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arXiv 1902.00176cs.CVcs.DM
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基于格林函数卷积的快速最优Laplacian解算器

Fast and Optimal Laplacian Solver for Gradient-Domain Image Editing using Green Function Convolution

Dominique Beaini, Sofiane Achiche, Fabrice Nonez, Olivier Brochu Dufour, Cédric Leblond-Ménard, Mahdis Asaadi, Maxime Raison

摘要:

在计算机视觉中,图像的梯度和拉普拉斯算子用于不同的应用,例如边缘检测、特征提取和无缝图像克隆。计算图像的梯度很简单,因为大多数计算机视觉工具箱中都有数值导数。然而,反向问题更困难,因为从其梯度计算图像需要求解拉普拉斯方程,也称为泊松方程。目前的离散方法要么速度慢,要么需要大量的并行计算。本文的目的是提出一种新的快速和鲁棒的方法来解决图像梯度或拉普拉斯算子的最小误差,可用于梯度域编辑。通过使用基于数值格林函数的单个卷积,整个过程可以更快,更直接地使用不同的计算机视觉库来实现。它也可以使用快速傅立叶变换在GPU上进行优化,并且可以很容易地推广到n维图像。测试结果表明,对于分辨率为801x1200的图像,建议的GFC可以解决100拉普拉斯并行在1.0毫秒ms左右,这是数量级快于我们最近的竞争对手,需要294毫秒为一个单一的图像。此外,我们从数学上证明和经验证明,所提出的方法是梯度域编辑的最小误差求解器。所开发的方法也验证了泊松混合,梯度去除,并建议梯度域合并GDM的例子。最后,我们提出了如何在未来的卷积神经网络CNN工作中利用GDM。

英文摘要:

In computer vision, the gradient and Laplacian of an image are used in different applications, such as edge detection, feature extraction, and seamless image cloning. Computing the gradient of an image is straightforward since numerical derivatives are available in most computer vision toolboxes. However, the reverse problem is more difficult, since computing an image from its gradient requires to solve the Laplacian equation, also called Poisson equation. Current discrete methods are either slow or require heavy parallel computing. The objective of this paper is to present a novel fast and robust method of solving the image gradient or Laplacian with minimal error, which can be used for gradient domain editing. By using a single convolution based on a numerical Green's function, the whole process is faster and straightforward to implement with different computer vision libraries. It can also be optimized on a GPU using fast Fourier transforms and can easily be generalized for an n dimension image. The tests show that, for images of resolution 801x1200, the proposed GFC can solve 100 Laplacian in parallel in around 1.0 milliseconds ms. This is orders of magnitude faster than our nearest competitor which requires 294ms for a single image. Furthermore, we prove mathematically and demonstrate empirically that the proposed method is the least error solver for gradient domain editing. The developed method is also validated with examples of Poisson blending, gradient removal, and the proposed gradient domain merging GDM. Finally, we present how the GDM can be leveraged in future works for convolutional neural networks CNN.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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