STABLE DISCRETIZATION OF SCALAR AND CONSTRAINED VECTORIAL

PERONA-MALIK EQUATION

SÖREN BARTELS∗ AND ANDREAS PROHL†

Abstract. We study the scalar Perona-Malik equation and vectorial Perona-Malik equations into spheres. Discrete energylaws in both cases for fully discrete finite element formulations are shown and computational experiments illustrate quantitativebehaviors.

Key words. total variation, Perona-Malik, p-harmonic map, finite elements, full discretization, discrete energy law

AMS subject classifications. 65M12, 65M60, 35K55, 35Q35

1. Introduction. Regularization of multivalued images with PDE’s is an active research area, whichincludes denoising/edge enhancement, segmentation, and inpainting of digital color images, and restorationof optical flow and direction fields, or fields of diffusion tensors in magnetic resonance imaging, for example.The goal in image segmentation and edge detection is to decompose a given image into regions that areessentially homogeneous (with little variation in color or brightness); these regions should be separated bysharp boundaries (edges). In this paper, we study directional diffusion, where a unit vector field u0 : Ω →SN−1 is anisotropically diffused by vectorial Perona-Malik equation into the sphere SN−1; this programis motivated by recent models, which suggest independent processing of brightness |u0 | : Ω → R andchromaticity u0/|u0 | : Ω → S2 of colored RGB-images u0 : Ω → R3, cf. [51, 52, 50, 21].

Originally, the model of Perona and Malik [46] was developed to anisotropically diffuse gray valuesu0 : Ω → R, i.e., u : ΩT → R solves for u(0, ·) = u0 ∈ L∞(Ω), and all 0 < t ≤ T ,

(1.1) ut − div(gκ(| ∇u |2

)∇u)

= 0 in ΩT := Ω × (0, T ) , ∂nu(t, ·) = 0 on ∂Ω .

Let Ω ⊂ RM be bounded, Lipshitz; a standard example for the smooth non-increasing positive functiongκ : R → R, with gκ(0) = 1 and sgκ(s2) → 0 at infinity is gκ(s2) =

(1 + s

2

κ

)−1, κ > 0, and (1.1)

may then formally be considered as the L2-gradient flow related to the nonconvex, noncoercive energy

Gκ(u) = κ∫Ωφκ(| ∇u |2) dx, for φκ(s2) = 12 log(1 + s

2

κ ); dropping the index κ > 0 in the sequel means to setκ = 1. The motivation for this model is to suppress diffusion at regions of large gradient to preserve sharpedges; it is in these regions where the gradient exceeds some threshold 0 < z = z(κ), that (1.1) becomesbackward parabolic, whereas (1.1) is forward parabolic else. As a consequence of the additional noncoercivity,well-posedness of (1.1) is delicate [40, 39, 34], and recent literature mainly focuses on modified versions of(1.1), which are partly outlined in [40], together with the proposal of a ‘reasonable’ concept of solutions: in[25, 30], solutions for a spatial discretization (h > 0) are characterized (M = 1), and a system of PDE’scoupled together via nonlinear boundary conditions is derived to control the limiting function admittingjumps as h → 0 in cases κ = O(h−1) at all times; see also [14]. These ansatzes are mainly motivatedto construct global solutions with possible (spatial) discontinuities, which enjoy further properties, likee.g. decrease of energy, and maximum principle. Another strategy introduces finite scales to the problem byspatial [18] or temporal [3, 45, 10] convolution of entries of gκ to verify (local/global) existence of Sobolevtype solutions. A third approach is closely related which adds regularizing terms, like +ε∆2u or −ε∆ut,ε > 0 to (1.1) to allow for Sobolev type solutions uε : ΩT → R, and derive governing PDE’s for appropriatelimits (as ε→ 0), which are of bounded variation to ensure edge-preserving regularization, cf. [11].

The growth-condition for the above functional at infinity is a crucial feature to keep and enhance edgesand corners of input images. In [47], the total variation functional J(u) =

∫Ω | ∇u | dx is introduced to

denoise and restore inputs u0 : Ω → R, and the L2-gradient flow formally reads

(1.2) ut − div( ∇u| ∇u |

)= 0 in ΩT := Ω × (0, T ) , ∂nu(t, ·) = 0 on ∂Ω .

∗Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany. (Supportedby Deutsche Forschungsgemeinschaft through the DFG Research Center Matheon ‘Mathematics for key technologies’ in Berlin.)

†Department of Mathematics, ETHZ, CH-8092 Zürich, Switzerland

1

Advantages of this model over (1.1) are well-posedness in the practically relevant space of functions withbounded variation, and further analytical [4, 5], and numerical [1, 24, 20, 27, 28, 19] understanding. However,computational experiments for a stable discretization in Section 2.2 evidence improved flexibility of the modelof Perona-Malik over the TV -model, due to the asymptotic sublinear growth of related energies: fading awayof bulk features at logarithmic time-scale (in contrast to linear one), invariance of nonconvex supports (incontrast to variance of nonconvex supports), or conservation of non-Lipshitz boundaries (instead of change);see also [41, 28].

Processing colored images is a more complex task, where brightness and directions/chromaticity of anRGB vector field u : Ω → R3 are independently and anisotropically diffused [50, 52]: the scalar brightnessis regularized by (1.1) or (1.2), for example, whereas heat flow p-harmonic map, 1 ≤ p ≤ 2 is used to diffusedirections; cf. [37, 32], and [3, 17, 6, 8], and cited references therein for analytical and numerical resultsfor heat flow p-harmonic map. In particular, computational studies in [6, 9] evidence interesting dynamicsbehavior for cases 1 < p ≤ 2, like finite-time blow-ups, or geometric changes. In this paper we continue thisprogram by numerically studying anisotropic diffusion of directions by Perona-Malik model,

(1.3) ut − u×(u×DGDGDG(u)

)= 0 in ΩT , ∂nu = 0 on ∂ΩT .

Again, solvability of this model is a critical issue, and related problems definitely exceed those mentionedbefore in the context of scalar problem (1.1); here, we focus on numerical aspects, i.e., construction of astable fully discrete method — which is a nontrivial task due to sphere constraint, and nonconvexity of theunderlying potential —, and computational studies to motivate properties of certain limits of approximatingsequences {UUU}k,h, for UUU ∈ C

([0, T ];Vh

)defined by

UUU(t,x) := t− tmk

Um+1(x) +tm+1 − t

kUm(x) t ∈ [tm, tm+1] .

Therefore, let Ik := {tm}m≥0 be a net of mesh size k > 0 to discretize [0, T ], and iterates Um+1 ∈ Vh arefinite element solutions to the subsequent algorithm.

Algorithm 1.1. For m ≥ 0, let Um ∈ Vh, and determine(Um+1,LLLm+1

)∈[Vh]2

from

(dtU

m+1,ψψψ)h−(U

m+1/2 × (Um+1/2 ×LLLm+1),ψψψ)

h= 0 ∀ψψψ ∈ Vh ,(1.4)

(LLLm+1,ψψψ)h =(γ̃γγ(∇Um+1,∇Um),∇ψψψ

)∀ψψψ ∈ Vh ,(1.5)

with γ̃γγ(ααα,βββ) =(γ̃(αααij ,βββij)

)

ij∈ R3M , where

γ̃(a, b) =

φ(a2) − φ(b2)a− b if a 6= b ,

g(a2)a if a = b .

Here, we denote dtϕm := k−1

(ϕm+1 − ϕm

), and ϕm+1/2 := 12

(ϕm+1 + ϕm

), for m ≥ 1, and a sequence

{ϕm}m≥0, and (·, ·)h denotes a discrete version (reduced integration) of the inner product in L2(Ω;R3);cf. Section 2.2 for used notation.

The remainder of this paper is as follows: Section 2.1 surveys recent strategies to solve (1.1); a fulldiscretization is proposed in Section 2.2, where solutions satisfy a discrete energy law (M ≥ 1); in order toexclude additional spatio-temporal perturbation effects, which may e.g. affect dynamics of staircasing [40] (atfinite scales), we consider a direct discretization of (1.1). Computational experiments for the total variationflow and the Perona-Malik evolution show very different quantitative behaviors. In Section 3, we study the(fully discrete) Perona-Malik evolution on the sphere, and show its stability; computational studies showdifferent dynamics in comparison with regularized heat flow 1-harmonic map.

2

2. Gradient flow of a nonconvex functional: scalar case. Nonstationary equations

(2.1) ut(s) = −DF(u(s)

)in ΩT , u(0) = u0 ,

for nonconvex F (u) =∫Ω ψ(| ∇u |) dx are forward-backward parabolic, which in case of coercivity of ψ : R →

R possess global Young measure solutions [23]: such solutions are not unique, even not for convex choicesof F ; however, the first moment of the Young measure is unique, and its stability is reflected by maximum,comparison, and energy decay property. Numerical experiments in [29] give further insight into dynamics of(2.1) in the coercive case: (i) the gradient flow for the convexified density (e.g., no ‘wrinkling’, see below)does not approximate the above Young measure solution of (2.1); (ii) formation of wrinkles of size of spatialdiscretization h > 0 only appears in the locally concave regions LUS =

{s ∈ R

∣∣ψ′′(s) < 0}

of the energies,

and not in the containing globally unstable regions GUS ={s ∈ R

∣∣ψ(s) > ψ∗∗(s)}, and movement of this

wrinkling region at later times is observed.

The study of (2.1) for nonconvex, noncoercive ψ is much less understood, and the approach in [23] toconstruct Young-measure solutions via Rothe’s method for the above setup does not apply here. Recently[49], the construction of Young-measure solutions to (1.1) for M = 1 succeeded, which is based on a reformu-lation of the problem; it is pointed out [49], that constructed solutions also suffer from severe instabilities;instabilities occur in the evolution for ‘mixed’ initial data u0, where u0,x belongs to both, convex and con-cave regions of ψ. Discretization of the governing PDE, or approximation respectively regularization of Fare common strategies to obtain well-posed problems, to construct and identify possible limits to controlasymptotical dynamics of the regularized model. The work [33] is a nice example for this program, where anotion of gradient flow of the nonconvex Mumford-Shah functional F : SBV (Ω) → R (M = 1) is given aslimit of the existing gradient flow (h > 0)

(2.2) uht (s) = −DF̂h(uh(s)

)in ΩT , uh(0) = u0,h ,

for the functional F̂h : PC2h → R,

F̂h(u) =1

h

∫

R

arctan( |u(x+ h) − u(x) |2

h

)dx ,

with PC2h :={u ∈ L2(R)

∣∣u∣∣K∈ P0(Ki)}, and which in variational sense converges to F . Here, we use a

nonoverlapping covering⋃

iKi = R, which is ensembled of triangles, such that Ki ∩Kj = ∅, for 0 ≤ i 6= j,and supi |Ki | ≤ h; it is shown in [33] that well-defined limits u : ΩT → R solve local heat equations withhomogeneous Neumann boundary conditions at (a finite number of) starting discontinuities, which keepposition and exist for finite (‘merging’) or infinite times (‘surface energy monotonicity’ ddtH0(Su(t)) ≤ 0,i.e., the singularity set Su :=

{x ∈ R

∣∣u+(x) 6= u−(x)}

is non-increasing), and enjoys desirable properties,

like maximum principle, energy decay ddtF(u(t)

)≤ 0, and Hölder continuity in time. It is due to some

connection between Mumford-Shah and Perona-Malik problems that those results may serve as a motivationto better understand the latter; cf. [38], and the literature cited therein. — Another example for unexpecteddynamics which follows a similar program is studied in [14] for the nonconvex φ(ξ) = min{ξ2, 1} (M =1), where (non-unique) limits of restricted functions of bounded variation {uh} ⊂ Vh ⊂ BV (Ω) (whichsolve a problem analogous to (2.2)) are in L∞

([0,∞), BV (0, 1)

)∩ AC2

([0,∞);L2(0, 1)

), and satisfy (i)

a free boundary value problem (local linear heat equations, with homogeneous Neumann conditions atinterfaces), (ii) maximum principle (no comparison principle), and (iii) monotonicity for total variation,i.e. |Du(t) |

((0, 1)

)≤ |Du0 |

((0, 1)

), t ≥ 0; moreover, the discontinuity set S ≡ Su(t) may move and merge

in space-time.

2.1. Scalar Perona Malik – an overview. Analytical studies [5] of the scalar-valued total variation(TV ) flow (p = 1) −ut ∈ ∂J(u), u(0) = u0 ∈ L2(Ω), for J(u) = |Du |(Ω) show interesting characterizations ofthe strong solution in the sense of semigroup theory: (i) finite extinction time (M = 2), (ii) u(t, ·) ∈ L∞(Ω),t > 0, if u0 ∈ LM (Ω), and no L1 − L2-regularizing effect for L1(Ω)-initial data, in general, (iii) C1,α-regularity of level sets ∂∗

[u(t) > λ

]for u0 ∈ LM (Ω) of decreasing size, i.e., ddtHM−1

(∂∗[u(t) > λ

])≤ 0,

and (iv) invariance of supports, provided e.g. the curvature of the smooth boundary of the simply connected

3

convex domain is not too large. The more practical notion of weak solution is established and studied in[36, 27], and convergent finite element methods are studied in [27, 28]. In order to study convergent finiteelement discretizations of total variation flow, let Th be a regular triangulation of Ω, and

V h ={vh ∈ C0(Ω)

∣∣ vh∣∣K∈ P1(K) ∀K ∈ Th

}.

Consider Uh,k

(·, t) := t−tm−1k Um(·) + tm−tk Um−1(·) ∈ V h in t ∈ [tm−1, tm], where iterates {Um} solve theregularized, fully discrete problem

(2.3)(dtU

m, vh)

+( ∇Um√

| ∇Um |2 + ε2,∇vh

)= 0 ∀ vh ∈ V h .

Then, convergence of {Uh,k} towards strong solutions of the TV -problem at rate

esssupt∈[0,T ]‖u(t) − Uε,h,k

(t) ‖L2 ≤ ‖u0 − uε0 ‖L2 + 2√|Ω |T√ε+ C1(ε)k + C2(ε)h2

is verified in [28, 27] provided that used spatial meshes are quasiuniform, and k = O(h2). Moreover, constantsCi(ε) > 0, depend on ε

−1 in some low polynomial order.

A mathematical study of Perona-Malik’s model (1.1) started with Kichenassamy’s work [40], whereweak solutions for (1.1) were excluded, and a concept of generalized solutions is motivated, which allowsfor (energetically favorable) step functions in finite and infinite time, to analytically describe numericallyobserved formation, merging, and segmentation (t → ∞) during a ‘staircasing’ process; cf. also Figure 2.1below. The following general assumptions from [40] for gκ : R → R apply to the upper prototypic example,in particular.

Assumption 2.1. 1. gκ(ξ) > 0 for all ξ ≥ 0.2. The parameter κ > 0 defines a positive critical value z(κ) such that ∂ξ

(ξgκ(ξ

2))> 0 for | ξ | < z(κ), and

∂ξ(ξgκ(ξ

2))< 0 for | ξ | > z(κ).

3. Both, gκ(ξ) and ∂ξ(ξgκ(ξ

2)))

tend to zero as ξ goes to infinity.In Esedoglu’s work [25], solutions uh : ΩT → R for a spatial semi-discretization of (1.1) in the case

κ = κ(h) are shown to converge (h → 0) to solutions of the following system of heat equations coupled toeach other through nonlinear boundary conditions that become singular at finite times (M = 1),

∂ui∂t

=∂2ui∂x2

in (pi−1, pi) ,∂ui∂x

(pi, t) =∂ui+1∂x

(pi, t) =1

ui+1(pi, t) − ui(pi, t)on pi ,

for all 1 ≤ i ≤ N − 1, and ∂u1∂x (p0, t) = ∂uN∂x (pN , t) = 0. Here, p0 < p1 < ... < pN denote jumps, whichare shown not to move during the evolution process, but will vanish at finite time (energy argument).(Variance of the jump set for M = 2 is computationally evidenced in [26].) Locally existing solutionssatisfy maximum principle (but no comparison principle, [25]), are Hölder continuous in time, and decrease

the energy PME(t) :=12

∑Ni=1

∫ pipi−1

| ∂ui∂x (t) |2 dx +∑N−1

i=1 log(|ui+1(pi, t) − ui(pi, t) |

), which, however, is

not bounded from below as jumps tend to zero. Canonical continuation of solutions beyond blow-upsat ‘quenching times’ is possible; this reflects merging of neighboring terraces (‘coarsening’) in the final stepduring evolution, which is preceeded by the formation of terraces (‘staircasing’). Hence, this realized programfollows the proposal [40] to explain formation and merging of terraces; see [25] for illustrative simulations in1D; see also 1D-simulations respectively 2D-simulations in [25] respectively [42, pp. 84ff],[38].

A different line of research uses recent results [4, 17] of Γ-convergent approximations of energies F :BV (Ω) → R of the form [44]

(2.4) F (u) =

∫

Ω

f(| ∇u |) dx +∫

Su

ϕ(u+ − u−) dHM−1 + C |Dcu | ,

by sequences Fε : W2,2(Ω) → R,

(2.5) Fε(u) =

∫

Ω

fε(| ∇u |) dx + | r(ε) |3∫

Ω

| ∇2u |2 dx .4

Here, C1(√z−1) ≤ ϕ(z) ≤ C2(z+1), for all z ≥ 0, and C1, C2 > 0 (which excludes Mumford-Shah functional;

cf. [16] for (M = 1)), and |Dcu | denotes the total variation of the Cantor part of the measure derivativeDu; moreover, {fε} is any family of positive, nondecreasing functions of nonconvex or convex-concave shape,and f respectively ϕ are defined as proper limits of fε respectively r(ε)fε(

·r(ε) ). As is pointed out in [44], by

using the right rescaling r(ε) = εlog 1

ε

for the Perona-Malik functional, there holds

Gε(u) +( ε

log 1ε

)3 ∫

Ω

| ∇2u |2 dx Γ→∫

Ω

| ∇u |2 dx + c∫

Su

|u+ − u− |1/2 dHM−1 ,

for some computable c > 0. Then, the general results in [44] allow to construct generalized Perona-Malikequations, where solutions of formal L2-gradient flows to (2.4) are interpreted as limits (ε → 0) of existingL2-gradient flows of (2.5); in [11, 12], the authors study L2-gradient flow for the Γ-limit of the slightlydifferent scaling (cf. (1.3)) Fν : W 2,2

((0, 1)

)→ R,

(2.6) Fν(u) :=1

2

∫ 1

0

φ(ux)

νφ( 1ν )+ ν3 |uxx |2 dx (ν > 0) ,

which is proved to be F : P((0, 1)

)→ R, with

F(u) :=∑

x∈Su

|u+(x) − u−(x) |1/2 , for P((0, 1)

):={u ∈ SBV

((0, 1)

)∣∣Dau = 0},

with Dau denoting the absolutely continuous part of the derivative Du; a motivation for the scaling in (2.6)comes from studying solutions of the L2-gradient flow (for Ω := (0, 1))

uεt +1

2

(φ′(uεx)

)x

+ ε2 uεxxxx = 0 in ΩT , uεx = u

εxxx = 0 on ∂ΩT

at large times O(

1νφ( 1

ν)

), with ε2 = ν4φ( 1ν ), where coarsening takes place (after an initial period of rapid

formation of microstructures, followed by a longer coarsening period). Then, the L2-gradient dynamicsdefined by F that is identified with the global L2-minimizing movement in the sense of DeGiorgi leads toa coupled system of nonlinear ODE’s, which controls dynamics of local heights of the piecewise constantfunction P

((0, 1)

)3 u(t) = s0 +

∑Nj=1 sj(t)χ[pj ,1); again, initial places of jumps do not move, and merging

is admitted; see also [29, Example 1] for computational evidence and discussion of formation and coarseningof piecewise constant mappings at times t > 0, for cases {u0,x ∈ LUS} 6= ∅.

Finally, we mention [30], where results help to better compare TV -flow and (1.1), and illustrate differentbehaviors of (1.1) in different spatial dimensions (M ≥ 1): for M = 1, the total variation of solutions to(1.1) is shown to be non-increasing in time, while counterexamples show that this assertion fails for M = 2.In addition, the authors in [30] study solutions to a semi-discretization in space for M = 1, 2, where onlymaximum and energy decay property are validated (M = 1), due to failure of tools which apply only forcontinuum model (1.1), like chain rule, for example.

Computational experiments for (1.1) nourish the above analytical studies; unfortunately, it is difficult todesign fully discrete schemes that satisfy all properties mentioned above at the same time, like energy, posi-tivity, and maximum principle for M ≥ 2. In [26], S. Esedoglu verifies a restricted comparison principle for aspatial semidiscretization (M ≥ 2), with additional control over gradients in the case M = 1; computationalexperiments are provided to evidence failure of these properties for M = 2. In [10], a discrete maximumprinciple is established for a fully discrete finite difference scheme to solve Nitzberg-Shiota’s [45] temporalregularization of (1.1) (M = 2). Unconditional convergence in C

(0, T ;L2(Ω)

)∩ L2

(0, T ;W 1,2(Ω)

)of the

whole sequence {Uh,k} (see construction as before (2.3)) of solutions {Um} of a semi-implicit finite elementdiscretization (M = 2) for the regularized version [18] with smoothing kernel Gσ(x) = 14πσ exp

(− |x |

2

4σ

), σ > 0

in the form

(2.7) (dtUm, vh) +

(g(| ∇Gσ ∗ Um−1 |2)∇Um,∇vh

)= 0 ∀ vh ∈ Vh , m ≥ 1 ,

5

for u0 = u0 ∈ L2(Ω) to a unique limit which solves its continuous version is verified in [41]: its proof restsupon the (a priori) bound

(2.8)1

2‖Um ‖2L2 +

k2

2

m∑

j=1

‖ dtU j ‖2L2 + km∑

j=1

∫

Ω

αj−1 | ∇U j |2 dx =1

2‖u0 ‖2L2 , m ≥ 1 ,

with αm−1 := g(| ∇Gσ ∗Um−1 |

)≥ νσ > 0. The properties of g(| ∇Gσ ∗ · |), together with equation (2.8) are

key tools for a compactness argument, and to establish solution character of a proper limit in L2(ΩT ) for thecontinuum version of (2.7). Interesting computations are also included in [41] for values h = k ≈ 10−3...10−2,and σ ≈ 10−8...10−5, where cusp-like edges are preserved/enhanced, while gross parts of the solution aresmoothened. A corresponding program was realized for a semi-implicit finite volume scheme in [43]; more

recently, suboptimal rates ‖Uh,k−u ‖L2(ΩT ) ≤ CT√h are verified for the case k = h in [35], using a (discrete)

Gronwall argument.

The works [41, 43, 35] employ properties of the special regularization of the elliptic PDE (1.1) given in[18]; in Section 2.2, we consider a full discretization of the original problem (1.1) which exploits its characteras gradient flow for the nonconvex functional G, for M ≥ 2, and whose iterates {Um} satisfy the discreteenergy law

(2.9) k

m∑

j=1

‖ dtU j ‖2L2 +G(Um) = G(u0) , m ≥ 1 (κ > 0) .

To reach this goal, the Fréchet derivative of Gκ : Vh → R is replaced by its difference quotient in Algo-rithm 2.1. As outlined above, convergence behavior has to remain unclear, but may easily by concluded —based on the discrete energy identity (2.9) — for different regularizations of (1.1).

2.2. Scalar Perona Malik — energy decreasing discretization. We propose a stable, implicit fi-nite element discretization of a slightly modified version of (1.1) (M = 2); a discrete energy law is established,for an implicit discretization in time. Numerical simulations are performed to directly compare propertieswith total variation flow. This section also serves as preparatory step and motivation for Section 3. In thesequel, we follow [46, 25, 26] and evolve u0 ∈ L∞(Ω) via (M = 2)

(2.10) ut − div(g(u2x)ux

g(u2y)uy

)= 0 in ΩT , ∂nu = 0 on ∂ΩT .

By evidence, (2.10) is formally the gradient flow to G(u) =∫Ω φ(u

2x) + φ(u

2y) dxdy.

Let Th be a regular triangulation of the polygonal or polyhedral bounded Lipschitz domain Ω ⊂ RMinto triangles or tetrahedra of maximal diameter h > 0 for M = 2 or M = 3, respectively. Given the set ofall nodes (or vertices) N in Th and letting

{ϕz : z ∈ N

}denote the nodal basis in Vh, we define the nodal

interpolation operator Ih : C(Ω) → Vh by Ihψ :=∑

z∈N ψ(z)ϕz, for ψ ∈ C(Ω). For functions φ, ψ ∈ C(Ω),a discrete inner product is defined by

(φ, ψ

)h

:=

∫

Ω

Ih(φψ)dx =

∑

z∈N

βzφ(z), ψ(z) ,

where βz =∫Ω ϕz dx, for all z ∈ N ; we define ||ψ||2h := (ψ, ψ)h. We remark that there holds

||ψh||L2 ≤ ||ψh||h ≤ (M + 2)1/2||ψh||L2

for all ψh ∈ Vh. Basic interpolation estimates yield that

(2.11)∣∣(φh, ψh)h − (φh, ψh)

∣∣ ≤ Ch ||φh||L2 ||∇ψh||L26

for all φh, ψh ∈ Vh, where here and throughout this paper C > 0 denotes an (h, k )-independent constant.Algorithm 2.1. For J ≥ 0, and {U j}0≤j≤J ⊂ Vh, determine UJ+1 ∈ Vh from

(2.12)(dtU

J+1, ψ)h

+(γ̃γγ(∇UJ+1,∇UJ ),∇ψ

)= 0 ∀ψ ∈ Vh ,

where γ̃γγ(ααα,βββ) =(γ̃(ααα1,βββ1), γ̃(ααα2,βββ2)

)T∈ R2, for ααα,βββ ∈ R2, with γ̃(·, ·) from Algorithm 1.1.

We may now use ψ = dtUj+1 to verify the discrete energy law for iterates of Algorithm 2.1; by Brouwer

fixed point theorem, this also implies existence of solutions to the problem.Theorem 2.1. Let k, h > 0. i) For every 0 ≤ m1 ≤ m2 0, for ‖∇u0 ‖L∞ > 1 sufficientlylarge.

Example 2.1. Let Ω := {(x, y) ∈ R2 : x2 + y2 < 1}, T = 1, and u0(x, y) = f(r), where r = (x2 + y2)1/2and

f(r) =1

9(2r − 1)9 − 2

5(2r − 1)5 + (2r − 1) .

7

Triangulations T (j)h consist of 4096 and 16384 triangles for j = 1 and j = 2, respectively, and define interiorapproximations of the domain Ω. We use hj = 2

−(4+j) and k = h2j/10, for j = 1, .., 5. Throughout thisexample, we employ κ = 1.

Example 2.1 has been constructed in [30, Example 3] in such a way that if there exists a radial C2

solution of the Perona-Malik equation in Ω subject to the given initial data then there exists δ > 0 such thatfor all t ∈ (0, δ) there holds

||∇u(t, ·)||L∞ < ||∇u0(·)||L∞ ,

which contradicts validity of a supercritical reverse maximum principle for the gradient. Figure 2.1 displayssnapshots of the numerical approximation obtained with Algorithm 2.1 for t = 0, 0.01, 0.02. The numericalsolution does not seem to be radially symmetric for t = 0.01 and t = 0.02, and we observe a ‘staircasingeffect’. Nevertheless, as predicted in [30], the W 1,∞ semi-norm is decreasing at small times as can be seenin Figure 2.2; more precisely, for the discrete time derivative dt‖∇U1 ‖L∞ , we successively compute values

dt‖∇U1 ‖L∞ = −23.039, −22.4700, −20.0971, −18.5080, −17.9046,

for hj = 2−(4+j), 1 ≤ j ≤ 5, respectively.

We remark that even though this example is critical in the sense that it leads to large gradients, ourapproximation scheme guarantees decreasing energy, cf. Figure 2.2. We also displayed the W 1,1 semi-normof the numerical approximations as functions of time and observe in the same Figure 2.2 that they aredecreasing.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

5

10

15

20

25

t

Gκ(U(t,⋅)) (h=2−5)

|U(t,⋅)|1,∞ (h=2

−5

|U(t,⋅)|1,1

(h=2−5)

Gκ(U(t,⋅)) (h=2−6)

|U(t,⋅)|1,∞ (h=2

−6

|U(t,⋅)|1,1

(h=2−6)

Fig. 2.2. Energy G`

uh(t, ·)´

, W 1,1 semi-norm ||∇Uj ||1, and W 1,∞-semi-norm ||∇Uj ||∞ of the numerical approximation

in Example 2.1 on the triangulations T(1)

h(h = 2−5) and T

(2)h

(h = 2−6).

We conclude this section with numerical experiments displaying advantages of the Perona-Malik evolutionover (regularized) TV -flow, putting ε = h in (2.3). The following example compares evolution of geometricobjects by Perona-Malik and TV flow.

Example 2.2. Let Ω = (− 12 , 12 )2 and u0(x) = χK(x), where(a) for d = 1/5 and Br(a, b) = {(x, y) ∈ R2 : (x− a)2 + (y − b)2 < r2}

K = B 18

(0,2d√

3) ∪ B 1

8

(−d,± d√3);

8

(b) for d = 14 and B∞r (a, b) = {(x, y) ∈ R2 : |x− a|, |y − b| < r}

K = B∞18

(0,±d

)∪ B∞1

8

(± d, 0

);

(c) for s = 27 and r =213

K = B∞s (0, 0) \(Br(0, 0) ∪ Br(±s, s) ∪ Br(−s,±s) ∪ Br(0,±s) ∪ Br(±s, 0)

).

We let Th be a triangulation of Ω consisting of halved squares along the direction (1, 1) and of maximaldiameter h =

√22−7.

We ran Algorithm 2.1 for the initial data specified in (a)-(c) in Example 2.2. The first row displays theinitial data, i.e., the nodal interpolant of the functions defined in Example 2.2 (a)-(c). The second row inFigure 2.3 shows the numerical approximation at t = 0.25. Structures disappear at times t > 0.35 (thirdrow), when drops a certain threshold value(t = 0.35 in our case). Diffusion of structures occurs much fasterfor the regularized TV -flow, for which snapshots of the numerical solutions at time t = 0.035 for the differentinitial data specified in Example 2.2 are shown in the last row of Figure 2.3; see also [28].

3. Mappings into the sphere: heat flow 1-harmonic map vs. Perona Malik evolution. Weverify stability and energy law for existing solutions to Algorithm 1.1. Let Ĝ(u) =

∑1≤k≤M

1≤l≤3

∫Ωφ(| ∂kul |2) dx.

Lemma 3.1. Let k, h > 0. Suppose that |U0(z)| = 1 for all z ∈ N . Then the sequence {Um} fromAlgorithm 1.1 satisfies for all 0 ≤ m 0, the fully practical linear scheme reads as follows.Algorithm 3.1. 1. Set Ũ0 := U0 and W1,0 := Ũ0. Set j := 0 and ` := 0.

2. Compute Wj+1,`+1 ∈ Vh, such that for all ψψψ ∈ Vh,2

k

(Wj+1,`+1,ψψψ

)h−(Wj+1,`+1 ×

(Wj+1,` ×LLLj+1,`

),ψψψ)

h=

2

k

(Ũj ,ψψψ

)h

∀ψψψ ∈ Vh ,(3.3)

(LLLj+1,`,ψψψ)h = −(γ̃γγ(∇{2Wj+1,` − Ũj},∇Ũj

),∇ψψψ

)∀ψψψ ∈ Vh .(3.4)

9

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Fig. 2.3. Nodal interpolant of the initial data defined in (a)-(c) of Example 2.2 (first row). Numerical approximation ofPerona-Malik equation for t = 0.25 (second row) and t = 0.75, t = 0.71, and = 0.37, respectively, (third row) in (a)-(c) ofExample 2.2. Developed initial data for the approximation of regularized TV -flow for t = 0.035 in (a)-(c) of Example 2.2 forε = h (last row).

3. Given εj > 0, stop if

(3.5) ‖Wj+1,`+1 −Wj+1,` ‖L2 ≤ εj ,10

and set Ũj+1 := 2Wj+1,`+1 − Ũj ∈ Vh.4. Set ` := `+ 1 and got to Step 2.

Unconditional unique solvability of (3.3) for Wj+1,`+1 ∈ L2 follows from Lax-Milgram theorem.Theorem 3.2. Let Ĝ(Ũj) ≤ C, and | Ũj(z) | = 1, for all z ∈ N , and j ≥ 0 in Algorithm 3.1. For all

` ≥ 0, there exists a unique solution Wj+1,`+1 ∈ L2 to (3.3)–(3.4), such that ‖Wj+1,`+1 ‖L∞ ≤ 1. Moreover,there exists C > 0, such that

(3.6) ‖LLLj+1,` ‖L∞ ≤ Ch−1 .

Additionally, there exists 0 ≤ C̃ < 1, such that for k ≤ C̃h2, there exists 0 < Γ < 1, and

(3.7) ‖Wj+1,`+1 −Wj+1,` ‖L2 ≤ Γ ‖Wj+1,` −Wj+1,`−1 ‖L2 ` ≥ 1 ,

and | Ũj+1(z) | = 1, for all z ∈ N .Proof. Control of ‖Wj+1,`+1 ‖L∞ by 1 follows from (3.3) on choosing ψψψ = Wj+1,`+1(z)ϕz.To verify ‖LLLj+1,` ‖L∞ ≤ Ch−1, choose z ∈ N , such that ‖LLLj+1,` ‖L∞ = |LLLj+1,`(z) |. Upon choosing

ψψψ = LLLj+1,`(z)ϕz in (3.2), there holds

βz |LLLj+1,`(z) |2 ≤∣∣∣(γ̃γγ(∇{2Wj+1 − Ũj},∇Ũj

),∇ϕz

)∣∣∣|LLLj+1,`(z) |

≤ |LLLj+1,`(z) |∑

K∈Th,z∈K

|K | | ∇ϕz | | γ̃γγ(∇{2Wj+1 − Ũj},∇Ũj

)| ,

with chM ≤ βz, |K | ≤ ChM . Since the sum is finite, and γ̃γγ ≤ C this implies |LLLj+1,`(z) | ≤ Ch−1.Subtraction of two subsequent equations in the fixed-point iteration yields for every ` ≥ 1, and all

ψψψ ∈ Vh1

k

(E`+1j+1,ψψψ

)h

+(E`+1j+1 ×

(Wj+1,` ×LLLj+1,`

),ψψψ)

h+(Wj+1,` ×

(E`j+1 ×LLLj+1,`

),ψψψ)

h

+(Wj+1,` ×

(Wj+1,` ×

[LLLj+1,` −LLLj+1,`−1

],ψψψ)

h= 0 .

Choosing ψψψ = E`+1j+1 leads to

(3.8) ‖E`+1j+1 ‖h ≤ k ‖Wj+1,` ‖L∞ ‖LLLj+1,` ‖L∞ ‖E`j+1 ‖h + k ‖Wj+1,` ‖2L∞ ‖LLLj+1,` −LLLj+1,`−1 ‖h .

Thanks to (3.2), mean value theorem together with uniform boundedness of |D2φ(| · |2) | which impliesuniform Lipschitz continuity of γ̃(·, X) when X is fixed, and inverse inequality, the last term in (3.8) maybe bounded by

≤ Ckh−1 ‖∇E`j+1 ‖L2 ≤ Ckh−2 ‖E`j+1 ‖L2 .

A combination of the estimates then yields assertion (3.7). Finally, if Ũj+1 = 2Wj+1,`+1 − Ũj , equation(3.3) can be written as

(dtŨj+1,ψψψ)h −

(Ũ

j+1/2

× (Ũj+1/2

×LLLj+1,`),ψψψ)h

= 0 .

The choice ψψψ = Ũj+1/2

(z)ϕz then implies dt| Ũj+1(z) |2 = 0.3.2. Computational experiments. Numerical experiments with Algorithm 1.1 show that evolved

edges/structures are preserved for a long time.Example 3.1. (a) Let Ω := (−1, 1)2, T = 1/2, and u0 be as in the left upper plot in Figure 3.1. We

choose a triangulation Th consisting of 2048 triangles which are halved squares (along the direction (1, 1))and with 1089 nodes. Hence, h =

√22−4 and we set k = h2/10.

(b) Let Ω, T , and Th be as in (a) and let u0 be the perturbation of the initial data of (a) shown in the leftplot of Figure 3.4.

11

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Fig. 3.1. First two components of the numerical solutions u(t, ·) for t = 0, 0.085938, 0.15469, 0.44688, 0.77344, 0.99688in Example 3.1 (a) obtained with Perona Malik evolution on the sphere.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Fig. 3.2. First two components of the numerical solution u(t, ·) for t = 0, 0.085938, 0.15469 in Example 3.1 (a) obtainedwith regularized one-harmonic map heat flow (ε = h).

Figure 3.1 shows the evolution defined by the Perona-Malik equation into the sphere in Example 3.1 (a)and computed with Algorithm 1.1. We observe that the sharp interfaces separating regions in which u =(1, 0, 0) from those in which u = (0, 1, 0) are preserved for a long time t = 0.44688 before all vectors start toalign. This is different if regularized one-harmonic flow (ε = h) is applied to the initial data of Example 3.1 (a)as one can deduce from the snapshots of the numerical solutions for different times shown in Figure 3.2; forits calculation, we use a consistent, stable scheme constructed in [9], where iterates satisfy an analogue ofLemma 3.1. The vectors immediately start to align, and at t = 0.15469 no sharp interfaces are observable.In Figure 3.3 we displayed the first component of the vectors u(t,x) for x ∈ {(− 14 , 0), ( 12 , 0)} as functionsof t > 0. This allows for a quantitative comparison of the speed of the alignment of the vectors displayedin Figures 3.1 and 3.2. We observe an approximately linear change in the direction for 0 ≤ t ≤ 2 in thecase of (regularized) one-harmonic map heat flow into the unit sphere, while the rotation is slowed downfor Perona-Malik evolution on the sphere. The numerical results for the latter in Example 3.1 (b) motivate

12

good performance of Perona-Malik for applications in image processing. The perturbed initial data shownin the left plot of Figure 3.4 is denoised at t = 0.020313 but (practical) discontinuities are preserved.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

U(1) (t,A) (one harmonic)

U(1) (t,B) (one harmonic)

U(1) (t,A) (Perona−Malik)

U(1) (t,B) (Perona−Malik)

Fig. 3.3. First components u(1)(t,A) and u(1)(t, B) of the numerical solutions in Example 3.1 (a) obtained with regularizedone-harmonic map heat flow and regularized one-harmonic map heat flow into the sphere for A = (− 1

4, 0) and B = ( 1

2, 0); for

ε = h.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Fig. 3.4. First two components of the numerical solution u(t, ·) for t = 0, 0.010156, 0.020313 in Example 3.1 (b) obtainedwith Perona Malik evolution on the sphere.

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