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The paper presents an analysis of sources of errors when estimating derivatives of numerical or noisy functions. A method of minimizing the errors is suggested. When being applied to the estimation of the curvature of digital curves, the analysis shows that under the conditions typical for digital image processing the curvature can rarely be estima...
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... pixel pairs it suffices to implicitly binarize the given gray value image with the threshold as specified above and to trace the crack boundary as described e.g. in [Kov92]. Each boundary crack found during the tracing separates two pixels composing a pair. Thus we can find for each boundary crack a corresponding sub-pixel point SP as shown in Fig. ...
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Citations
... Ideally, one should use parallel projection for imaging, for which scanning is the closest widely available method. Scanning was used also by Kovalevsky (2001) to analyse the contour's curvature for various objects [1]. At the time of the measurements, no proper imaging device was on hand, thus a DSLr camera was used instead. ...
The verification of different plate bending problems require a punctual measurement method of the bent shape. However, with a proper curvature measurement procedure the calculations can be made more accurate. This is due to the workaround this method provides by neglecting the inaccuracies of the beam theory and the tensile tests, measuring directly the function between the curvature and bending moment for a given sheet metal. The measurements in this paper are made with the help of a digital camera and telephotographic lens. The evaluations of these images are compared to the results obtained from the Euler-Bernoulli beam theory. While the results regarding the curvature measurements have a significant deviation, the shape of the plate is in good agreement with the numerical calculations.
... Several studies regarding the calculation of a digital path's curvature in computer vision in 2D have shown that it is very challenging to keep the curvature error below 10% and errors of 30-50% are not unusual. 11,16,17 The challenges increase even more in 3D space. 7 In order to overcome these challenges in this study, the 3D coordinates were calculated using epipolar geometry in combination with an applied smooth cubic spline. ...
Better understanding of the lead curvature, movement and their spatial distribution may be beneficial in developing lead testing methods, guiding implantations and improving life expectancy of implanted leads.
Objective
The aim of this two-phase study was to develop and test a novel biplane cine-fluoroscopy-based method to evaluate input parameters for bending stress in leads based on their in vivo 3D motion using precisely determined spatial distributions of lead curvatures. Potential tensile, compressive or torque forces were not subjects of this study.
Methods
A method to measure lead curvature and curvature evolution was initially tested in a phantom study. In the second phase using this model 51 patients with implanted ICD leads were included. A biplane cine-fluoroscopy recording of the intracardiac region of the lead was performed. The lead centerline and its motion were reconstructed in 3D and used to define lead curvature and curvature changes. The maximum absolute curvature C max during a cardiac cycle, the maximum curvature amplitude C amp and the maximum curvature C max@amp at the location of C amp were calculated. These parameters can be used to characterize fatigue stress in a lead under cyclical bending.
Results
The medians of C amp and C max@amp were 0.18 cm ⁻¹ and 0.42 cm ⁻¹ , respectively. The median location of C max was in the atrium whereas the median location of C amp occurred close to where the transit through the tricuspid valve can be assumed. Increased curvatures were found for higher slack grades.
Conclusion
Our results suggest that reconstruction of 3D ICD lead motion is feasible using biplane cine-fluoroscopy. Lead curvatures can be computed with high accuracy and the results can be implemented to improve lead design and testing.
... Ebből a szempontból kiemelkedően jó a vékony lemez, hiszen minél vékonyabb, a középvonal pontjai annál nagyobb pontossággal határozhatóak meg. A képalkotás során ideálisan párhuzamos vetítést kellene alkalmazni, amihez a szkennelés áll a legközelebb, ahogy Kovalevsky (2001) is azt használta tárgyak kontúrjának görbületi elemzéséhez [1]. A mérések elvégzésekor nem állt rendelkezésemre megfelelő képalkotó eszköz, így a lemezek leképezése digitális kamera segítségével történt. ...
... Seule la dernière méthode estime la courbure locale avec une technique basée sur l'estimation du cercle oscillant (Landau, 1987 ;Thomas et Chan, 1989), qui est plus proche de la dénition numérique. Kovalevsky (2001) a discrétisé une dénition continue considérant, dans le référentiel approprié, que la courbure est égale à la valeur de la dérivée seconde au point courant. ...
Le monde est face à une crise environnementale majeure, ceci alors que la consommation d'énergie ne cesse d'augmenter. Quelles solutions pouvons-nous trouver pour fournir l'énergie demandée, tout en réduisant les émissions de gaz à effet de serre ? L'une d'entre elles consiste à améliorer l'efficacité énergétique des procédés industriels, notamment par le biais de la catalyse hétérogène. Les catalyseurs hétérogènes, ici des solides poreux, sont utilisés en raffinage et pétrochimie, en particulier pour la génération de biocarburant.La question se pose de caractériser l'efficacité de ces catalyseurs. Une description morphologique fournit des informations clés, au sens où des corrélations ont été établies entre propriétés structurales et performances de ces matériaux. Néanmoins, la catalyse hétérogène est un processus complexe et les descripteurs traditionnellement utilisés sont insuffisants, dans l'optique d'une aide aux choix de ces matériaux.Ce travail de thèse vise à développer de nouveaux descripteurs numériques de microstructures, facilement interprétables, efficients et complémentaires à l'état de l'art, afin d'aider in fine à choisir les catalyseurs appropriés à une application donnée. Ces descripteurs permettent une caractérisation géométrique et topologique du réseau poreux, quelle que soit sa complexité et indépendamment des phénomènes physico-chimiques pouvant être en jeu.Nous mettons en œuvre à la fois la percolation, capacité à traverser une microstructure ; la tortuosité géométrique, sinuosité et interconnectivité d'un réseau ; et l'hétérogénéité. Ces descripteurs relèvent pour l'essentiel de méthodes morphologiques. Les caractéristiques géométriques et topologiques liées aux fonctionnelles de Minkowski en 3D sont adaptées au domaine d'intérêt par estimation de l'accessibilité à une microstructure pour une sphère de taille donnée (A-protocole), décrite par érosion morphologique calculée de façon efficace grâce aux cartes de distances. Nous caractérisons ensuite la topologie des pores via un nouvel opérateur, la M-tortuosité, applicable à tout volume segmenté, sans définition arbitraire de points ou de plans sources. Nous proposons un estimateur efficace de la M-tortuosité par calcul de cartes de distances ; dont une généralisation par facteurs puissances est définie. Cet opérateur est ensuite étendu de différentes façons.Tout d'abord, au cas d'une sonde de taille finie (M-tortuosité-par-érosions-itératives), caractérisant les goulots d'étranglement, habituellement décrits par la constrictivité. Puis, pour caractériser la dépendance en échelle spatiale de la tortuosité (H-tortuosité), ce qui quantifie, entre autres, l'hétérogénéité de la structure. Enfin, ces deux aspects sont regroupés dans la H-tortuosité-par-érosions-itératives.Dans un second temps, ces opérateurs ensemblistes, valables pour des images binaires, reçoivent une extension fonctionnelle permettant de décrire des images à niveaux de gris (F-tortuosité et HF-tortuosité). Les objectifs sont multiples : tenir compte d'informations locales lors de l'estimation de la tortuosité globale de microstructures, et discriminer des images tomographiques, sans segmentation précise de la structure.Le pouvoir discriminant de ces opérateurs ensemblistes et fonctionnels, et leur comportement, sont évalués au fil des définitions, sur des cas d'école et sur des modèles booléens multi-échelles de Cox. Leurs similitudes et complémentarités sont analysées sur ces mêmes images synthétiques.En catalyse et biocatalyse, trois types de microstructures sont considérés : les zéolithes, les MOFs (Metal-Organic Framework) et les alumines. Ces applications montrent l'étendue du champ applicatif, par l'adaptabilité et la complémentarité des descripteurs proposés, entrainant la considération de leur utilisation en dehors du cadre de la catalyse, notamment pour certains domaines des neurosciences et l'analyse des milieux turbides.
... Seule la dernière méthode estime la courbure locale avec une technique basée sur l'estimation du cercle oscillant (Landau, 1987 ;Thomas et Chan, 1989), qui est plus proche de la dénition numérique. Kovalevsky (2001) a discrétisé une dénition continue considérant, dans le référentiel approprié, que la courbure est égale à la valeur de la dérivée seconde au point courant. ...
Characterization of complex materials using new topological descriptors, mainly based on tortuosity concept. Description and discrimination of heterogeneous catalysts used in petrochimicals. Extension of descriptors to grayscale cases to characterize unsegmented images, or mix local features, by means of 3D maps, with global assessment. Application over various current projects in petrochemicals, with distinct purposes as relative classification, identification of intuitive features of paramount interest for a given application, and improvement of the understanding of complex materials. Finally, extension of our functional descriptors to other fields, different from materials science, as neurosciences and biology, for the characterization of living cells.
... Finally, local shape features measure the curvature of each ventricle boundary. 21 To account for inter-patient variability in head size, these features were normalized by the size of the manually determined bounding box encompassing the cranium (i.e., maximum length of ventricles' circumscribed ellipses, asymmetry of centroids between ventricles, and circularity of ventricles) or by the size of ventricle's circumscribed ellipse (i.e., minimum area of the nonhemorrhagic ventricles and maximum medial length of ventricles). To account for the differences between the two lateral ventricles, predictive parameters were determined from the overall maximum, minimum, and ratio between features computed on the left and right lateral ventricles. ...
Background:
To compare the ability of ventricular morphology on cranial ultrasound (CUS) versus standard clinical variables to predict the need for temporizing cerebrospinal fluid drainage in newborns with intraventricular hemorrhage (IVH).
Method:
This is a retrospective study of newborns (gestational age <29 weeks) diagnosed with IVH. Clinical variables known to increase the risk for post-hemorrhagic hydrocephalus were collected. The first CUS with IVH was identified and a slice in the coronal plane was selected. The frontal horns of the lateral ventricles were manually segmented. Automated quantitative morphological features were extracted from both lateral ventricles. Predictive models of the need of temporizing intervention were compared.
Results:
Sixty-two newborns met inclusion criteria. Fifteen out of the 62 had a temporizing intervention. The morphological features had a better accuracy predicting temporizing interventions when compared to clinical variables: 0.94 versus 0.85, respectively; p < 0.01 for both. By considering both morphological and clinical variables, our method predicts the need of temporizing intervention with positive and negative predictive values of 0.83 and 1, respectively, and accuracy of 0.97.
Conclusion:
Early cranial ultrasound-based quantitative ventricular evaluation in premature newborns can predict the eventual use of a temporizing intervention to treat post-hemorrhagic hydrocephalus. This may be helpful for early monitoring and treatment.
... The main reason for choosing to fit a circle rather than an arbitrary curve to the available data points is that it is essentially impossible to calculate the radius of curvature from a digital 2D image with good accuracy, mainly because this parameter is inversely proportional to the second derivative of the curve in question, and the estimation of the higher derivatives of functions with values specified with limited precision is very problematic. 10 In addition, it has been shown experimentally that for most purposes considering a spherical shock profile is a good approximation for the geometry of a wavefront radiated from an instrument bell, 6,7 although there is also compelling evidence that this approximation is far from perfect. [11][12][13] Thus, in order to estimate the radii of curvature of the shock fronts along the central instrument axis, we first located the shock region in a particular frame and placed a large amount of points on the left boundary of this region. ...
The Schlieren method has been used before to visualize weak shock waves radiated from the open ends of brass instruments, but no attempt has previously been undertaken, however, to measure the geometry of the radiated wavefronts using the Schlieren images. In this paper Schlieren visualization is used to estimate the geometry of the two-dimensional shock wavefronts radiated from the bell of a trumpet at different frequencies. It is observed that the geometry of the shocks does change with frequency, in the expected manner. The propagation speeds of these shocks are also calculated, and they too exhibit the anticipated behavior.
... Before estimation, it is a common practice of this field to digitize a smooth input curve into a number of characteristic pixels and then connect them into a polygonal line to form the approximate input. This process of digitization incurs substantial approximation errors in the input polygon, which seriously limit the accuracy of curvature estimation in computer imaging [34,17]. ...
... The input curve in a real-world application always contains approximation errors. For computer imaging, Kovalevsky [17] showed that if the input curve is not exact, the curvature estimation error will be unbounded as the sampling size h κ approaches zero. Furthermore, for a given inexact curve, errors of curvature estimation on uniform grids have a global minimum called the best attainable accuracy. ...
... which is due to Kovalevsky [17]. Taking j = 1, the triangle inequality and (4.6) yield ...
The author proposes a height function (HF) method for estimating curvature and unit vectors of planar curves.
The new method is named HFES since its signature consists of explicit input parameters (instead of implicit volume fractions) and signed output estimates. Due to this signature and its formal description, HFES has a number of appealing advantages. First, the convergence rate of HFES can be 4, 6, or a greater even integer. Second, local arcs can be rotated to further reduce leading truncation errors. Third, the sampling size of estimation is completely independent from the Eulerian grid size of the main flow. Fourth, signs of the signed output can be conveniently chosen by orientations of input Jordan curves, e.g., surface tension of a tracked material can be determined in a simple and worry-free manner. Last, HFES is applicable both to numerical simulation of multiphase flows and shape analysis in computer imaging. Furthermore, the author analyzes the effect of input perturbations upon output estimates, derives expressions of the best attainable accuracy afforded by a given input on uniform grids, and explains how to achieve this best accuracy with a concrete example. Results of numerical experiments with both exact and inexact input demonstrate that HFES can be much more accurate than previous HF methods. For the vortex-shear test of a circular disk, the eighth-order HFES coupled with the recent cubic iPAM method is millions of times more accurate than a fourth-order HF method coupled with volume-of-fluid methods! This drastic accuracy improvement is mainly due to the purely explicit spline representation of the curve and the rotation of local arcs.
... A multiscale, curvature-based shape representation for planar curves has been proposed in Mokhtarian and Mackworth [MM92], see also Marcondes Cesar Jr. and Da Fontoura Costa [MD96] who have proposed effective shape representations based on curvature multiscale methods. The occuring numerical problems in the curvature estimation have been analyzed in Kovalevsky [Kov01]. Due to these numerical issues, Utcke [Utc03] has discussed error bounds. ...
This thesis is divided into two parts. The first part is devoted to the curvature estimation of piecewise smooth curves using variation diminishing splines. The variation diminishing property combined with the ability to reconstruct linear functions leads to a convexity preserving approximation that is crucial if additional sign changes in the curvature estimation have to be avoided. To this end, we will first establish the foundations of variation diminishing transforms and introduce the Bernstein and the Schoenberg operator on the space of continuous functions and its generalization to the Lp-spaces. In order to be able to detect C2-singularities in piecewise smooth curves, we establish lower estimates for the approximation error in terms of the second order modulus of smoothness for Schoenberg’s variation diminishing operator. Afterwards, we consider smooth curve approximations using only finitely many samples of the curve, where the approximation, its first, and its second derivative converge uniformly to its corresponding part of the curve to be approximated. In this case, we can show that the estimated curvature converges uniformly to the real curvature if the number of samples goes to infinity. Based on the lower estimates that relates the decay rate of the approximation error with smoothness we propose a multi-scale algorithm to estimate the curvature and to detect C2-singularities. We numerically evaluate our algorithm and compare it to others to show that our algorithm achieves competitive accuracy while our curvature estimations are significantly faster to compute.
The second part deals with generalizations of the established lower estimates for the Schoenberg operator. We will show that such estimates can be obtained for linear operators on a general Banach function space with smooth range provided that the iterates of the operator converge uniformly and a semi-norm defined on the range of the operator annihilates the fixed points of the operator. To this end, we will prove by spectral properties that the iterates of every positive finite-rank operator converge uniformly. As highlight of this thesis, we show a constructive way using a Gramian matrix where the dual fixed points operate on the fixed points of an operator to derive the limit of the iterates for an arbitrary quasi-compact operator defined on a general Banach space.
... Recent results [32] indicate that the theoretical limit of accuracy of determining curvature in digitized images could be as high as 40 %, and this should be kept in mind. Especially here where most neck widths are in the range of a few pixels only. ...
In general terms, sintering describes the bonding of particles into a more coherent structure, where joins form between packed particles, usually as a result of heating. Characterization of sintering is an important topic in the fields of metallurgy, steel, iron ore pellets, ceramics, and snow for understanding material properties and material strength. Characterization using image analysis has been applied in a number of these fields but is either semi-automatic, requiring human interaction in the analysis, or based on statistical sampling and stereology to characterize the sample. This paper presents a novel fully automatic image analysis algorithm to analyze and determine the degree of sintering based on analysis of the particle joins and structure. Quantitative image analysis of the sintering degree is demonstrated for samples of iron ore pellets but could be readily applied to other packed particle materials. Microscope images of polished cross-sections of iron ore pellets have been imaged in their entirety and automated analysis of hundreds of images has been performed. Joins between particles have been identified based on morphological image processing and features have been calculated based on the geometric properties and curvature of these joins. The features have been analyzed and determined to hold discriminative power by displaying properties consistent with sintering theory and results from traditional pellet diameter measurements on the heated samples, and a statistical evaluation using the Welch t-test.
