Confidence bands in nonparametric time series regression

We consider nonparametric estimation of mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The imposed dependence structure allows applications in many linear and nonlinear auto-regressive processes. The results are applied to the S&P 500 Index data.Comment: Published in at http://dx.doi.org/10.1214/07-AOS533 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

View point robust visual search technique

In this thesis, we have explored visual search techniques for images taken from diferent view points and have tried to enhance the matching capability under view point changes. We have proposed the Homography based back-projection as post-processing stage of Compact Descriptors for Visual Search (CDVS), the new MPEG standard; moreover, we have deined the aine adapted scale space based aine detection, which steers the Gaussian scale space to capture the features from aine transformed images; we have also developed the corresponding gradient based aine descriptor. Using these proposed techniques, the image retrieval robustness to aine transformations has been signiicantly improved. The irst chapter of this thesis introduces the background on visual search. In the second chapter, we propose a homography based back-projection used as the postprocessing stage of CDVS to improve the resilience to view point changes. The theory behind this proposal is that each perspective projection of the image of 2D object can be simulated as an aine transformation. Each pair of aine transformations are mathematically related by homography matrix. Given that matrix, the image can be back-projected to simulate the image of another view point. In this way, the real matched images can then be declared as matching because the perspective distortion has been reduced by the back-projection. An accurate homography estimation from the images of diferent view point requires at least 4 correspondences, which could be ofered by the CDVS pipeline. In this way, the homography based back-projection can be used to scrutinize the images with not enough matched keypoints. If they contain some homography relations, the perspective distortion can then be reduced exploiting the few provided correspondences. In the experiment, this technique has been proved to be quite efective especially to the 2D object images. The third chapter introduces the scale space, which is also the kernel to the feature detection for the scale invariant visual search techniques. Scale space, which is made by a series of Gaussian blurred images, represents the image structures at diferent level of details. The Gaussian smoothed images in the scale space result in feature detection being not invariant to aine transformations. That is the reason why scale invariant visual search techniques are sensitive to aine transformations. Thus, in this chapter, we propose an aine adapted scale space, which employs the aine steered Gaussian ilters to smooth the images. This scale space is lexible to diferent aine transformations and it well represents the image structures from diferent view points. With the help of this structure, the features from diferent view points can be well captured. In practice, the scale invariant visual search techniques have employed a pyramid structure to speed up the construction. By employing the aine Gaussian scale space principles, we also propose two structures to build the aine Gaussian scale space. The structure of aine Gaussian scale space is similar to the pyramid structure because of the similiar sampling and cascading iii properties. Conversely, the aine Laplacian of Gaussian (LoG) structure is completely diferent. The Laplacian operator, under aine transformation, is hard to be aine deformed. Diferently from a simple Laplacian operation on the scale space to build the general LoG construction, the aine LoG can only be obtained by aine LoG convolution and the cascade implementations on the aine scale space. Using our proposed structures, both the aine Gaussian scale space and aine LoG can be constructed. We have also explored the aine scale space implementation in frequency domain. In the second chapter, we will also explore the spectrum of Gaussian image smoothing under the aine transformation, and propose two structures. General speaking, the implementation in frequency domain is more robust to aine transformations at the expense of a higher computational complexity. It makes sense to adopt an aine descriptor for the aine invariant visual search. In the fourth chapter, we will propose an aine invariant feature descriptor based on aine gradient. Currently, the state of the art feature descriptors, including SIFT and Gradient location and orientation histogram (GLOH), are based on the histogram of image gradient around the detected features. If the image gradient is calculated as the diference of the adjacent pixels, it will not be aine invariant. Thus in that chapter, we irst propose an aine gradient which will contribute the aine invariance to the descriptor. This aine gradient will be calculated directly by the derivative of the aine Gaussian blurred images. To simplify the processing, we will also create the corresponding aine Gaussian derivative ilters for diferent detected scales to quickly generate the aine gradient. With this aine gradient, we can apply the same scheme of SIFT descriptor to generate the gradient histogram. By normalizing the histogram, the aine descriptor can then be formed. This aine descriptor is not only aine invariant but also rotation invariant, because the direction of the area to form the histogram is determined by the main direction of the gradient around the features. In practice, this aine descriptor is fully aine invariant and its performance for image matching is extremely good. In the conclusions chapter, we draw some conclusions and describe some future work