On the discrete linear L1 approximation and L1 solutions of overdetermined linear equations

Abdelmalek, Nabih N.

doi:10.1016/0021-9045(74)90037-9

Cited by 51 publications

(24 citation statements)

References 8 publications

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“…6 Both share the reconstruction using the l1 minimization for solving the under determined linear system. 7 Compression and reconstruction can be separated operations without sharing the same matrix or even the same compression-reconstruction algorithm. To achieve this, suitable orthogonal base functions must be known as a priori knowledge.…”

Section: Fragmented L1-norm Transformmentioning

confidence: 99%

Portability of the fragmented l1-norm transform for massively parallel processing

Scholz

Rosenberger

Notni

2019

Photonics and Education in Measurement Science 2019

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White light interferometry is a major optical non-contact and therefore nondestructive testing method for nanostructures and surface reconstruction. By the reason of scanning, the data throughput is high and the resulting data stack can exceed gigabytes of raw data. Effective data compression was realized in an FPGA early in the signaling cascade. On the one hand this can significantly boost the achievable data throughput from the sensor, on the other hand the compression results in fragmented raw data with non-equidistant sampling steps and is therefore incompatible with FFT based reconstruction algorithms. In order to face this issue the fragmented l1norm transform (flot) was developed. The flot reconstruction algorithm is a symbiosis of the l1-norm known from compressive sensing and additionally the wavelet-transform. In contrast to the traditional wavelet-transform the flot algorithm has no dependence on FFT and can quickly handle non-equidistant sampled data. Raw data is heavily independent between pixels in white light interferometry by design. Therefore, implementing the reconstruction algorithm on massively parallel hardware is promising. In the last decade this usually meant data processing on GPUs. Nowadays alternatives in the form of affordable CPU clusters or easy to program FPGAs gain importance. OpenCL is a framework to accelerate highly parallel problems on all of the three platforms. In this paper the implementation of the flot algorithm in OpenCL will be explained, compared by speed and power consumption and categorized for suitable use-cases.

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Section: Fragmented L1-norm Transformmentioning

confidence: 99%

Portability of the fragmented l1-norm transform for massively parallel processing

Scholz

Rosenberger

Notni

2019

Photonics and Education in Measurement Science 2019

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“…In [1], a dual simplex algorithm for solving problem (4) is given, where no artificial variables are used. It was also shown that the algorithm in [1] is completely equivalent to a modified version of a method due to Usow [9] for solving the discrete linear L. approximation problem. Except that one iteration in the latter is equivalent to one or more iterations in the former.…”

Section: =1mentioning

confidence: 99%

An efficient method for the discrete linear 𝐿₁ approximation problem

Abdelmalek¹

1975

Math. Comp.

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An improved dual simplex algorithm for the solution of the discrete linear Lj approximation problem is described. In this algorithm certain intermediate iterations are skipped. This method is comparable with an improved simplex method due to Barrodale and Roberts, in both speed and number of iterations. It also has the advantage that in case of ill-conditioned problems, the basis matrix can lend itself to triangular factorization and can thus ensure a stable solution. Numerical results are given.

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“…(1) and (2) was demonstrated by Chames, Cooper and Ferguson [8], it has generally been agreed that linear programming is computationally the most efficient method for obtaining an optimal a value. Wagner [15] showed that the linear programming dual of (2) can be solved with simple upper bounding techniques which requires a working basis of size m by m. This would be as opposed to the « by « basis if (1) were to be solved with a standard primal simplex algorithm. Generally, m is substantially less than « and, thus, Wagner's approach was considered the most efficient for some time.…”

mentioning

confidence: 99%

“…Abdelmalek [2] also presents a special purpose linear programming algorithm to solve (1). He employs a transformation on the dual of (2) and solves it with a modification of the dual simplex algorithm for bounded variables [14].…”

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confidence: 99%

Two linear programming algorithms for the linear discrete 𝐿₁ norm problem

Armstrong¹,

Godfrey²

1979

Math. Comp.

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Computational studies by several authors have indicated that linear programming is currently the most efficient procedure for obtaining L, norm estimates for a discrete linear problem. However, there are several linear programming algorithms, and the "best" approach may depend on the problem's structure (e.g., sparsity, triangularity, stability). In this paper we shall compare two published simplex algorithms, one referred to as primal and the other referred to as dual, and show that they are conceptually equivalent.

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On the discrete linear L1 approximation and L1 solutions of overdetermined linear equations

Cited by 51 publications

References 8 publications

Portability of the fragmented l1-norm transform for massively parallel processing

Portability of the fragmented l1-norm transform for massively parallel processing

An efficient method for the discrete linear 𝐿₁ approximation problem

Two linear programming algorithms for the linear discrete 𝐿₁ norm problem

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