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Register a new userSegmentation of Facial Expressions Using Semi-Definite Programming and Generalized Principal Component Analysis
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Behnood Gholami
Publication date
2009
fields
Informatics Engineering
and research's language is
English
Authors
Behnood Gholami
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In this paper, we use semi-definite programming and generalized principal component analysis (GPCA) to distinguish between two or more different facial expressions. In the first step, semi-definite programming is used to reduce the dimension of the image data and unfold the manifold which the data points (corresponding to facial expressions) reside on. Next, GPCA is used to fit a series of subspaces to the data points and associate each data point with a subspace. Data points that belong to the same subspace are claimed to belong to the same facial expression category. An example is provided.
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A principal component analysis based on the generalized Gini correlation index is proposed (Gini PCA). The Gini PCA generalizes the standard PCA based on the variance. It is shown, in the Gaussian case, that the standard PCA is equivalent to the Gini PCA. It is also proven that the dimensionality reduction based on the generalized Gini correlation matrix, that relies on city-block distances, is robust to outliers. Monte Carlo simulations and an application on cars data (with outliers) show the robustness of the Gini PCA and provide different interpretations of the results compared with the variance PCA.
Cryo-electron microscopy nowadays often requires the analysis of hundreds of thousands of 2D images as large as a few hundred pixels in each direction. Here we introduce an algorithm that efficiently and accurately performs principal component analysis (PCA) for a large set of two-dimensional images, and, for each image, the set of its uniform rotations in the plane and their reflections. For a dataset consisting of $n$ images of size $L times L$ pixels, the computational complexity of our algorithm is $O(nL^3 + L^4)$, while existing algorithms take $O(nL^4)$. The new algorithm computes the expansion coefficients of the images in a Fourier-Bessel basis efficiently using the non-uniform fast Fourier transform. We compare the accuracy and efficiency of the new algorithm with traditional PCA and existing algorithms for steerable PCA.
In this paper, we propose a new approach for facial expression recognition using deep covariance descriptors. The solution is based on the idea of encoding local and global Deep Convolutional Neural Network (DCNN) features extracted from still images, in compact local and global covariance descriptors. The space geometry of the covariance matrices is that of Symmetric Positive Definite (SPD) matrices. By conducting the classification of static facial expressions using Support Vector Machine (SVM) with a valid Gaussian kernel on the SPD manifold, we show that deep covariance descriptors are more effective than the standard classification with fully connected layers and softmax. Besides, we propose a completely new and original solution to model the temporal dynamic of facial expressions as deep trajectories on the SPD manifold. As an extension of the classification pipeline of covariance descriptors, we apply SVM with valid positive definite kernels derived from global alignment for deep covariance trajectories classification. By performing extensive experiments on the Oulu-CASIA, CK+, and SFEW datasets, we show that both the proposed static and dynamic approaches achieve state-of-the-art performance for facial expression recognition outperforming many recent approaches.
In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.
Using a semi-analytical model developed by Choudhury & Ferrara (2005) we study the observational constraints on reionization via a principal component analysis (PCA). Assuming that reionization at z>6 is primarily driven by stellar sources, we decompose the unknown function N_{ion}(z), representing the number of photons in the IGM per baryon in collapsed objects, into its principal components and constrain the latter using the photoionization rate obtained from Ly-alpha forest Gunn-Peterson optical depth, the WMAP7 electron scattering optical depth and the redshift distribution of Lyman-limit systems at z sim 3.5. The main findings of our analysis are: (i) It is sufficient to model N_{ion}(z) over the redshift range 2<z<14 using 5 parameters to extract the maximum information contained within the data. (ii) All quantities related to reionization can be severely constrained for z<6 because of a large number of data points whereas constraints at z>6 are relatively loose. (iii) The weak constraints on N_{ion}(z) at z>6 do not allow to disentangle different feedback models with present data. There is a clear indication that N_{ion}(z) must increase at z>6, thus ruling out reionization by a single stellar population with non-evolving IMF, and/or star-forming efficiency, and/or photon escape fraction. The data allows for non-monotonic N_{ion}(z) which may contain sharp features around z sim 7. (iv) The PCA implies that reionization must be 99% completed between 5.8<z<10.3 (95% confidence level) and is expected to be 50% complete at z approx 9.5-12. With future data sets, like those obtained by Planck, the z>6 constraints will be significantly improved.
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