Abstract. Nonlinear manifold learning from unorganized data points is a very challenging unsupervised learning and data visualization problem with a great variety of applications. In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction. Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood of each data point. Those tangent spaces are aligned to give the internal global coordinates of the data points with respect to the underlying manifold by way of a partial eigendecomposition of the neighborhood connection matrix. We present a careful error analysis of our algorithm and show that the reconstruction errors are of second-order accuracy. We illustrate our algorithm using curves and surfaces both in 2D/3D and higher dimensional Euclidean spaces, and 64-by-64 pixel face images with various pose and lighting conditions. We also address several theoretical and algorithmic issues for further research and improvements.

In this paper, we develop methods for outlier removal and noise reduction based on weighted local linear smoothing for a set of noisy points sampled from a nonlinear manifold. The methods can be used by manifold learning methods such as Isomap, LLE and LTSA as a preprocessing procedure so as to obtain a more accurate reconstruction of the underlying nonlinear manifolds. Weighted principal component analysis is used as a building block of our methods and we develop an iterative weight selection scheme that leads to robust local linear fitting. We also develop an e#cient and e#ective bias-reduction method to deal with the trim the peak and fill the valley phenomenon in local linear smoothing. Several illustrative examples are presented to show that nonlinear manifold learning methods combined with weighted local linear smoothing give more accurate reconstruction of the underlying nonlinear manifolds.

Abstract: A novel system with high level of retrieval accuracy has been presented in this paper. Color as one of the most important discriminators in CBIR (content-based image retrieval) is utilized through calculating some of the primitive color features. The indexing of image database is performed with SOM (self-organizing map) which identified the BMU's (best matching units). Subsequently, Fuzzy Color Histogram (FCH) and subtractive fuzzy clustering algorithms have been utilized to identify the cluster for which the query image is belonging. Furthermore, the paper presents an enhanced edge detection algorithm to remove unwanted pixels and to solidify objects within images which ease similarity measures based on extracted shape features. The proposed approach overcomes the computational complexity of applying bin-to-bin comparison as a multi dimensional feature vectors in the original color histogram approach and improves the retrieval accuracy based on shape as compared with the most dominant approaches in this filed of study.