Infinitesimal generators of the Alvarez–Guichard–Lions–Morel axiomatic scale spaces: existence without a regularity axiom, and how far it can be weakened

The axiomatic framework of Alvarez, Guichard, Lions and Morel (1993) derives the fundamental equations of image processing from elementary axioms imposed on a multiscale analysis: a one-parameter family of image operators satisfying the semigroup (recursivity) law, translation invariance, grey-scale-shift invariance, a comparison principle, continuity of the scale-space map, and locality. Under these structural axioms plus a local regularity estimate, the framework proves that the analysis admits an infinitesimal generator of the form $A[u]=F(D^2u,Du)$ with $F$ degenerate elliptic, so every axiom-compatible multiscale analysis is the viscosity-solution flow of a fully nonlinear parabolic PDE. The sharpest question the framework left open is whether the structural axioms alone, with no regularity-type estimate, already force the existence of an infinitesimal generator and hence the PDE representation — and if not, how far below the original estimate the regularity hypothesis can be weakened while the theorem is retained? This is a boundary problem in nonlinear monotone semigroup theory on lattices of continuous functions. The problem is open.

Background

In mathematical image processing, a multiscale analysis (or scale space) is a one-parameter family of operators $(T_t)_{t\ge 0}$ acting on images, viewed as real-valued functions on $\mathbb{R}^N$; $T_tu$ is the image observed at scale $t$, and increasing $t$ progressively simplifies it. The axiomatic program asks which evolutions can be derived from a short list of elementary principles — invariance and physical-admissibility requirements — rather than postulated. For the linear case, causality-type axioms single out Gaussian smoothing; the nonlinear theory was founded by the 1993 paper of Alvarez, Guichard, Lions and Morel (AGLM), which derives the fundamental nonlinear equations of image processing from axioms on $(T_t)$.

AGLM state their axioms for $(T_t)$ acting on the space of bounded uniformly continuous functions. The structural axioms are: (i) recursivity (the semigroup law), $T_0=\mathrm{Id}$ and $T_{t+s}=T_t\circ T_s$, so that passing from scale $s$ to scale $s+t$ always acts the same way; (ii) translation invariance, $T_t$ commutes with every spatial shift of the image; (iii) grey-scale-shift invariance, $T_t(u+c)=T_tu+c$ for every real constant $c$, expressing that only relative grey levels matter; (iv) the comparison principle, $u\le v$ pointwise implies $T_tu\le T_tv$, i.e. each operator is order-preserving (monotone) on the lattice of images; (v) continuity, the scale-space map $(t,x)\mapsto T_t[u_0]\,(x)$ is continuous for every initial image $u_0$; and (vi) locality, if two smooth images $f,g$ agree on a neighborhood of $x$, then $T_t[f]\,(x)-T_t[g]\,(x)=o(t)$ as $t\downarrow 0$, i.e. the analysis reacts at first order only to the local germ of the image. The publisher abstract of the 1993 paper classifies requirements such as locality, recursivity and causality as "architectural requirements" and the comparison principle as a "stability requirement"; continuity and locality appear among the hypotheses of Biton (2001) as well, whose own assumptions are stated to be "more or less slightly weak versions of these used in" the 1993 paper. To the structural axioms the paper adds a regularity axiom, a local estimate $\|T_t(f+hg)-(T_tf+hg)\|_\infty\le Cht$ for small $t$ and $h$, which bounds how sensitively the analysis reacts to adding a small multiple $h$ of a smooth function $g$ to an image $f$; Biton (2001) relaxes this estimate to $\le m(\lambda)\,t$ with an arbitrary modulus $m$ ($m(0^+)=0$) and compactly supported smooth $g$.

Under the full axiom set, the generator theorem of AGLM states that $(T_t)$ admits an infinitesimal generator — the instantaneous velocity $A[u]=\lim_{t\downarrow 0}(T_tu-u)/t$ of the image at zero scale, well-defined for smooth images $u$ — and that $A$ has the local form

$$A[u]=F(D^2u,Du),$$

where $F$ is degenerate elliptic (nondecreasing in its matrix argument under the symmetric matrix ordering) and continuous away from $Du=0$. Because a one-parameter semigroup obeying the recursivity law has a time-homogeneous generator, $F$ carries no explicit $t$-argument; explicitly time-dependent functions $F$ arise only for two-parameter families $(T_{t,s})$. Degenerate ellipticity is exactly the condition under which viscosity solution theory — the notion of generalized solution for fully nonlinear degenerate PDEs, defined through sub- and supersolution inequalities at contact points with smooth test functions — gives existence, uniqueness and continuous dependence for $\partial_tu=F(D^2u,Du)$; the semigroup is then identified with the viscosity solution flow. This theorem is what turns axiom systems into PDEs: adding further invariances prunes the admissible $F$. Under contrast invariance (invariance under monotone grey-scale changes, also called morphological invariance) the generators reduce to the flat morphological operations of mathematical morphology — sup- and inf-convolutions with a structuring set (dilation and erosion) — and adding affine invariance of the image plane yields, after scale normalization, the affine morphological scale space $\partial_tu=|\nabla u|\operatorname{curv}(u)^{1/3}$, where $\operatorname{curv}(u)$ is the curvature of the level line of $u$ through the point; this is the scale-space counterpart of the affine curve-shortening flow of Sapiro and Tannenbaum.

The load-bearing step is the generator theorem, and its proof uses the regularity axiom in an essential way: per Biton's restatement, AGLM obtain the existence of the nonlinear infinitesimal generator "using only the regularity assumption on the semigroup together with a contraction property", and then obtain the local degenerate-elliptic representation "using the monotonicity and the locality assumption". The structural axioms carry no such quantitative infinitesimal estimate, although continuity and locality do impose qualitative regularity content. Whether the structural axioms alone already force the existence of an infinitesimal generator — necessarily of local degenerate-elliptic form — or whether genuinely pathological monotone semigroups satisfying them can defeat generator existence, is a question of nonlinear monotone semigroup theory that the 1993 paper does not settle: its generator theorem assumes a regularity estimate rather than deriving it from the structural axioms. The direction pursued by the direct follow-up literature is weakening: Biton (2001) replaces the linear-modulus estimate by an arbitrary-modulus estimate and generalizes the underlying function spaces, but still retains a regularity-type assumption; whether regularity can be dispensed with altogether, and how far it can be weakened, remains open. Classical linear semigroup theory (Hille–Yosida type, which characterizes the generators of strongly continuous contraction semigroups on Banach spaces) and its nonlinear extensions of Crandall–Liggett type (which build semigroups from accretive operators, i.e. operators satisfying a one-sided metric contraction estimate) do not apply directly, since they presuppose norm-continuity or accretivity structure that the image axioms neither provide nor trivially imply. The known partial results are the following: the direct extension by Biton (2001) weakens the hypotheses but retains regularity-type assumptions, and adjacent work on convex monotone semigroups (published 2023, from a 2020–2021 preprint, and 2026) shows that generator questions of this kind remain subtle without answering them for the image axioms.

Problem Statement

Let $X$ be a Banach lattice of real-valued continuous functions on $\mathbb{R}^N$ — that is, a real Banach space of functions that is simultaneously a lattice under the pointwise order, with a norm compatible with that order — and assume $X$ contains the smooth compactly supported functions; for definiteness, take the space of bounded uniformly continuous functions used in the 1993 source, or the space of continuous functions with prescribed growth at infinity used in the 2001 extension. Let $(T_t)_{t\ge 0}$ be a family of operators on $X$ satisfying the structural axioms of the 1993 framework:

(i) recursivity: $T_0=\mathrm{Id}$ and $T_{t+s}=T_t\circ T_s$ for all $s,t\ge 0$;

(ii) translation invariance: $T_t(u(\cdot-\xi))=(T_tu)(\cdot-\xi)$ for every spatial shift $\xi$ of $\mathbb{R}^N$;

(iii) grey-scale-shift invariance: $T_t(u+c)=T_tu+c$ for every real constant $c$;

(iv) comparison principle: $u\le v$ pointwise implies $T_tu\le T_tv$;

(v) continuity: for every $u_0\in X$, the map $(t,x)\mapsto T_t[u_0]\,(x)$ is continuous on $[0,\infty)\times\mathbb{R}^N$;

(vi) locality: for every smooth $f,g$, if $f\equiv g$ on a neighborhood of $x$, then $T_t[f]\,(x)-T_t[g]\,(x)=o(t)$ as $t\downarrow 0$.

Hypotheses (v) and (vi) belong to the original 1993 framework: locality is listed among the "architectural requirements" in the publisher abstract of the 1993 paper, and Biton (2001) states his assumptions — including continuity and locality, under the same terminology — to be "more or less slightly weak versions of these used in" that paper.

Say that an infinitesimal generator exists if $A[f]=\lim_{t\downarrow 0}(T_tf-f)/t$ exists for every smooth $f\in X$, uniformly on compact sets — the convergence mode of the 1993 and 2001 generator theorems, in which the difference quotients form a Cauchy sequence for smooth data.

Question: is a regularity-type hypothesis genuinely necessary in the generator theorem?

(A) Sufficiency. Prove that structural axioms (i)–(vi) alone, with no regularity estimate, imply that the infinitesimal generator exists in the above sense and has the local form $A[f]\,(x)=F(D^2f(x),Df(x))$ with $F$ degenerate elliptic and continuous away from $Df=0$, so that $u(t,x)=T_t[u_0]\,(x)$ is the viscosity solution of $\partial_tu=F(D^2u,Du)$ for every $u_0\in X$.

(B) Necessity. Construct an explicit family $(T_t)$ satisfying all of (i)–(vi) for which this conclusion fails — the generator does not exist in the stated sense, or is not of local degenerate-elliptic form — thereby showing that some regularity-type hypothesis is genuinely needed.

If (B) holds, the answer should determine how far the regularity hypothesis can be weakened — with the weakest such hypothesis (the sharpest modulus class) as the target — below the 1993 regularity axiom $\|T_t(f+hg)-(T_tf+hg)\|_\infty\le Cht$ and below the relaxation of Biton (2001), $|T_t[f+\lambda g]\,(x)-T_t[f]\,(x)-\lambda g(x)|\le m(\lambda)\,t$ with $m$ an arbitrary modulus ($m(0^+)=0$) and $g$ smooth and compactly supported, while retaining the generator theorem and the degenerate-elliptic PDE representation.

No restriction on the dimension $N$, on the specific lattice $X$ beyond the above, or on the form of the answer is intended.

References

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