Diffusion models are now widely used in Bayesian inverse problems in imaging as priors, where latent diffusion models are often used for larger scale problems to keep the computational complexity and model-size manageable. Unfortunately, the auto-encoder based compression results in loss of spatial detail. In addition, the optimization is converted to a non-linear problem. In this paper, we introduce a posterior sampling algorithm customized for the pyramidal/cascaded architecture, which relies on a coarse to fine hierarchical strategy to generate images in the pixel domain. We present CLIMB-Flow which alternates between three steps: an end-point estimation from the current coarse and noisy image, data-consistent update of the clean image, and re-noising it back to the level the network expects. Together these steps sample the posterior at that scale using an approximate Gibbs sampling from two conditional distributions. Experiments on ImageNet, CelebA, AFHQ and fastMRI span inpainting, deblurring, super-resolution and accelerated MRI, with PSNR gains of 1.37-7.66 dB over the strongest competing method on CelebA and pixel-domain reconstruction up to 512x512.

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