76 research outputs found
Plug-and-Play Methods Provably Converge with Properly Trained Denoisers
Plug-and-play (PnP) is a non-convex framework that integrates modern
denoising priors, such as BM3D or deep learning-based denoisers, into ADMM or
other proximal algorithms. An advantage of PnP is that one can use pre-trained
denoisers when there is not sufficient data for end-to-end training. Although
PnP has been recently studied extensively with great empirical success,
theoretical analysis addressing even the most basic question of convergence has
been insufficient. In this paper, we theoretically establish convergence of
PnP-FBS and PnP-ADMM, without using diminishing stepsizes, under a certain
Lipschitz condition on the denoisers. We then propose real spectral
normalization, a technique for training deep learning-based denoisers to
satisfy the proposed Lipschitz condition. Finally, we present experimental
results validating the theory.Comment: Published in the International Conference on Machine Learning, 201
On the Construction of Averaged Deep Denoisers for Image Regularization
Plug-and-Play (PnP) and Regularization by Denoising (RED) are recent
paradigms for image reconstruction that can leverage the power of modern
denoisers for image regularization. In particular, these algorithms have been
shown to deliver state-of-the-art reconstructions using CNN denoisers. Since
the regularization is performed in an ad-hoc manner in PnP and RED,
understanding their convergence has been an active research area. Recently, it
was observed in many works that iterate convergence of PnP and RED can be
guaranteed if the denoiser is averaged or nonexpansive. However, integrating
nonexpansivity with gradient-based learning is a challenging task -- checking
nonexpansivity is known to be computationally intractable. Using numerical
examples, we show that existing CNN denoisers violate the nonexpansive property
and can cause the PnP iterations to diverge. In fact, algorithms for training
nonexpansive denoisers either cannot guarantee nonexpansivity of the final
denoiser or are computationally intensive. In this work, we propose to
construct averaged (contractive) image denoisers by unfolding ISTA and ADMM
iterations applied to wavelet denoising and demonstrate that their
regularization capacity for PnP and RED can be matched with CNN denoisers. To
the best of our knowledge, this is the first work to propose a simple framework
for training provably averaged (contractive) denoisers using unfolding
networks
Learned reconstruction methods with convergence guarantees
In recent years, deep learning has achieved remarkable empirical success for image reconstruction. This has catalyzed an ongoing quest for precise characterization of correctness and reliability of data-driven methods in critical use-cases, for instance in medical imaging. Notwithstanding the excellent performance and efficacy of deep learning-based methods, concerns have been raised regarding their stability, or lack thereof, with serious practical implications. Significant advances have been made in recent years to unravel the inner workings of data-driven image recovery methods, challenging their widely perceived black-box nature. In this article, we will specify relevant notions of convergence for data-driven image reconstruction, which will form the basis of a survey of learned methods with mathematically rigorous reconstruction guarantees. An example that is highlighted is the role of ICNN, offering the possibility to combine the power of deep learning with classical convex regularization theory for devising methods that are provably convergent. This survey article is aimed at both methodological researchers seeking to advance the frontiers of our understanding of data-driven image reconstruction methods as well as practitioners, by providing an accessible description of useful convergence concepts and by placing some of the existing empirical practices on a solid mathematical foundation
Provably Convergent Plug-and-Play Quasi-Newton Methods
Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as non-expansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2--8x faster convergence as compared to other provable PnP methods with similar reconstruction quality
Provable Convergence of Plug-and-Play Priors with MMSE denoisers
Plug-and-play priors (PnP) is a methodology for regularized image
reconstruction that specifies the prior through an image denoiser. While PnP
algorithms are well understood for denoisers performing maximum a posteriori
probability (MAP) estimation, they have not been analyzed for the minimum mean
squared error (MMSE) denoisers. This letter addresses this gap by establishing
the first theoretical convergence result for the iterative
shrinkage/thresholding algorithm (ISTA) variant of PnP for MMSE denoisers. We
show that the iterates produced by PnP-ISTA with an MMSE denoiser converge to a
stationary point of some global cost function. We validate our analysis on
sparse signal recovery in compressive sensing by comparing two types of
denoisers, namely the exact MMSE denoiser and the approximate MMSE denoiser
obtained by training a deep neural net
Provably Convergent Plug-and-Play Quasi-Newton Methods
Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as non-expansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2--8x faster convergence as compared to other provable PnP methods with similar reconstruction quality
SIMBA: scalable inversion in optical tomography using deep denoising priors
Two features desired in a three-dimensional (3D) optical tomographic image reconstruction algorithm are the ability to reduce imaging artifacts and to do fast processing of large data volumes. Traditional iterative inversion algorithms are impractical in this context due to their heavy computational and memory requirements. We propose and experimentally validate a novel scalable iterative mini-batch algorithm (SIMBA) for fast and high-quality optical tomographic imaging. SIMBA enables highquality imaging by combining two complementary information sources: the physics of the imaging system characterized by its forward model and the imaging prior characterized by a denoising deep neural net. SIMBA easily scales to very large 3D tomographic datasets by processing only a small subset of measurements at each iteration. We establish the theoretical fixedpoint convergence of SIMBA under nonexpansive denoisers for convex data-fidelity terms. We validate SIMBA on both simulated and experimentally collected intensity diffraction tomography (IDT) datasets. Our results show that SIMBA can significantly reduce the computational burden of 3D image formation without sacrificing the imaging quality.https://arxiv.org/abs/1911.13241First author draf
Plug-and-Play gradient-based denoisers applied to CT image enhancement
Blur and noise corrupting Computed Tomography (CT) images can hide or distort
small but important details, negatively affecting the diagnosis. In this paper,
we present a novel gradient-based Plug-and-Play algorithm, constructed on the
Half-Quadratic Splitting scheme, and we apply it to restore CT images. In
particular, we consider different schemes encompassing external and internal
denoisers as priors, defined on the image gradient domain. The internal prior
is based on the Total Variation functional. The external denoiser is
implemented by a deep Convolutional Neural Network (CNN) trained on the
gradient domain (and not on the image one, as in state-of-the-art works). We
also prove a general fixed-point convergence theorem under weak assumptions on
both internal and external denoisers. The experiments confirm the effectiveness
of the proposed framework in restoring blurred noisy CT images, both in
simulated and real medical settings. The achieved enhancements in the restored
images are really remarkable, if compared to the results of many
state-of-the-art methods.Comment: Submitted to journa
Provably Convergent Plug-and-Play Quasi-Newton Methods
Plug-and-Play (PnP) methods are a class of efficient iterative methods that
aim to combine data fidelity terms and deep denoisers using classical
optimization algorithms, such as ISTA or ADMM. Provable PnP methods are a
subclass of PnP methods with convergence guarantees, such as fixed point
convergence or convergence to critical points of some energy function. Many
existing provable PnP methods impose heavy restrictions on the denoiser or
fidelity function, such as non-expansiveness or strict convexity, respectively.
In this work, we propose a novel algorithmic approach incorporating
quasi-Newton steps into a provable PnP framework based on proximal denoisers,
resulting in greatly accelerated convergence while retaining light assumptions
on the denoiser. By characterizing the denoiser as the proximal operator of a
weakly convex function, we show that the fixed points of the proposed
quasi-Newton PnP algorithm are critical points of a weakly convex function.
Numerical experiments on image deblurring and super-resolution demonstrate
significantly faster convergence as compared to other provable PnP methods with
similar convergence results
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