WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

We propose a new framework for toolpath generation for five-axis machining of part surfaces represented by the StereoLithography (STL) format. The framework is based on flattening the STL part and generation of adaptive curvilinear toolpaths. The corresponding cost functions, designed to represent the accuracy and the efficiency of the toolpath, are scalar functions, such as the curvature, kinematic error, rotation angles, machining strip or material removal rate or a vector field when the tool moves along a curvilinear path partly or even entirely aligned with directions considered to be optimal. The adaptive toolpath exploits grid generation methods and biased space-filling curves, combined with adaptation to the boundary and the domain decomposition. The proposed methodology of the adaptive curvilinear toolpath (ACT) has been tested on a variety of STL surfaces, including a case study of STL dental parts. Machining crowns/implants for four basic types of human teeth, molar, premolar, canine and incisor...

Five-axis machining of STL surfaces by adaptive curvilinear toolpaths

Public visualization of high quality medical information has been wildly available since the creation of the Visible
Human Project in the late 90´s. We discuss the extraction of information from 3D volumes along curved slices with
emphasis on those that can be displayed on the plane without deformation. Special attention is given to a dental volume
containing the sixteen teeth of the upper human jaw. We review several approaches to display information along curved
slices contained within the 3D data set.

Between developable surfaces and circular cone splines: curved slices of 3D volumes

The assessment of rib bone fractures and lesions consists of many images that have to be thoroughly inspected slice-by-slice and rib-by-rib. Existing visualization methods, such as curved planar reformation (CPR), reduce the number of images to inspect and, in turn, the time spent per case. However, this task remains time-consuming and exhausting. In this paper, we propose a novel rib unfolding strategy that considers the cross-sectional shape of each rib individually and independently. This leads to shape-adaptive slices through the ribs. By aggregating these slices into a single image, we support radiologists with a concise overview visualization of the entire rib cage for fracture and lesion assessment. We present results of our approach along different cases of rib and spinal fractures as well as lesions. To assess the applicability of our method, we separately evaluated the segmentation (with 954 data sets) and the visualization (with two clinical coaches). CCS Concepts •Human-centered computing → Scientific visualization; •Computing methodologies → Shape analysis; Image processing; •Applied computing → Health informatics;

Bone Fracture and Lesion Assessment using Shape-Adaptive Unfolding.

Interpolation problems have been widely studied in Computer Aided Geometric Design (CAGD). They consist in the construction of curves and surfaces that pass exactly through a given data set, such as point clouds, tangents, curvatures, lines/planes, etc. In general, these curves and surfaces are represented in a parametrized form. This representation is independent of the coordinate system, it adapts itself well to geometric transformations and the differential geometric properties of curves and surfaces are invariant under reparametrization. In this context, the main goal of this thesis is to present 2D and 3D data interpolation schemes by means of Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) curves. The latter are parametric curves defined in a mixed algebraic-trigonometric space, whose hodograph satisfies a Pythagorean condition. This representation allows to analytically calculate the curve's arc-length as well as the rational-trigonometric parametrization of the offsets curves. These properties are usable for the design of geometric models in many applications including manufacturing, architectural design, shipbuilding, computer graphics, and many more. In particular, we are interested in the geometric modeling of odontological objects. To this end, we use the spatial ATPH curves for the construction of developable patches within 3D odontological volumes. This may be a useful tool for extracting information of interest along dental structures. We give an overview of how some similar interpolating problems have been addressed by the scientific community. Then in chapter 2, we consider the construction of planar C2 ATPH spline curves that interpolate an ordered sequence of points. This problem has many solutions, its number depends on the number of interpolating points. Therefore, we employ two methods to find them. Firstly, we calculate all solutions by a homotopy method. However, it is empirically observed that only one solution does not have any self-intersections. Hence, the Newton-Raphson iteration method is used to directly compute this \good" solution. Note that C2 ATPH spline curves depend on several free parameters, which allow to obtain a diversity of interpolants. Thanks to these shape parameters, the ATPH curves prove to be more exible and versatile than their polynomial counterpart, the well known Pythagorean-Hodograph (PH) quintic curves and polynomial curves in general. These parameters are optimally chosen through a minimization process of fairness measures. We design ATPH curves that closely agree with well-known trigonometric curves by adjusting the shape parameters. We extend the planar ATPH curves to the case of spatial ATPH curves in chapter 3. This characterization is given in terms of quaternions, because this allows to properly analyze their properties and simplify the calculations. We employ the spatial ATPH curves to solve the first-order Hermite interpolation problem. The obtained ATPH interpolants depend on three free angular values. As in the planar case, we optimally choose these parameters by the minimization of integral shape measures. This process is also used to calculate the C1 interpolating ATPH curves that closely approximate well-known 3D parametric curves. To illustrate this performance, we present the process for some kind of helices. In chapter 4 we then use these C1 ATPH splines for guiding developable surface patches, which are deployed within odontological computed tomography (CT) volumes, in order to visualize information of interest for the medical professional. Particularly, we construct piecewise conical surfaces along smooth ATPH curves to display information related to the anatomical structure of human jawbones. This information may be useful in clinical assessment, diagnosis and/or treatment plan. Finally, the obtained results are analyzed and conclusions are drawn in chapter 5.

https://tel.archives-ouvertes.fr/tel-01789081/document

Algebraic-trigonometric Pythagorean hodograph curves for solving planar and spatial interpolation problems and their use for visualizing dental information within 3D tomographic volumes

In the last years, three-dimensional (3D) medical imaging techniques have taken an increasing importance in patient care and medical research. Volume images provide medical specialists with a direct access to the interior of a patient's body and reduce the need for invasive exploration. The use of volume imaging modalities such as X-ray CT, PET or MRI has therefore become essential for medical diagnosis and surgical planning. Computer visualization techniques such as extraction of planar slices of arbitrary orientation (Multiplanar Reprojection), surface rendering of anatomic structures, and volume rendering provide medical users with the tools for exploiting 3D volume images. Surface or respectively volume rendering provides information about the 3D geometry and 3D context of the structures of interest but does not allow to directly visualize original intensities, respectively colors located within the 3D structures. In addition, surface rendering requires the segmentation of the volume data and volume rendering often requires a classification of the volume image pixels. In contrast, the extraction of planar sections provides interactivity, requires no pre-processing and the original intensity, respectively color of each slice element may be directly inspected. However, it does not allow the visualization of curved anatomic structures within a single slice. In this thesis, we propose to overcome this limitation by generalizing the concept of planar section to the extraction of curved cross-sections. In the first part, we focus on the interactive extraction of curved surfaces from volume images. Unlike planar slices, curved cross-sections may follow the trajectory of tubular structures such as the Aorta or follow structures with an irregular shape such as the Pelvis. In the second part of this work, we focus on the visualization of curved surfaces. We would like to offer the possibility of carrying out distance measurements along a structure of interest both for medical applications and for anatomical studies. Orthogonal or perspective projection of curved surfaces induces angular and metric distortions as well as surface overlapping. In order to enable measurements, we propose to use surface flattening methods, which preserve distances along specific orientations and minimize distortions around a focus point. Flattening of curved cross-sections enables inspecting spatially complex relationship between anatomic structures and their neighbourhood. They also allow the visualization of a curved anatomic structure within a single planar view and therefore to precisely inspect the original intensity, respectively color at each surface point. Thanks to a multi-resolution approach, surfaces are flattened at interactive rates, allowing users to displace the focus point during the visualization of the flattened surface. We also propose a new efficient method for computing a flattened surface minimizing global distortions and still preserving distances along one orientation. Surface extraction and flattening techniques are integrated into an interactive visualization Java applet. This Java applet enables anyone to precisely and interactively inspect the Visible Human anatomy. Besides medical visualization, the presented methods may also be useful for creating new interesting views of anatomic structures for didactic purposes.

/pdf/surface-extraction-and-flattening-for-anatomical-lpqopy6aji.pdf

Surface extraction and flattening for anatomical visualization

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Surface extraction and flattening for anatomical visualization