New PDF release: A Sampler of Useful Computational Tools for Applied

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 picture Processing indicates easy methods to use a set of mathematical thoughts to resolve very important difficulties in utilized arithmetic and machine technological know-how components. The e-book discusses primary instruments in analytical geometry and linear algebra. It covers a variety of subject matters, from matrix decomposition to curvature research and valuable part research to dimensionality reduction.

Written via a group of hugely revered professors, the publication can be utilized in a one-semester, intermediate-level path in desktop technological know-how. It takes a realistic problem-solving procedure, averting designated proofs and research. compatible for readers with out a deep educational history in arithmetic, the textual content explains easy methods to remedy non-trivial geometric difficulties. It speedy will get readers on top of things on various instruments hired in visible computing and utilized geometry.

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Additional info for A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

Example text

Finally, we locally apply an LS polynomial fit using the computed normal tangent as the reference frame (right). , p − qi ≤ r. Let n be the unit normal to S at p, and let the local tangent or the reference plane be denoted by n x + d = 0. We can solve for this reference plane by looking for the plane that best fits the given set of point Nr (p). Thus, our goal is to solve minn,d (n qi + d)2 such that n = 1. The solution will be discussed in detail in the next chapter; essentially, n is the smallest eigenvector of the covariance ¯ )(qi − q ¯ ) , where q ¯ = i qi /|Nr (p)|.

PCA defines a new orthogonal coordinate system for our data. First of all, we define the origin of this coordinate system to be the center of mass of all the data points: m= 1 n n xi . i=1 The new origin m is the best zero-order approximation for our data, in the sense that it is the point that has the minimal sum of square distances to all the data points: n xi − x 2 . m = argmin x i=1 50 PCA and SVD Next, let us find the directions that represent our data best. For this, the scatter (or covariance) matrix S ∈ Rd×d is defined.

That is Ax = λx ⇔ Ax − λx = 0 ⇔ Ax − λIx = 0 ⇔ (A − λI)x = 0 . So, a nontrivial solution exists if and only if det(A − λI) = 0. The expression det(A − λI) can be developed into a polynomial of degree n called the characteristic polynomial of A. The roots of this characteristic polynomial are the eigenvalues of A. Therefore, there are always n eigenvalues (some or all of which may be complex). If n is odd, there is at least one real eigenvalue. Let us look at an example:   1 0 2 A =  3 0 −3 .

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A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing by Daniel Cohen-Or, Chen Greif, Tao Ju, Niloy J. Mitra, Ariel Shamir, Olga Sorkine-Hornung, Hao (Richard) Zhang

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