A Hierarchical On-Line Path Planning Scheme

“... instead of the great number of precepts of which logic is composed, I believed that the four following would prove perfectly sufficient for me, provided I took the firm and unwavering resolution never in a single instance to fail in observing them. The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt. The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution. The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence. And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.” “Discourse on the Method of Rightly Conducting One’s Reason and of Seekin

Unbalanced Haar technique for nonparametric function estimation

The discrete unbalanced Haar (UH) transform is a decomposition of one-dimensional data with respect to an orthonormal Haar-like basis where jumps in the basis vectors do not necessarily occur in the middle of their support. We introduce a multiscale procedure for estimation in Gaussian noise that consists of three steps: a UH transform, thresholding of the decomposition coefficients, and the inverse UH transform. We show that our estimator is mean squared consistent with near-optimal rates for a wide range of functions, uniformly over UH bases that are not “too unbalanced.” A vital ingredient of our approach is basis selection. We choose each basis vector so that it best matches the data at a specific scale and location, where the latter parameters are determined by the "parent" basis vector. Our estimator is computable in O(n log n) operations. A simulation study demonstrates the good performance of our estimator compared with state-of-the-art competitors. We apply our method to the estimation of the mean intensity of the time series of earthquake counts occurring in northern California. We discuss extensions to image data and to smoother wavelets

Projectable Multivariate Refinable Functions and Biorthogonal Wavelets

A biorthogonal wavelet... In this paper, we introduce the concept of projectable refinable functions and demonstrate that many multivariate refinable functions are projectable; that is, they essentially carry the tensor product (separable) structure though themselves may be non-tensor product (nonseparable) refinable functions. For any pair of biorthogonal refinable functions (φ, φ^d) in L_2(R^n), when the refinable function φ is projectable, we prove that without loss of several desirable properties such as spatial localization, smoothness and approximation order, from the pair of biorthogonal refinable functions (φ, φ^d), we can easily obtain another pair of biorthogonal refinable functions in L_2(R^n) which are tensor product separable refinable functions. As an application, we show that there is no dual refinable function φ^d to the refinable basis function in the Loop scheme such that..

Surfaces: Today {Totaldegreepatches.

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