  We re-examine the problem of reconstructing a high-dimensional signal from a
small set of linear measurements, in combination with image prior from a
diffusion probabilistic model. Well-established methods for optimizing such
measurements include principal component analysis (PCA), independent component
analysis (ICA) and compressed sensing (CS), all of which rely on axis- or
subspace-aligned statistical characterization. But many naturally occurring
signals, including photographic images, contain richer statistical structure.
To exploit such structure, we introduce a general method for obtaining an
optimized set of linear measurements, assuming a Bayesian inverse solution that
leverages the prior implicit in a neural network trained to perform denoising.
We demonstrate that these measurements are distinct from those of PCA and CS,
with significant improvements in minimizing squared reconstruction error. In
addition, we show that optimizing the measurements for the SSIM perceptual loss
leads to perceptually improved reconstruction. Our results highlight the
importance of incorporating the specific statistical regularities of natural
signals when designing effective linear measurements.
