By Daniel Cohen-Or, Chen Greif, Tao Ju, Niloy J. Mitra, Ariel Shamir, Olga Sorkine-Hornung, Hao (Richard) Zhang
ISBN-10: 1498706304
ISBN-13: 9781498706308
A Sampler of beneficial Computational instruments for utilized Geometry, special effects, and snapshot Processing indicates the right way to use a suite of mathematical options to unravel very important difficulties in utilized arithmetic and desktop technological know-how components. The ebook discusses basic instruments in analytical geometry and linear algebra. It covers quite a lot of themes, from matrix decomposition to curvature research and important part research to dimensionality reduction.
Written via a staff of hugely revered professors, the publication can be utilized in a one-semester, intermediate-level path in desktop technological know-how. It takes a pragmatic problem-solving procedure, averting unique proofs and research. appropriate for readers with no deep educational heritage in arithmetic, the textual content explains tips on how to remedy non-trivial geometric difficulties. It speedy will get readers in control on a number of instruments hired in visible computing and utilized geometry.
Read Online or Download A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing PDF
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Additional info for A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing
Sample text
One important issue is that of the magnitude or norm of a vector. The size or magnitude of a number is given by its absolute value; this is very intuitive. For vectors there are a few commonly used possibilities. The norm of a vector u is a scalar denoted by u , which satisfies the following properties: • u ≥ 0, and u = 0 if and only if u = 0, namely all of the components of u are zero. • For any scalar α and a vector u, αu = |α| · u . • The triangle inequality holds: u + v ≤ u + v for any two vectors u and v.
In Chapter 4 we will review the singular value decomposition (SVD), which will tell us how general linear transformations behave, among other things. Finding the coordinates of a vector Suppose we are given a basis v1 , v2 , v3 in R3 . An arbitrary vector a ∈ R3 has three coordinates denoted by, say, α1 , α2 , α3 and associated with the basis vectors: n a= α i vi . i=1 Linear Algebra? 27 To extract the αi , we can use the linearity of the space and form inner products that give us n a, vj = αi vi , vj , i=1 and thus we have a, v1 = α1 v1 , v1 + α1 v2 , v1 + α3 v3 , v1 ; a, v2 = α2 v1 , v2 + α1 v2 , v2 + α3 v3 , v2 ; a, v3 = α3 v1 , v3 + α1 v2 , v3 + α3 v3 , v3 .
In the weighted formulation, 2 we minimize i wi zi − F (xi , yi ) , where wi ≥ 0. Typically, we choose a weight function such that weights get smaller as the distance from the point p increases. A popular choice of such a weight function is exponential decay. Least-Squares Solutions 43 Concluding remarks In this chapter, we have learned about Least-Squares fitting and solution. Recall that formulating the optimization using quadratic or square terms allows us to get a linear system after taking partial derivatives.