130 research outputs found
Spiral Complete Coverage Path Planning Based on Conformal Slit Mapping in Multi-connected Domains
Generating a smooth and shorter spiral complete coverage path in a
multi-connected domain is an important research area in robotic cavity
machining. Traditional spiral path planning methods in multi-connected domains
involve a subregion division procedure; a deformed spiral path is incorporated
within each subregion, and these paths within the subregions are interconnected
with bridges. In intricate domains with abundant voids and irregular
boundaries, the added subregion boundaries increase the path avoidance
requirements. This results in excessive bridging and necessitates longer
uneven-density spirals to achieve complete subregion coverage. Considering that
conformal slit mapping can transform multi-connected regions into regular disks
or annuluses without subregion division, this paper presents a novel spiral
complete coverage path planning method by conformal slit mapping. Firstly, a
slit mapping calculation technique is proposed for segmented cubic spline
boundaries with corners. Then, a spiral path spacing control method is
developed based on the maximum inscribed circle radius between adjacent
conformal slit mapping iso-parameters. Lastly, the spiral path is derived by
offsetting iso-parameters. The complexity and applicability of the proposed
method are comprehensively analyzed across various boundary scenarios.
Meanwhile, two cavities milling experiments are conducted to compare the new
method with conventional spiral complete coverage path methods. The comparation
indicate that the new path meets the requirement for complete coverage in
cavity machining while reducing path length and machining time by 12.70% and
12.34%, respectively.Comment: This article has not been formally published yet and may undergo
minor content change
Subset Warping: Rubber Sheeting with Cuts
Image warping, often referred to as "rubber sheeting" represents the deformation of a domain image space into a range image space. In this paper, a technique is described which extends the definition of a rubber-sheet transformation to allow a polygonal region to be warped into one or more subsets of itself, where the subsets may be multiply connected. To do this, it constructs a set of "slits" in the domain image, which correspond to discontinuities in the range image, using a technique based on generalized Voronoi diagrams. The concept of medial axis is extended to describe inner and outer medial contours of a polygon. Polygonal regions are decomposed into annular subregions, and path homotopies are introduced to describe the annular subregions. These constructions motivate the definition of a ladder, which guides the construction of grid point pairs necessary to effect the warp itself
Conformal Mapping and Brain Flattening
In this dissertation we study some of the main results concerning conformal mappings in the complex plane and between Riemann surfaces and we apply those results to the so-called brain flattening problem. In the first part of this thesis we prove the Riemann Mapping Theorem and we provide an introduction to the Uniformization Theorem for simply connected Riemann surfaces. The second part of the thesis is focused on the brain flattening problem, which deals with how to construct a conformal mapping from the brain's cortical surface to the unitary sphere. This procedure leads to a possible definition of the discrete mean curvature on a triangulated closed surface of genus zero. This flattening method has several applications in neuroscience
Surface Comparison with Mass Transportation
We use mass-transportation as a tool to compare surfaces (2-manifolds). In
particular, we determine the "similarity" of two given surfaces by solving a
mass-transportation problem between their conformal densities. This mass
transportation problem differs from the standard case in that we require the
solution to be invariant under global M\"obius transformations. Our approach
provides a constructive way of defining a metric in the abstract space of
simply-connected smooth surfaces with boundary (i.e. surfaces of disk-type);
this metric can also be used to define meaningful intrinsic distances between
pairs of "patches" in the two surfaces, which allows automatic alignment of the
surfaces. We provide numerical experiments on "real-life" surfaces to
demonstrate possible applications in natural sciences
Structural Surface Mapping for Shape Analysis
Natural surfaces are usually associated with feature graphs, such as the cortical surface with anatomical atlas structure. Such a feature graph subdivides the whole surface into meaningful sub-regions. Existing brain mapping and registration methods did not integrate anatomical atlas structures. As a result, with existing brain mappings, it is difficult to visualize and compare the atlas structures. And also existing brain registration methods can not guarantee the best possible alignment of the cortical regions which can help computing more accurate shape similarity metrics for neurodegenerative disease analysis, e.g., Alzheimer’s disease (AD) classification. Also, not much attention has been paid to tackle surface parameterization and registration with graph constraints in a rigorous way which have many applications in graphics, e.g., surface and image morphing.
This dissertation explores structural mappings for shape analysis of surfaces using the feature graphs as constraints. (1) First, we propose structural brain mapping which maps the brain cortical surface onto a planar convex domain using Tutte embedding of a novel atlas graph and harmonic map with atlas graph constraints to facilitate visualization and comparison between the atlas structures. (2) Next, we propose a novel brain registration technique based on an intrinsic atlas-constrained harmonic map which provides the best possible alignment of the cortical regions. (3) After that, the proposed brain registration technique has been applied to compute shape similarity metrics for AD classification. (4) Finally, we propose techniques to compute intrinsic graph-constrained parameterization and registration for general genus-0 surfaces which have been used in surface and image morphing applications
Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory
Conformal mapping, a classical topic in complex analysis and differential
geometry, has become a subject of great interest in the area of surface
parameterization in recent decades with various applications in science and
engineering. However, most of the existing conformal parameterization
algorithms only focus on simply-connected surfaces and cannot be directly
applied to surfaces with holes. In this work, we propose two novel algorithms
for computing the conformal parameterization of multiply-connected surfaces. We
first develop an efficient method for conformally parameterizing an open
surface with one hole to an annulus on the plane. Based on this method, we then
develop an efficient method for conformally parameterizing an open surface with
holes onto a unit disk with circular holes. The conformality and
bijectivity of the mappings are ensured by quasi-conformal theory. Numerical
experiments and applications are presented to demonstrate the effectiveness of
the proposed methods
Circular slit maps of multiply connected regions with application to brain image processing
In this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is O((M+ 1) n) , where M+ 1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require O((M+ 1) 3n3) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented
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