[Studies in Computational Intelligence 481] Artur Babiarz, Robert Bieda, Karol Jędrasiak, Aleksander Nawrat (auth.), Aleksander Nawrat, Zygmunt Kuś (eds.) - Vision Based Systemsfor UAV Applications (2013, Sprin

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236 H. Josiński et al. Principal component analysis is one of the classic linear methods of dimensionality reduction. Given a set of high-dimensional feature vectors, PCA projects them onto low-dimensional subspace spanned by principal components. The first principal component is the direction which maximizes variance. The second principal component maximizes variance in a direction orthogonal to (i.e., uncorrelated with) the first component. Generally, the k-th principal component (k > 1) indicates the direction along which variance is maximum among all directions orthogonal to the preceding k-1 components. PCA is most useful in the case when data lie on or close to a linear subspace of the data set [22]. Isometric features mapping and locally linear embedding are algorithms proposed for manifold learning which is an approach to non-linear dimensionality reduction. Manifold learning is the process of estimating a low-dimensional structure which underlies a collection of high-dimensional data [23]. In other words, manifold learning algorithms essentially attempt to duplicate the behavior of PCA but on manifolds instead of linear subspaces [22]. The Isomap algorithm [24] comprises the following stages: 1. Estimate pairwise the geodesic distances (distances along a manifold) between all data points. To perform this estimation, first construct a k-nearest neighbor graph weighted by the Euclidean distances. Then compute the shortest-path distances using Dijkstra’s or Floyd-Warshall’s algorithm. As the number of data points increases, this estimate converges to the true geodesic distances. 2. Use the Multidimensional Scaling (MDS) [23] to find points in lowdimensional Euclidean space whose interpoint distances match the distances estimated in the previous step. A D-dimensional manifold can be arbitrarily well-approximated by a d- dimensional linear subspace by taking a sufficiently small region about any point. Consequently, the LLE algorithm finds a local representation of each point based on its k nearest neighbors (measured by Euclidean distance) assuming that within the neighborhood the manifold is approximately flat and then reconstructs a lowdimensional representation with a similar configuration (see Fig. 1 – an often cited illustration from Saul and Roweis [25]). The LLE algorithm consists of the following three steps: 1. Assign k nearest neighbors to each data point X i , 1 ≤ i ≤ n in D-dimensional space. 2. Compute the weights W ij in order to reconstruct linearly each data point from its k nearest neighbors. The weights are chosen to minimize the reconstruction error: ( ) ε W = X − W X i ij j i j∈N ( i) 2 , (1) where N(i) denotes the set of k nearest neighbors of the data point X i . The reconstruction weights sum to 1 and W ij = 0 if j∉ N() i .
Feature Extraction and HMM-Based Classification of Gait Video Sequences 237 3. Under assumption that the same weights W ij that reconstruct the data point X i in D dimensions should also reconstruct its embedded manifold coordinates in d dimensions, compute the outputs Y i which are best reconstructed by the weights W ij in d-dimensional space (d
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Studies in Computational Intelligen
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Aleksander Nawrat · Zygmunt Kuś E
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“In God We Trust, All others we o
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VIII Preface valuable information i
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Contents Part I: Design of Object D
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Contents XIII Technology Developmen
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XVI List of Contributors Adam Czorn
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XVIII List of Contributors Henryk M
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XX List of Contributors Józef Wron
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2 Design of Object Detection, Recog
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4 A. Babiarz et al. 1 Introduction
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6 A. Babiarz et al. The description
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8 A. Babiarz et al. a) b) Fig. 5. T
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10 A. Babiarz et al. o o o “micvo
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12 A. Babiarz et al. a) b) Fig. 7.
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14 A. Babiarz et al. • The camera
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16 A. Babiarz et al. the camera ser
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18 A. Babiarz et al. a) b) Fig. 11.
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24 A. Babiarz et al. The sample cod
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Recognition and Location of Objects
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48 P. Demski et al. Fig. 1. Automat
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52 P. Demski et al. 4 Automatic Tar
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54 P. Demski et al. implementing su
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Object Tracking for Rapid Camera Mo
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94 Construction of Image Acquisitio
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96 G. Bieszczad et al. effectivenes
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98 G. Bieszczad et al. such a way t
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100 G. Bieszczad et al. Table 1. No
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110 G. Bieszczad et al. has to make
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112 G. Bieszczad et al. MTF functio
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114 G. Bieszczad et al. [11] Krupi
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116 D. Bereska et al. Presented in
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118 D. Bereska et al. Fig. 2. Image
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120 D. Bereska et al. Fig. 4. Ortho
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Omnidirectional Video Acquisition D
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Part III Design of Vision Based Con
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Vision System for Group of Mobile R
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A Distributed Control Group of Mobi
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344 Conclusions Control algorithms
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[Studies in Computational Intelligence 481] Artur Babiarz, Robert Bieda, Karol Jędrasiak, Aleksander Nawrat (auth.), Aleksander Nawrat, Zygmunt Kuś (eds.) - Vision Based Systemsfor UAV Applications (2013, Sprin

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