New constraints on data-closeness and needle map consistency for shape-from-shading

This paper makes two contributions to the problem of needle-map recovery using shape-from-shading. First, we provide a geometric update procedure which allows the image irradiance equation to be satisfied as a hard constraint. This not only improves the data closeness of the recovered needle-map, but also removes the necessity for extensive parameter tuning. Second, we exploit the improved ease of control of the new shape-from-shading process to investigate various types of needle-map consistency constraint. The first set of constraints are based on needle-map smoothness. The second avenue of investigation is to use curvature information to impose topographic constraints. Third, we explore ways in which the needle-map is recovered so as to be consistent with the image gradient field. In each case we explore a variety of robust error measures and consistency weighting schemes that can be used to impose the desired constraints on the recovered needle-map. We provide an experimental assessment of the new shape-from-shading framework on both real world images and synthetic images with known ground truth surface normals. The main conclusion drawn from our analysis is that the data-closeness constraint improves the efficiency of shape-from-shading and that both the topographic and gradient consistency constraints improve the fidelity of the recovered needle-map

Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey

This paper provides a tutorial and survey for a specific kind of illustrative visualization technique: feature lines. We examine different feature line methods. For this, we provide the differential geometry behind these concepts and adapt this mathematical field to the discrete differential geometry. All discrete differential geometry terms are explained for triangulated surface meshes. These utilities serve as basis for the feature line methods. We provide the reader with all knowledge to re-implement every feature line method. Furthermore, we summarize the methods and suggest a guideline for which kind of surface which feature line algorithm is best suited. Our work is motivated by, but not restricted to, medical and biological surface models.Comment: 33 page

Invariant Reconstruction of Curves and Surfaces with Discontinuities with Applications in Computer Vision

The reconstruction of curves and surfaces from sparse data is an important task in many applications. In computer vision problems the reconstructed curves and surfaces generally represent some physical property of a real object in a scene. For instance, the sparse data that is collected may represent locations along the boundary between an object and a background. It may be desirable to reconstruct the complete boundary from this sparse data. Since the curves and surfaces represent physical properties, the characteristics of the reconstruction process differs from straight forward fitting of smooth curves and surfaces to a set of data in two important areas. First, since the collected data is represented in an arbitrarily chosen coordinate system, the reconstruction process should be invariant to the choice of the coordinate system (except for the transformation between the two coordinate systems). Secondly, in many reconstruction applications the curve or surface that is being represented may be discontinuous. For example in the object recognition problem if the object is a box there is a discontinuity in the boundary curve at the comer of the box. The reconstruction problem will be cast as an ill-posed inverse problem which must be stabilized using a priori information relative to the constraint formation. Tikhonov regularization is used to form a well posed mathematical problem statement and conditions for an invariant reconstruction are given. In the case where coordinate system invariance is incorporated into the problem, the resulting functional minimization problems are shown to be nonconvex. To form a valid convex approximation to the invariant functional minimization problem a two step algorithm is proposed. The first step forms an approximation to the curve (surface) which is piecewise linear (planar). This approximation is used to estimate curve (surface) characteristics which are then used to form an approximation of the nonconvex functional with a convex functional. Several example applications in computer vision for which the invariant property is important are presented to demonstrate the effectiveness of the algorithms. To incorporate the fact that the curves and surfaces may have discontinuities the minimizing functional is modified. An important property of the resulting functional minimization problems is that convexity is maintained. Therefore, the computational complexity of the resulting algorithms are not significantly increased. Examples are provided to demonstrate the characteristics of the algorithm

Quantifying Membrane Topology at the Nanoscale

Changes in the shape of cellular membranes are linked with viral replication, Alzheimer\u27s, heart disease and an abundance of other maladies. Some membranous organelles, such as the endoplasmic reticulum and the Golgi, are only 50 nm in diameter. As such, membrane shape changes are conventionally studied with electron microscopy (EM), which preserves cellular ultrastructure and achieves a resolution of 2 nm or better. However, immunolabeling in EM is challenging, and often destroys the cell, making it difficult to study interactions between membranes and other proteins. Additionally, cells must be fixed in EM imaging, making it impossible to study mechanisms of disease. To address these problems, this thesis advances nanoscale imaging and analysis of membrane shape changes and their associated proteins using super-resolution single-molecule localization microscopy. This thesis is divided into three parts. In the first, a novel correlative orientation-independent differential interference contrast (OI-DIC) and single-molecule localization microscopy (SMLM) instrument is designed to address challenges with live-cell imaging of membrane nanostructure. SMLM super-resolution fluorescence techniques image with ~ 20 nm resolution, and are compatible with live-cell imaging. However, due to SMLM\u27s slow imaging speeds, most cell movement is under-sampled. OI-DIC images fast, is gentle enough to be used with living cells and can image cellular structure without labelling, but is diffraction-limited. Combining SMLM with OI-DIC allows for imaging of cellular context that can supplement sparse super-resolution data in real time. The second part of the thesis describes an open-source software package for visualizing and analyzing SMLM data. SMLM imaging yields localization point clouds, which requires non-standard visualization and analysis techniques. Existing techniques are described, and necessary new ones are implemented. These tools are designed to interpret data collected from the OI-DIC/SMLM microscope, as well as from other optical setups. Finally, a tool for extracting membrane structure from SMLM point clouds is described. SMLM data is often noisy, containing multiple localizations per fluorophore and many non-specific localizations. SMLM\u27s resolution reveals labelling discontinuities, which exacerbate sparsity of localizations. It is non-trivial to reconstruct the continuous shape of a membrane from a discrete set of points, and even more difficult in the presence of the noise profile characteristic of most SMLM point clouds. To address this, a surface reconstruction algorithm for extracting continuous surfaces from SMLM data is implemented. This method employs biophysical curvature constraints to improve the accuracy of the surface

Massively Parallel Approach to Modeling 3D Objects in Machine Vision

Electrical Engineerin

Assessment of plot-scale sediment transport on young moraines in the Swiss Alps using a fluorescent sand tracer

Glacial retreat uncovers large bodies of unconsolidated sediment that are prone to erosion. However, our knowledge of overland flow (OF) generation and sediment transport on moraines that have recently become ice-free is still limited. To investigate how the surface characteristics of young moraines affect OF and sediment transport, we installed five bounded runoff plots on two moraines of different ages in a proglacial area of the Swiss Alps. On each plot we conducted three sprinkling experiments to determine OF characteristics (i.e., total OF and peak OF flow rate) and measured sediment transport (turbidity, sediment concentrations, and total sediment yield). To determine and visualize where sediment transport takes place, we used a fluorescent sand tracer with an afterglow as well as ultraviolet (UV) and light-emitting diode (LED) lamps and a high-resolution camera. The results highlight the ability of this field setup to detect sand movement, even for individual fluorescent sand particles (300–500 µm grain size), and to distinguish between the two main mechanisms of sediment transport: OF-driven erosion and splash erosion. The higher rock cover on the younger moraine resulted in longer sediment transport distances and a higher sediment yield. In contrast, the higher vegetation cover on the older moraine promoted infiltration and reduced the length of the sediment transport pathways. Thus, this study demonstrates the potential of the use of fluorescent sand with an afterglow to determine sediment transport pathways as well as the fact that these observations can help to improve our understanding of OF and sediment transport processes on complex natural hillslopes

Sediment Dynamics and Channel Connectivity on Hillslopes

The pattern, magnitude, and frequency of hillslope erosion and deposition are spatially varied under the influence of micro-topography and channel geometry. This research investigates the interrelationships between erosion/deposition, micro-topography, and channel connectivity on a hillslope in Loudon, Tennessee using the centimeter (cm) level temporal Digital Elevation Models collected using laser scanning. This research addressed (1) the effect of spatial resolution on the erosion/deposition quantification, and rill delineation; (2) the influences of micro-topographic factors (e.g. slope, roughness, aspect) on erosion and deposition; (3) the relationship between the structural connectivity -- depressions and confluence of rills -- and the sedimentological connectivity. I conducted (1) visual and quantitative assessments for the erosion and deposition, and the revised automated proximity and conformity analysis for the rill network; (2) quantile regression for micro-topographic factors using segmented rill basins; and (3) cross-correlation analysis using erosion and deposition series along the channels.Overall, rills are sedimentologically more dynamic than the interrill areas. A larger grid size reduces the detectable changes in both areal and volumetric quantities, and also decreases the total length and number of rills. The offset between delineated rills and the reference increases with larger grid sizes. A larger rill basin has higher erosion and deposition with the magnitude of erosion greater than deposition. The slope has a positive influence on erosion and a negative one on deposition; roughness has a positive influence on deposition and a negative one on erosion. Areas that are more north-facing experience higher erosion and lower deposition. Rill length explains 46% of the variability for erosion and 24% for deposition. The depressions are associated with higher erosion in the downslope direction. The correlations between the erosion and the confluence are positive; the correlation between the deposition and the sink is positive. Overall, the influence of structural connectivity on the sedimentological connectivity is within 25 cm in both upstream and downstream directions. This research contributes to the understanding in how the sediment movement on hillslopes is governed by topographic variations and channel connectivity, and future work may explore hillslope channels at broader geographical and temporal scales

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd

Superquadric representation of scenes from multi-view range data

Object representation denotes representing three-dimensional (3D) real-world objects with known graphic or mathematic primitives recognizable to computers. This research has numerous applications for object-related tasks in areas including computer vision, computer graphics, reverse engineering, etc. Superquadrics, as volumetric and parametric models, have been selected to be the representation primitives throughout this research. Superquadrics are able to represent a large family of solid shapes by a single equation with only a few parameters. This dissertation addresses superquadric representation of multi-part objects and multiobject scenes. Two issues motivate this research. First, superquadric representation of multipart objects or multi-object scenes has been an unsolved problem due to the complex geometry of objects. Second, superquadrics recovered from single-view range data tend to have low confidence and accuracy due to partially scanned object surfaces caused by inherent occlusions. To address these two problems, this dissertation proposes a multi-view superquadric representation algorithm. By incorporating both part decomposition and multi-view range data, the proposed algorithm is able to not only represent multi-part objects or multi-object scenes, but also achieve high confidence and accuracy of recovered superquadrics. The multi-view superquadric representation algorithm consists of (i) initial superquadric model recovery from single-view range data, (ii) pairwise view registration based on recovered superquadric models, (iii) view integration, (iv) part decomposition, and (v) final superquadric fitting for each decomposed part. Within the multi-view superquadric representation framework, this dissertation proposes a 3D part decomposition algorithm to automatically decompose multi-part objects or multiobject scenes into their constituent single parts consistent with human visual perception. Superquadrics can then be recovered for each decomposed single-part object. The proposed part decomposition algorithm is based on curvature analysis, and includes (i) Gaussian curvature estimation, (ii) boundary labeling, (iii) part growing and labeling, and (iv) post-processing. In addition, this dissertation proposes an extended view registration algorithm based on superquadrics. The proposed view registration algorithm is able to handle deformable superquadrics as well as 3D unstructured data sets. For superquadric fitting, two objective functions primarily used in the literature have been comprehensively investigated with respect to noise, viewpoints, sample resolutions, etc. The objective function proved to have better performance has been used throughout this dissertation. In summary, the three algorithms (contributions) proposed in this dissertation are generic and flexible in the sense of handling triangle meshes, which are standard surface primitives in computer vision and graphics. For each proposed algorithm, the dissertation presents both theory and experimental results. The results demonstrate the efficiency of the algorithms using both synthetic and real range data of a large variety of objects and scenes. In addition, the experimental results include comparisons with previous methods from the literature. Finally, the dissertation concludes with a summary of the contributions to the state of the art in superquadric representation, and presents possible future extensions to this research

A saddle in a corner - a model of collinear triatomic chemical reactions

A geometrical model which captures the main ingredients governing atom-diatom collinear chemical reactions is proposed. This model is neither near-integrable nor hyperbolic, yet it is amenable to analysis using a combination of the recently developed tools for studying systems with steep potentials and the study of the phase space structure near a center-saddle equilibrium. The nontrivial dependence of the reaction rates on parameters, initial conditions and energy is thus qualitatively explained. Conditions under which the phase space transition state theory assumptions are satisfied and conditions under which these fail are derived