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Graph Algorithm Unrolling with Douglas-Rachford Iterations for Image Interpolation with Guaranteed Initialization

Xue Zhang, Bingshuo Hu, Gene Cheung

TL;DR

The paper tackles image interpolation with a focus on parameter efficiency and reliable initialization. It leverages a theorem that maps a pseudo-linear interpolator $Θ$ to a directed graph filter solving a MAP problem with a graph shift variation (GSV) prior, initializing the graph adjacency from $Θ$ to guarantee baseline performance. It then learns two perturbation matrices $P$ and $P^{(2)}$ to augment the graph and uses Douglas-Rachford (DR) iterations unrolled into a lightweight neural network to realize the restoration. Experiments show state-of-the-art interpolation while using only a fraction of the parameters of strong baselines, demonstrating the effectiveness of structured initialization plus dual-perturbation learning in an interpretable unrolled framework.

Abstract

Conventional deep neural nets (DNNs) initialize network parameters at random and then optimize each one via stochastic gradient descent (SGD), resulting in substantial risk of poor-performing local minima.Focusing on the image interpolation problem and leveraging a recent theorem that maps a (pseudo-)linear interpolator Θ to a directed graph filter that is a solution to a MAP problem regularized with a graph shift variation (GSV) prior, we first initialize a directed graph adjacency matrix A based on a known interpolator Θ, establishing a baseline performance.Then, towards further gain, we learn perturbation matrices P and P(2) from data to augment A, whose restoration effects are implemented via Douglas-Rachford (DR) iterations, which we unroll into a lightweight interpretable neural net.Experimental results demonstrate state-of-the-art image interpolation results, while drastically reducing network parameters.

Graph Algorithm Unrolling with Douglas-Rachford Iterations for Image Interpolation with Guaranteed Initialization

TL;DR

The paper tackles image interpolation with a focus on parameter efficiency and reliable initialization. It leverages a theorem that maps a pseudo-linear interpolator to a directed graph filter solving a MAP problem with a graph shift variation (GSV) prior, initializing the graph adjacency from to guarantee baseline performance. It then learns two perturbation matrices and to augment the graph and uses Douglas-Rachford (DR) iterations unrolled into a lightweight neural network to realize the restoration. Experiments show state-of-the-art interpolation while using only a fraction of the parameters of strong baselines, demonstrating the effectiveness of structured initialization plus dual-perturbation learning in an interpretable unrolled framework.

Abstract

Conventional deep neural nets (DNNs) initialize network parameters at random and then optimize each one via stochastic gradient descent (SGD), resulting in substantial risk of poor-performing local minima.Focusing on the image interpolation problem and leveraging a recent theorem that maps a (pseudo-)linear interpolator Θ to a directed graph filter that is a solution to a MAP problem regularized with a graph shift variation (GSV) prior, we first initialize a directed graph adjacency matrix A based on a known interpolator Θ, establishing a baseline performance.Then, towards further gain, we learn perturbation matrices P and P(2) from data to augment A, whose restoration effects are implemented via Douglas-Rachford (DR) iterations, which we unroll into a lightweight interpretable neural net.Experimental results demonstrate state-of-the-art image interpolation results, while drastically reducing network parameters.

Paper Structure

This paper contains 16 sections, 1 theorem, 24 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Shift Variation
  4. Interpolation Theorem
  5. Problem Formulation
  6. Feed-Forward Network Construction
  7. Computing Interpolated Pixels
  8. Cascading Filter Interpretation
  9. Multiple Perturbations via Cascades
  10. Second Perturbation Matrix via Denoising
  11. Experiments
  12. Experimental Setup
  13. Experimental Results
  14. Conclusion
  15. Proof of (21)
  16. ...and 1 more sections

Key Result

Theorem 1

Interpolator $[{\mathbf I}_M;\mathbf{{\boldsymbol \Theta}}]$ is the solution filter to the MAP problem eq:obj if $M=N$, ${\boldsymbol \Theta}$ is invertible, and ${\mathbf A}_{M,N}={\boldsymbol \Theta}^{-1}$.

Figures (2)

  • Figure 1: Network Architecture from Unrolling of Douglas-Rachford iterations into neural layers. Note that a graph learning module (self-attention mechanism) is periodically inserted, as done in Do2024.
  • Figure 2: Visual interpolation results for the image "Urban062".

Theorems & Definitions (1)

  • Theorem 1