Vanishing-regularization limit of the Catté–Lions–Morel–Coll selective-smoothing equation and well-posedness of the unregularized Perona–Malik equation

The 1992 paper of Catté, Lions, Morel and Coll stabilized the Perona–Malik edge-preserving diffusion model by evaluating the diffusivity on the gradient of a Gaussian-smoothed image, proving existence, uniqueness and smoothness of the solution for every positive smoothing scale. The central degenerate regime is the limit in which the smoothing scale tends to zero: the model then formally reduces to the unregularized Perona–Malik equation, a forward-backward parabolic equation that is classically ill-posed, and whether the regularized evolutions converge to a well-defined, stable evolution — with approach to step-function (piecewise-constant) image states, as suggested by Kichenassamy's result that the Heaviside step function is a stationary generalized solution of the unregularized equation — remains open in the subsequent literature. That literature — non-existence of global classical solutions, severe non-uniqueness of weak solutions, Young-measure and generalized-solution concepts, and convergence theorems for alternative regularizations — identifies as the surviving open core the construction (or rigorous impossibility) of a vanishing-scale limit theory for the convolution regularization itself, with the qualitative edge, staircasing and piecewise-constant behavior of the limit evolution as motivating targets. No such theory is known in any space dimension; the problem remains open.

Background

Since the 1980s, multiscale image smoothing has been modeled by diffusion equations: the heat equation applied to a noisy image for a time $t$ is equivalent to convolution with a Gaussian of width proportional to $t^{1/2}$ and produces the classical linear "scale space", but it blurs perceptually important edges. Perona and Malik (1990) proposed replacing the constant diffusivity by a decreasing function of the local gradient magnitude, $g:[0,\infty)\to(0,1]$ with $g(0)=1$ and $g(s)\to 0$ as $s\to\infty$ (the standard choice is $g(s)=1/(1+(s/\lambda)^{2})$ with contrast parameter $\lambda>0$), giving the nonlinear diffusion equation $\partial_{t}u=\operatorname{div}(g(|\nabla u|)\nabla u)$ with homogeneous Neumann boundary conditions: small gradients (object interiors, noise) diffuse fast, while large gradients (edges) diffuse slowly and are even sharpened. Writing the one-dimensional equation as $u_{t}=\Phi'(u_{x})\,u_{xx}$ with flux $\Phi(s)=s\,g(s)$, one sees that $\Phi'(s)<0$ wherever $|s|$ exceeds the threshold $s_{*}>0$ at which the flux $\Phi$ attains its maximum (for the standard diffusivity, $s_{*}=\lambda$), so the equation becomes a backward heat equation there; initial data are called subcritical when $|\nabla u_{0}|\le s_{*}$ everywhere, and transcritical when the gradient magnitude lies below $s_{*}$ in some regions and above it in others, so that forward and backward diffusion coexist. Such forward-backward parabolic equations are classically ill-posed: Höllig and Nohel (1983) had already exhibited a diffusion equation with a nonmonotone constitutive function possessing infinitely many solutions. Nevertheless, numerical implementations of the Perona–Malik model are strikingly stable and effective at denoising while keeping edges sharp; this clash between analysis and computation was named the Perona–Malik paradox by Kichenassamy (1997).

Catté, Lions, Morel and Coll (1992) — "CLMC" — proposed restoring well-posedness by evaluating the diffusivity not on the raw gradient but on the gradient of the image pre-smoothed with a Gaussian: their selective-smoothing equation is $\partial_{t}u=\operatorname{div}(g(|\nabla G_{\sigma}*u|)\nabla u)$, where $G_{\sigma}$ is the Gaussian kernel of scale $\sigma$ ($G_{0}$ being the Dirac delta) and $*$ denotes convolution. They proved (their Theorem 2.1) that for every $\sigma>0$, initial data $u_{0}\in L^{2}$ on the unit square and Neumann conditions, this equation has a unique solution in $C([0,T];L^{2})\cap L^{2}(0,T;H^{1})$ that is $C^{\infty}$ for positive times; they gave a convergent fixed-point iteration, an unconditionally stable implicit finite-difference scheme, and numerical experiments with ad hoc choices of stopping time and threshold. A companion paper (Alvarez, Lions and Morel, 1992) treated a geometric variant based on mean curvature motion. Several related questions surround this theory: a principled rule for the stopping time, a theory for choosing the diffusivity and its threshold, spatially anisotropic (tensor-valued) diffusion that better preserves corners (as in the coupled anisotropic system of Nitzberg and Shiota, 1992, or the biased forced model of Nordström, 1990), forcing terms of Nordström type $u_{0}-u$, and — most fundamentally — the degenerate limit $\sigma\to 0$: as the Gaussian tends to the Dirac delta, the model formally becomes the Perona–Malik equation, and whether the regularized evolutions converge to a well-defined limit evolution approaching step-function (piecewise-constant) states remains open in the subsequent literature (Gobbino and Picenni, 2024).

The later mathematical literature developed several notions needed to state the current frontier. A generalized (ultraweak) solution is a limit of Lipschitz functions whose equation residuals tend to zero in the sense of distributions; Kichenassamy (1997) showed that the Heaviside step function is a stationary solution of this kind. Young-measure solutions (Chen and Zhang, 2006) describe limits whose gradients are only statistically determined. Functions of bounded variation (BV) are integrable functions whose distributional derivative is a finite measure — the natural class for piecewise-constant images. Convex integration (Kim and Yan, 2015) is a differential-inclusions technique that produces infinitely many "wild" weak solutions. Staircasing denotes the experimentally observed tendency of Perona–Malik-type evolutions to develop piecewise-constant regions. Known partial results on the unregularized equation include: non-existence of global $C^{1}$ solutions for one-dimensional transcritical initial data (Kawohl and Kutev, 1998; Gobbino, 2007); density of local classical solutions in one dimension and existence of global $C^{2,1}$ radial transcritical solutions in dimension at least two (Ghisi and Gobbino, 2009, 2011); infinite weak-solution families in one dimension (Zhang, 2006) and in all dimensions (Kim and Yan, 2015); and a nearly complete slow-time description of the one-dimensional semidiscrete (finite-difference) scheme, whose generalized solutions are limits of discrete evolutions and evolve piecewise-constant functions by moving their plateaus vertically according to a system of ordinary differential equations (Colombo and Gobbino, 2011; Gobbino and Picenni, 2024). For the convolution regularization itself no uniform-in-$\sigma$ a priori estimates are known; estimates of this kind have been established only for the one-dimensional semidiscrete scheme (Gobbino and Picenni, 2024).

Problem Statement

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded Lipschitz image domain (the problem is equally meaningful in general dimension $n\ge 1$, the two-dimensional case being that of the originating paper), let $u_{0}$ be an initial image (the admissible data class is part of the answer; in the default two-dimensional case it should at least include data of bounded variation, which on bounded domains lie in the $L^{2}$ class covered by the Catté–Lions–Morel–Coll theory), and let $g$ be a smooth decreasing diffusivity with $g(0)=1$ and $g(+\infty)=0$, such as $g(s)=1/(1+(s/\lambda)^{2})$. For each $\sigma>0$, let $u_{\sigma}$ be the unique CLMC solution of

$$\partial_{t}u_{\sigma}=\operatorname{div}\big(g(|\nabla G_{\sigma}*u_{\sigma}|)\,\nabla u_{\sigma}\big),\qquad \partial_{n}u_{\sigma}=0 \ \text{on}\ \partial\Omega,\qquad u_{\sigma}(0)=u_{0},$$

whose existence and uniqueness is guaranteed by CLMC's Theorem 2.1 (here $G_{\sigma}$ is the Gaussian kernel of scale $\sigma$ and $*$ denotes convolution). As $\sigma\to 0$ the equation degenerates formally to the Perona–Malik equation $\partial_{t}u=\operatorname{div}(g(|\nabla u|)\nabla u)$, which is forward-backward parabolic wherever the flux derivative $\Phi'(s)=(s\,g(s))'$ is negative.

The open problem is the construction — or the rigorous impossibility — of a satisfactory vanishing-scale limit theory for this degenerate regime, where "satisfactory" has the precise, standard meaning in the literature (Gobbino and Picenni, 2024): a notion of solution for the unregularized equation that (i) exists for a reasonable class of initial data, (ii) depends in a reasonable way on the initial condition, and (iii) to which solutions of approximating models converge. A satisfactory limit theory would consist of a solution concept together with an existence theorem covering the limits of the CLMC evolutions $u_{\sigma}$ as $\sigma\to 0$ (subsequential or full limits, in a stated topology); a stability or uniqueness statement strong enough to exclude the known infinite families of wild weak solutions (Zhang, 2006; Kim and Yan, 2015) yet weak enough to admit transcritical data — data whose gradient magnitude lies below the threshold $s_{*}$ (the maximum point of the flux $\Phi(s)=s\,g(s)$) in some regions and above it in others — for which global classical solutions provably do not exist; and a convergence theorem for the CLMC convolution regularization itself. Qualitative features of the expected limit evolution — edge formation and preservation, whether evolutions approach piecewise-constant (step-function-like) states near large gradients, the presence or absence of staircasing, and consistency with the classical subcritical regime in which $|\nabla u_{0}|$ lies below the threshold everywhere and global classical solutions exist — motivate the problem and are natural objects for a limit theory to address. Rigorous impossibility could likewise be established, for example, by constructing a single initial datum for which the CLMC evolutions admit two genuinely different subsequential $\sigma\to 0$ limits, or by proving that no solution concept satisfying (i)–(iii) simultaneously can exist. Results restricted to one space dimension, to special data, or to other regularization families (semidiscrete schemes, higher-order singular perturbations, nonlocal or time-delayed regularizations) constitute recognized partial progress toward such a theory while leaving the general problem open.

References

1. Francine Catté, Pierre-Louis Lions, Jean-Michel Morel, Tomeu Coll, "Image Selective Smoothing and Edge Detection by Nonlinear Diffusion", SIAM Journal on Numerical Analysis 29(1):182–193, 1992. DOI 10.1137/0729012, https://doi.org/10.1137/0729012
2. Pietro Perona, Jitendra Malik, "Scale-space and edge detection using anisotropic diffusion", IEEE Transactions on Pattern Analysis and Machine Intelligence 12(7):629–639, 1990. DOI 10.1109/34.56205, https://doi.org/10.1109/34.56205
3. Niklas Nordström, "Biased anisotropic diffusion: a unified regularization and diffusion approach to edge detection", Image and Vision Computing 8(4):318–327, 1990. DOI 10.1016/0262-8856(90)80008-H, https://doi.org/10.1016/0262-8856(90)80008-H
4. M. Nitzberg, T. Shiota, "Nonlinear image filtering with edge and corner enhancement", IEEE Transactions on Pattern Analysis and Machine Intelligence 14(8):826–833, 1992. DOI 10.1109/34.149593, https://doi.org/10.1109/34.149593
5. Klaus Höllig, John A. Nohel, "A Diffusion Equation with a Nonmonotone Constitutive Function", in Systems of Nonlinear Partial Differential Equations, pp. 409–422, 1983. DOI 10.1007/978-94-009-7189-9_26, https://doi.org/10.1007/978-94-009-7189-9_26
6. Luis Alvarez, Pierre-Louis Lions, Jean-Michel Morel, "Image Selective Smoothing and Edge Detection by Nonlinear Diffusion. II", SIAM Journal on Numerical Analysis 29(3):845–866, 1992. DOI 10.1137/0729052, https://doi.org/10.1137/0729052
7. Satyanad Kichenassamy, "The Perona–Malik Paradox", SIAM Journal on Applied Mathematics 57(5):1328–1342, 1997. DOI 10.1137/S003613999529558X, https://doi.org/10.1137/S003613999529558X
8. Bernd Kawohl, Nikolai Kutev, "Maximum and comparison principle for one-dimensional anisotropic diffusion", Mathematische Annalen 311(1):107–123, 1998. DOI 10.1007/s002080050179, https://doi.org/10.1007/s002080050179
9. Massimo Gobbino, "Entire Solutions of the One-Dimensional Perona-Malik Equation", Communications in Partial Differential Equations 32(5):719–743, 2007. DOI 10.1080/03605300600634981, https://doi.org/10.1080/03605300600634981
10. Marina Ghisi, Massimo Gobbino, "A class of local classical solutions for the one-dimensional Perona-Malik equation", Transactions of the American Mathematical Society 361(12):6429–6446, 2009. DOI 10.1090/S0002-9947-09-04793-X, https://doi.org/10.1090/S0002-9947-09-04793-X
11. Marina Ghisi, Massimo Gobbino, "An Example of Global Classical Solution for the Perona-Malik Equation", Communications in Partial Differential Equations 36(8):1318–1352, 2011. DOI 10.1080/03605302.2010.542672, https://doi.org/10.1080/03605302.2010.542672 (preprint arXiv:0907.0772, https://arxiv.org/abs/0907.0772)
12. Marina Ghisi, Massimo Gobbino, "On the evolution of subcritical regions for the Perona–Malik equation", Interfaces and Free Boundaries 13(1):105–125, 2011. DOI 10.4171/ifb/250, https://doi.org/10.4171/ifb/250 (preprint arXiv:0909.0219, https://arxiv.org/abs/0909.0219)
13. Marina Ghisi, Massimo Gobbino, "Gradient estimates for the Perona–Malik equation", Mathematische Annalen 337(3):557–590, 2007. DOI 10.1007/s00208-006-0047-1, https://doi.org/10.1007/s00208-006-0047-1
14. Selim Esedoğlu, "An analysis of the Perona-Malik scheme", Communications on Pure and Applied Mathematics 54(12):1442–1487, 2001. DOI 10.1002/cpa.3008, https://doi.org/10.1002/cpa.3008
15. Selim Esedoğlu, "Stability Properties of the Perona-Malik Scheme", SIAM Journal on Numerical Analysis 44(3):1297–1313, 2006. DOI 10.1137/S0036142903424817, https://doi.org/10.1137/S0036142903424817
16. Kewei Zhang, "Existence of infinitely many solutions for the one-dimensional Perona-Malik model", Calculus of Variations and Partial Differential Equations 26(2):171–199, 2006. DOI 10.1007/s00526-005-0363-4, https://doi.org/10.1007/s00526-005-0363-4
17. Yan Chen, Kewei Zhang, "Young measure solutions of the two-dimensional Perona-Malik equation in image processing", Communications on Pure and Applied Analysis 5(3):617–637, 2006. DOI 10.3934/cpaa.2006.5.617, https://doi.org/10.3934/cpaa.2006.5.617
18. Seonghak Kim, Baisheng Yan, "Convex Integration and Infinitely Many Weak Solutions to the Perona-Malik Equation in All Dimensions", SIAM Journal on Mathematical Analysis 47(4):2770–2794, 2015. DOI 10.1137/15M1012220, https://doi.org/10.1137/15M1012220 (preprint arXiv:1503.00772, https://arxiv.org/abs/1503.00772)
19. Seonghak Kim, Baisheng Yan, "Radial weak solutions for the Perona-Malik equation as a differential inclusion", Journal of Differential Equations 258(6):1889–1932, 2015. DOI 10.1016/j.jde.2014.11.017, https://doi.org/10.1016/j.jde.2014.11.017 (preprint arXiv:1309.6646, https://arxiv.org/abs/1309.6646)
20. Patrick Guidotti, "A New Well-posed Nonlocal Perona-Malik Type Equation", preprint, 2007. arXiv:math/0702715, https://arxiv.org/abs/math/0702715
21. Patrick Guidotti, Yuanzhen Shao, "Wellposedness of a nonlocal nonlinear diffusion equation of image processing", Nonlinear Analysis: Theory, Methods and Applications 150:114–137, 2017. DOI 10.1016/j.na.2016.11.003, https://doi.org/10.1016/j.na.2016.11.003 (preprint arXiv:1604.02319, https://arxiv.org/abs/1604.02319)
22. Maria Colombo, Massimo Gobbino, "Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow", Mathematical Models and Methods in Applied Sciences 22(8):1250017, 2012. DOI 10.1142/S0218202512500170, https://doi.org/10.1142/S0218202512500170 (preprint arXiv:1103.3365, https://arxiv.org/abs/1103.3365)
23. Maria Colombo, Massimo Gobbino, "Slow Time Behavior of the Semidiscrete Perona-Malik Scheme in One Dimension", SIAM Journal on Mathematical Analysis 43(6):2564–2600, 2011. DOI 10.1137/100818698, https://doi.org/10.1137/100818698 (preprint arXiv:1012.4406, https://arxiv.org/abs/1012.4406)
24. Massimo Gobbino, Nicola Picenni, "Monotonicity Properties of Limits of Solutions to the Semidiscrete Scheme for a Class of Perona-Malik Type Equations", SIAM Journal on Mathematical Analysis 56(2):2034–2062, 2024. DOI 10.1137/23M1569873, https://doi.org/10.1137/23M1569873 (preprint arXiv:2304.04729, https://arxiv.org/abs/2304.04729)
25. Massimo Gobbino, Nicola Picenni, "A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functional", Transactions of the American Mathematical Society 376(8):5307–5375, 2023. DOI 10.1090/tran/8841, https://doi.org/10.1090/tran/8841 (preprint arXiv:2205.02467, https://arxiv.org/abs/2205.02467)
26. Massimo Gobbino, Nicola Picenni, "Multi-scale analysis of minimizers for a second order regularization of the Perona-Malik functional", Calculus of Variations and Partial Differential Equations 64(5):156, 2025. DOI 10.1007/s00526-025-03013-6, https://doi.org/10.1007/s00526-025-03013-6 (preprint arXiv:2311.14565, https://arxiv.org/abs/2311.14565)
27. Nicola Picenni, "Staircasing effect for minimizers of the one-dimensional discrete Perona-Malik functional", ESAIM: Control, Optimisation and Calculus of Variations 30:44, 2024. DOI 10.1051/cocv/2024035, https://doi.org/10.1051/cocv/2024035 (preprint arXiv:2306.08652, https://arxiv.org/abs/2306.08652)
28. Giovanni Bellettini, Antonin Chambolle, Michael Goldman, "The Γ-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension", Mathematical Models and Methods in Applied Sciences 24(6):1091–1113, 2014. DOI 10.1142/S0218202513500772, https://doi.org/10.1142/S0218202513500772 (preprint arXiv:1301.5170, https://arxiv.org/abs/1301.5170)
29. Andrea Braides, Valerio Vallocchia, "Static, Quasistatic and Dynamic Analysis for Scaled Perona-Malik Functionals", Acta Applicandae Mathematicae 156(1):79–107, 2018. DOI 10.1007/s10440-018-0155-4, https://doi.org/10.1007/s10440-018-0155-4 (preprint arXiv:1610.08295, https://arxiv.org/abs/1610.08295)
30. Maciek Wielgus, "Perona-Malik equation and its numerical properties", survey, 2014. arXiv:1412.6291, https://arxiv.org/abs/1412.6291
31. Pavel Mrázek, Mirko Navara, "Selection of Optimal Stopping Time for Nonlinear Diffusion Filtering", International Journal of Computer Vision 52(2/3):189–203, 2003. DOI 10.1023/A:1022908225256, https://doi.org/10.1023/A:1022908225256
32. Joachim Weickert, "Coherence-Enhancing Diffusion Filtering", International Journal of Computer Vision 31(2/3):111–127, 1999. DOI 10.1023/A:1008009714131, https://doi.org/10.1023/A:1008009714131
33. Lars M. Mescheder, Dirk A. Lorenz, "An extended Perona-Malik model based on probabilistic models", Journal of Mathematical Imaging and Vision 60(1):128–144, 2017. DOI 10.1007/s10851-017-0746-0, https://doi.org/10.1007/s10851-017-0746-0 (preprint arXiv:1612.06176, https://arxiv.org/abs/1612.06176)
34. Y. Chen, C. A. Z. Barcelos, B. A. Mair, "Smoothing and Edge Detection by Time-Varying Coupled Nonlinear Diffusion Equations", Computer Vision and Image Understanding 82(2):85–100, 2001. DOI 10.1006/cviu.2001.0903, https://doi.org/10.1006/cviu.2001.0903
35. Yihui Tong, Wenjie Liu, Zhichang Guo, Wenjuan Yao, "A class of forward-backward regularizations of the Perona-Malik equation with variable exponent", Zeitschrift für angewandte Mathematik und Physik 77(9):254, 2026. DOI 10.1007/s00033-026-02908-4, https://doi.org/10.1007/s00033-026-02908-4 (preprint arXiv:2510.23982, https://arxiv.org/abs/2510.23982)
36. Javier Sanguino, Carlos Platero, Olga Velasco, "Pre-process for segmentation task with nonlinear diffusion filters", preprint, 2026. arXiv:2604.21422, https://arxiv.org/abs/2604.21422