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– A geometric perspective is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and t-designs.

Unimodular waveforms x are constructed on the integers with the property that the autocorrelation of x is one at the origin and zero elsewhere. There are three different constructions: exponentials of the form e 2πin α θ , sequences taken from roots of unity, and sequences constructed from the elements of real Hadamard matrices. The first is expected and… (More)

We explore the stability of image reconstruction algorithms under deterministic compressed sensing. Recently, we have proposed [1-3] deterministic compressed sensing algorithms for 2D images. These algorithms are suitable when Daube-chies wavelets are used as the sparsifying basis. In the initial work, we have shown that the algorithms perform well for… (More)

A recently proposed approach for compressed sensing, or compressive sampling, with deterministic measurement matrices made of chirps is applied to images that possess varying degrees of sparsity in their wavelet representations. The " fast reconstruction " algorithm enabled by this deterministic sampling scheme as developed by Applebaum et al. [1] produces… (More)

- John J. Benedetto, Somantika Datta, Anna Graeber

Bounded codes or waveforms are constructed whose autocorrelation is the inverse Fourier transform of certain positive functions. For the positive function F ≡ 1 the corresponding unimodular waveform of infinite length, whose autocorrelation is the inverse Fourier transform of F, is constructed using real Hadamard matrices. This waveform has a… (More)

An image reconstruction algorithm using compressed sensing (CS) with deterministic matrices of second-order Reed-Muller (RM) sequences is introduced. The 1D algorithm of Howard et al. using CS with RM sequences suffers significant loss in speed and accuracy when the degree of sparsity is not high, making it inviable for 2D signals. This paper describes an… (More)

Frames have become standard tools in signal processing due to their robustness to transmission errors and their resilience to noise. Equiangular tight frames (ETFs) are particularly useful and have been shown to be optimal for transmission under a certain number of erasures. Unfortunately, ETFs do not exist in many cases and are hard to construct when they… (More)

- Somantika Datta
- 2014

The goal is to construct random frames and study properties of such frames. Starting with the construction of unimodular random sequences whose expected autocorrelations can be made arbitrarily low outside the origin, these random sequences are used to construct frames for ℂ. Using recent theory of non-asymptotic analysis of random matrices, the eigenvalue… (More)

This article gives a procedure to convert a frame which is not a tight frame into a Parseval frame for the same space, with the requirement that each element in the resulting Parseval frame can be explicitly written as a linear combination of the elements in the original frame. Several examples are considered, such as a Fourier frame on a spiral. The… (More)