Efficient Lip-1 Spline Networks for Convergent PnP Image Reconstruction

M. Unser, S. Ducotterd, P. Bohra

Proceedings of the International BASP Frontiers Conference (BASP'23), Villars-sur-Ollon, Swiss Confederation, February 5-10, 2023, pp. 18.

Download PDFDownload PSDownload Global BibTeX

Neural networks are constructed via the composition of simple modules: affine transformations (e.g. convolutions) and pointwise non-linearities. PnP schemes for image reconstruction make use of such convolutional neural networks for the so-called denoising step. The condition for convergence is that the denoising CNN should be non-expansive. It is achieved by imposing a tight Lipchitz-1 constraint on each module; for instance, by using ReLU non-linearities and by spectrally normalizing each linear layer. The downside of this simple normalization approach is that the resulting network generally loses expressivity. Concretely, this means that PnP with current Lip-1 CNN fall short of reaching the best possible (state-of-the-art) image quality, which is price to pay for the consistency and stability of the reconstruction. The approach that is investigated in this work is to replace the ReLU activations by more expressive trainable non-linearities, subject to the Lip-1 constraint. The foundation for our approach is a representer theorem for the design of deep neural networks with Lipchitz-constrained free-form activations subject to TV² regularization to favor "simple" neuronal responses. It states that the global optimum of this constrained functional optimization problem can be achieved with a configuration where each neuron is a linear spline with a small number of adaptive knots (break points). We then describe a B-spline framework for the efficient implementation and training of such Lip-1 spline networks. Finally, we apply our framework to the reconstruction of magnetic resonance images and show that it compares favorably with other existing Lip-1 neuronal architectures.

Related work available on arXiv (2210.16222 [cs.LG], 1802.09210 [cs.LG])

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/unser2304.html,
AUTHOR="Unser, M. and Ducotterd, S. and Bohra, P.",
TITLE="Efficient {Lip-1} Spline Networks for Convergent {PnP} Image
	Reconstruction",
BOOKTITLE="Proceedings of the International {BASP} Frontiers Conference
	({BASP'23})",
YEAR="2023",
editor="",
volume="",
series="",
pages="1",
address="Villars-sur-Ollon, Swiss Confederation",
month="February 5-10,",
organization="",
publisher="",
note="")