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A Bijective Image Retargeting Algorithm Based on Conformal Energy

Chengyang Liu, Michael K. Ng

TL;DR

The paper addresses image retargeting with the requirement of a bijective warp to preserve content. It introduces a variational model that minimizes the conformal energy $E^C(f)$ of the deformation map $f: D1 -> D2$ under ROI/line and boundary constraints, followed by a bijection-correction step. It proves existence and uniqueness of the minimizer in both continuous and discrete settings and establishes convergence of discrete minimizers to the continuous solution under mesh refinement. Experimental results on the RetargetMe dataset show reduced distortion energy and zero flips, indicating robust, artifact-free retargeting and competitive efficiency due to the sparse linear systems involved. The work provides a rigorous, scalable foundation for bijective, structure-preserving image retargeting with clear theoretical guarantees and practical impact.

Abstract

Image retargeting, which resizes images to one with a prescribed aspect ratio by determining an optimal warping map, has gained substantial interest in imaging science. Despite significant advances, existing methods often fail to ensure bijective warping maps essential for preserving visual information. This paper introduces a novel bijective image retargeting model through conformal energy minimization of the deformation field. The proposed model establishes mathematical rigor by proving the well-posedness for the optimal warping map in both continuous and discrete settings and showing that the discrete solutions converge to their continuous counterpart under mesh refinement. Numerical experiments corroborate the model's efficacy and the convergence of discrete solutions during progressive mesh subdivision processes, validating both theoretical guarantees and practical performance.

A Bijective Image Retargeting Algorithm Based on Conformal Energy

TL;DR

The paper addresses image retargeting with the requirement of a bijective warp to preserve content. It introduces a variational model that minimizes the conformal energy of the deformation map under ROI/line and boundary constraints, followed by a bijection-correction step. It proves existence and uniqueness of the minimizer in both continuous and discrete settings and establishes convergence of discrete minimizers to the continuous solution under mesh refinement. Experimental results on the RetargetMe dataset show reduced distortion energy and zero flips, indicating robust, artifact-free retargeting and competitive efficiency due to the sparse linear systems involved. The work provides a rigorous, scalable foundation for bijective, structure-preserving image retargeting with clear theoretical guarantees and practical impact.

Abstract

Image retargeting, which resizes images to one with a prescribed aspect ratio by determining an optimal warping map, has gained substantial interest in imaging science. Despite significant advances, existing methods often fail to ensure bijective warping maps essential for preserving visual information. This paper introduces a novel bijective image retargeting model through conformal energy minimization of the deformation field. The proposed model establishes mathematical rigor by proving the well-posedness for the optimal warping map in both continuous and discrete settings and showing that the discrete solutions converge to their continuous counterpart under mesh refinement. Numerical experiments corroborate the model's efficacy and the convergence of discrete solutions during progressive mesh subdivision processes, validating both theoretical guarantees and practical performance.
Paper Structure (22 sections, 16 theorems, 45 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 16 theorems, 45 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Review and contributions
  3. Notation and outline
  4. Background
  5. Mathematical modeling
  6. Conformal Energy
  7. The Bijective Retargeting Algorithm
  8. Minimizer in the continuous setting
  9. Discretization
  10. Minimizer in the discrete setting
  11. Bijection correction algorithm
  12. ROIs, line structures and boundary condition
  13. The main algorithm
  14. Preliminaries
  15. Simplicial interpolation in Sobolev space
  16. ...and 7 more sections

Key Result

Theorem 1

Let $u \in L^q\left(\mathbb{R}^n\right)$. If $u$ is weakly differentiable and if $D u \in L^p\left(\mathbb{R}^n\right)$, then for every $\varepsilon>0$ there exists a triangulation $M$ of $\mathbb{R}^n$ such that

Figures (7)

  • Figure 1: A diagram to show the conceptions defined on mesh $M$.
  • Figure 2: Comparison with BR and our algorithm to resize a skiing image to 75% of the original width.
  • Figure 3: Comparison with BR and our algorithm to resize a lotus image to 50% of the original width.
  • Figure 4: Resizing a Canalhouse image to 50% of the original width.
  • Figure 5: Comparison of different methods to resize a car image to 75% of the original width.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Theorem 1: Theorem 1 in van2014approximation
  • Definition 1: Subdivision of simplicial complex, rourke2012introduction
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2: Alaoglu's Theorem, conway2019course
  • Theorem 3: Rellich-Kondrachov theorem, evans2010partial
  • Theorem 4: Existence and uniqueness of the minimizer in the continuous setting
  • proof
  • ...and 20 more