The time and resource costs of obtaining pristine training data in machine learning are high. In signal recovery tasks, Noise2Noise proposed by Lehtinen et al. aims to reduce the data cost by learning the regression over two noisy measurements corresponding to the same latent variable. Close examination shows that Lehtinen’s original derivation requires a strictly "frequentist" conjecture—i.e. deterministic treatment of the latent variable. This paper presents a Bayesian counter-piece to the original Noise2Noise formulation, with a fully stochastic treatment of the latent variable. We propose to extend Noise2Noise further to unbiased estimate of risk (Noise2Noise2MSE), covariance analysis (Noise2Noise2Cov), and minimum mean squared error estimate (Noise2Noise2MMSE), all derived from pairs of noisy measurements only.

We propose “convolutional distance transform”- efficient implementations of distance transform. Specifically, we leverage approximate minimum functions to rewrite the distance transform in terms of convolution operators. Thanks to the fast Fourier transform, the proposed convolutional distance transforms have O(N log N) complexity, where N is the total number of pixels. The proposed acceleration technique is “distance metric agnostic.” In the special case that the distance function is a p-norm, the distance transform can be further reduced to separable convolution filters; and for Euclidean norm, we achieve O(N) using constant-time Gaussian filtering.

Bilateral filter is a popular edge-preserving nonlinear filter. It prevents smoothing across object boundaries by limiting the contribution of pixels dissimilar in appearance. Despite its usefulness, the complexity of the filter makes it unattractive for high dimensional data and large window size. We develop a novel technique to accelerate this filter so that the execution time is nearly constant with respect to the color dimensionality and the window size-a hyperspectral image can be processed almost as fast as a grayscale image.

We propose stochastic bilateral filter (SBF) and stochastic non-local means (SNLM), efficient randomized processes that agree with conventional bilateral filter (BF) and non-local means (NLM) on average, respectively. By Monte-Carlo, we repeat this process a few times with different random instantiations so that they can be averaged to attain the correct BF/NLM output. The computational bottleneck of the SBF and SNLM are constant with respect to the window size and the color dimension of the edge image, meaning the execution times for color and hyperspectral images are nearly the same as for the grayscale images. In addition, for SNLM, the complexity is constant with respect to the block size. The proposed stochastic filter implementations are considerably faster than the conventional and existing "fast" implementations for high dimensional image data.

We propose a stochastic bilateral filter (SBF) - fast image filtering aimed at processing high dimensional images (such as color and hy-perspectral images). SBF is comprised of an efficient randomized process, where it agrees with conventional bilateral filter (BF) on average. By Monte-Carlo, we repeat this process a few times with different random instantiations so that they can be averaged to attain the correct BF output. The computational bottleneck of the SBF is constant with respect to the color dimension, meaning the complexity for filtering hyperspectral images is nearly the same as the grayscale images. It is considerably faster than the conventional and existing “fast” bilateral filter implementations.