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A Sampler of Useful Computational Tools for Applied by Daniel Cohen-Or, Chen Greif, Tao Ju, Niloy J. Mitra, Ariel

By Daniel Cohen-Or, Chen Greif, Tao Ju, Niloy J. Mitra, Ariel Shamir, Olga Sorkine-Hornung, Hao (Richard) Zhang

A Sampler of priceless Computational instruments for utilized Geometry, special effects, and photograph Processing exhibits easy methods to use a suite of mathematical innovations to unravel very important difficulties in utilized arithmetic and desktop technology parts. The booklet discusses basic instruments in analytical geometry and linear algebra. It covers quite a lot of issues, from matrix decomposition to curvature research and significant part research to dimensionality reduction.

Written through a workforce of hugely revered professors, the publication can be utilized in a one-semester, intermediate-level path in machine technological know-how. It takes a realistic problem-solving process, averting certain proofs and research. appropriate for readers and not using a deep educational historical past in arithmetic, the textual content explains the way to resolve non-trivial geometric difficulties. It speedy will get readers on top of things on quite a few instruments hired in visible computing and utilized geometry.

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Extra resources for A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

Sample text

LS solutions owe their popularity to the fact that they can be solved using linear systems. 4), let us take a careful look at it. Do you see a problem with this formulation? Suppose Least-Squares Solutions 35 the point sets are given with respect to another coordinate system. 4 (right). Thus, in the current form, our solution depends on the choice of coordinate system. This should not be the case. The best-fitting line should only depend on the arrangement of the points, and remain unchanged with respect to the point set as long as its arrangement remains fixed.

20) Least-Squares Solutions Thus n is one of the eigenvectors of the covariance matrix C (see Chapter 4). 19), the objective function reduces to simply λ. , the eigenvector corresponding to the smallest ¯. eigenvalue. Once we have n, we obtain d as d = −n p 45 This page intentionally left blank Chapter 4 PCA and SVD Olga Sorkine-Hornung In this chapter, we introduce two related tools from linear algebra that have become true workhorses in countless areas of scientific computing: principal component analysis (PCA) and singular value decomposition (SVD).

We stop iterating once the process converges. 6 (right). Notice that there are many different choices for weight functions that lead to similar behavior. The idea of weighted LS is also applicable for a weighted leastsquares solution. Recall that, corresponding to the n unknowns {xi } stacked together as vector x, we had m equations. , nonnegative value wi . Thus we have m weights. 14) where W is an m×m diagonal matrix with the respective diagonal entries being the weights {wi }. Least-Squares Solutions 41 Local surface fitting to 3D points We have already seen how LS fitting is useful for fitting a polynomial to a set of points coming from a curve.

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