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This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.

Notas/Comentarios de Juan Ignacio Godino:
Los autores proponen un tipo de muestreo que establece que las señales e imágenes de carácter disperso se pueden reconstruir a partir de lo que previamente se creía que era información incompleta (por debajo de lo establecido en el teorema de Nyquist-Shannon). Este tipo de muestreo ha dado lugar a la técnica que hoy se conoce como Detección Comprimida (Compressed Sensing- CS) y que ha encontrado aplicaciones en una gran cantidad de dominios, como el enfoque y la eliminación de ruido en imágenes, la reconstrucción de imágenes en resonancia magnética, el desarrollo de la cámara de un solo píxel, y muchas otras. El término de CS se acuña en el artículo D. L. Donoho; Compressed sensing. IEEE Transactions on Information Theory (Volume: 52, Issue: 4, April 2006, Pages: 1289-1306).

Especificaciones

  • Autor/es: Emmanuel J. Candès, Justin Romberg, Terence Tao.
  • Fecha: 2006-02
  • Publicado en: IEEE Transactions on Information Theory (Volume: 52, Issue: 2, Feb. 2006, Pages: 489-509).
  • Idioma: Inglés
  • Formato: PDF
  • Contribución: Juan Ignacio Godino Llorente.
  • Palabras clave: Imágenes, Matemáticas, Proceso de señal
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