Newton's Iteration for Matrix Inversion, Advances and Extensions

2010 ◽  
pp. 364-381 ◽  
Author(s):  
Victor Y. Pan
Keyword(s):  
Matrix Inversion ◽  
Newton's Iteration

Certification of Algorithm 58: Matrix inversion

1962 ◽  
Vol 5 (8) ◽  
pp. 438-439
Author(s):  
Richard George
Keyword(s):  
Matrix Inversion

scholarly journals Fast Split Bregman Based Deconvolution Algorithm for Airborne Radar Imaging

2020 ◽  
Vol 12 (11) ◽  
pp. 1747 ◽  
Author(s):  
Yin Zhang ◽  
Qiping Zhang ◽  
Yongchao Zhang ◽  
Jifang Pei ◽  
Yulin Huang ◽  
...  
Keyword(s):  
Real Time ◽  
Computing Time ◽  
Real Data ◽  
Radar Imaging ◽  
Matrix Inversion ◽  
Airborne Radar ◽  
Split Bregman ◽  
Regularization Problem ◽  
Azimuth Resolution ◽  
Split Bregman Algorithm

Deconvolution methods can be used to improve the azimuth resolution in airborne radar imaging. Due to the sparsity of targets in airborne radar imaging, an L 1 regularization problem usually needs to be solved. Recently, the Split Bregman algorithm (SBA) has been widely used to solve L 1 regularization problems. However, due to the high computational complexity of matrix inversion, the efficiency of the traditional SBA is low, which seriously restricts its real-time performance in airborne radar imaging. To overcome this disadvantage, a fast split Bregman algorithm (FSBA) is proposed in this paper to achieve real-time imaging with an airborne radar. Firstly, under the regularization framework, the problem of azimuth resolution improvement can be converted into an L 1 regularization problem. Then, the L 1 regularization problem can be solved with the proposed FSBA. By utilizing the low displacement rank features of Toeplitz matrix, the proposed FSBA is able to realize fast matrix inversion by using a Gohberg–Semencul (GS) representation. Through simulated and real data processing experiments, we prove that the proposed FSBA significantly improves the resolution, compared with the Wiener filtering (WF), truncated singular value decomposition (TSVD), Tikhonov regularization (REGU), Richardson–Lucy (RL), iterative adaptive approach (IAA) algorithms. The computational advantage of FSBA increases with the increase of echo dimension. Its computational efficiency is 51 times and 77 times of the traditional SBA, respectively, for echoes with dimensions of 218 × 400 and 400 × 400 , optimizing both the image quality and computing time. In addition, for a specific hardware platform, the proposed FSBA can process echo of greater dimensions than traditional SBA. Furthermore, the proposed FSBA causes little performance degradation, when compared with the traditional SBA.

THE RESOLUTION CORRECTION IN THE SCINTILLATION SPECTROMETRY OF CONTINUOUS X RAYS

1958 ◽  
Vol 36 (12) ◽  
pp. 1624-1633 ◽  
Author(s):  
W. R. Dixon ◽  
J. H. Aitken
Keyword(s):  
Integral Equation ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Analytical Procedure ◽  
Pulse Height ◽  
Matrix Inversion ◽  
Height Distribution ◽  
X Rays ◽  
Smooth Solutions ◽  
Pulse Height Distribution

The problem of making resolution corrections in the scintillation spectrometry of continuous X rays is discussed. Analytical solutions are given to the integral equation which describes the effect of the statistical spread in pulse height. The practical necessity of making some kind of numerical analysis is pointed out. Difficulties with numerical methods arise from the fact that the observed pulse-height distribution cannot be defined precisely. As a result it is possible in practice only to find smooth "solutions". Additional difficulties arise if the numerical method is based on an invalid analytical procedure. For example matrix inversion is of doubtful value in making the resolution correction because there does not appear to be an inverse kernel for the integral equation in question.

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