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Local features of smooth shapes: ridges and courses
Abstract
If one direction of (three-dimensional) space is singled out, it makes sense to formulate the description of embedded curves and surfaces in a frame that is adapted both to the embedded manifold and to the special direction, rather than a frame based upon the curvature landscape. Such a case occurs often in computer vision, where the image plane plays a role that differs essentially from the direction of view. The classical case is that of geomorphology, where the vertical is the singled out dimension. In computer vision the `ridges' and `(water-)courses' are recognized as important entities and attempts have been made to make the intuitive notions precise. These attempts repeat the unfortunate misunderstandings that marked the course of the late 19th century struggle to define the `Talweg' (equals `valley path' or `(water-)course'). We elucidate the problems and their solution via novel examples.
... The ridge-and dale-lines have been studied by the computer scientists working on imaging and vision [37][38][39], where the ridge-and dale-lines have rich applications, especially in two-dimensional Euclidean geometry. In physical configuration spaces having a higherdimensional (pseudo-)Riemannian geometry, the ridge-and dale-lines have not been much used, to our knowledge. ...
... (5.3) in eq. (5.2) results in the de Saint-Venant equation for ridges (dSVr) [37,40] ...
... The stream approach is also adapted by modern computer scientists in image processing and computer vision [37,38]. The mathematics behind this approach is the inverse integral factor and inverse Jacobi multiplier, which work for two-and higher-dimensional cases, respectively [48,49]. ...
We introduce a concept of ridge-lines to investigate the semi-classical prediction from wave-packets with arbitrary width in conventional quantum mechanics and the Wheeler--DeWitt quantum cosmology. Two primary approaches are applied to the exact calculation of the ridge-lines, namely the contour and the stream approach. Moreover, aspects of these are discussed and compared to other scenarios and approaches, i.e. the narrow WKB wave-packets and the first-derivative test. As the main result, we show that the semi-classical predictions in toy models have more abundant solutions than in the classical theory, and most interestingly they may deviate from classical solutions due to the quantum corrections.
... Cette condition tombe en échec en détectant des slopelines droites, comme montrée plus tard par Boussinesq [5] et Breton de Champ [13]. Or, les vallées sur le terrain sont des slopelines [35]. ...
... Discutons 35 proches, les points détectés sont restés discrets, ce qui demande une étape de chaînage avant d'aboutir à une interprétation plus significative de la surface. ...
... Since then, numerous authors have addressed the issue often in confusing and contradictory ways. This historical body of work includes Boussinesq [8,9], Breton de Champ [10], Jordan [11] and Rothe [12] and we encourage the reader to review Koenderink and van Doorn's [13] excellent overview. ...
... For a detailed discussion on some of the more popular approaches, we refer the reader to [16,2,17,18,19]. It has been previously noted that all of the ridges and valleys in the height ridge definition, must be points where the gradient of the height function and the gradient of the magnitude are aligned [17,20,19,13], but the subtle consequences of this observation have not been examined, which is one of the results of this paper. In their recent paper, Sadlo and Peikert [19] take advantage of this fact to extract "raw features," which are then filtered to produce ridges and valleys. ...
Ridges are one of the key features of interest in areas such as computer vision and image processing. Even though a significant amount of research has been directed to defining and extracting ridges some fundamental challenges remain. For example, the most popular ridge definition (height ridge) is not invariant under monotonic transformations and its global structure is typically ignored during numerical computations. Furthermore, many existing algorithms are based on numerical heuristics and are rarely guaranteed to produce consistent results. This paper reexamines a slightly different ridge definition that is consistent with all desired invariants. Nevertheless, we show that this definition results in similar structures compared to height ridges and that both formulations are equivalent for quadratic functions. Furthermore, this definition can be cast in the form of a degenerate Jacobi set, which allows insights into the global structure of ridges. In particular, we introduce the Ridge–Valley graph as the complete description of all ridges in an image. Finally, using the connection to Jacobi sets we describe a new combinatorial algorithm to extract the Ridge–Valley graph from sampled images guaranteed to produce a valid structure.
... Canny's local maxima along the gradient direction and the zero crossings of the Laplacian are two common examples. A comparison of alternatives can be found in [2] . Unfortunately, it turns out that local definitions only work well for 1-dimensional ridges, but fail near junctions. ...
... Sorting is complicated by the phenomenon that watersheds often converge tangentially towards the maximum, cf.fig. 2 . This is not an artifact of our implementation , but a well-known watershed property [2, 9]. In theory, watersheds do not meet before the maximum, even if they converge tangentially; however, due to the finite accuracy of flowline tracing, the computed polygonal arcs do cross in practice when they are very close to each other. ...
Discrete algorithms for low-level boundary detection are geometrically inaccurate and topologically unreliable. C or- responding continuous methods are often more accurate and need fewer or no heuristics. Thus, we transfer discrete boundary indicators into a continuous form by means of differentiable spline interpolation and detect boundarie s us- ing the exact watershed transform. We demonstrate that this significantly improves the obtained segmentations. keywords: sub-pixel segmentation, watersheds, splines
... For example, roads typically comply with known engineering limits for slope, side slope, radius of curvature, and so forth. Mountain ridges exhibit well-known di erential properties that are comparable to those satis ed by river valleys Koenderink and van Doorn, 1993 . We view the contribution of this paper as proposing a general approach to accurately modeling terrain and features from several information sources that may be in error and inconsistent with one another. ...
... Following standard snake practices Kass et al., 1988 , w e model C as a list of regularly spaced 3 D vertices S 3 of the form S 3 = fx i y i z i ; i = 1 ; : : : ; n g; 7 and we write E D C = 1 2 X 2x i , x i,1 , x i+1 2 + 2 y i , y i , 1 , y i +1 2 + 2 z i , z i , 1 , z i +1 2 E P C = X z i : 8 ...
We propose an automated approach to modeling drainage channels—and, more generally, linear features that lie on the terrain—from multiple images, which results not only in high-resolution, accurate and consistent models of the features, but also of the surrounding terrain. In our specific case, we have chosen to exploit the fact that rivers flow downhill and lie at the bottom of local depressions in the terrain, valley floors tend to be U shaped, and the drainage pattern appears as a network of linear features that can be visually detected in single gray-level images.
Different approaches have explored individual facets of this problem. Ours unifies these elements in a common framework. We accurately model terrain and features as 3-dimensional objects from several information sources that may be in error and inconsistent with one another. This approach allows us to generate models that are faithful to sensor data, internally consistent and consistent with physical constraints. We have proposed generic models that have been applied to the specific task at hand—river delineation and data elevation model (DEM) refinement—and show that the constraints can be expressed in a computationally effective way and, therefore, enforced while initializing the models and then fitting them to the data. We will also argue that the same techniques are robust enough to work on other features that are constrained by predictable forces.
... Early definitions of such extremal structures in the context of image analysis were given in the works by Haralick [14], Koenderink and van Doorn [23], and Eberly et al. [11]. A good summary of these early definitions is given by Schultz et al. [35]. ...
Ridge surfaces represent important features for the analysis of 3-dimensional (3D) datasets in diverse applications and are often derived from varying underlying data including flow fields, geological fault data, and point data, but they can also be present in the original scalar images acquired using a plethora of imaging techniques. Our work is motivated by the analysis of image data acquired using micro-computed tomography (Micro-CT) of ancient, rolled and folded thin-layer structures such as papyrus, parchment, and paper as well as silver and lead sheets. From these documents we know that they are 2-dimensional (2D) in nature. Hence, we are particularly interested in reconstructing 2D manifolds that approximate the document's structure. The image data from which we want to reconstruct the 2D manifolds are often very noisy and represent folded, densely-layered structures with many artifacts, such as ruptures or layer splitting and merging. Previous ridge-surface extraction methods fail to extract the desired 2D manifold for such challenging data. We have therefore developed a novel method to extract 2D manifolds. The proposed method uses a local fast marching scheme in combination with a separation of the region covered by fast marching into two sub-regions. The 2D manifold of interest is then extracted as the surface separating the two sub-regions. The local scheme can be applied for both automatic propagation as well as interactive analysis. We demonstrate the applicability and robustness of our method on both artificial data as well as real-world data including folded silver and papyrus sheets.
... The conceptual framework developed here stems from observing the complex ridges and valleys patterns in topographic landscapes, and the related work in the fields of image processing, geomorphology and hydrology related to the duality between the interlocking network of ridges and valleys [29][30][31]. For its mathematical formulation, we draw inspiration from the landscape evolution models (LEMs), which have been successful in describing the formation of river and stream networks [11,[32][33][34][35]. Generalizing these models, we develop a simple system consisting of three nonlinear coupled PDEs with an essential parametrization. ...
Numerous complex systems, both natural and artificial, are characterized by the presence of intertwined supply and/or drainage networks. Here, we present a minimalist model of such coevolving networks in a spatially continuous domain, where the obtained networks can be interpreted as a part of either the counter-flowing drainage or co-flowing supply and drainage mechanisms. The model consists of three coupled, nonlinear partial differential equations that describe spatial density patterns of input and output materials by modifying a mediating scalar field, on which supply and drainage networks are carved. In the two-dimensional case, the scalar field can be viewed as the elevation of a hypothetical landscape, of which supply and drainage networks are ridges and valleys, respectively. In the three-dimensional case, the scalar field serves the role of a chemical signal, according to which vascularization of the supply and drainage networks occurs above a critical ‘erosion’ strength. The steady-state solutions are presented as a function of non-dimensional channelization indices for both materials. The spatial patterns of the emerging networks are classified within the branched and congested extreme regimes, within which the resulting networks are characterized based on the absolute as well as the relative values of two non-dimensional indices.
... The conceptual framework developed here stems from observing the complex ridges and valleys patterns in topographic landscapes, and the related work in the fields of image processing, geomorphology, and hydrology to formalize the duality between the interlocking network of ridges and valleys [29,30,31]. For mathematical formulization, we draw inspiration from landscape evolution models (LEMs) which have been successful in describing the formation of river and stream networks [32,33,11,34]. ...
Numerous complex systems, both natural and artificial, are characterized by the presence of intertwined supply and/or drainage networks. Here we present a minimalist model of such co-evolving networks in a spatially continuous domain, where the obtained networks can be interpreted as a part of either the counter-flowing drainage or co-flowing supply and drainage mechanisms. The model consists of three coupled, nonlinear partial differential equations that describe spatial density patterns of input and output materials by modifying a mediating scalar field, on which supply and drainage networks are carved. In the 2-dimensional case, the scalar field can be viewed as the elevation of a hypothetical landscape, of which supply and drainage networks are ridges and valleys, respectively. In the 3-dimensional case, the scalar field serves as the chemical signal strength, in which vascularization of the supply and drainage networks occurs above a critical 'erosion' strength. The steady-state solutions are presented as a function of non-dimensional channelization indices for both materials. The spatial patterns of the emerging networks are classified within the branched and congested extreme regimes, within which the resulting networks are characterized based on the absolute as well as the relative values of two non-dimensional indices.
... [46,47,49]). However, they can be broadly applied to digital terrain analysis [41,48,[50][51][52][53] for the detection of ridge and valley lines (representing watershed divides and channels, respectively) and, as will be investigated in this work, for developing and testing models and algorithms for the numerical calculation of the drainage area at non-regular points. In addition, skeleton construction techniques have been shown to allow a fully automated recognition in contour-based DEMs of the complex topographic structures that can be found in real landscapes [54]. ...
The drainage area is an important, nonlocal property of a landscape, which controls surface and subsurface hydrologic fluxes. Its role in numerous ecohydrological and geomorphological applications has given rise to several numerical methods for its computation. However, its theoretical analysis has lagged behind. Only recently, an analytical definition for the specific catchment area has been proposed [1], with the derivation of a differential equation whose validity is limited to regular points of the watershed. Here we show that such a differential equation can be derived from a continuity equation [2] and extend the theory to critical and singular points both applying Gauss’ theorem and by means of a dynamical systems approach to define basins of attraction of local surface minima. Simple analytical examples as well as applications to more complex topographic surfaces are examined. The theoretical description of topographic features and properties, such as the drainage area, channel lines, and watershed divides can be broadly adopted to develop and test the numerical algorithms currently used in digital terrain analysis for the computation of the drainage area, as well as for the theoretical analysis of landscape evolution and stability.
... Rothe (1915) argued that crests and thalwegs are singular solutions of the differential equation of the slope lines. Koenderink and van Doorn (1993) supposed that crests and thalwegs are the special type of slope lines where other ones converge, and they also argued that local differential geometric criteria for crests and thalwegs cannot exist. Since the problem of analytical description of structural lines has not been resolved, practical derivation of crests and thalwegs is mainly carried out by flow routing algorithms, similar to calculations of non-local variables (see reviews: Clarke and Romero, 2017;López et al., 1999;Tribe, 1992). ...
Geomorphometry is widely used to solve various multiscale geoscientific problems. For the successful application of geomorphometric methods, a researcher should know the basic mathematical concepts of geomorphometry and be aware of the system of morphometric variables, as well as understand their physical, mathematical and geographical meanings. This paper reviews the basic mathematical concepts of general geomorphometry. First, we discuss the notion of the topographic surface and its limitations. Second, we present definitions, formulae and meanings for four main groups of morphometric variables, such as local, non-local, two-field specific and combined topographic attributes, and we review the following 29 fundamental morphometric variables: slope, aspect, northwardness, eastwardness, plan curvature, horizontal curvature, vertical curvature, difference curvature, horizontal excess curvature, vertical excess curvature, accumulation curvature, ring curvature, minimal curvature, maximal curvature, mean curvature, Gaussian curvature, unsphericity curvature, rotor, Laplacian, shape index, curvedness, horizontal curvature deflection, vertical curvature deflection, catchment area, dispersive area, reflectance, insolation, topographic index and stream power index. For illustrations, we use a digital elevation model (DEM) of Mount Ararat, extracted from the Shuttle Radar Topography Mission (SRTM) 1-arc-second DEM. The DEM was treated by a spectral analytical method. Finally, we briefly discuss the main paradox of general geomorphometry associated with the smoothness of the topographic surface and the non-smoothness of the real topography; application of morphometric variables; statistical aspects of geomorphometric modelling, including relationships between morphometric variables and roughness indices; and some pending problems of general geomorphometry (i.e. analysis of inner surfaces of caves, analytical description of non-local attributes and structural lines, as well as modelling on a triaxial ellipsoid). The paper can be used as a reference guide on general geomorphometry.
... The blue lines in Fig. 7 highlight the primary geological structures identified by the lineament Evans-Young method (Florinsky, 1998;Schmidt et al., 2003). Finally, the lineament elements are automatically extracted from the digital map after the digital image processing (Haralick, 1983;Koenderink and van Doorn, 1993;Eberly et al., 1994;Rieger, 1997;López et al., 1998), where the degree of contrast is visually chosen to guide the lineament extraction. ...
The Honam-Jeju, Korea-Japan, and Korea-China subsea tunnel construction projects have drawn significant attention since the early 2000s. These subsea tunnels are much deeper than most existing natural shallow sea tunnels linking coastal areas. Thus, the need for developing new technologies for the site selection and construction of deep subsea tunnels has recently emerged, with the launch of a research project titled "Development of Key Subsea Tunnelling Technology" in 2013. A component of this research, an analysis of deep subsea geological structure, is currently underway. A ground investigation, such as a borehole or geophysical investigation, is generally carried out for tunnel design. However, when investigating a potential site for a deep subsea tunnel, borehole drilling requires equipment at the scale of offshore oil drilling. The huge cost of such an undertaking has raised the urgent need for methods to indirectly assess the local geological structure as much as possible to limit the need for repeated borehole investigations. This study introduces an indirect approach for assessing the geological structure of the seafloor through a submarine bathymetry analysis. The ultimate goal here is to develop an automated approach to the analysis of submarine geological structures, which may prove useful in the selection of future deep subsea tunnel sites.
... One should note that our definition of ridges is intrinsic to the isophote surface, without privileging any particular direction of the 3D space. One should refer to [84] for a rigorous and historical analysis of oriented ridges, i.e. ridges depending on a privileged direction in space. ...
The automated analysis of 3D medical images can improve significantly both diagnosis and therapy. This automation raises a number of new fascinating research problems in the fields of computer vision, graphics and robotics. In this article, (a complete version is published by the Int. Journal of Image and Vision Computing [4]) I propose a list of such problems after a review of the current major 3D imaging modalities, and a description of the related medical needs. I then present some of past and current work done in our research group EPIDAURE at INRIA, on the following topics: segmentation of 3D images, 3D shape modeling, 3D rigid and nonrigid registration, 3D motion analysis and 3D simulation of therapy.
... This makes it impossible for different researchers to reach the same result, and there is no objective measure of correctness. Several researchers have realized this difficulty, and tried to remedy it [10,17,46]. ...
... The rst insight is that directed and oriented circuit or path integrals of physical observations on the detector array reveal the set of construction rules. The second insight is that a dynamic scale-space of this set yields a stable and reproducible description of it that is slightly a ected by noise e ects, that preserves the most salient coherent structures such as watersheds, crest lines, edges, plateaus, divides and ridges, channels and ruts, and defect lines (see 37,29,21,11,30] for de nitions of these physical objects), and that is invariant under certain transformation groups 27]. Among these transformation groups are those of Euclidean, a ne, projective, Galilean or Lorentzian transformations, that of anamorphoses or contrast transfor-2 A. H. SALDEN mations (monotonic and order preserving grey-value transformations) and that of gauge eld transformations (di eomorphisms of the image in a spatio-temporal but also dynamical sense) containing contrast transformations. ...
The geometric and statistical physical concepts of dynamic scale-space paradigms are presented and juxtaposed to those of mathematical morphology. It turns out that the dynamic paradigms can be applied to, substantiate and even generalize the morphological techniques and paradigms. In particular the importance of the dynamic scale-space concepts in granulometry by means of size densities or statistical morphological operations, and in morphological scale-space theories by means of parabolic dilations and watersheds is pointed out.
... In fact, it is often even overlooked that the watershed definition is a valid ridge definition and even among the first ones to be proposed, see Maxwell [4]. The relationship between these mathematically different approaches (local vs. global definition) was debated vigorously in the computer vision community in the early 1990s (see, e.g., Koenderink and van Doorn [5]). Nowadays, there is a consensus that both approaches have their merits, and López et al. [6] provide an exhaustive evaluation of the equated local and global definitions. ...
The finite-time Lyapunov exponent (FTLE) has become a standard tool for analyzing unsteady flow phenomena, partly since its ridges can be interpreted as Lagrangian Coherent Structures (LCS). While there are several definitions for ridges, a particular one called second derivative ridges has been introduced in the context of LCS, but subsequently received criticism from several researchers for being over-constrained. Among the critics are Norgard and Bremer (2012) [3], who suggest furthermore that the widely used definition of height ridges was a part of the definition of second derivative ridges, and that topological separatrices were ill-suited for describing ridges. We show that (a) the definitions of height ridges and second derivative ridges are not directly related, and (b) there is an interdisciplinary consensus throughout the literature that topological separatrices describe ridges. Furthermore, we provide pointers to practically feasible and numerically stable ridge extraction schemes for FTLE fields.
... For ridge detection on surfaces, we utilize the ridge definition first introduced by Rothe (1915), more recently described by Koenderink and van Doorn (1993). Intuitively a ridge of a height function can be imagined as the way one would take when walking up a mountain. ...
Deformable surface models are often represented as triangular meshes in image segmentation applications. For a fast and easily regularized deformation onto the target object boundary, the vertices of the mesh are commonly moved along line segments (typically surface normals). However, in case of high mesh curvature, these lines may not intersect with the target boundary at all. Consequently, certain deformations cannot be achieved. We propose omnidirectional displacements for deformable surfaces (ODDS) to overcome this limitation. ODDS allow each vertex to move not only along a line segment but within the volumetric inside of a surrounding sphere, and achieve globally optimal deformations subject to local regularization constraints. However, allowing a ball-shaped instead of a linear range of motion per vertex significantly increases runtime and memory. To alleviate this drawback, we propose a hybrid approach, fastODDS, with improved runtime and reduced memory requirements. Furthermore, fastODDS can also cope with simultaneous segmentation of multiple objects. We show the theoretical benefits of ODDS with experiments on synthetic data, and evaluate ODDS and fastODDS quantitatively on clinical image data of the mandible and the hip bones. There, we assess both the global segmentation accuracy as well as local accuracy in high curvature regions, such as the tip-shaped mandibular coronoid processes and the ridge-shaped acetabular rims of the hip bones.
... However, its higher-dimensional generalizations cannot be defined in a canonical way, i.e., several equated definitions exist for extremal lines or surfaces. This is documented throughout the literature [Kv93,Dam99,LLSV99,SWTH07,PS08,SPFT12]. Besides their global definition as topological separatrices [Max70], another frequently used concept are Height Ridges, which goes back to De Saint-Venant [dSV52]. ...
Extremal lines and surfaces are features of a 3D scalar field where the scalar function becomes minimal or maximal with respect to a local neighborhood . These features are important in many applications, e.g. computer tomography, fluid dynamics, cell biology . We present a novel topological method to extract these features using discrete Morse theory. In particular, we extend the notion of ‘separatrix persistence’ from 2D to 3D, which gives us a robust estimation of the feature strength for extremal lines and surfaces. Not only does it allow us to determine the most important (parts of) extremal lines and surfaces, it also serves as a robust filtering measure of noise-induced structures. Our purely combinatorial method does not require derivatives or any other numerical computations . © 2012 Wiley Periodicals, Inc.
... The results on a variety of other types of ODTs are omitted for space considerations. Despite much debate in the literature on the proper way to compute ridges (e.g., [12,43,27,20,9], here we use a local method by López et al. [24] that provides excellent numerical stability. Results of our theory on selected ODTs are demonstrated in Fig. 8 (also in Fig. 7) and illustrate the outstanding match between perception and the proposed computational theory. ...
The analysis of texture patterns, and texture segregation in particular, are at the heart of perceptual organization. In this paper we question the widely accepted view that the de-tection (both perceptual and computational) of salient per-ceptual singularities between perceptually coherent texture regions is tightly dependent upon feature gradients. Specif-ically, we study smooth orientation-defined textures (ODTs) and show that they exhibit striking perceptual singularities even without any outstanding gradients in their defining fea-ture, namely orientation. We further show how these generic singularities are not only unpredictable from the orienta-tion gradient, but that they also defy popular segmentation algorithms and neural models. We then examine smooth ODTs from a (differential) geometric point of view and de-velop a theory that fully predicts their perceptual singular-ities from two ODT curvatures. The computational results exhibit striking correspondence to segregation performed by human subjects and provide a conclusive evidence for the role of curvature in texture segregation. Extensions and im-plications of our results are developed for various aspects of visual processing.
... Schmitt and Chen [12] updated Fowler and Little's method by first identifying surface and slope discontinuities and including these lines in the resulting approximation, which uses their own triangulation criterion. They choose lines based on the local differential structure of the surface, which is independent of the choice of coordinate system and is not necessarily coincident with the paths determined by the flow of water [13]. [14,15] also insert "crest" lines into adaptive meshes to improve stereo-driven surface approximation. ...
The standard method of building compact triangulated surface approximations to terrain surfaces (TINs) from dense digital
elevation models (DEMs) adds points to an initial sparse triangulation or removes points from a dense initial mesh. Instead,
we find structural lines to act as the initial skeleton of the triangulation. These lines are based on local curvature of
the surface, not on the flow of water. We build TINs from DEMs with points and structural lines. These experiments show that
initializing the TIN with structural lines at the correct scale creates a TIN with fewer points given a particular approximation
error. Structural lines are especially effective for small numbers of points and correspondingly rougher approximations.
Key words. TIN, DEM, Structural lines, Constrained Delaunay triangulation, Curvature.
... These neighborhoods form the two sides of the edge. The geometric difference between folds and cuts results in an observable photometric difference [33][24][22]. If we define the shading flow field to be the vector field of tangents to isoluminance contours in the image [4], we can differentiate between the fold and cut neighborhoods of edges as follows. ...
A natural sequel to edge detection is the interpretation of edges. This interpretation can provide useful information to various computer vision processes, including recognition, reconstruction, and tracking. In this paper we consider the problem of identifying occlusion edges in a single image. We examine the appearance of occlusion edges under variable illumination, both analytically and empirically, and find that the pattern of shading in the neighborhood of occlusion edges is a stable feature. Finally, we derive a filter for detecting occlusion and present the results of its application.
... The terrain model, however, is almost always assumed to be error-free, which, in practice, only rarely is the case. Furthermore, Koenderink and Van Doorn have shown [9] that the local differential criteria many of these systems use to detect valleys have inherent problems. Much work has also been devoted to the extraction of linear patterns from single images using techniques such as dynamic programming [4,10] or graph-based techniques [3]. ...
We propose an automated approach to modeling dra- inage channels—and, more generally, linear features that lie on the terrain—from multiple images, which results not only in high-resolution, accurate and consistent models of the features, but also of the surrounding terrain. In our specific case, we have chosen to exploit the fact that rivers flow downhill and lie at the bottom of local depressions in the terrain, valley floors tend to be "U" sha- ped, and the drainage pattern appears as a network of li- near features that can be visually detected in single gray- level images. Different approaches have explored individual facets of this problem. Ours unifies these elements in a common fra- mework. We accurately model terrain and features as 3- dimensional objects from several information sources that may be in error and inconsistent with one another. This ap- proach allows us to generate models that are faithful to sen- sor data, internally consistent and consistent with physical constraints.
... Ridges generalize local maxima, while valleys correspond to local minima. Even though the most adequate crease definition is not undisputed [64], the so-called "height crease definition" [43] has become a well-researched tool to find medial axes in grayscale images [96] and has been extended towards applications as diverse as medical image analysis, molecular modeling, and analysis of fluid flows [35]. ...
Ridge surfaces represent important features for the analysis of 3-dimensional (3D) datasets in diverse applications and are often derived from varying underlying data including flow fields, geological fault data, and point data, but they can also be present in the original scalar images acquired using a plethora of imaging techniques. Our work is motivated by the analysis of image data acquired using micro-computed tomography (
$\mu\text{CT}$
) of ancient, rolled and folded thin-layer structures such as papyrus, parchment, and paper as well as silver and lead sheets. From these documents we know that they are 2-dimensional (2D) in nature. Hence, we are particularly interested in reconstructing 2D manifolds that approximate the document's structure. The image data from which we want to reconstruct the 2D manifolds are often very noisy and represent folded, densely-layered structures with many artifacts, such as ruptures or layer splitting and merging. Previous ridge-surface extraction methods fail to extract the desired 2D manifold for such challenging data. We have therefore developed a novel method to extract 2D manifolds. The proposed method uses a local fast marching scheme in combination with a separation of the region covered by fast marching into two sub-regions. The 2D manifold of interest is then extracted as the surface separating the two sub-regions. The local scheme can be applied for both automatic propagation as well as interactive analysis. We demonstrate the applicability and robustness of our method on both artificial data as well as real-world data including folded silver and papyrus sheets.
We introduce a concept of ridgelines to investigate the semiclassical predictions from wave packets with arbitrary widths in conventional quantum mechanics and the Wheeler–DeWitt quantum cosmology. Two primary approaches are applied to the exact calculation of the ridgelines, namely the contour and the stream approaches. Moreover, aspects of these are discussed and compared to other scenarios and approaches, i.e., the narrow WKB wave packets and the first-derivative test. As the main result, we show that the semiclassical predictions in toy models have more abundant solutions than the ones in the classical theory, and most interestingly they may deviate from classical solutions due to the quantum corrections.
Image formation of a two-dimensional input image can be quantified by imposing an image induced connection and computing the associated torsion and curvature. The latter aspects of image formation are especially nonvanishing at sets of discontinuities and non-isolated singularities, such as ridges and ruts. Next dynamic scale-space theories for the input image are constructed on the basis of an image induced connection. Finally dynamic scale-space theories for the image formation are constructed that are coupled to the image formation itself.
Based on TINs DEM, a new single-flow direction algorithm for topographic patterns extraction is proposed. The algorithm contains three main steps:(1)Flow direction is determined by calculating the steepest descent of each triangle and flow area of each triangle edge;(2)River networks are tracked for both channel flow and cross-triangle flow;(3)Depressions are extracted as local individual drainage basins and they can be merged into their outflow drainage basins if necessary. The present algorithm does not need pre-processing of depressions and the extracted drainage networks are well structured with high efficiency. The proposed method has been tested in a case study for Dongyang City, Zhejiang, and the topographic patterns extracted are compared with official drainage map and those extracted by ArcGIS. This algorithm is suitable for automatic topographic extraction, and it can be applied in parallel computation for distributed hydrological model that use an irregular spatial discretization as it can split a giant mesh and recombine it in high efficiency, without changing mesh structure.
The purpose of this chapter is to examine how changing images constitute information for vision. The concept of information is critical to understanding the relationship between objects, images, and perceptions. Environments and objects are visible by virtue of the information their images contain. Differences among theories of perception usually stem from differing assumptions about the form of image information. The term information has become ubiquitous in psychology, but all too often the term has a little or no meaning or is even misleading. Before considering how information may be contained in changing images, it is useful to clarify the concept of information.
This chapter reviews the empirical literature on the quality of perceived pictorial space under a variety of viewing conditions both ideal and ordinary. The effects of viewing pictures away from their center of projection are reviewed and evaluated to address theoretical issues in perceiving pictorial space. The empirical evidence supports the idea that pictures bear a close structural relation to the real scenes they represent; it is argued that the concept of constraint is an essential one. Pictures have great potential to be ambiguous and to present systematically distorted spaces. The fact that they are rarely perceived as ambiguous or distorted is due to traditional artistic practice that provides constraints on the construction and observation of pictures and that almost guarantee their success.
A new algorithm to automatically extract drainage networks and catchments based on triangulation irregular networks (TINs) digital elevation model (DEM) was developed. The flow direction in this approach is determined by computing the spatial gradient of triangle and triangle edges. Outflow edge was defined by comparing the contribution area that is separated by the steepest descent of the triangle. Local channels were then tracked to build drainage networks. Both triangle edges and facets were considered to construct flow path. The algorithm has been tested in the site for Hawaiian Island of Kaho’olawe, and the results were compared with those calculated by ARCGIS as well as terrain map. The reported algorithm has been proved to be a reliable approach with high efficiency to generate well-connected and coherent drainage networks.
‘Pictorial relief’ is a surface in 3D ‘pictorial space’. It is perceived in single flat pictures and clearly has nothing t do with binocular stereopsis but with the interpretation of image structure in terms of relations in the external world. Way to perform geometrical measurements in pictorial space are presented and a number of empirical results are reviewed. Application to the theory of optical instruments aiding human vision are discussed.
In the present work we deal with the assistance to the diagnostic of coronaries stenosis from X-rays angiographies. Our goal is a 3D-reconstruction of the coronarian tree, therefore the extraction of some 2D characteristics is necessary. Here, we treat the problem of the 2D vessels medial axis extraction. The vessels geometry looks like valleys embedded in the image surface. Using differential geometry we can locally characterize medial axis as bottom lines of valleys. However, we have to calculate the image local derivatives, which is an ill-posed and noise sensitive problem. To overcome this drawback, we use a PDE based approach. We first consider the PDE's numerical scheme as an iterative method known as fixed point search. So, we obtain a new method which assure the stability of the resolution process. The combinaison of this method an appropriate PDE generates a scale-space where we can detect arteries of various diameters. We use then the eigenvalues and eigenvectors of the Weingarten endomorphism to define a new valley-ness measure. We have tested this technique on several angiographies, where the medial axis have well been extracted, even in presence of strong stenosis. Furthermore, the extracted axis are one pixel large and quite continuous.
User Guide for developer who use ITK in their applications. Everything you need to install, use, and extend the Insight Segmentation and Registration Toolikit ITK. Includes detailed examples, installation procedures, and system overview for ITK version 2.4. (The included examples are taken directly from the ITK source code repository and are designed to demonstrate the essential features of the software.) The book comes with a CD-ROM that contains a complete hyperlinked version of the book plus ITK source code, data, Windows binaries, and extensive class documentation. Also includes CMake binaries for managing the ITK build process on a variety of compiler and operating system configurations. U.S. National Library of Medicine (NLM).
It is difficult to generate high-definition time-frequency maps for rapidly changing transient signals. New details of the theory of harmonic wavelet analysis are described which provide the basis for computational algorithms designed to improve map definition. Features of these algorithms include the use of ridge identification and phase gradient as diagnostic features.
Topology was introduced in the visualization literature some 15 years ago as a mathematical language to describe and capture the salient structures of symmetric second-order tensor fields. Yet, despite significant theoretical and algorithmic advances, this approach has failed to gain wide acceptance in visualization practice over the last decade. In fact, the very idea of a versatile visualization methodology for tensor fields that could transcend application domains has been virtually abandoned in favor of problem-specific feature definitions and visual representations. We propose to revisit the basic idea underlying topology from a different perspective. To do so, we introduce a Lagrangian metaphor that transposes to the structural analysis of eigenvector fields a perspective that is commonly used in the study of fluid flows. Building upon the strong theoretical connections that exist between vector and eigenvector fields, we show that the separatrices of 3D tensor field's topology can in fact be captured in a fuzzy and numerically more robust setting as ridges of a trajectory coherence measure. As a result, we propose an alternative structure characterization strategy for the visual analysis of practical 3D tensor fields, which we demonstrate on several synthetic and computational datasets.
This thesis presents a novel computational framework that allows for a robust extraction and quantification of the Morse-Smale complex of a scalar field given on a 2- or 3-dimensional manifold. The proposed framework is based on Forman’s discrete Morse
theory, which guarantees the topological consistency of the computed complex. Using
a graph theoretical formulation of this theory, we present an algorithmic library that
computes the Morse-Smale complex combinatorially with an optimal complexity of
O(n^2) and efficiently creates a multi-level representation of it. We explore the discrete
nature of this complex, and relate it to the smooth counterpart. It is often necessary to
estimate the feature strength of the individual components of the Morse-Smale complex
– the critical points and separatrices. To do so, we propose a novel output-sensitive
strategy to compute the persistence of the critical points. We also extend this wellfounded
concept to separatrices by introducing a novel measure of feature strength
called separatrix persistence. We evaluate the applicability of our methods in a wide
variety of application areas ranging from computer graphics to planetary science to
computer and electron tomography.
Vector fields are a common concept for the representation of many different kinds of flow phenomena in science and engineering. Methods based on vector field topology are known for their convenience for visualizing and analysing steady flows, but a counterpart for unsteady flows is still missing. However, a lot of good and relevant work aiming at such a solution is available. We give an overview of previous research leading towards topology-based and topology-inspired visualization of unsteady flow, pointing out the different approaches and methodologies involved as well as their relation to each other, taking classical (i.e. steady) vector field topology as our starting point. Particularly, we focus on Lagrangian methods, space–time domain approaches, local methods and stochastic and multifield approaches. Furthermore, we illustrate our review with practical examples for the different approaches.
The importance of plaque component classi .cation and vessel wall quantification has been well established by several research
groups (see Refs. [1–30]).
We measured local surface attitude for monocular pictorial relief and performed pairwise depthcomparison judgments on the
same picture. Both measurements were subject to internal consistency checks. We found that both measurements were consistent
with a relief (continuous pictorial surface) interpretation within the session-to-session scatter. We reconstructed the pictorial
relief from both measurements separately, and found results that differed in detail but were quite similar in their basic
structures. Formally, one expects certain geometrical identities that relate range and attitude data. Because we have independent
measurements of both, we can attempt an empirical verification of such geometrical identities. Moreover, we can check whether
the statistical scatter in the data indicates that, for example, the surface attitudes are derivable from a depth map or vice
versa. We estimate that pairwise depth comparisons are an order of magnitude less precise than might be expected from the
attitude data. Thus, the surface attitudes cannot be derived from a depth map as operationally defined by our methods, although
the reverse is a possibility.
Creases are a type of ridge/valley structures of an image characterized by local conditions. As creases tend to be at the center of anisotropic grey-level shapes, creaseness can be considered a measure of medialness, and therefore as useful in many image analysis problems. Among the several possibilities, a priori the creaseness based on the level-set extrinsic curvature (LSEC) is especially interesting due to its invariance properties. However, in practice, it produces a discontinuous response with a badly dynamic range. The same problems arise with other related creaseness measures proposed in the literature. In this paper, we argue that these problems are due to the very local definition of the LSEC. Therefore, rather than designing an ad hoc solution, we propose two new multilocal creaseness measures that we will show to be free of discontinuities and to have a meaningful dynamic range of response. Still, these measures are based on the LSEC idea, to preserve its invariance properties. We demonstrate the usefulness of the new creaseness measures in the context of two applications that we are currently developing in the field of 3D medical image analysis, the rigid registration of CT and MR head volumes and the orientation analysis of trabecular bone patterns.
Automatic fish fillet inspection, i.e. detection of flaws and irregularities, is often done on fast moving conveyer belts using line scan imaging. Recent work shows that imaging spectroscopy is a promising method for automatic fish fillet inspection. This requires fast algorithms that can handle large amount of data and make reliable decisions in fractions of a second. One important step in these algorithms is segmentation, where different regions in the image are labelled. For cod fillets, this can be to identify which part of the fillet belongs to the loin, belly flap, centre cut and tail. How severe a detected flaw is depends on which part of the fillet it is located in. Segmentation requires a robust spatial reference system which is invariant to rotation and warping of the fillet. The centreline, consisting of veins and arteries cut off during filleting, is always visible on cod fillets and hence a good reference for segmentation. We show how to enhance the centreline by using the absorption characteristics of haemoglobin, and how a novel ridge detection method can detect the centreline in cod fillets. The results show that the centreline can be detected with an average accuracy of 1 mm from the tail and 77% into the fillet relative to its total length. The average error increases rapidly in the neck region and typical errors of 4 mm is reported.
Surfaces play an important role in visual perception. They are perceived as ‘(perceptual) reliefs’, that are surfaces in 2 + 1D perceptual space, that is the product space of the 2D visual field and the 1D ‘depth dimension’. It is in many respects irrelevant whether the observer views a true 3D scene or a flat (2D) picture of a scene. In both cases, the percepts are reliefs in 2 + 1D perceptual space. In the latter case, one speaks of ‘pictorial relief’. We discuss how perceptual reliefs can be measured and which aspects of these reliefs are especially robust against day-to-day intraobserver variations, changes of viewing conditions and interobserver differences. It turns out that only aspects of the partial depth order (based on depth precedence in infinitesimal regions) are stable. Thus, features of the relief are invariants of general ‘relief preserving transformations’ that may actually scramble depth values at different locations. This is evident from the fact that human observers can only judge depth precedence with some degree of certainty for points that are on a single slope. We discuss the formal structure of these relief invariants. Important ones are the Morse critical points and the ridges and courses of the relief.
The automated analysis of 3D medical images can improve both diagnosis and therapy significantly. This automation raises a number of new fascinating research problems in the fields of computer vision, graphics and robotics. In this paper, I propose a list of such problems after a review of the current major 3D imaging modalities, and a description of the related medical needs. I then present some of the past and current work done in our research group EPIDAURE∗ at INRIA, on the following topics: segmentation of 3D images; 3D shape modelling; 3D rigid and nonrigid registration; 3D motion analysis; and 3D simulation of therapy. Most topics are discussed in a synthetic manner, and illustrated by results. Rigid matching is treated more thoroughly as an illustration of a transfer from computer vision towards 3D image processing. The later topics are illustrated by preliminary results, and a number of promising research tracks are suggested.
Recent work has shown that sparse lines defined on 3D shapes, including occluding contours and suggestive contours, are effective at conveying shape. We introduce two new families of lines called suggestive highlights and principal highlights, based on definitions related to suggestive contours and geometric creases. We show that when these are drawn in white on a gray background they are naturally interpreted as highlight lines, and complement contours and suggestive contours. We provide object-space definitions and algorithms for extracting these lines, explore several stylization possibilities, and compare the lines to ridges and valleys of intensity in diffuse-shaded images.
Different kinds of digital images can be modelled as the sampling of a continuous surface, being described and analyzed through the extraction of geometric features from the underlying surface. Among them, ridges and valleys or, generically, creases, have deserved special interest. The computer vision community has been relying on different crease definitions, some of them equivalent. Although they are quite valuable in a number of applications, they usually do not correspond to the real creases of a topographic relief. These definitions give rise either to algorithms that label pixels as crease points, and then focus on the problem of grouping them into curves, or to operators whose outcome is a creaseness image. We draw our attention to the real crease definition for a landscape, due to Rudolf Rothe, which is based on the convergence of slopelines. They are computed by numerically solving a system of differential equations. Afterwards, we extract Rothe creases which are parts of slopelines where others converge, avoiding in such a way any pixel-grouping step. At the same time we compute a creaseness image according to this definition.
This paper discusses some problems that should be addressed by future object recognition systems.
In particular, there are things that we know how to do today, for example:
1.
Computing the pose of a free-form three-dimensional object from its outline (e.g. [106]).
2.
Identifying a polyhedral object from point and line features found in an image (e.g., [46, 89]).
3.
Recognizing a solid of revolution from its outline (e.g., [59]).
4.
Identifying a face with a fixed pose in a photograph (e.g., [10, 111]).
Triangulated surfaces are often used to represent terrains in Geographic Information Systems (GIS); one of the primary computations on terrains is determining drainage networks. Under natural definitions of the flow of water on a terrain represented by n triangles, we show that the river network has \Theta(n 3 ) worst-case complexity, where complexity is measured in the number of line segments that make up the network. 1 Introduction Terrain drainage characteristics provide important information on water resources, possible flood areas, erosion and other natural processes. In natural resource management, for example, the basic management unit is the watershed, the area around a stream that drains into the stream. Road building, logging, or other activities carried out in a watershed all have the potential to affect the defining stream. Manual quantification of terrain drainage characteristics is a tedious and time consuming job. Fortunately, through spatial analysis of digital repre...
While vortex region quantities are Galilean invariant, most methods for extracting vortex cores depend on the frame of reference. We present an approach to extracting vortex core lines independently of the frame of reference by extracting ridge and valley lines of Galilean invariant vortex region quantities. We discuss a generalization of this concept leading to higher dimensional features. For the visualization of extracted line features we use an iconic representation indicating their scale and extent. We apply our approach to datasets from numerical simulations and experimental measurements.
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