A Combined First and Second Order Variational Approach for Image Reconstruction

Abstract

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler’s elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images—a known disadvantage of the ROF model—while being simple and efficiently numerically solvable.

Appendix: Some Useful Theorems

Proposition A.1

Suppose that $g:\mathbb{R}^{m}\to\mathbb{R}$ is a continuous function, positively homogeneous of degree 1 and let $\mu\in[\mathcal{M}(\varOmega )]^{m}$. Then for every positive measure Radon measure ν such that μ is absolutely continuous with respect to ν, we have

$$g(\mu)=g \biggl(\frac{\mu}{\nu} \biggr)\nu. $$

Moreover, if g is a convex function, then $g:[\mathcal{M}(\varOmega )]^{m}\to\mathcal{M}(\varOmega)$ is a convex function as well.

Proof

Since μ≪ν, we have that |μ|≪ν. Using the fact that g is positively homogeneous and the fact that |μ|/ν is a positive function, we get

Assuming that g is convex and using the first part of the proposition we get for 0≤λ≤1, μ, $\nu\in[\mathcal{M}(\varOmega)]^{m}$:

 □

The following theorem which is a special case of a theorem that was proved in [20] and can be also found in [4] establishes the lower semicontinuity of convex functionals of measures with respect to the weak^∗ convergence.

Theorem A.2

(Buttazzo and Freddi [20])

Let Ω be an open subset of $\mathbb{R}^{n}$, ν, $(\nu _{k})_{k\in\mathbb{N}}$ be $\mathbb{R}^{m}$-valued finite Radon measures and μ, $(\mu_{k})_{k\in\mathbb{N}}$ be positive Radon measures in Ω. Let $g:\mathbb{R}^{m}\to\mathbb{R}$ be a convex function and suppose that ν _k →ν and μ _k →μ weakly ^∗ in Ω. Consider the Lebesgue decompositions ν=(ν/μ)μ+ν ^s, $\nu_{k}=(\nu_{k}/\mu_{k})\mu_{k}+\nu_{k}^{s}$, $k\in\mathbb{N}$. Then

In particular, if $\mu=\mu_{k}=\mathcal{L}^{n}$ for all $k\in\mathbb {N}$ then according to the definition (2.1) the above inequality can be written as follows:

$$g(\nu) (\varOmega)\le\liminf_{k\to\infty}g(\nu_{k}) ( \varOmega). $$

The following theorem is a special case of Theorem 2.3 in [33].

Theorem A.3

(Demengel and Temam [33])

Suppose that $\varOmega\subseteq\mathbb{R}^{n}$ is open, with Lipschitz boundary and let g be a convex function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ with at most linear growth at infinity. Then for every u∈BH(Ω) there exists a sequence $(u_{k})_{k\in\mathbb{N}}\subseteq C^{\infty }(\varOmega)\cap W^{2,1}(\varOmega)$ such that

Lemma A.4

(Kronecker’s lemma)

Suppose that $(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$ are two sequences of real numbers such that $\sum_{n=1}^{\infty }a_{n}<\infty$ and 0<b ₁≤b ₂≤⋯ with b _n →∞. Then

$$\frac{1}{b_{n}}\sum_{k=1}^{n}b_{k}a_{k} \to0,\quad\textit{as } n\to\infty. $$

In particular, if $(c_{n})_{n\in\mathbb{N}}$ is a decreasing positive real sequence such that $\sum_{n=1}^{\infty}c_{n}^{2}<\infty$, then

$$c_{n}\sum_{k=1}^{n}c_{k} \to0,\quad\textit{as } n\to\infty. $$