The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with an easy-to-handle numerical scheme. So far, existing stabilisation strategies in the literature require sophisticated numerics to solve the underlying initial value problem. We derive a class of space-discrete one-dimensional backward diffusion as gradient descent of energies where we gain stability by imposing range constraints. Interestingly, these energies are even convex. Furthermore, we establish a comprehensive theory for the time-continuous evolution and we show that stability carries over to a simple explicit time discretisation of our model. Finally, we confirm the stability and usefulness of our technique in experiments in which we enhance the contrast of digital greyscale and colour images.

中文翻译：

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稳定的向后扩散模型，可将凸能量最小化

已知向后扩散的逆问题是不适定的并且高度不稳定。向后扩散过程自然出现在图像增强和去模糊应用中。因此，非常需要建立一种向后扩散模型，该模型实现了可以与易于处理的数值方案结合使用的智能稳定方法。迄今为止，文献中现有的稳定策略需要复杂的数值来解决潜在的初始值问题。我们导出一类空间离散的一维向后扩散作为能量的梯度下降，在其中我们通过施加范围约束来获得稳定性。有趣的是，这些能量甚至是凸的。此外，我们建立了一个关于时间连续演化的综合理论，并且我们表明稳定性可以延续到我们模型的简单显式时间离散化。最后，我们在增强数字灰度和彩色图像对比度的实验中证实了我们技术的稳定性和实用性。