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Constructing an Interpretable Deep Denoiser by Unrolling Graph Laplacian Regularizer

Seyed Alireza Hosseini, Tam Thuc Do, Gene Cheung, Yuichi Tanaka

TL;DR

Experimental results show that GDD achieves competitive image denoising performance compared to competitors, but employing far fewer parameters, and is more robust to covariate shift.

Abstract

An image denoiser can be used for a wide range of restoration problems via the Plug-and-Play (PnP) architecture. In this paper, we propose a general framework to build an interpretable graph-based deep denoiser (GDD) by unrolling a solution to a maximum a posteriori (MAP) problem equipped with a graph Laplacian regularizer (GLR) as signal prior. Leveraging a recent theorem showing that any (pseudo-)linear denoiser $\boldsymbol Ψ$, under mild conditions, can be mapped to a solution of a MAP denoising problem regularized using GLR, we first initialize a graph Laplacian matrix $\mathbf L$ via truncated Taylor Series Expansion (TSE) of $\boldsymbol Ψ^{-1}$. Then, we compute the MAP linear system solution by unrolling iterations of the conjugate gradient (CG) algorithm into a sequence of neural layers as a feed-forward network -- one that is amenable to parameter tuning. The resulting GDD network is "graph-interpretable", low in parameter count, and easy to initialize thanks to $\mathbf L$ derived from a known well-performing denoiser $\boldsymbol Ψ$. Experimental results show that GDD achieves competitive image denoising performance compared to competitors, but employing far fewer parameters, and is more robust to covariate shift.

Constructing an Interpretable Deep Denoiser by Unrolling Graph Laplacian Regularizer

TL;DR

Experimental results show that GDD achieves competitive image denoising performance compared to competitors, but employing far fewer parameters, and is more robust to covariate shift.

Abstract

An image denoiser can be used for a wide range of restoration problems via the Plug-and-Play (PnP) architecture. In this paper, we propose a general framework to build an interpretable graph-based deep denoiser (GDD) by unrolling a solution to a maximum a posteriori (MAP) problem equipped with a graph Laplacian regularizer (GLR) as signal prior. Leveraging a recent theorem showing that any (pseudo-)linear denoiser , under mild conditions, can be mapped to a solution of a MAP denoising problem regularized using GLR, we first initialize a graph Laplacian matrix via truncated Taylor Series Expansion (TSE) of . Then, we compute the MAP linear system solution by unrolling iterations of the conjugate gradient (CG) algorithm into a sequence of neural layers as a feed-forward network -- one that is amenable to parameter tuning. The resulting GDD network is "graph-interpretable", low in parameter count, and easy to initialize thanks to derived from a known well-performing denoiser . Experimental results show that GDD achieves competitive image denoising performance compared to competitors, but employing far fewer parameters, and is more robust to covariate shift.
Paper Structure (17 sections, 1 theorem, 17 equations, 7 figures, 1 table)

This paper contains 17 sections, 1 theorem, 17 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GSP Definitions
  4. Graph Laplacian Regularizer
  5. Algorithm Development
  6. Review of Theorem 1
  7. Generalizing Bilateral Filter
  8. Matrix Normalization
  9. Initializing Graph Laplacian
  10. Algorithm Unrolling
  11. Laplacian Sub-network
  12. Unrolling Conjugate Gradient Descent
  13. Loss Function
  14. Experiments
  15. Experimental Setup
  16. ...and 2 more sections

Key Result

Theorem 1

Denoiser ${\boldsymbol \Psi}$eq:denoiser is the solution filter for the MAP problem eq:MAP_denoise if ${\mathcal{L}} = \mu^{-1} ({\boldsymbol \Psi}^{-1} - {\mathbf I}_N)$, assuming matrix ${\boldsymbol \Psi}$ is non-expansive, symmetric, and PD.

Figures (7)

  • Figure 1: Feed-forward sub-network to implement a truncated TSE of $f({\boldsymbol \Psi}) = {\boldsymbol \Psi}^{-1}$ according to \ref{['eq:TSR']}. TSE coefficients $a_k$ for each ${\boldsymbol \Psi}_i$ are learned through end-to-end training.
  • Figure 1: PSNR versus noise SD for test set. Both models were trained on small dataset with noise SD $\sigma=10$.
  • Figure 2: Overview of proposed network architecture, here $y$ is the noisy patch, $f_i$'s are hand-crafted features and $M$ is initialized metric matrix.
  • Figure 3: PSNR versus noise SD for variants of trained unrolled network, initialized network, and total variation.
  • Figure 4: Examples of noisy and denoised wall images by competing methods and their quality in PSNR.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1