Kaczmarz holography: holographic projection in the Fraunhofer region

Guendy, Miguel De V. El; Mouthaan, Ralf; Wetherfield, Benjamin; Kadis, Andrew; Christopher, Peter J.; Kaczorowski, Andrzej; Wilkinson, Timothy D.

doi:10.1117/1.oe.60.10.103101

Cited by 2 publications

(6 citation statements)

References 24 publications

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“…From the work previously conducted on Kaczmarz holographic projections [12], it is noticeable the image quality for Kaczmarz outperforms Gerchberg-Saxton in 10 iterations. However, the main drawbacks of implementing Kaczmarz are its slow running time and ability to solely be adopted on smaller images 128x128 pixels since it relies on discrete Fourier transform (DFT) matrices.…”

Section: Algorithm Replay Field Image Qualitymentioning

confidence: 96%

“…Our earlier investigation, on Kaczmarz holographic projections in the Fraunhofer region [12] revealed that while very high quality replay fields are generated, it is still comparatively slower than the more widely used Gerchberg-Saxton. The Kaczmarz algorithm iterates the Ax component in a linear equation on a row-by-row basis and therefore explains the slower running of this phase retrieval method.…”

Section: Fig 1 Diffraction and Replay Fields Coordinate Systemmentioning

confidence: 99%

“…More recently, the Kaczmarz algorithm was shown to be effective in computer-generated holography, projecting holographic images in the Fraunhofer region [12]. It is therefore very suitable to resolving the phase retrieval problem associated with holographic projections.…”

Section: Algebraic Reconstruction Techniquesmentioning

confidence: 99%

“…Our previous work on using the Kaczmarz algorithm to project replay fields in the Fraunhofer region [12] demonstrates that while high image quality is obtainable, its row-action approach makes it much slower than alternative methods that compute the product of Ax simultaneously. Furthermore, iterative algorithms such as Gerchberg-Saxton use Fast Fourier Transforms (FFTs), a significantly more time-effective approach to resolve this phase retrieval problem, O(NlogN) as opposed to O(N 2 ).…”

Section: Algebraic Reconstruction Techniquesmentioning

confidence: 99%

“…As well as only being capable of reconstructing replay fields of smaller dimensions, the Kaczmarz algorithm is also slower than fast Fourier transform (FFT) based algorithms such as the Gerchberg-Saxton, as seen in our previous work [12]. As a result, alternative algorithms that compute the full image simultaneously rather than relying on row-by-row iteration, as seen by Kaczmarz, were evaluated.…”

Section: Algorithm Running Timesmentioning

confidence: 99%

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Cimmino simultaneously iterative holographic projection

et al. 2022

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The Cimmino algorithm is an iterative algorithm used for solving linear equations (Ax = b) and is part of a wider family of algebraic reconstruction algorithms. Despite being used in computed tomography and digital signal processing, we present an approach for using it in holographic projections in the Fraunhofer region. Following the work carried out by our group on Kaczmarz holography, an earlier algebraic linear equation solver, we compare the performance and speed of Cimmino against this algorithm. Three versions of the Cimmino algorithm, Cimmino full, Cimmino eye, and Cimmino FFT, are evaluated and compared to both Kaczmarz and Gerchberg-Saxton. Interestingly, the derivation and simplification of Kaczmarz and Cimmino leads to the input-output algorithm originally derived by Fienup and while the solution is well-known, the link between these algorithms is not and despite being around since the 1930s, it was originally used to solve linear equations and not phase retrieval. While the Cimmino full and Cimmino eye reflect fairly poorly against Cimmino FFT and Kaczmarz in terms of image quality and speed, the Cimmino FFT delivers better results in terms of image quality than all algorithms considered. It is significantly faster than Kaczmarz, being only a few milliseconds slower than Gerchberg-Saxton.

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Section: Algorithm Replay Field Image Qualitymentioning

confidence: 96%

Section: Fig 1 Diffraction and Replay Fields Coordinate Systemmentioning

confidence: 99%

Section: Algebraic Reconstruction Techniquesmentioning

confidence: 99%

Section: Algebraic Reconstruction Techniquesmentioning

confidence: 99%

Section: Algorithm Running Timesmentioning

confidence: 99%

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Cimmino simultaneously iterative holographic projection

et al. 2022

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Holographic phase retrieval via Wirtinger flow: Cartesian form with auxiliary amplitude

Uchiyama,

Tsutake,

Takahashi

et al. 2024

Opt. Express

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We propose a new gradient method for holography, where a phase-only hologram is parameterized by not only the phase but also amplitude. The key idea of our approach is the formulation of a phase-only hologram using an auxiliary amplitude. We optimize the parameters using the so-called Wirtinger flow algorithm in the Cartesian domain, which is a gradient method defined on the basis of the Wirtinger calculus. At the early stage of optimization, each element of the hologram exists inside a complex circle, and it can take a large gradient while diverging from the origin. This characteristic contributes to accelerating the gradient descent. Meanwhile, at the final stage of optimization, each element evolves along a complex circle, similar to previous state-of-the-art gradient methods. The experimental results demonstrate that our method outperforms previous methods, primarily due to the optimization of the amplitude.

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Kaczmarz holography: holographic projection in the Fraunhofer region

Cited by 2 publications

References 24 publications

Cimmino simultaneously iterative holographic projection

Cimmino simultaneously iterative holographic projection

Holographic phase retrieval via Wirtinger flow: Cartesian form with auxiliary amplitude

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