Download Guide to Computational Geometry Processing: Foundations, by Jakob Andreas Bærentzen, Jens Gravesen, François Anton, PDF
By Jakob Andreas Bærentzen, Jens Gravesen, François Anton, Henrik Aanæs
This publication reports the algorithms for processing geometric info, with a pragmatic concentrate on vital thoughts now not coated via conventional classes on computing device imaginative and prescient and special effects. positive factors: provides an summary of the underlying mathematical conception, overlaying vector areas, metric area, affine areas, differential geometry, and finite distinction tools for derivatives and differential equations; experiences geometry representations, together with polygonal meshes, splines, and subdivision surfaces; examines suggestions for computing curvature from polygonal meshes; describes algorithms for mesh smoothing, mesh parametrization, and mesh optimization and simplification; discusses element place databases and convex hulls of element units; investigates the reconstruction of triangle meshes from element clouds, together with tools for registration of aspect clouds and floor reconstruction; presents extra fabric at a supplementary web site; contains self-study workouts through the text.
From the again Cover
Optical scanning is swiftly changing into ubiquitous. From business laser scanners to scientific CT, MR and 3D ultrasound scanners, various businesses now have easy accessibility to optical acquisition units that offer large volumes of picture facts. even though, the uncooked geometry facts obtained needs to first be processed earlier than it truly is helpful. This consultant to Computational Geometry Processing studies the algorithms for processing geometric information, with a realistic specialise in very important suggestions now not lined via conventional classes on machine imaginative and prescient and special effects. this can be balanced with an creation to the theoretical and mathematical underpinnings of every method, permitting the reader not to in basic terms enforce a given technique, but additionally to appreciate the information in the back of it, its boundaries and its benefits. issues and contours: provides an outline of the underlying mathematical concept, protecting vector areas, metric area, affine areas, differential geometry, and finite distinction equipment for derivatives and differential equationsReviews geometry representations, together with polygonal meshes, splines, and subdivision surfacesExamines thoughts for computing curvature from polygonal meshesDescribes algorithms for mesh smoothing, mesh parametrization, and mesh optimization and simplificationDiscusses aspect position databases and convex hulls of element setsInvestigates the reconstruction of triangle meshes from element clouds, together with equipment for registration of element clouds and floor reconstructionProvides extra fabric at a supplementary websiteIncludes self-study routines in the course of the textGraduate scholars will locate this article a worthwhile, hands-on consultant to constructing key talents in geometry processing. The booklet also will function an invaluable reference for pros wishing to enhance their competency during this sector.
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Extra resources for Guide to Computational Geometry Processing: Foundations, Algorithms, and Methods
Example text
For all u ∈ V , u, u ≥ 0. 5. For all u ∈ V , u, u = 0 ⇐⇒ u = 0. 35) is an Euclidean vector space. 44) is the usual angle. 39 The set Rn equipped with inner product (x1 , . . , xn ), (y1 , . . 36) is an Euclidean vector space. 37) a is an Euclidean vector space. The corresponding norm is called the L2 -norm. 38) is an Euclidean vector space. 42 If (V , ·, · ) is an Euclidean vector space and U ⊆ V is a subspace then U equipped with the restriction ·, · |U ×U of ·, · to U × U is an Euclidean vector space too.
We are interested in the surface given by f (x, y, z) = 0. Observe that another level set f (x, y, z) = c can be given as f (x, y, z) − c = 0. Let h denote the signed distance function for the surface. 58) where λ : R3 → R+ is some positive function. 58) yields ⎡ ∂f ⎤ ⎡ ⎤ ⎡ ⎤ ∂λ ∂h ∂x ∂x ⎢ ∂x ⎥ ⎢ ⎥ ⎢ ∂f ⎥ ⎢ ⎥ = h ⎢ ∂λ ⎥ + λ ⎢ ∂h ⎥ . , λ = ∇f , 1 ∇h = ∇f.
19 A basis e1 , . . , en for an Euclidean vector space is called orthonormal if ei , ej = δij = 1 if i = j , 0 if i = j . 46) That is, the elements of the basis are pairwise orthogonal and have norm 1. If u1 , . . , un is a basis for an Euclidean vector space V then we can construct an orthonormal basis e1 , . . , en by Gram–Schmidt orthonormalization. The elements of that particular orthonormal basis is defined as follows: −1 v =u − u , e k ek , e = k=1 v v , = 1, . . , n. 20 A linear map L : U → V between two Euclidean vector spaces is called an isometry if it is bijective and L(u), L(v) V = u, v U for all u, v ∈ U .