We consider solving ill-posed imaging inverse problems without access to an image prior or ground-truth examples. An overarching challenge in these inverse problems is that an infinite number of images, including many that are implausible, are consistent with the observed measurements. Thus, image priors are required to reduce the space of possible solutions to more desirable reconstructions. However, in many applications it is difficult or potentially impossible to obtain example images to construct an image prior. Hence inaccurate priors are often used, which inevitably result in biased solutions. Rather than solving an inverse problem using priors that encode the spatial structure of any one image, we propose to solve a set of inverse problems jointly by incorporating prior constraints on the collective structure of the underlying images. The key assumption of our work is that the underlying images we aim to reconstruct share common, low-dimensional structure. We show that such a set of inverse problems can be solved simultaneously without the use of a spatial image prior by instead inferring a shared image generator with a low-dimensional latent space. The parameters of the generator and latent embeddings are found by maximizing a proxy for the Evidence Lower Bound (ELBO). Once identified, the generator and latent embeddings can be combined to provide reconstructed images for each inverse problem. The framework we propose can handle general forward model corruptions, and we show that measurements derived from only a small number of ground-truth images ($\leqslant 150$) are sufficient for image reconstruction. We demonstrate our approach on a variety of convex and non-convex inverse problems, including denoising, phase retrieval, and black hole video reconstruction.

翻译：我们考虑在没有图像先验或真实样本的情况下求解病态成像逆问题。这类逆问题的核心挑战在于：存在无穷多幅图像（包括大量不可信的图像）均与观测测量值一致。因此，需要图像先验将可能的解空间缩减至更理想的复原结果。然而在许多应用中，获取用于构建图像先验的示例图像极其困难甚至不可能，这使得不准确的先验常被采用，从而导致不可避免的偏差解。我们提出联合求解一组逆问题的方法，通过约束底层图像的集体结构而非单一图像的空间结构来引入先验信息。本研究的关键假设是：待重建的底层图像共享共同的低维结构。我们证明，通过推断具有低维隐空间的共享图像生成器，无需空间图像先验即可同时求解这类逆问题。该生成器的参数和隐编码通过最大化证据下界（ELBO）的代理函数获得。确定后，生成器与隐编码可组合为每个逆问题提供重建图像。所提框架能处理通用的前向模型破坏，实验表明仅需少量真实图像（≤150幅）的测量值即可完成图像重建。我们在去噪、相位恢复和黑洞视频重建等凸/非凸逆问题上验证了该方法。