Topic

# Orthogonality

About: Orthogonality is a(n) research topic. Over the lifetime, 3723 publication(s) have been published within this topic receiving 59256 citation(s). The topic is also known as: orthogonal.

##### Papers

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01 Nov 1993-

TL;DR: A modification to the matching pursuit algorithm of Mallat and Zhang (1992) that maintains full backward orthogonality of the residual at every step and thereby leads to improved convergence is proposed.

Abstract: We describe a recursive algorithm to compute representations of functions with respect to nonorthogonal and possibly overcomplete dictionaries of elementary building blocks e.g. affine (wavelet) frames. We propose a modification to the matching pursuit algorithm of Mallat and Zhang (1992) that maintains full backward orthogonality of the residual (error) at every step and thereby leads to improved convergence. We refer to this modified algorithm as orthogonal matching pursuit (OMP). It is shown that all additional computation required for the OMP algorithm may be performed recursively. >

4,245 citations

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TL;DR: The results demonstrate that there exist ideal binary time-frequency masks that can separate several speech signals from one mixture and show that the W-disjoint orthogonality of speech can be approximate in the case where two anechoic mixtures are provided.

Abstract: Binary time-frequency masks are powerful tools for the separation of sources from a single mixture. Perfect demixing via binary time-frequency masks is possible provided the time-frequency representations of the sources do not overlap: a condition we call W-disjoint orthogonality. We introduce here the concept of approximate W-disjoint orthogonality and present experimental results demonstrating the level of approximate W-disjoint orthogonality of speech in mixtures of various orders. The results demonstrate that there exist ideal binary time-frequency masks that can separate several speech signals from one mixture. While determining these masks blindly from just one mixture is an open problem, we show that we can approximate the ideal masks in the case where two anechoic mixtures are provided. Motivated by the maximum likelihood mixing parameter estimators, we define a power weighted two-dimensional (2-D) histogram constructed from the ratio of the time-frequency representations of the mixtures that is shown to have one peak for each source with peak location corresponding to the relative attenuation and delay mixing parameters. The histogram is used to create time-frequency masks that partition one of the mixtures into the original sources. Experimental results on speech mixtures verify the technique. Example demixing results can be found online at http://alum.mit.edu/www/rickard/bss.html.

1,472 citations

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Abstract: On propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations, etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

1,220 citations

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TL;DR: It is shown how a variant of PLS can be used to achieve a signal correction that is as close to orthogonal as possible to a given Y-vector or Y-matrix and is applied to four different data sets of multivariate calibration.

Abstract: Near-infrared (NIR) spectra are often pre-processed in order to remove systematic noise such as base-line variation and multiplicative scatter effects. This is done by differentiating the spectra to first or second derivatives, by multiplicative signal correction (MSC), or by similar mathematical filtering methods. This pre-processing may, however, also remove information from the spectra regarding Y (the measured response variable in multivariate calibration applications). We here show how a variant of PLS can be used to achieve a signal correction that is as close to orthogonal as possible to a given Y-vector or Y-matrix. Thus, one ensures that the signal correction removes as little information as possible regarding Y. In the case when the number of X-variables (K) exceeds the number of observations (N), strict orthogonality is obtained. The approach is called orthogonal signal correction (OSC) and is here applied to four different data sets of multivariate calibration. The results are compared with those of traditional signal correction as well as with those of no pre-processing, and OSC is shown to give substantial improvements. Prediction sets of new data, not used in the model development, are used for the comparisons.

952 citations

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TL;DR: There are at least three different definitions of cross polarization used in the literature, and the definition which corresponds to one standard measurement practice is proposed as the best choice.

Abstract: There are at least three different definitions of cross polarization used in the literature. The alternative definitions are discussed with respect to several applications, and the definition which corresponds to one standard measurement practice is proposed as the best choice.

799 citations