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A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution

Jianping Zhang, Ke Chen

arXiv:1509.04237v1cs.CVcs.NAmath.NA

TL;DR

Image-restoration models based on total variation and earlier fractional-order formulations leave limitations in handling smooth images, theoretical analysis, and boundary conditions. This paper analyzes a total α-order-variation model, develops a boundary regularization method and four algorithms, and reports that PDE-SB outperforms currently competitive variational models in restoration quality.

  • Problem

    Earlier fractional-order variation work had not established rigorous theory, while zero Dirichlet conditions used in tested formulations were unrealistic for imaging applications.

    Source: §1.2 · p. 3, §4 · p. 9

  • Method

    The paper analyzes a total α-order-variation denoising model, regularizes nonzero boundary conditions, and develops one Split-Bregman and three optimization-based algorithms.

    Source: §4 · p. 9, §8 · pp. 19–20

  • Results

    PDE-SB outperforms currently competitive variational models in restoration quality and performs similarly to, though more stably than, optimization-based approaches.

    Source: §8 · p. 20

  • Takeaways & Limitations

    The proposed boundary regularization method is essential for obtaining good image-denoising results with the model.

    Source: §8 · p. 20

    Optimal selection of λ remains unresolved; the tests use the best numerical λ for all models rather than a systematic study.

    Source: §7 · p. 16, §8 · p. 20

Abstract

from arXiv · show

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however first no analysis is done yet and second all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while non-zero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem, one based on the variational Split-Bregman idea and three based on direct solution of the discretise-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.

1 Introduction

The introduction motivates a fractional-order regularizer for image restoration by linking inverse-problem ill-posedness and TV’s shortcomings to gaps in fractional-model analysis and boundary treatment. It presents the paper’s theory, nonzero-boundary method, and four solution algorithms.

1.1 Image inverse problem

  • Image restoration is framed as an ill-posed inverse problem, and TV’s staircase effects and contrast loss make smooth-image restoration difficult.The introduction also notes that Tikhonov smoothing cannot preserve image edges.Source: §1.1 · p. 2

1.2 High-order regularization

  • Earlier fractional-derivative methods included diffusion and denoising models; reported improvements lacked analysis, leaving a need for rigorous theory.Fractional models are described as less nonlinear than high-order alternatives, which can be difficult to solve efficiently.Source: §1.2 · pp. 2–3

1.3 Our contributions

  • The paper studies a continuous total α-order-variation model and analyzes its properties as a regularizer for image inverse problems.The authors distinguish this focus from prior discrete formulations.Source: §1.3 · p. 3
  • The proposed nonzero Dirichlet boundary method generalizes related one-dimensional results to two dimensions.Standard fractional-derivative definitions require zero Dirichlet conditions, which the introduction identifies as unrealistic for imaging.Source: §1.3 · p. 3
  • The paper establishes convexity, solvability, and a solution theory for the model, contrasting with high-order and non-convex counterparts.The introduction characterizes those counterparts as having little known theory on their solutions.Source: §1.3 · p. 3
  • Four algorithms address the model: Split-Bregman, forward-backward, Nesterov accelerated, and FISTA methods.The introduction says the methods are tested and compared with related models.Source: §1.3 · p. 3

2 Review of fractional-order derivatives

The section reviews Riemann–Liouville, Grünwald–Letnikov, and Caputo fractional derivatives and properties used in the model. It also explains why their boundary requirements matter for image applications.

Definitions of fractional derivatives

  • Riemann–Liouville, Grünwald–Letnikov, and Caputo are the three reviewed definitions of fractional-order derivatives.For integer α, the left-sided Riemann–Liouville definition reduces to the usual derivative.Source: §2 · p. 4

Properties and assumptions

  • The reviewed derivative definitions are equivalent under suitable regularity and homogeneous boundary conditions, but the paper does not require that equivalence for its image function.The section also notes linearity and a property used later to prove convexity and derive first-order optimal conditions.Source: §2 · p. 5

Zero fractional derivatives

  • For 1 < α < 2, the left Riemann–Liouville derivative is zero for non-constant functions of the stated power forms, including x^0.6 and x^-0.4 when α = 1.6.Source: §2 · p. 5

Boundary conditions

  • Standard fractional-derivative definitions require zero endpoint values to avoid singularities, constraining their direct use with nonzero image boundary conditions.The Riesz Riemann–Liouville derivative requires zero values at both endpoints.Source: §2 · p. 5

3 The total α-order variation and its related model

The section defines total α-order variation and the function space BVα(Ω) suited to the model, then establishes key properties of the denoising energy. For λ > 0, the energy has a unique minimizer.

The space BVα(Ω)

  • BVα(Ω), the space of functions with finite total α-order variation, is a Banach space.The proof uses lower semi-continuity of total α-order variation under L1 convergence.Source: §3 · pp. 6–7

Total α-order variation

  • The section defines total α-order variation using fractional derivatives and identifies BVα(Ω) as the suitable function space for the model.The authors motivate this choice by the model’s expected preservation of edges and smoothness.Source: §3 · p. 6

Convexity

  • The total α-order variation functional is convex, supporting the convexity result for the denoising energy.The paper attributes this property to linearity of fractional derivatives and positive homogeneity and subadditivity of total α-order variation.Source: §3 · p. 8

Theory for a total α-order variation model

  • The model’s energy is convex for λ ≥ 0 and strictly convex for λ > 0, and it has a unique minimizer when λ > 0.The paper also establishes that the energy attains a minimum in BVα(Ω).Source: §3 · p. 8

4 Nonzero Dirichlet boundary conditions and regularization

The section addresses the mismatch between zero-boundary requirements of standard fractional derivatives and nonzero Dirichlet conditions common in imaging. It reduces nonzero conditions to zero ones using auxiliary functions.

Motivation

  • Standard fractional derivatives require zero Dirichlet conditions, which are unrealistic for imaging; inaccurate boundary conditions can cause oscillations near image boundaries.The paper therefore treats boundary conditions as a crucial part of fractional-derivative computations.Source: §4 · p. 9

Boundary regularization

  • Boundary regularization introduces an auxiliary function satisfying zero boundary conditions, moving the nonzero conditions into a known right-hand-side quantity.The resulting image with zero Dirichlet conditions can use standard fractional-derivative definitions.Source: §4 · pp. 9–10

Two-dimensional construction

  • The two-dimensional construction uses a bilinear auxiliary function based on corner values, which must be known or accurately estimated.The paper proposes estimating corner values from the observed image and reconstructing the four edges with one-dimensional inverse problems.Source: §4 · pp. 9–10

5 Discretization and Split-Bregman algorithm

The section reformulates the model for Split-Bregman solution, discretizes fractional derivatives with finite differences, and presents an iterative algorithm. The algorithm regularizes boundaries, updates auxiliary variables, and solves the image subproblem by conjugate gradients.

5 Discretization and Split-Bregman algorithm

5.1 A Split-Bregman formulation

  • The Split-Bregman formulation introduces d = (d1, d2) to create a constrained optimization problem, then enforces the constraint through a Bregman formulation.The d subproblem has a closed-form solution; the u subproblem follows an Euler–Lagrange equation.Source: §5.1 · p. 10

5.2 Discretization of the fractional derivative

  • The fractional derivative is discretized at interior grid points with finite differences, using a linear combination of image values along each direction.The stated discretization applies to both Riemann–Liouville and Caputo derivatives.Source: §5.2 · pp. 11–12

5.3 The Split-Bregman algorithm

  • The discretized derivatives are assembled in structured matrix form, and the positive-definite operator in the image subproblem supports conjugate-gradient solution.The coefficient matrix for the one-dimensional discretization is described as symmetric negative definite Toeplitz.Source: §5.2 · p. 11, §5.3 · p. 12

Algorithm 1 (Split-Bregman iterations (PDE-SB))

  • Algorithm 1 regularizes the observed-image boundary, updates the auxiliary variable, solves the image subproblem by conjugate gradients, and stops when successive solutions differ by less than ε.The boundary-corrected solution is returned after the stopping condition is met.Source: Algorithm 1 (Split-Bregman iterations… · pp. 12–13

6 Optimization based numerical methods

The section discretizes the model and presents three optimization-based solvers: forward-backward, Nesterov’s accelerated method, and FISTA. These methods use gradient and proximal steps to solve the resulting problem.

6 Optimization based numerical methods

  • Computing the proximal point for f1 is a major nontrivial step, addressed here with a dual formulation and gradient projection.The discussion adapts ideas from discrete dual methods used for TV regularization.Source: §6 · pp. 13–14

6.1 Forward-backward algorithm

  • Forward-backward splitting alternates an explicit gradient step for f2 with an implicit proximal step for f1.The scheme decouples the two function contributions in a gradient-descent step.Source: §6.1 · p. 14

6.2 Nesterov’s method

  • Nesterov’s accelerated method uses a composite gradient mapping and converges at O(1/r^2), where r is the iteration number.The paper presents it as an accelerated multistep method for the composite problem.Source: §6.2 · p. 15

6.3 FISTA method

  • FISTA applies Nesterov’s idea within forward-backward splitting, retaining the optimal convergence rate with wider applicability.The authors apply it to the discretized problem as Algorithm 4.Source: §6.3 · p. 15

7 Numerical results

The numerical tests compare boundary treatment, four solvers, parameter sensitivity, and denoising models. Overall, the fractional-order model performs competitively, with results depending on image type, parameter choice, and stopping criteria.

7 Numerical results

  • Tests initialize from the noisy image and stop at specified residual or error thresholds, with limits of 1000 outer and 15 inner iterations.The four datasets are P1 parabolic surface, P2 saddle surface, P3 Pepper, and P4 Penguin.Source: §7 · p. 16
  • The evaluation presents denoising results only, and its PSNR/SNR measures require a known true image and may not track human perception.The authors also leave a systematic study of regularization-parameter selection for future work and use the best numerical λ values in these tests.Source: §7 · p. 16

7.1 Performance comparisons of boundary regularization

  • For the nonzero-boundary P1 test, treatment yields PSNR 47.53 and SNR 35.43, versus 23.69 and 10.38 without treatment.The authors conclude that boundary regularization is effective; direct use of the fractional model gives incorrect boundary restoration.Source: Figure 2 · p. 17, §7.1 · p. 17

7.2 Comparisons of Algorithms 1–4

  • Across the four test datasets, PDE-SB is the most consistently good performer; Opti-Nesterov and PDE-SB have similar restoration quality, while Opti-FISTA and PDE-SB are fastest.The comparison reports PSNR, SNR, and computation time under dataset-specific parameter settings.Source: §7.2 · p. 17, Table 1 · p. 18

7.3 Sensitivity tests for α and λ

  • The model gives satisfactory solutions across a broad λ range; the tests suggest α = 1.6 for smooth problems and α = 1.4 for nonsmooth problems.Smaller α is associated with staircase effects, while larger α makes solutions too smooth along both spatial directions.Source: §7.3 · p. 18

7.4 Comparisons with other non-fractional variational models

  • With empirically optimal λ values, the new convex model is slightly better; without them it performs more robustly, and on noisy smooth Pepper it significantly outperforms the compared models.For that Pepper test, PDE-SB is also reported much faster than mean curvature and TGV.Source: §7.4 · p. 19, Table 2 · p. 19

8 Conclusions

The paper rigorously analyzes a total α-order variation model for denoising and develops algorithms with boundary regularization. Its PDE-SB method outperforms competitive variational models in restoration quality.

8 Conclusions

  • PDE-SB outperforms competitive variational models in restoration quality; Figure 4 compares it with TV, mean-curvature, and TGV models.The conclusion reports the overall comparison without giving quantitative values.Source: §8 · p. 20, Figure 4 · p. 20
  • The paper rigorously analyzes a total α-order variation model for image denoising and proposes boundary regularization to handle fractional derivatives.The authors describe the boundary regularization method as essential for good denoising results.Source: §8 · pp. 19–20
  • One Split-Bregman algorithm and three optimization-based algorithms solve the model; PDE-SB performs similarly to, but more stably than, the optimization approaches.Source: §8 · p. 20

Appendix: Proof of Theorem 4

The appendix derives variational expressions and examines boundary-condition cases. It also notes practical difficulties with the boundary-condition sets considered for imaging.

Appendix: Proof of Theorem 4

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