Characterizing statistical properties of solutions of inverse problems is essential for decision making. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for most realistic imaging applications in the clinic. We introduce two novel deep learning based methods for solving large-scale inverse problems using Bayesian inversion: a sampling based method using a WGAN with a novel mini-discriminator and a direct approach that trains a neural network using a novel loss function. The performance of both methods is demonstrated on image reconstruction in ultra low dose 3D helical CT. We compute the posterior mean and standard deviation of the 3D images followed by a hypothesis test to assess whether a "dark spot" in the liver of a cancer stricken patient is present. Both methods are computationally efficient and our evaluation shows very promising performance that clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.
Regarding sampling from a posterior, conditional generative models @cite_63 @cite_34 have been widely used in the machine learning literature for this purpose. Typical use cases is to sample from a posterior where an image is conditioned on a text, like the bird is yellow'' @cite_24 @cite_18 , but also for simple applications in imaging, including image super-resolution and in-painting @cite_4 @cite_1 @cite_43 . These approaches do not consider sampling from the posterior for more elaborate inverse problems that involve a physics driven data likelihood. An approach in this direction is presented in @cite_50 where variational auto-encoders are used to sample from the posterior of possible segmentations (model parameter) given CT images (data).
@cite_18 @cite_4 @cite_1 @cite_24 @cite_43 @cite_63 @cite_50 @cite_34
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In this paper we present a Bayesian framework for parsing images into their constituent visual patterns. The parsing algorithm optimizes the posterior probability and outputs a scene representation as a "parsing graph", in a spirit similar to parsing sentences in speech and natural language. The algorithm constructs the parsing graph and re-configures it dynamically using a set of moves, which are mostly reversible Markov chain jumps. This computational framework integrates two popular inference approaches--generative (top-down) methods and discriminative (bottom-up) methods. The former formulates the posterior probability in terms of generative models for images defined by likelihood functions and priors. The latter computes discriminative probabilities based on a sequence (cascade) of bottom-up tests filters. In our Markov chain algorithm design, the posterior probability, defined by the generative models, is the invariant (target) probability for the Markov chain, and the discriminative probabilities are used to construct proposal probabilities to drive the Markov chain. Intuitively, the bottom-up discriminative probabilities activate top-down generative models. In this paper, we focus on two types of visual patterns--generic visual patterns, such as texture and shading, and object patterns including human faces and text. These types of patterns compete and cooperate to explain the image and so image parsing unifies image segmentation, object detection, and recognition (if we use generic visual patterns only then image parsing will correspond to image segmentation (Tu and Zhu, 2002. IEEE Trans. PAMI, 24(5):657--673). We illustrate our algorithm on natural images of complex city scenes and show examples where image segmentation can be improved by allowing object specific knowledge to disambiguate low-level segmentation cues, and conversely where object detection can be improved by using generic visual patterns to explain away shadows and occlusions. We show that a generative random field model, which we call generative ConvNet, can be derived from the commonly used discriminative ConvNet, by assuming a ConvNet for multicategory classification and assuming one of the categories is a base category generated by a reference distribution. If we further assume that the non-linearity in the ConvNet is Rectified Linear Unit (ReLU) and the reference distribution is Gaussian white noise, then we obtain a generative ConvNet model that is unique among energy-based models: The model is piecewise Gaussian, and the means of the Gaussian pieces are defined by an auto-encoder, where the filters in the bottom-up encoding become the basis functions in the top-down decoding, and the binary activation variables detected by the filters in the bottom-up convolution process become the coefficients of the basis functions in the top-down deconvolution process. The Langevin dynamics for sampling the generative ConvNet is driven by the reconstruction error of this autoencoder. The contrastive divergence learning of the generative ConvNet reconstructs the training images by the auto-encoder. The maximum likelihood learning algorithm can synthesize realistic natural image patterns. We show that a generative random field model, which we call generative ConvNet, can be derived from the commonly used discriminative ConvNet, by assuming a ConvNet for multi-category classification and assuming one of the categories is a base category generated by a reference distribution. If we further assume that the non-linearity in the ConvNet is Rectified Linear Unit (ReLU) and the reference distribution is Gaussian white noise, then we obtain a generative ConvNet model that is unique among energy-based models: The model is piecewise Gaussian, and the means of the Gaussian pieces are defined by an auto-encoder, where the filters in the bottom-up encoding become the basis functions in the top-down decoding, and the binary activation variables detected by the filters in the bottom-up convolution process become the coefficients of the basis functions in the top-down deconvolution process. The Langevin dynamics for sampling the generative ConvNet is driven by the reconstruction error of this auto-encoder. The contrastive divergence learning of the generative ConvNet reconstructs the training images by the auto-encoder. The maximum likelihood learning algorithm can synthesize realistic natural image patterns. We introduce a method to infer a variational approximation to the posterior distribution of solutions in computational imaging inverse problems. Machine learning methods applied to computational imaging have proven very successful, but have so far largely focused on retrieving a single optimal solution for a given task. Such retrieval is arguably an incomplete description of the solution space, as in ill-posed inverse problems there may be many similarly likely reconstructions. We minimise an upper bound on the divergence between our approximate distribution and the true intractable posterior, thereby obtaining a probabilistic description of the solution space in imaging inverse problems with empirical prior. We demonstrate the advantage of our technique in quantitative simulations with the CelebA dataset and common image reconstruction tasks. We then apply our method to two of the currently most challenging problems in experimental optics: imaging through highly scattering media and imaging through multi-modal optical fibres. In both settings we report state of the art reconstructions, while providing new capabilities, such as estimation of error-bars and visualisation of multiple likely reconstructions.