Principal arc analysis on direct product manifolds

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

True-color 3D rendering of human anatomy using surface-guided color sampling from cadaver cryosection image data: A practical approach Jon Jatsu Azkue

Three-dimensional computer graphics are increasingly used for scientific visualization and for communicating anatomical knowledge and data. This study presents a practical method to produce true-color 3D surface renditions of anatomical structures. The procedure involves extracting the surface geometry of the structure of interest from a stack of cadaver cryosection images, using the extracted surface as a probe to retrieve color information from cryosection data, and mapping sampled colors back onto the surface model to produce a true-color rendition. Organs and body parts can be rendered separately or in combination to create custom anatomical scenes. By editing the surface probe, structures of interest can be rendered as if they had been previously dissected or prepared for anatomical demonstration. The procedure is highly flexible and nondestructive, offering new opportunities to present and communicate anatomical information and knowledge in a visually realistic manner. The technical procedure is described, including freely available open-source software tools involved in the production process, and examples of color surface renderings of anatomical structures are provided

Medial/skeletal linking structures for multi-region configurations

We consider a generic configuration of regions, consisting of a collection of distinct compact regions $\{\Omega_i\}$ in $\mathbb{R}^{n+1}$ which may be either smooth regions disjoint from the others or regions which meet on their piecewise smooth boundaries $\mathcal{B}_i$ in a generic way. We introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the "positional geometry" of the collection. The linking structure extends in a minimal way the individual "skeletal structures" on each of the regions, allowing us to significantly extend the mathematical methods introduced for single regions to the configuration. We prove for a generic configuration of regions the existence of a special type of Blum linking structure which builds upon the Blum medial axes of the individual regions. This requires proving several transversality theorems for certain associated "multi-distance" and "height-distance" functions for such configurations. We show that by relaxing the conditions on the Blum linking structures we obtain the more general class of skeletal linking structures which still capture the geometric properties. In addition to yielding geometric invariants which capture the shapes and geometry of individual regions, the linking structures are used to define invariants which measure positional properties of the configuration such as: measures of relative closeness of neighboring regions and relative significance of the individual regions for the configuration. These invariants, which are computed by formulas involving "skeletal linking integrals" on the internal skeletal structures, are then used to construct a "tiered linking graph," which identifies subconfigurations and provides a hierarchical ordering of the regions.Comment: 135 pages, 36 figures. Version to appear in Memoirs of the Amer. Math. So

The Blum medial linking structure for multi-region analysis

The Blum medial axis of a region with smooth boundary in Rsuperscript{n+1} is a skeleton-like topological structure that captures shape and geometric properties of the region and its boundary. We introduce a structure, called the Blum medial linking structure, which extends the advantages of the medial axis to configurations of multiple disjoint regions in order to capture both their individual and positional or relative geometry. We use singularity theory to classify the generic local normal forms of the medial linking structure for generic configurations of regions in dimensions n is less than or equal to 6, which requires proving a transversality theorem for families of multi--distance functions. We show how invariants of the geometry of the regions and their complement may be computed directly from the linking structure. We conclude with applications of the linking structure to the analysis of multiple objects in medical images

Modeling and Simulation in Engineering

This book provides an open platform to establish and share knowledge developed by scholars, scientists, and engineers from all over the world, about various applications of the modeling and simulation in the design process of products, in various engineering fields. The book consists of 12 chapters arranged in two sections (3D Modeling and Virtual Prototyping), reflecting the multidimensionality of applications related to modeling and simulation. Some of the most recent modeling and simulation techniques, as well as some of the most accurate and sophisticated software in treating complex systems, are applied. All the original contributions in this book are jointed by the basic principle of a successful modeling and simulation process: as complex as necessary, and as simple as possible. The idea is to manipulate the simplifying assumptions in a way that reduces the complexity of the model (in order to make a real-time simulation), but without altering the precision of the results

AN AUTOMATED, DEEP LEARNING APPROACH TO SYSTEMATICALLY & SEQUENTIALLY DERIVE THREE-DIMENSIONAL KNEE KINEMATICS DIRECTLY FROM TWO-DIMENSIONAL FLUOROSCOPIC VIDEO

Total knee arthroplasty (TKA), also known as total knee replacement, is a surgical procedure to replace damaged parts of the knee joint with artificial components. It aims to relieve pain and improve knee function. TKA can improve knee kinematics and reduce pain, but it may also cause altered joint mechanics and complications. Proper patient selection, implant design, and surgical technique are important for successful outcomes. Kinematics analysis plays a vital role in TKA by evaluating knee joint movement and mechanics. It helps assess surgery success, guides implant and technique selection, informs implant design improvements, detects problems early, and improves patient outcomes. However, evaluating the kinematics of patients using conventional approaches presents significant challenges. The reliance on 3D CAD models limits applicability, as not all patients have access to such models. Moreover, the manual and time-consuming nature of the process makes it impractical for timely evaluations. Furthermore, the evaluation is confined to laboratory settings, limiting its feasibility in various locations. This study aims to address these limitations by introducing a new methodology for analyzing in vivo 3D kinematics using an automated deep learning approach. The proposed methodology involves several steps, starting with image segmentation of the femur and tibia using a robust deep learning approach. Subsequently, 3D reconstruction of the implants is performed, followed by automated registration. Finally, efficient knee kinematics modeling is conducted. The final kinematics results showed potential for reducing workload and increasing efficiency. The algorithms demonstrated high speed and accuracy, which could enable real-time TKA kinematics analysis in the operating room or clinical settings. Unlike previous studies that relied on sponsorships and limited patient samples, this algorithm allows the analysis of any patient, anywhere, and at any time, accommodating larger subject populations and complete fluoroscopic sequences. Although further improvements can be made, the study showcases the potential of machine learning to expand access to TKA analysis tools and advance biomedical engineering applications

Homeostatic Elastic States and the Stability of Elastic Arteries

Vascular mechanics has undergone significant growth within the last 50 years owing to the rapid development of nonlinear continuum mechanics occurring roughly within the same period and motivated primarily by rubber materials. However, one important distinction of blood vessels, in contrast to typical engineering materials is that, through a variety of physiological mechanisms, they seek to maintain constant a preferred mechanical state (mechanical homeostasis) thereby exhibiting a remarkable mechanical stability in response to temporal evolution and alterations in blood pressure, vessel tethering forces and geometry and material properties. The mechanical state experienced by blood vessels plays a critical role in mechanical homeostasis and mechanical stability, and there remains a pressing need for mechanical/mathematical analysis to i) understand/predict the stretch/stress states within vessels and how they evolve with increasing blood pressure and tethering forces, ii) understand/predict the mechanical stability of arteries in response to diverse stimuli such as inhomogeneities in geometry and material properties. This dissertation seeks to add to this vibrant field by conducting a rigorous analysis of i) the mechanics of the homeostatic states of uniform circumferential stress and uniform stretch in an N-layer cylindrical artery subject to circumferential prestress, axial tethering force and the pressure of blood and ii) the local mechanical stability by imperfection growth in a solid body subject to inhomogeneities in geometry and material properties. In order to make these results relevant to a blood vessel, a micromechanics based constitutive relation is proposed based on the more or less regular architecture of a large elastic artery composed of collagen, elastin and vascular smooth muscle. Although the primary focus of the work is on the healthy artery, the effect on imperfection growth of diseased tissue constituents is accounted for in a simple model of damaged elastin and collagen

The Inhuman Overhang: On Differential Heterogenesis and Multi-Scalar Modeling

As a philosophical paradigm, differential heterogenesis offers us a novel descriptive vantage with which to inscribe Deleuze’s virtuality within the terrain of “differential becoming,” conjugating “pure saliences” so as to parse economies, microhistories, insurgencies, and epistemological evolutionary processes that can be conceived of independently from their representational form. Unlike Gestalt theory’s oppositional constructions, the advantage of this aperture is that it posits a dynamic context to both media and its analysis, rendering them functionally tractable and set in relation to other objects, rather than as sedentary identities. Surveying the genealogy of differential heterogenesis with particular interest in the legacy of Lautman’s dialectic, I make the case for a reading of the Deleuzean virtual that departs from an event-oriented approach, galvanizing Sarti and Citti’s dynamic a priori vis-à-vis Deleuze’s philosophy of difference. Specifically, I posit differential heterogenesis as frame with which to examine our contemporaneous epistemic shift as it relates to multi-scalar computational modeling while paying particular attention to neuro-inferential modes of inductive learning and homologous cognitive architecture. Carving a bricolage between Mark Wilson’s work on the “greediness of scales” and Deleuze’s “scales of reality”, this project threads between static ecologies and active externalism vis-à-vis endocentric frames of reference and syntactical scaffolding