2 A one-dimensional nonlocal nonlinear model

There is a vast literature concerning nonlinear anisotropic diffusions with applications to image processing, which dates back to the seminal paper by Perona and Malik [Reference Perona and Malik12], who considered a discrete version of the problem

(2.1) $$ \begin{align} \begin{cases} \dfrac{\partial u}{\partial t} - \nabla\cdot (g(|\nabla u|)\nabla u) = 0 \quad &\textrm{in } \Omega_{T}=(0,T)\times \Omega , \\[5pt] u(x,0)=u_{0}(x) \quad &\textrm{on } \Omega, \\[2pt] \dfrac{\partial u}{\partial \vec{n}} (x,t) =0 \quad &\textrm{on } \Gamma\times (0,T), \end{cases} \end{align} $$

where $ \Gamma = \partial \Omega $ , the image domain $\Omega \subset \mathbb {R}^{2}$ is an open regular set (typically a rectangle), $\vec {n}$ denotes the unit outer normal to its boundary $\Gamma $ , $\nabla \cdot $ is the divergence operator, and $u(x,t)$ denotes the (scalar) image analysed at time (scale) t and point x. The initial condition $u_{0}(x)$ is, as in the linear case, the original image. To reduce smoothing at the edges, the diffusivity g is chosen as a decreasing function of the ‘edge detector’ $|\nabla u|$ . Here, we introduce a nonlocal diffusive coefficient that considers the ‘monotonicity’ of the signal. In other words, a high modulus of the gradient may lead to a small diffusion if the function is also locally monotone. At the same time, we want to reduce the noise present, as in the case of linear diffusion. We focus on the one-dimensional case, more precisely, where $u:[a,b] \rightarrow \mathbb {R}$ is a real function defined on a bounded interval $[a,b]$ , and on a subinterval $[c,d] \subset [a,b ]$ . We define the local variation $LV_{[c,d]}(u)$ of u on the interval $[c,d]$ by

$$ \begin{align*} LV_{[c,d]}(u) = |u(d) - u(c)|. \end{align*} $$

We also define the total local variation $TV_{[c,d]}(u)$ of u on the interval $[c,d]$ by

$$ \begin{align*} TV_{[c,d]}(u) = \sup_{\mathcal{P}} \sum_{i=0}^{n_{P}-1} | u ( x_{i+1}) - u( x_{i}) | \end{align*} $$

where $\mathcal {P} = \{ P = \{ x_{0}, \ldots , x_{n_{P}} \} \mid P \textrm { is a partition of } [ c,d ] \} $ is the set of all possible finite partitions of the interval $[c,d]$ .

Let $\varepsilon \in \mathbb {R}^{+}$ , $\varepsilon \ll 1, \, \varepsilon>0$ and let $\delta \in \mathbb {R}^{+}$ . We define the ratio,

$$ \begin{align*} R_{\delta ,u} = \frac{\varepsilon + LV_{[ x-\delta ,x+\delta ]}(u)}{\varepsilon + TV_{[x-\delta ,x+\delta ]}(u)}. \end{align*} $$

If the parameter $\delta $ is chosen appropriately, we can distinguish between oscillations caused by noise contained in a range of amplitude $\delta $ . As in the Perona–Malik model given by (2.1), we adapt the diffusivity coefficient by using the above ratio $R_{\delta ,u}$ . For small values of the latter, we have to reduce the noise, while for values close to $1$ , the upper bound of $R_{\delta ,u}$ , we have to preserve the signal variation (as the edges in the image). The resulting diffusivity coefficient $g(R_{\delta ,u})$ becomes nonlocal. We assume that $g:[0,+\infty )\rightarrow \mathbb {R}$ is a positive, nonincreasing, Lipschitz continuous function such that $g(0)=1$ and $g(1)=\alpha>0$ . In the following, we assume that the parameter $\varepsilon $ ( $0<\varepsilon \ll 1$ ) is fixed. In Figure 1, we show an illustrative example of a denoised signal using our nonlocal and nonlinear diffusion filter. In particular, we have numerically simulated (2.2) by adopting a semi-implicit method based on central finite differences (see [Reference Aletti, Moroni and Naldi2]) and with the following numeric values of the parameters (see also (2.3)):

$$ \begin{align*} g(s) = \begin{cases} 1 \quad &\text{ if }s=0\text{,} \\[2pt] 1 - s^{2} e^{-3.315/(s/\lambda)^{8})} \quad &\text{ if }s\neq 0\text{,} \end{cases}\quad \lambda= 4,\ \varepsilon =10^{-3},\ \delta = 0.075, \end{align*} $$

and the time domain $t\in [0,0.6]$ and space domain $x\in [0, 255]$ . The signal in Figure 1 was obtained from a simulation of a biophysical model of a neuron with an additive Gaussian noise (mean equal to $0$ and variance equal to $3$ ) (see [Reference Aletti, Lonardoni, Naldi and Nieus1] for more details). The MATLAB code and the details are available from the authors.

Figure 1 An illustrative example of signal denoising using the new nonlocal and nonlinear diffusion equation (2.2) with data from [Reference Aletti, Lonardoni, Naldi and Nieus1]. Solid line, original signal; grey line, signal with noise; dotted line, reconstructed signal.

In the following, $I=(a,b)\subset \mathbb {R}$ denotes a bounded open interval and $H^{k}(I)$ , $k\in \mathbb {N}$ , the Sobolev space of all functions u defined in I such that u and its distributional derivatives of order $1,\ldots ,k$ all belong to $L^{2}(I)$ . Let $D^{s}$ denote the distributional derivative. Then $H^{k}(I)$ is a Hilbert space for the norm

$$ \begin{align*} \| u \|_{k} = \| u \|_{H^{k}} = \bigg( \sum_{|s|\leq k} \int_{I} | D^{s} u(x) |^{2} \,dx \bigg)^{1/2} , \quad \| u \|_{0} = \| u \|_{L^{2}}. \end{align*} $$

Let $L^{p}(0, T; H^{k}(I))$ be the set of all functions u, such that, for almost every t in $(0, T)$ with $T>0$ , $u(t)$ belongs to $H^{k}(I)$ . Then $L^{p}(0, T; H^{k}(I))$ is a normed space for the norm

$$ \begin{align*} \|u \|_{L^{p}(0, T; H^{k}(I ))} = \bigg( \int_{0}^{T}\| u \|^{p}_{k} \,dt \bigg)^{1/p}, \end{align*} $$

where $p \geq 1$ and $k\in \mathbb {N}$ . Finally, we denote by $(\cdot \, , \cdot )$ the scalar product in $L^{2}(I)$ .

We now establish our existence result. As initial conditions, we take the original signal $u_{0}$ but with some regularisation obtained with a standard smoothing filter, for example, a Gaussian filter, and we assume homogeneous Neumann conditions at the boundary.

Theorem 2.1 (Existence).

Let $u_{0} \in H^{1}(I)$ and $T>0$ , $\delta>0$ . Then there exists $u\in L^{2}(0,T;H^{1}(I))\bigcap C^{0}([0,T]; L^{2}(I))$ , satisfying u(x, 0) = u ₀ (x) on I, ${\partial u}/{\partial x} =0$ at $x=a,\, b$ , and

(2.2) $$ \begin{align} \frac{\partial u}{\partial t} - \frac{\partial}{\partial x} \bigg( g( R_{\delta ,u}) \frac{\partial u}{\partial x} \bigg)=0, \end{align} $$

on $(0,T]\times I$ in the distributional sense.

Proof. We show the existence of a weak solution of (2.2) by a classical fixed point theorem of Schauder (see, for example, [Reference Bonsall7, Theorem 2.2]). We introduce the space

$$ \begin{align*} V(0,T)=\bigg\{ v\in L^{2}(0,T;H^{1}(I)),\,\frac{dv}{dt}\in L^{2}(0,T;(H^{1}(I))^{\prime} ) \bigg\}. \end{align*} $$

The space $V(0,T)$ is a Hilbert space with the graph norm. Let v be a function in $V(0,T)\bigcap L^{\infty } (0,T; L^{2}(I))$ such that

$$ \begin{align*} \| v \|_{L^{\infty} (0,T; L^{2}(I))} \leq \| u_{0}\|_{L^{2}(I)}. \end{align*} $$

We consider the following variational problem $(P_{v})$ :

$$ \begin{align*} \bigg\langle \frac{\partial u}{\partial t}(t),w \bigg\rangle + \int_{I} g(R_{\delta ,v}) \frac{\partial u(t)}{\partial x}\frac{\partial w}{\partial x} \,dx & =0\quad \mbox{for all } w\in H^{1}(I)\,\,\text{(a.e.) in}\,\,[0,T] \\[2pt] u(0) & \in H^{1}(I). \nonumber \end{align*} $$

Here $\langle \cdot \, , \cdot \rangle $ represents the duality product. A function $u\in H^{1}(I)$ has locally bounded variation (see, for example, [Reference Evans and Gariepy10, Theorem 5.1]) and, moreover, is equal almost everywhere (a.e.) to an absolutely continuous function and $u^{\prime }$ exists a.e. and belongs to $L^{2}(I)$ . The term $R_{\delta ,v}$ can be represented as

(2.3) $$ \begin{align} R_{\delta ,v} = \frac{\varepsilon + \Big|\int_{x-\delta}^{x+\delta} u^{\prime} (s) \,ds \Big|}{\varepsilon + \int_{x-\delta}^{x+\delta} | u^{\prime} (s)| \,ds} \end{align} $$

and $0<R_{\delta ,v}\leq 1$ . So, $g(R_{\delta ,v})\geq \alpha>0$ .

Using classical results about parabolic equations (see, for example, [Reference Brezis8, Theorem 10.1] and [Reference Evans9, Theorem 7.3]), the problem $(P_{v})$ has a unique solution $U(v)$ in $V(0,T)$ . We can deduce the following estimates:

(2.4) $$ \begin{align} \| U(v) \|_{L^{\infty} (0,T;L^{2}(I))} & \leq \| u_{0}\|_{L^{2}(I)}, \nonumber\\[2pt] \| U(v) \|_{L^{2} (0,T;H^{1}(I))} & \leq C_{1}, \\[2pt] \| U(v) \|_{L^{2} (0,T;(H^{1}(I))^{\prime} )} & \leq C_{2}, \nonumber \end{align} $$

for suitable constants $C_{1}$ and $C_{2}$ depending only on $u_{0}$ , T and the Lipschitz constant of the function g. We introduce the subset $V_{0}$ of $V(0,T)$ defined by functions $v\in V(0,T)$ such that these estimates are satisfied and $v(0)=u_{0}$ . Then U is a mapping from $V_{0}$ to $V_{0}$ . Moreover, $V_{0}$ is a nonempty, convex and weakly compact subset of $V(0,T)$ .

To use the Schauder theorem, we have to prove that the mapping $v \rightarrow U(v)$ is weakly continuous from $V_{0}$ to $V_{0}$ . Then, since $V(0,T)$ is contained in $L^{2}(0,T; L^{2}(I))$ with compact inclusion, this yields the existence of $u\in V_{0}$ such that $u=U(u)$ .

Let $(v_{j})$ be a sequence in $V_{0}$ which converges weakly to $v\in V_{0}$ and $u_{j}=U(v_{j})$ . From the classical theorems of compact inclusion (see, for example, [Reference Brezis8, Theorem 9.16]), up to sub-sequences,

$$ \begin{align*} u_{j} & \rightarrow u\quad\ \ \text{weakly in }\ L^{2}(0,T; H^{1}(I)), \\[2pt] \frac{du_{j}}{dt} & \rightarrow \frac{du}{dt} \quad \text{weakly in}\ L^{2}(0,T; (H^{1}(I))^{\prime} ),\\[2pt] \frac{\partial u_{j}}{\partial x} & \rightarrow \frac{\partial u}{\partial x} \quad \text{weakly in}\ L^{2}(0,T; L^{2}(I) ). \end{align*} $$

Moreover, $u_{j} \rightarrow u$ in $L^{2}(0,T; L^{2}(I))$ and a.e. on $I\times (0,T)$ and $u_{j}(0) \rightarrow u(0)$ in $ (H^{1}(I))^{\prime }$ . For the $(v_{j})$ , from (2.4), there is a subsequence such that $v_{j} \rightarrow v$ in $L^{2}(0,T; L^{2}(I))$ and, from the Rellich–Kodrachov theorem (see, for example, [Reference Evans9, Theorem 5.1], and (2.3)), $g(R_{\delta ,v_{j}}) \rightarrow g(R_{\delta ,v})$ in $L^{2}(0,T; L^{2}(I))$ . By the uniqueness of the solution of $(P_{v})$ , the whole sequence $u_{j}=U(v_{j})$ converges weakly in $V(0,T)$ . Thus, the mapping U is weakly continuous from $V_{0}$ into $V_{0}$ and we can apply the Schauder theorem.

Under the hypotheses of Theorem 2.1, we have the following uniqueness result.

Proof. Let $\bar {u}$ and $\hat {u}$ be two solutions of (2.2) and let $u=\bar {u}-\hat {u}$ . Then for almost all t in $[0,T]$ ,

(2.5) $$ \begin{align} \frac{{d}\bar{u}}{dt}-\frac{\partial}{\partial x} \bigg( g ( R_{\delta ,\bar{u}}) \frac{\partial \bar{u}}{\partial x} \bigg) & = 0, \quad \bar{u}(0)=u_{0}, \end{align} $$

(2.6) $$ \begin{align} \frac{{d}\hat{u}}{dt}-\frac{\partial}{\partial x} \bigg( g ( R_{\delta ,\hat{u}}) \frac{\partial \hat{u}}{\partial x} \bigg) & = 0, \quad \hat{u}(0)=u_{0}. \end{align} $$

By subtracting (2.6) from (2.5),

$$ \begin{align*} \frac{d(\bar{u}-\hat{u})}{dt}-\frac{\partial}{\partial x} \bigg( g ( R_{\delta ,\bar{u}}) \frac{\partial \bar{u}}{\partial x}\bigg)+ \frac{\partial}{\partial x} \bigg( g ( R_{\delta ,\hat{u}}) \frac{\partial \hat{u}}{\partial x} \bigg)=0. \end{align*} $$

Adding and subtracting the quantity ${\partial _{x}}( g ( R_{\delta ,\bar {u}}) {\partial _{x}} \hat {u})$ , we can rewrite the equation as

(2.7) $$ \begin{align} \frac{du}{dt}-\frac{\partial}{\partial x} \bigg( g ( R_{\delta ,\bar{u}}) \frac{\partial {u}}{\partial x}\bigg)= \frac{\partial}{\partial x} \bigg( [ g ( R_{\delta ,\bar{u}})- g ( R_{\delta ,\hat{u}})] \frac{\partial \hat{u}}{\partial x} \bigg). \end{align} $$

Multiplying (2.7) by $u=(\bar {u}-\hat {u})$ , integrating on the interval I, using the properties of the function g and the lower bound $g(1)=\alpha>0$ and the estimates (2.4), we obtain

$$ \begin{align*} \frac{1}{2}\frac{d}{dt}\|u(t)\|_{L^{2}(I)}^{2} + \alpha \bigg\| \frac{\partial}{\partial x}u(t) \bigg\|_{L^{2}(I)}^{2} \leq C \| u(t) \|_{L^{2}(I)} \bigg\| \frac{\partial}{\partial x}\hat{u}(t) \bigg\|_{L^{2}(I)} \bigg\| \frac{\partial}{\partial x}u(t) \bigg\|_{L^{2}(I)}, \end{align*} $$

for a suitable constant C. The term on the right-hand side can be estimated, using Young’s inequality, by

$$ \begin{align*} \frac{2}{\alpha} C^{2} \| u(t) \|_{L^{2}(I)}^{2} \bigg\| \frac{\partial}{\partial x}\hat{u}(t) \bigg\|_{L^{2}(I)}^{2} + \frac{\alpha}{2} \bigg\| \frac{\partial}{\partial x}u(t) \bigg\|_{L^{2}(I)}^{2}. \end{align*} $$

Subtracting the term $({\alpha }/{2}) \| {(\partial }/{\partial x)}u(t) \|_{L^{2}(I)}^{2}$ on both sides and using the a priori estimates (2.4), we get the inequality

(2.8) $$ \begin{align} \frac{1}{2}\frac{d}{dt}\|u(t)\|_{L^{2}(I)}^{2} +\frac{\alpha}{2} \bigg\| \frac{\partial}{\partial x}u(t) \bigg\|_{L^{2}(I)}^{2} \leq C^{*} \| u(t) \|_{L^{2}(I)}^{2}, \end{align} $$

where $C^{*}=2C^{2}C_{1}/\alpha $ . Since $\bar {u}(0)=\hat {u}(0)=u_{0}$ , by the inequality (2.8) and Gronwall’s lemma (see, for example, [Reference Piccinini, Stampacchia and Vidossich13, Theorem 1.8], we obtain the uniqueness of the solution.