Formalization of continuous Fourier transform in verifying applications for dependable cyber-physical systems

Guan, Yong; Zhang, Jie; Shi, Zhiping; Wang, Yi

doi:10.1016/j.sysarc.2020.101707

Cited by 11 publications

(2 citation statements)

References 39 publications

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“…One of the core ideas is to generalize the Fourier transform to an arbitrary locally compact abelian group. The special case of the continuous Fourier transform has been formalized in HOL4 [10], but the abstract Fourier transform has never been formalized. One specific goal could be to formallize Pontryagin duality.…”

Section: Concluding Thoughtsmentioning

confidence: 99%

Formalized Haar Measure

Doorn¹

2021

Preprint

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We describe the formalization of the existence and uniqueness of Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and it has not been formalized in a proof assistant before. We will also discuss the measure theory library in Lean's mathematical library mathlib, and discuss the construction of product measures and the proof of Fubini's theorem for the Bochner integral.

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Section: Concluding Thoughtsmentioning

confidence: 99%

Formalized Haar Measure

Doorn¹

2021

Preprint

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“…Fourier transform, which means that a function satisfying certain conditions can be expressed as a trigonometric function (sine and/or cosine function) or a linear combination of their integrals [24]. In different research fields, Fourier transform has many different variant forms, such as con-tinuous Fourier transform and discrete Fourier transform.…”

Section: A Fourier Transform Of Imagesmentioning

confidence: 99%

A Fast Fusion Method for Visible and Infrared Images Using Fourier Transform and Difference Minimization

2020

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Images of different modalities play important roles in the fields of military, navigation, and target detection, including visible and infrared images. Existing fusion methods can reach relatively good fusion effects, but often make the processing speed slow. To achieve the purpose of faster fusion, this paper proposes a fast fusion of visible and infrared images (FFVI) based on Fourier transform and difference minimization. First, both visible and infrared images are transformed using Fourier transform, and the contour information of infrared image is extracted out at the same time. Then the two transformed images are added together to obtain a composite image, the inverse Fourier transform and the grayscale normalization are performed on the composite transformation image, which will generate an image that reflects the target features. Finally, the resulting image is reconstructed by fusing visible and the generated images according to the principle of "difference minimization". The experimental results show that the novel method FFVI proposed in this paper can attain remarkable fusion effects in terms of the subjective qualitative and the objective quantitative analysis. In addition, this method outperforms several representative image fusion algorithms in the computational speed, which will be beneficial to practical application.

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