The authors consider extensions of the 1D Perona-Malik functional of the type \(\mathbb{PMF}_{\varepsilon }(\beta ,f,\Omega ,u)=\int_{\Omega }\{\varepsilon ^{6}\omega (\varepsilon )^{4}u^{\prime \prime }(x)^{2}+\log (1+u^{\prime }(x)^{2})+\beta (u(x)-f(x))^{2}\}dx\), where \(\beta >0\) is a real number, \(\Omega \subseteq \mathbb{R}\) an open set, \(\varepsilon \in (0,1)\), and \(f\in L^{2}(\Omega )\) a given function called forcing term. The authors first prove the existence of minimizers to the problem \(\min\{\mathbb{ PMF}_{\varepsilon }(\beta ,f,(0,1),u):u\in H^{2}(0,1)\}=m(\varepsilon ,\beta ,f)\). Every minimizer to this problem is proved to belong to \(H^{4}(0,1)\) and in particular to \(C^{2}([0,1])\). When \(\varepsilon \rightarrow 0^{+}\), the minimum \(m(\varepsilon ,\beta ,f)\) converges to \(0\). The proof is a standard application of the direct method in the calculus of variations and of the fact that the convex envelope of the function \(p\rightarrow \log(1+p^{2})\) is identically equal to \(0\).
The first main result proves that if \(\omega (\varepsilon )=\varepsilon \left\vert \log \varepsilon \right\vert ^{1/2}\) and \(f\in C^{1}([0,1])\), the minimum value \( m(\varepsilon ,\beta ,f)\) satisfies \(\lim_{\varepsilon \rightarrow 0^{+}} \frac{m(\varepsilon ,\beta ,f)}{\omega (\varepsilon )^{2}}=10(\frac{2\beta }{ 27})^{1/5}\int_{0}^{1}\left\vert f^{\prime }(x)\right\vert ^{4/5}dx\). The second main result proves blow-up properties of minimizers \(u_{\varepsilon }\) at standard resolution, the sequences \(w_{\varepsilon }(y)=\frac{ u_{\varepsilon }(x_{\varepsilon }+\omega (\varepsilon )y)-f(x_{\varepsilon }) }{\omega (\varepsilon )}\) and \(v_{\varepsilon }(y)=\frac{u_{\varepsilon }(x_{\varepsilon }+\omega (\varepsilon )y)-u_{\varepsilon }(x_{\varepsilon }) }{\omega (\varepsilon )}\) being relatively compact in the sense of locally of locally strict convergence and all their limit points being suitable staircases. These \(w_{\varepsilon }\) and \(v_{\varepsilon }\) correspond to zooms of the graph of a minimizer \(u_{\varepsilon }\) in a neighborhood of \( (x_{\varepsilon },f(x_{\varepsilon })\) and \((x_{\varepsilon },u_{\varepsilon }(x_{\varepsilon }))\) at scale \(\omega (\varepsilon )\). A sequence of functions \(\{u_{n}\}\subseteq BV_{loc}(\mathbb{R})\) converges locally strictly to some \(u_{\infty }\in BV_{loc}(\mathbb{R})\) if \( u_{n}\) converges to \(u_{\infty }\) in \(BV(a,b)\) for every interval \( (a,b)\subseteq \mathbb{R}\) whose endpoints are not jump points of the limit \( u_{\infty }\). The authors also prove that the family \(\{u_{\varepsilon }\}\) of minimizers converges to \(f\) in the strict sense and also as varifolds.
In a further part of their paper, the authors consider the functional defined on \(H^{2}(\Omega )\) by \(\mathbb{RMF}_{\varepsilon }(a,b,u)=\int_{a}^{b}\{\varepsilon ^{6}u^{\prime \prime }(x)^{2}+\frac{1}{ \omega (\varepsilon )^{2}}\log (1+u^{\prime }(x)^{2})\}dx\) and the functional \(\mathbb{J}^{1/2}(a,b,u)=\sum_{s\in S_{u}\cap (a,b)}\left\vert \lim_{x\rightarrow s^{+}}u(x)-\lim_{x\rightarrow s^{-}}u(x)\right\vert ^{1/2} \), defined for every pure jump function that is having a finite or countable jump set \(S_{u}\).Both functionals are extended by \(+\infty \) to \(L^{2}(a,b) \). They prove that the \(\Gamma\)-\(\lim_{\varepsilon \rightarrow 0^{+}}\mathbb{RMF}_{\varepsilon }(a,b,u)\) is equal to \(\frac{16}{\sqrt{3}}\mathbb{J} ^{1/2}(a,b,u)\) for every \(u\in L^{2}(a,b)\), with respect to the metric of \( L^{2}(a,b)\). They also prove a compactness result for minimizers and the strict convergence of recovery sequences. The paper ends with some perspectives and open problems.