Unit circle | EPFL Graph Search

In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere. If (x, y) is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x^2 + y^2 = 1. Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed u

The perceptual relevance of balance, evenness, and entropy in musical rhythms

Steffen Alexander Herff

There is an uncountable number of different ways of characterizing almost any given real-world stimulus. This necessitates finding stimulus features that are perceptually relevant - that is, they have distinct and independent effects on the perception and cognition of the stimulus. Here, we provide a theoretical framework for empirically testing the perceptual relevance of stimulus features through their association with recognition, memory bias, and asthetic evaluation. We deploy this framework in the auditory domain to explore the perceptual relevance of three recently developed mathematical characterizations of periodic temporal patterns: balance, evenness, and interonset interval entropy. By modelling recognition responses and liking ratings from 177 participants listening to a total of 1252 different musical rhythms, we obtain very strong evidence that all three features have distinct effects on the memory for, and the liking of, musical rhythms. Interonset interval entropy is a measure of the unpredictability of a rhythm derived from the distribution of its durations. Balance and evenness are both obtained from the discrete Fourier transform (DFT) of periodic patterns represented as points on the unit circle, and we introduce a teleological explanation for their perceptual relevance: the DFT coefficients representing balance and evenness are relatively robust to small random temporal perturbations and hence are coherent in noisy environments. This theory suggests further research to explore the meaning and relevance of robust coefficients such as these to the perception of patterns that are periodic in time and, possibly, space.

ELSEVIER2020

Bit Precision Analysis for Compressed Sensing

Mahdi Cheraghchi Bashi Astaneh, Mohammad Amin Shokrollahi

This paper studies the stability of some reconstruction algorithms for compressed sensing in terms of the bit precision. Considering the fact that practical digital systems deal with discretized signals, we motivate the importance of the total number of accurate bits needed from the measurement outcomes in addition to the number of measurements. It is shown that if one uses a

2k \times n

Vandermonde matrix with roots on the unit circle as the measurement matrix,

O(\ell + k \log \frac{n}{k})

bits of precision per measurement are sufficient to reconstruct a

k

-sparse signal

x \in \R^n

with dynamic range (i.e., the absolute ratio between the largest and the smallest nonzero coefficients) at most

2^\ell

within

\ell

bits of precision, hence identifying its correct support. Finally, we obtain an upper bound on the total number of required bits when the measurement matrix satisfies a restricted isometry property, which is in particular the case for random Fourier and Gaussian matrices. For very sparse signals, the upper bound on the number of required bits for Vandermonde matrices is shown to be better than this general upper bound.

2009

Exact Algorithms for $ L ^{ 1 } $ -TV Regularization of Real-Valued or Circle-Valued Signals

Martin Kurt Storath, Michaël Unser

We consider

L ^{ 1 }

-TV regularization of univariate signals with values on the real line or on the unit circle. While the real data space leads to a convex optimization problem, the problem is nonconvex for circle-valued data. In this paper, we derive exact algorithms for both data spaces. A key ingredient is the reduction of the infinite search spaces to a finite set of configurations, which can be scanned by the Viterbi algorithm. To reduce the computational complexity of the involved tabulations, we extend the technique of distance transforms to nonuniform grids and to the circular data space. In total, the proposed algorithms have complexity

\mathcal{O}(KN)

, where

N

is the length of the signal and

K

is the number of different values in the data set. In particular, the complexity is

\mathcal{O}(N)

for quantized data. It is the first exact algorithm for total variation regularization with circle-valued data, and it is competitive with the state-of-the-art methods for scalar data, assuming that the latter are quantized.

SIAM2016