By René Vidal, Yi Ma, Shankar Sastry

ISBN-10: 0387878106

ISBN-13: 9780387878102

ISBN-10: 0387878114

ISBN-13: 9780387878119

This booklet presents a complete creation to the most recent advances within the mathematical conception and computational instruments for modeling high-dimensional facts drawn from one or a number of low-dimensional subspaces (or manifolds) and very likely corrupted through noise, gross error, or outliers. This tough activity calls for the advance of recent algebraic, geometric, statistical, and computational equipment for effective and powerful estimation and segmentation of 1 or a number of subspaces. The booklet additionally provides fascinating real-world purposes of those new equipment in snapshot processing, picture and video segmentation, face attractiveness and clustering, and hybrid approach id and so forth.

This booklet is meant to function a textbook for graduate scholars and starting researchers in info technological know-how, laptop studying, machine imaginative and prescient, picture and sign processing, and structures concept. It includes abundant illustrations, examples, and workouts and is made principally self-contained with 3 Appendices which survey uncomplicated thoughts and rules from data, optimization, and algebraic-geometry utilized in this book.

**René**** Vidal** is a Professor of Biomedical Engineering and Director of the imaginative and prescient Dynamics and studying Lab on the Johns Hopkins college.

Yi Ma is government Dean and Professor on the college of knowledge technological know-how and expertise at ShanghaiTech collage. **S. Shankar Sastry** is Dean of the school of Engineering, Professor of electric Engineering and laptop technological know-how and Professor of Bioengineering on the collage of California, Berkeley.

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**Additional resources for Generalized Principal Component Analysis**

**Example text**

7) Z D S1 [ S2 [ [ Sn ; also called a piecewise linear model, to approximate many nonlinear manifolds or piecewise smooth topological spaces. This is the standard model considered in geometric approaches to generalized principal component analysis (GPCA), which will be studied in Part II of this book. x/gniD1 , where x 2 RD but each distribution has mass concentrated near a subspace. Â1 ; : : : ; Ân ; 1 ; : : : ; n / are the model parameters and i > 0 are mixing weights with 1 C 2 C C n D 1. x/ is a nearly degenerate Gaussian distribution.

The image on the right shows the displacement of these feature points from the first to the second image as well as the segmentation of these displacement vectors using a mixture model of two linear subspaces. We will describe in detail the motion segmentation method used to achieve this result in Chapter 11. Temporal Video Segmentation and Event Detection Another example arises in the context of detecting events from video sequences. A typical video sequence contains multiple activities or events separated in time.

For computational efficiency, we first down-sample each image to a size of 30 40 pixels. 12 By modeling the projected data with a mixture model of linear subspaces in R3 , we obtain three affine subspaces of dimension 2, 1, and 1, respectively. 3(b). Since face images are rather intuitive real data and have good subspace structures, we will use them to produce many running examples in the book to help demonstrate certain abstract concepts or to evaluate certain methods. Image Representation and Segmentation The next set of examples arises in the context of image processing, especially image representation and segmentation.

### Generalized Principal Component Analysis by René Vidal, Yi Ma, Shankar Sastry

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