[Paper Review] Single image super-resolution by approximated Heaviside functions

Liang-Jian Deng, Weihong Guo+1 more|arXiv (Cornell University)|Mar 12, 2015

Advanced Image Processing Techniques55 references3 citations

TL;DR

This paper proposes a continuous-domain image super-resolution method using two classes of approximated Heaviside functions (AHFs) to model smooth and non-smooth image components separately. By employing an L1-regularized iterative model solved via ADMM and applying it patchwise, the method achieves high-quality reconstruction at any upscaling factor without learning data, outperforming existing methods in both visual quality and quantitative metrics like RMSE.

ABSTRACT

Image super-resolution is a process to enhance image resolution. It is widely used in medical imaging, satellite imaging, target recognition, etc. In this paper, we conduct continuous modeling and assume that the unknown image intensity function is defined on a continuous domain and belongs to a space with a redundant basis. We propose a new iterative model for single image super-resolution based on an observation: an image is consisted of smooth components and non-smooth components, and we use two classes of approximated Heaviside functions (AHFs) to represent them respectively. Due to sparsity of the non-smooth components, a $L_{1}$ model is employed. In addition, we apply the proposed iterative model to image patches to reduce computation and storage. Comparisons with some existing competitive methods show the effectiveness of the proposed method.

Motivation & Objective

• Address the challenge of single image super-resolution (SISR) when only one low-resolution image is available, particularly in scenarios with hardware or cost constraints.
• Overcome limitations of discrete-domain models by modeling the image intensity function on a continuous domain using redundant basis functions.
• Develop a method that preserves fine image details and edges by separately representing smooth and non-smooth components through two classes of approximated Heaviside functions (AHFs).
• Reduce computational cost and storage by applying the iterative model to image patches rather than the full image.
• Minimize ring artifacts in smooth regions by introducing a gradient-based strategy to refine edge reconstruction.

Proposed method

• Model the unknown high-resolution image intensity function as a sum of two classes of approximated Heaviside functions (AHFs): one for smooth components and one for non-smooth components like edges.
• Formulate an L1-regularized optimization problem to exploit the sparsity of non-smooth components in the AHF representation.
• Use the alternating direction method of multipliers (ADMM) to solve the iterative model efficiently, enabling convergence to a high-resolution reconstruction.
• Apply the coefficient computation process patchwise to reduce memory usage and computational load, especially for large images.
• Incorporate image gradient information in smooth regions to suppress ring artifacts around large-scale edges.
• Allow arbitrary upscaling factors by directly applying the learned coefficients to reconstruct HR images at any desired resolution.

Experimental results

Research questions

• RQ1 Can a continuous-domain model based on approximated Heaviside functions effectively represent both smooth and non-smooth components in single image super-resolution?
• RQ2 How does the use of L1 regularization on AHF coefficients improve edge and detail preservation compared to traditional methods?
• RQ3 To what extent does patch-based processing reduce computational cost while maintaining reconstruction quality in the proposed method?
• RQ4 Can the proposed method achieve high-quality super-resolution without requiring external training data, unlike most learning-based approaches?
• RQ5 How effective is the gradient-based strategy in reducing ring artifacts on smooth image regions?

Key findings

• The proposed method achieves an average RMSE of 12.08 on tested images, outperforming existing competitive methods both quantitatively and qualitatively.
• The iterative AHF model with L1 regularization produces sharper edges and better-preserved image details compared to non-iterative counterparts.
• Computation time remains under 10 seconds for a 60×60 low-resolution image across upscaling factors from 2 to 9, indicating scalability.
• The method is robust to parameter variations, with minimal performance degradation when tuning λ₁ and λ₂ around 10⁻² and 10⁻⁶ respectively.
• The patchwise application of the model reduces memory usage significantly, enabling efficient processing of large images.
• The gradient-based artifact reduction strategy effectively suppresses ring artifacts in smooth regions, improving visual quality.