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Real Equiangular Frames
"... Real equiangular tight frames can be especially useful in practice because of their structure. The problem is that very few of them are known. We will look at recent advances on the problem of classifying the equiangular tight frames and as a consequence give a classification of this family of frame ..."
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Real equiangular tight frames can be especially useful in practice because of their structure. The problem is that very few of them are known. We will look at recent advances on the problem of classifying the equiangular tight frames and as a consequence give a classification of this family of frames for all real Hilbert spaces of dimension less than or equal to 50.
The geometry of structured . . . Frame Potentials
, 2015
"... In this dissertation, we study the geometric character of structured Parseval frames, which are families of vectors that provide perfect Hilbert space reconstruction. Equiangular Parseval frames (EPFs) satisfy that the magnitudes of the pairwise inner products between frame vectors are constant. ..."
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In this dissertation, we study the geometric character of structured Parseval frames, which are families of vectors that provide perfect Hilbert space reconstruction. Equiangular Parseval frames (EPFs) satisfy that the magnitudes of the pairwise inner products between frame vectors are constant. These types of frames are useful in many applications. However, EPFs do not always exist and constructing them is often difficult. To address this problem, we consider two generalizations of EPFs, equidistributed frames and Grassmannian equalnorm Parseval frames, which include EPFs when they exist. We provide several examples of each type of Parseval frame. To characterize and locate these classes of frames, we develop an optimization program involving families of real analytic frame potentials, which are realvalued functions of frames. With the help of the Łojasiewicz gradient inequality, we prove that the gradient descent of these functions on the manifold of Gram matrices of
A Sparse Auditory Envelope Representation with Iterative Reconstruction for Audio Coding
, 2011
"... 1933–2009 iiiii Modern audio coding exploits the properties of the human auditory system to efficiently code speech and music signals. Perceptual domain coding is a branch of audio coding in which the signal is stored and transmitted as a set of parameters derived directly from the modeling of the h ..."
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1933–2009 iiiii Modern audio coding exploits the properties of the human auditory system to efficiently code speech and music signals. Perceptual domain coding is a branch of audio coding in which the signal is stored and transmitted as a set of parameters derived directly from the modeling of the human auditory system. Often, the perceptual representation is designed such that reconstruction can be achieved with limited resources but this usually means that some perceptually irrelevant information is included. In this thesis, we investigate perceptual domain coding by using a representation designed to contain only the audible information regardless of whether reconstruction can be performed efficiently. The perceptual representation we use is based on a multichannel Basilar membrane model, where each channel is decomposed into envelope and carrier components. We assume that the information in the carrier is also present in the envelopes and therefore discard the carrier components. The envelope components are sparsified using a transmultiplexing masking model and form our basic
FINITE TWODISTANCE TIGHT FRAMES
"... ABSTRACT. A finite collection of unit vectors S ⊂ Rn is called a spherical twodistance set if there are two numbers a and b such that the inner products of distinct vectors from S are either a or b. We prove that if a 6 = −b, then a twodistance set that forms a tight frame for Rn is a spherical em ..."
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ABSTRACT. A finite collection of unit vectors S ⊂ Rn is called a spherical twodistance set if there are two numbers a and b such that the inner products of distinct vectors from S are either a or b. We prove that if a 6 = −b, then a twodistance set that forms a tight frame for Rn is a spherical embedding of a strongly regular graph. We also describe all twodistance tight frames obtained from a given graph. Together with an earlier work by S. Waldron on the equiangular case (Linear Alg. Appl., vol. 41, pp. 22282242, 2009) this completely characterizes twodistance tight frames. As an intermediate result, we obtain a classification of all twodistance 2designs. 1.