## Direct fast method for time-limited signal reconstruction

Applied Optics, Vol. 45, Issue 13, pp. 3111-3126 (2006)

http://dx.doi.org/10.1364/AO.45.003111

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### Abstract

We consider reconstruction of signals by a direct method for the solution of the discrete Fourier system. We note that the reconstruction of a time-limited signal can be simply realized by using only either the real part or the imaginary part of the discrete Fourier transform (DFT) matrix. Therefore, based on the study of the special structure of the real and imaginary parts of the discrete Fourier matrix, we propose a fast direct method for the signal reconstruction problem, which utilizes the numerically truncated singular value decomposition. The method enables us to recover the original signal in a stable way from the frequency information, which may be corrupted by noise and∕or some missing data. The classical inverse Fourier transform cannot be applied directly in the latter situation. The pivotal point of the reconstruction is the explicit computation of the singular value decomposition of the real part of the DFT for any order. Numerical experiments for 1D and 2D signal reconstruction and image restoration are given.

© 2006 Optical Society of America

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(100.3010) Image processing : Image reconstruction techniques

(100.3020) Image processing : Image reconstruction-restoration

(100.3190) Image processing : Inverse problems

**ToC Category:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: February 22, 2005

Revised Manuscript: July 8, 2005

Manuscript Accepted: September 12, 2005

**Citation**

Yanfei Wang, Zaiwen Wen, Zuhair Nashed, and Qiyu Sun, "Direct fast method for time-limited signal reconstruction," Appl. Opt. **45**, 3111-3126 (2006)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-13-3111

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