important for shape analysis and possess remarkable mathematical properties. For example, they correspond to the end points of shape skeletons. In this paper we derive formulas to detect the ridges and ravines on a surface given in implicit form. We also propose an algorithm for obtaining piecewise linear approximation of ridges and ravines as intersection curves of two implicit surfaces.

We present a method of stable extraction of curvature features from a scattered data obtained by measuring a Noh mask shape. First, since the Noh mask shape can be represented as the graph of a function, we obtain a height function representation of the data. Then, we smooth the height function data by a Gaussian filter and by an iterative nonlinear filter. Finally we compute curvature features using finite-difference approximations. We test our approach detecting convex/concave, saddle, cylindrical, and plane regions on the Noh mask scattered data.

We propose fast algorithms for the detection of view independent ridges on surfaces given by graphs of functions. The ridges are defined via extrema of the principal curvatures along the associated principal direction. These algorithms have the advantage to be fast and it appears that the so defined ridges deserve as a characteristic feature of the shape of the surface. Results on analytic surfaces and on real data are shown and discussed.

We introduce a new concept of ridges, ravines and related structures (skeletons) associated with surfaces in three-dimensional space that generalizes the medial axis transformation approach. The concept is based on singularity theory and involves both local and global geometric properties of the surface; it is invariant with respect to translations and rotations of the surface. It leads to a method of hierarchic description of surfaces that yields new approaches to shape coding, rendering and design. The extraction of the features is based on differential geometry of surfaces with consequent segregation via multiscale analysis. Terrain feature recognition, dental shape reconstruction and medical imagery are a partial list of applications.>