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where a data point xj is among xi s neighbors in the training set. Consequently, in LLE, local geometry is characterized by the neighbors of each image. In particular, LLE is useful for dimensionality reduction; by LLE, the data sampled from an underlying manifold are mapped into a lower-dimensional data space. Dimension reduction by LLE preserves the neighborhood relationships; the goal of
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LLE for dimension reduction is to nd a lower-dimensional embedding yi characterized by the same weighted linear combination of the neighbors with xi . For each point in the D-dimensional sample space, the LLE algorithm can be summarized as follows: 1. Using the Euclidean distance measure, nd K nearest neighbors of a data point xi among N training images: x1 , x2 , . . . , xj , . . . xK . 2. Calculate the optimal weights of the neighbors such that minimize the reconstruction error: w ij
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where Eq. (12.34) can be solved by Lagrange multiplier. 3. Compute the d-dimensional embedding which is best reconstructed by the same neighbors and weights where D d:
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The optimal solution of yi in Eq. (12.35) is the smallest eigenvectors of matrix T (I W) where I is the N N identity matrix and W is the matrix (I W) consisting of {wij }; wij is 0 when xj is not xi s neighbor.
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According to the assumption that low-resolution patches and high-resolution patches have the same neighborhood relationships, the high-resolution counterpart h
12.3 Subspace-Based Approaches
of the low-resolution patch l is reconstructed by the same weighted linear combination in the high-resolution data space:
wi hi ,
where hi is a high-resolution counterpart of li in the training set.
12.3.3 Super-Resolution Using LPP by Park and Savvides
Park and Savvides proposed a novel super-resolution method for face images focusing on the fact that it has been shown that face images lie on a nonlinear manifold [13 16]. Manifold learning algorithms are more powerful for face image analysis than other pattern recognition methods which analyze a Euclidean space because they can reveal the underlying nonlinear distribution of the face space. PCA and linear discriminant analysis (LDA) [26] effectively see only the Euclidean structure; they fail to discover the underlying structure when the data lie on a nonlinear manifold. Thus, it is expected that manifold learning methods can improve the tasks demanding face image analysis, such as face recognition, super-resolution, or face synthesis. However, almost all the methods for face super-resolution have not utilized the manifold in the distribution of face images. Park and Savvides [20, 21] applied another novel manifold learning method, locality preserving projections (LPP) [15, 25] to face super-resolution.
Locality Preserving Projections
LPP is to nd a linear projective mapping for dimensionality reduction. Compared to LPP, other manifold learning techniques such as isomap [14], LLE [13], or Laplacian eigenmap [16] de ne the mapping only on the training data. They successfully show the training data are distributed along manifolds, but it is unclear how to evaluate the maps for new test samples. On the other hand, by LPP, we obtain the wellde ned transformation matrix that is applicable to new test images absent from the training set. LPP is designed for optimally preserving the neighborhood structure of the data set while principal component analysis (PCA) utilizes only a global basis. LPP is a novel method for dimensionality reduction by using both the local structure and the global basis of the data set. LPP aims to nd a linear projection for dimensionality reduction such that the local structure of the data space is preserved. LPP utilizes a weight which represents how close any two data points are in the data space. Using a set of these weights, we can obtain a set of eigenvectors which represent both the global basis and the neighbor embedding in the data set. When the high-dimensional data lies on a low-dimensional manifold embedded in data space, the locality preserving projections are obtained by nding the optimal linear approximations to the