POINT-WISE BEHAVIOR OF THE GEMAN–McCLURE AND THE HEBERT–LEAHY IMAGE RESTORATION MODELS

We present new continuous variants of the Geman–McClure model and the Hebert–Leahy model for image restoration, where the energy is given by the nonconvex function x 7→ x2/(1+x2) or x 7→ log(1+x2), respectively. In addition to studying these models’ Γ-convergence, we consider their point-wise behaviour when the scale of convolution tends to zero. In both cases the limit is the Mumford-Shah functional.


Introduction.
A basic denoising problem encountered in image processing is obtaining an estimate u for an unknown image u 0 based on a corrupted observation f .In a variational approach, one seeks to solve this problem by minimizing an energy functional which typically consists of two parts, a fidelity term such as ´|u − f | 2 dx and a regularity term such as ´|∇u| dx.More generally, one can consider the class of problems: where Du denotes the gradient of the BV function u; if ϕ (t)  t → ϕ ∞ as t → ∞, then ˆΩ ϕ(|Du|) dx := ˆΩ ϕ(|∇ a u|) dx + ϕ ∞ |D s u|(Ω), where ∇ a u and D s u are the absolutely continuous and singular parts of Du.Note that the choice ϕ(t) = t leads to the Total Variation (ROF) denoising model of Rudin, Osher and Fatemi [24] in 1992.
In their discussion about variational approaches to image restoration, Aubert and Kornprobst [5, Section 3.2.6]point out that non-convex potentials such as ϕ(t) = t 2 1+t 2 give better results in numerical tests than convex potentials such as ϕ(t) = t.In fact, this choice of ϕ leads to a discrete model which was first proposed by Geman and McClure [18] in 1985, many years before the ROF model.This model has been used, for instance, in tomography, road scene analysis, magnetic resonance imaging, geophysical imaging, volume reconstruction, target detection, functional MRI and image segmentation.Hebert and Leahy [20] presented in 1989 a somewhat similar model whose potential is t → log(1 + t 2 ); in a differential equation form this potential corresponds to the Perona-Malik model.
However, it was shown by Chipot, March, Rosati and Vergara Caffarelli [11] that the functional corresponding to this choice of ϕ, (1) does not have a minimizer in BV and that the energy minimum of the functional equals 0. The corresponding discrete model (2) i where D h is the discrete gradient with step size h, performs very well, see [8,9,19,23].Rosati [23] has shown that this discrete model Γ-convergences to a modified Mumford-Shah functional as the step size tends to zero.This corresponds to sampling a continuous function at individual points.Another way to get from the continuous to the discrete is to take the average of a gradient of the function in some pixel.(After smoothing one can equivalently use integrals or sums.)This route was followed by Braides and Dal Maso [7] and Chambolle and Dal Maso [10]; more on this below.
We also follow an averaging approach.To be more precise, let G ∈ C 1 (R n ) be a non-negative, radial symmetric and decreasing function with spt G ⊂ B(0, 1) and For a (scalar-or vector valued) function u ∈ L 1 loc (R n ), we denote If u ∈ BV(R n ), then we denote and we write, for simplicity, for U ⊂⊂ Ω and σ > 0 sufficiently small that the convolution be defined.Further we set F σ (u) := F σ (u, Ω σ ) where Ω σ := {x ∈ Ω : d(x, ∂Ω) > σ}.
We extend F σ to GBV as a limit, and to L 1 (R n ) by ∞.For definitions of GBV and other terms, we refer to Section 2.
Assumptions 1.1.We need the following assumptions on g and : (G) g in increasing and concave, g(0) = 0 and lim (GL) lim Note that (G) yields that g(x) x, as the following deduction shows: Since g(0) = 0, concavity implies that g(ax) ag(x) for every a 1.
Note that ĝ is increasing in t and that ĝ(σ, t) t 2 .It turns out that the limit of this functional is intimately connected to the weak formulation of the Mumford-Shah (MS) functional.Let Ω ⊂ R n be a bounded open set and f ∈ L ∞ (Ω).The MS functional, in weak formulation and without fidelity term, is MS(u) := ˆΩ |∇ a u| 2 dx + 2H n−1 (S u ), for u ∈ GSBV(Ω) and MS(u) := ∞ for u ∈ L 1 (Ω) \ GSBV(Ω).This functional was introduced by De Giorgi and Ambrosio [16] to solve the existence problem for the strong (C 1 ) functional proposed by Mumford and Shah in 1989 [22].The lack of lower semicontinuity of H n−1 (S u ) means that it is difficult to deal with this functional and hence in many papers it has been approximated, in the sense of Γconvergence, by more regular functionals, see for example [4,14].For overviews of results on the MS functional we refer to [3,15].The Geman-McClure model is rather easy to deal with numerically, whereas the opposite is true for the Mumford-Shah model.Therefore there is a practical incentive to establish connections between the models.
Following Braides, Chambolle and Dal Maso [7,10], we study the Γ-convergence of our functional.We will need the following functional by Braides and Dal Maso [7]: we extend F BD σ to L 1 by ∞.Braides and Dal Maso have shown that F BD σ Γconverges to MS with respect to the L 1 -topology.In Section 5 we will use this to derive the Γ-converges of our functional in the case g(t) = t 1+t .Γ-convergence in the case g(t) = log(1 + t) has been considered by Tiirola in a separate paper [25].The previous result holds also for the previously known approximating functionals.However, the next result indicates that the new approximation is more precise, in that we also obtain point-wise convergence.In this sense this result is the main innovation and motivation of this paper.It is proved in Sections 3 and 4.
Theorem 1.3.Let Ω be a bounded open set.Suppose that u ∈ SBV 2 (Ω) and S u is contained H n−1 -a.e. in the union of a finite number of hyperplanes.Then Note that each u ∈ GSBV 2 (Ω) can be estimated by functions u i satisfying the conditions in the previous theorem such that MS(u i ) → MS(u).This result is by G. Cortesani [12,Corollary 3.11].
The functional lim σ→0 F BD σ (u) is finite only for H 1 -Sobolev functions and thus the previous theorem is not true for F BD σ .Therefore F σ is point-wise better behaved than F BD σ .The same is true for the functional of Chambolle and Dal Maso [10].
Remark 1.4.When sampling a continuous function by averaging in these type of models, there are three operations to be performed: convolution, derivative, and raising to the power two.By varying the point at which the convolution operator is applied, we arrive at three possible orders: derivative-power-convolution, derivative-convolution-power, and convolution-derivative-power.These give rise to the functionals ˆ1 σ g σ(|∇u| 2 ) σ dx, ˆ1 σ g σ(|∇u| σ ) 2 dx, and ˆ1 σ g σ|∇(u σ )| 2 dx.
The argument of the first is finite in H 1 , the second in BV, and the last in L 1 .Braides and Dal Maso opt for the first (largest), we consider the second one.Most closely related to the discrete model would be the third.Unfortunately, that model is not well behaved analytically.The model of Chambolle and Dal Maso [10] is of this type, but instead of the convolution u σ they consider regularizations which are piece-wise affine on a triangular grid of controlled geometry.The extra structure allows them to handle the complications arising.
In [6], Bourdin and Chambolle study the numerical implementation of the model from [10].In our case this remains for future work.We point out that the larger function space on which our functional is finite allows for many different approximation schemes.A natural starting point is with functions satisfying the condition of Theorem 1.3.On the theoretical side, it would be interesting to study whether the condition in the theorem can be removed.
Remark 1.5.Various indirect approaches have been proposed to deal with the Mumford-Shah functional, see [5,Section 4.2.4].In the terminology of Aubert and Kornprobst, our approach falls under the heading "Approximation by introducing non-local terms".

2.
Notation.By c we denote a generic constant whose value can change between each appearance.We denote the Lebesgue n-measure of (a measurable set) E ⊂ R n by |E|.The Hausdorff d-dimensional outer measure is denoted by H d .
We state some properties of BV-spaces that are needed.For further background and proofs see [3].Let Ω be an open set of R n and u ∈ L 1 (Ω), set and define the space of functions of bounded variation as We define S u as the subset of Ω where the function u does not have approximate limit: x ∈ Ω \ S u if and only if there exists z ∈ R such that lim |u(y) − z| dy = 0.
For u ∈ BV(Ω), the set S u is Borel, |S u | = 0 and where The distributional gradient Du can be divided into an absolutely continuous part ∇ a u (with respect to Lebesgue measure) and a singular part D s u.We say that u ∈ BV(Ω) is a special function of bounded variation and denote u ∈ SBV(Ω) if the singular part of the gradient D s u is concentrated on S u , i.e.D s u (Ω \ S u ) = 0.The space SBV was introduced by De Giorgi and Ambrosio [16].The spaces GBV and GSBV are defined to consist of functions u for which u λ belongs to BV and SBV, respectively, for all λ > 0 where u λ is the truncation of u at levels −λ and λ.We write The concepts of Γ-convergence, introduced by De Giorgi, has been systematically studied in [13].We present here only the definition.A family of functionals F σ : X → R is said to Γ-converge (in topology τ ) to F : X → R if the following hold for every positive sequence (σ i ) converging to zero: (a) for every u ∈ X and every sequence (u i ) ⊂ X τ -converging to u, we have (b) for every u ∈ X there exists (u i ) ⊂ X τ -converging to u such that The sequence in (b) is called the recovery sequence.
We conclude the introduction by showing that the Geman-McClure functions and the Hebert-Leahy functions satisfy the required conditions in the introduction.Proof.Condition (L) is clear.Since the second derivative of g is negative for every point in [0, ∞), the function is concave.Now (G) is clear.We check the (GL): for every fixed c > 0 and k > 1.
Proof.Condition (L) is clear.Since the second derivative of g is negative for every point in [0, ∞), the function is concave and (G) follows.For (GL) we note, with k = 2 or k = 2n, that The symbol ∼ means asymptotically equivalent.
3. Lower bound by the Hausdorff measure.In this section we start with an estimate for the Hausdorff measure of the singular set S u .This is done in several steps with increasingly more general sets S u .We denote by R n−1 also the subset where Proof.By [3, Lemma 3.76] there exists a non-negative function v ∈ L 1 (R n−1 ) representing the measure |D s u| restricted to R n−1 such that |Du| v dH n−1 .Then, by monotonicity, where For λ ∈ (−1, 1) and , where G λ (x) := G(x, λ) is an (n − 1)-dimensional section of G with R n−1 , and with scaling 1/σ n−1 .Now ´Gλ dH n−1 = c λ > 0 and so as σ → 0 + at all H n−1 -Lebesgue points of v.
Fix r ∈ (0, 1) and > 0 we define Note that E σ is the intersection of the closed sets {σ v σ (x, λσ ) }, so it is itself closed, in particular measurable.With this set, we find that ˆRn ĝ(σ, v σ ) dx Notice that the fraction in front of the Hausdorff measure tends to 1 as σ → 0 + , by assumption (GL).
We define the push-forward Φ # Du of a measure as in [3, Definition 1.70].
Lemma 3.2.Let u ∈ BV(R n ) and let Φ : R n → R n be an L-bilipschitz map, L 1.
Then, for open , Proof.Since Φ is bilipschitz, it is Lipschitz and proper.Hence by [3,Theorem 3.16], Since Φ is bilipschitz and G σ is radially decreasing and symmetric we further have where With this we may continue our previous estimate: Let us denote the constant in the inequality by d L .Then we conclude that Inverse Problems and Imaging Volume X, No. X (20xx), X-XX where we used that ĝ is increasing and the inequality g(ax) ag(x).The proof is concluded by a change of variables: where Proof.Every point x 0 ∈ S has a neighborhood U such that there exists a C 1isomorphism Φ : U → R n under which S ∩ U maps into a hyperplane and DΦ(x 0 ) = I.Since the mapping is C 1 , we conclude that it is bilipschitz with constant L; the constant can be chosen arbitrarily close to 1 by making the neighbourhood U small.Hence Then it follows from Lemma 3.1 that Let > 0. We can find a finite family of separated open sets U k such that the above claim holds in each and from which the claim follows as → 0 + and L → 1 + .
Next we leave the singular part and estimate the absolutely continuous part.
Proof.We assume the right hand side is finite since otherwise there is nothing to prove.Choose a decreasing sequence (σ k ) such that (σ k ) 2 −k and lim k F σ k (u) < c < ∞.Fix > 0 and set Then we calculate Inverse Problems and Imaging Volume X, No. X (20xx), X-XX Define Ω j := Ω \ G j .In Ω j , (σ k )|Du| 2 σ k for every k j.Since g(t)/t → 1 as t → 0 + (by assumption (G)), we conclude that Since ∇ a u(x) dx and D s are mutually singular measures [3, Proposition 3.92, p. 184], we obtain in every Borel set E that Du (E) equals Thus Hence ∇ a u ∈ L 2 (Ω j ) with norm independent of j.Then letting j → ∞ yields that ∇ a u ∈ L 2 (∪ j Ω j ).Since |Ω \ ∪ j Ω j | = 0 and ∇ a u is absolutely continuous with respect to the Lebesgue measure, we conclude that ∇ a u ∈ L 2 (Ω).The estimate for the norm follows as → 0.
Proof.If we show that for every U ⊂⊂ Ω, then the claim follows by taking the supremum over U .Let U ⊂⊂ V ⊂⊂ Ω; we define in Ω and Hence for all sufficiently small σ, F σ (u, U ) = F σ (u ext , U ). Therefore it suffices to prove the claim of the theorem under the assumption u ∈ SBV(R n ).
Let K i ⊂ U be a countable collection of compact sets, each contained in a C 1hypersurface, such that since the claim then follows as we take the limit m → ∞.
and dist(K i , K j ) > 0 for distinct i and j.Set σ 0 := min i =j dist(K i , K j ).Let N k s be the s-neighborhood of K k , s > 0. For σ < 1 6 σ 0 , the sets N j 3σ and N k 3σ are disjoint (j = k), and when σ < 1 3 dist(∂V, ∂U ) they are contained in V .Thus Corollary 3.3 can be applied to the hypersurfaces individually, and it gives that with c σ converging to 1 as before.In view of this and the absolute continuity of the last integral, To complete the proof we observe that the sets U \ ∪N k 4σ and N k 2σ are disjoint, even when dilated by σ, so that Together with the previous estimate, this completes the proof.
Next we generalize Theorem 3.5 for GSBV-function.We need the following lemma, which follows from the co-area formula [3,Theorem 3.40 Proof.Since G σ is radially symmetric and decreasing, we have Inverse Problems and Imaging Volume X, No. X (20xx), X-XX where h 0 is a suitably chosen weight function.This representation also holds for u λ .Since |Du λ |(B(x, r)) |Du|(B(x, r)) by Lemma 3.6, the claim follows.
We can now summarize the result for the lower bound.
Theorem 3.8.Let Ω ⊂ R n be a bounded open set.If u ∈ GSBV(Ω), then MS(u) lim inf σ→0 + F σ (u).Proof.By monotonicity of our functional (assumption (G)), it follows from Corollary 3.7 that F σ (u λ ) F σ (u) for every λ > 0. Theorem 3.5 yields that 4. Upper bound.In this section we prove the upper bound of lim sup F σ by the Mumford-Shah functional.We start with a lemma.Recall that the Besov space B s p,p (Ω), s ∈ (0, 1), is defined as 2,2 = H s , the fractional Sobolev space.We denote by f Q the average ffl Q f dx of f over the set Q and by Q(z, σ) a cube with center z and side-length σ.Lemma 4.1.Let Ω ⊂ R n be an open set, 1 p < ∞ and s 0. If v ∈ B s p,p (Ω), then lim Since the integrand is non-negative, we conclude that Thus x → |v(x)−v(z)| p |x−z| n+sp is an integrable function, and so Inverse Problems and Imaging Volume X, No. X (20xx), X-XX since |x − z| < 2σ when x ∈ Q(z, σ).The right hand side tends to zero, giving the claim.
Lemma 4.2.Suppose that u ∈ SBV 2 (Ω) in a rectangle Ω with S u ⊂ R n−1 .Then where Proof.As before, let v : R We have By assumptions (G) and (L) we have the following two estimates: By assumption (GL) g( (σ)/σ 2 ) (σ)/σ tends to 1 as σ → 0 + .Hence the latter upper bound can be estimated above by the function c (M v) 2 + 2 which belongs to L 1 (Ω ∩ R n−1 ) since v ∈ L 2 (Ω ∩ R n−1 ) and the maximal operator is bounded.Hence this upper bound gives for σ ĝ(σ, c σ V σ ) a majorant to be used in the dominated convergence theorem.
Proof of Theorem 1.  when σ σ 0 .By absolute continuity the last integral tends to zero when σ 0 → 0. Then the factor 1 + is gotten rid of when → 0. Thus it remains only to bound the integral over Σ σ by the Hausdorff measure.
By assumption the set S u lies H n−1 -a.e. in the finite union ∪S i of hyperplanes.Consider the components K of the hyperplane S 1 lying at distance at least δ from the other hyperplanes S i and from ∂Ω.In each of these we may apply Lemma 4.2 to conclude, for σ < 1  2 δ, that lim sup here we denote by K σ the points within distance σ of K.
In the complement C := (S 1 \ K) ∩ Ω σ of the components we estimate where we used the estimate |Du| σ cσ −n Du (Ω) for the first inequality and assumption (GL) for the second.When δ → 0, C approaches the set consisting of a the finite number of (n − 2)-planes; hence H n−1 (C) → 0. Thus we have estimated F σ in the set S σ 1 .The same argument can be repeated for each S i .This yields the desired estimate for Σ σ by additivity of F σ with respect to the set variable.
Then we take α → 1 and β → 1 and obtain the lower bound 2H n−1 (S u ∩ U ).This inequality holds also if we restrict the right hand side to a neighborhood of S u .If we let α → 0 and β → 0, we obtain on the other hand the lower bound ´U |∇ a u| 2 dx.
Combining the part near S u and the rest part as in the proof of Theorem 3.5, we obtain that MS(u, U ) lim inf i→∞ F σi (u i ).
Finally, we let U → Ω and obtain the result in the case u i ∈ BV(Ω).Assume then that u i ∈ GBV(Ω) for every i.Thus by the previous inequality we obtain MS(u λ ) lim inf i→∞ F σi ((u i ) λ ).
For the second condition of Γ-convergence, we note that Theorem 1.3 actually shows that lim sup F σ (u) MS(u) for every u ∈ L 1 which satisfies the regularity condition.Furthermore, by [12,Theorem 3.10] we can choose a sequence u i of functions satisfying the regularity condition of Theorem 1.3 such that MS(u i ) → MS(u).Since lim sup F σ (u i ) MS(u i ), we can find for each i a value σi ∈ (0, 1 i ) such that F σ (u i ) MS(u i ) + when σ < σi .Now we define a new sequence v i by setting v i = u k , where k ∈ N is the smallest number such that σ i ∈ (σ k , σk−1 ].If no such k exists we set v i = u 1 .Since σ i → 0, only a finite number of u 1 's will appear in (v i ).For all other terms we have F σ (v i ) MS(v i ) + by construction.For the same reason every other u k appears at most a finite number of times, so we have MS(v i ) → MS(u) as well, which concludes the proof.

Theorem 1 . 2 .
Let Ω ⊂ R n be a bounded open set with Lipschitz boundary.If g(t) = t 1+t then F σ Γ-converges to MS with respect to the L 1 topology.

− 1 →
R be the representative of the singular part of the derivative of u, D s u = v dH n−1 .Then |v| = |u + − u − |, where u ± is the approximately continuous extension of u to R n−1 from the positive and negative half-spaces [3, (4.1)].Since S u ∩ R n + = ∅, it follows that Du = ∇ a u in Ω ∩ R n + and since u ∈ SBV 2 (Ω), it follows that ∇u ∈ L 2 (Ω ∩ R n + ).As Ω ∩ R n + hasLipschitz boundary, we obtain by the Poincaré inequality that u ∈ L 2 (Ω ∩ R n + ) [21, Corollary on page 37], and thus u ∈ H 1 (Ω ∩ R n + ).Therefore v ± are boundary-values of a Sobolev function, and hence in the trace space H 1 2 (R n−1 ) [1, Theorem 7.39]; the same holds for their difference v.