This Note presents the resolution of a differential system on the plane that translates a geometrical problem about isotropic deformations of area and length. The system stems from a probability study on deformed random fields [1], which are the composition of a random field with invariance properties defined on the plane with a deterministic diffeomorphism. The explicit resolution of the differential system allows to prove that a weak notion of isotropy of the deformed field, linked to its excursion sets, in fact coincides with the strong notion of isotropy. The present Note first introduces the probability framework that gave rise to the geometrical issue and then proposes its resolution

Briant, Marc

Fournier, Julie

English

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ISOTROPIC DIFFEOMORPHISMS: SOLUTIONS TO
A DIFFERENTIAL SYSTEM FOR A DEFORMED
RANDOM FIELDS STUDY
Marc Briant, Julie Fournier
To cite this version:
Marc Briant, Julie Fournier. ISOTROPIC DIFFEOMORPHISMS: SOLUTIONS TO A DIFFEREN-
TIAL SYSTEM FOR A DEFORMED RANDOM FIELDS STUDY. 2019. ￿hal-02307952￿
ISOTROPIC DIFFEOMORPHISMS: SOLUTIONS TO A
DIFFERENTIAL SYSTEM FOR A DEFORMED RANDOM FIELDS
STUDY
MARC BRIANT AND JULIE FOURNIER
Abstract. This Note presents the resolution of a differential system on the plane
that translates a geometrical problem about isotropic deformations of area and
length. The system stems from a probability study on deformed random fields [1],
which are the composition of a random field with invariance properties defined
on the plane with a deterministic diffeomorphism. The explicit resolution of the
differential system allows to prove that a weak notion of isotropy of the deformed
field, linked to its excursion sets, in fact coincides with the strong notion of isotropy.
The present Note first introduces the probability framework that gave rise to the
geometrical issue and then proposes its resolution.
Contents
1. Characterization of isotropy in deformed random via excursion sets 2
2. Planar deformations modifying lengths and areas isotropically 3
References 8
The motivation for the result featured in the present article originates from a
probability problem about deformed random fields. Indeed, one of the main results of
[1] needed a complete characterization of isotropic deformed fields and it turned out
such a description was given by solutions to a system of nonlinear partial differential
equations. The resolution of that system is a major step in the proof, yet it is
completely independent. Moreover, its analytical flavour as well as the geometric
classification it contains makes it interesting on its own and out of step with the
probability nature of [1].
Geometrically, the aim is to investigate the class of planar transformations F :
R2 −→ R2 that are C2 and transform isotropically areas of rectangles and lengths
of segments:
(0.1) ∀ϕ ∈ SO(2), ∀i ∈ {1, 2} , li(F ◦ ϕ(E)) = li(F (E))
where li stands for the Lebesgue measure in R1 or R2 depending on E being a
segment (embedded in R1) or a rectangle (viewed as a surface embedded in R2).
Such a property boils down to the fact that both the norms of each of the columns
of the cartesian Jacobian matrix of the polar form of F and its determinant are
1
2 MARC BRIANT AND JULIE FOURNIER
radial: introducing T = R/2piZ the one-dimensional torus
∀(r, θ) ∈ R+ × T,
∥∥∥∥JacF (r, θ) [ 10
]∥∥∥∥ = g(r), ∥∥∥∥JacF (r, θ) [ 01
]∥∥∥∥ = h(r)
det (JacF ) (r, θ) = f(r),
(0.2)
where f , g and h are C2(R+∗,R).
In this Note we prove that the class (0.1) is exactly the family of “spiral deforma-
tions” described in polar coordinates by:
(0.3) ∀(r, θ) ∈ R+ × T, R(r, θ) = R(r) and Θ(r, θ) = ±θ + Θ(r).
The main goal of the present Note is therefore to introduce the background and
one of the main results of [1] in order to motivate the analytic problem and then to
solve it.
1. Characterization of isotropy in deformed random via excursion
sets
All the random fields mentioned in this introduction are defined on R2, take real
values and we furthermore assume that they are Gaussian.
A deformed random field is constructed with a regular, stationary and isotropic
random fieldX composed with a deterministic diffeomorphism F such that F (0) = 0.
The result of this composition is a random field X ◦ F . Stationarity, respectively
isotropy (refered to in the following as strong isotropy), consists in an invariance
of the law of a random field under translations, respectively under rotations in R2.
Even though the underlying field X is isotropic, the deformed random fields con-
structed with X are generally not. It is however possible to characterize explicitely
a diffeomorphism F such that for any underlying field X, the deformed field X ◦ F
is strongly isotropic. Such diffeomorphisms are exactly the spiral diffeomorphisms
introduced above (0.3).
The objective in [1] is to study a deformed field using sparse information, that is,
the information provided by excursion sets of the field over some basic subsets in
R2. If a real number u is fixed, the excursion set of the field X ◦ F above level u
over a compact set T is the random set
Au(X ◦ F, T ) := {t ∈ T /X(F (t)) > u}.
One useful functional to study the topology of sets is the Euler characteristic, de-
noted by χ. Heuristically, the Euler characteristic of a one-dimensional compact
regular set is simply the number of intervals in this set; the Euler characteristic of a
two-dimensional compact regular set is the number of connected components minus
the number of holes in this set.
A rotational invariance condition of the mean Euler characteristic of the excursion
sets of X ◦ F over rectangles is then introduced as a weak isotropy property. More
precisely, a random field X ◦ F is said to satisfy this weak isotropy property if for
any real u, for any rectangle T in R2 and for rotation ϕ,
(1.1) E[χ(Au(X ◦ F, ϕ(T )))] = E[χ(Au(X ◦ F, T ))].
The latter condition is in particular clearly true if the deformed field X◦F is strongly
isotropic or, in other words, if F is a spiral diffeomorphism. Provided that we add
some assumptions onX, for any rectangle T in R2, the expectation of χ(Au(X◦F, T ))
ISOTROPIC DIFFEOMORPHISMS FOR DEFORMED RANDOM FIELDS 3
can be expressed as a linear combination of the area of the set F (T ) and the length
of its frontier, with coefficients depending on u only and not on the precise law of
X.
Consequently, it occurs that Condition (1.1) is equivalent to Condition (0.1) and
therefore to Condition (0.2) on F in the present Note. Theorem 2.1 that we are
going to demonstrate in this Note therefore implies that the spiral diffeomorphisms
are the only solutions. This means that the associated deformed field is strongly
isotropic, as explained before. As a result, the weak definition of isotropy coincides
with the strong definition, as far as deformed fields are concerned.
From a practical point of view, a major consequence is that we only need infor-
mation contained in the excursion sets of a deformed random field X ◦ F (more
precisely, Condition (1.1) fulfilled) to decide the issue of isotropy.
2. Planar deformations modifying lengths and areas isotropically
We now turn to the study of the class of C2 planar transformations satisfying
(0.1). Using a polar representation for such F we translate the rotational invariant
property (0.2) into the following system of non-linear partial differential equations.
Theorem 2.1. Let two functions R : R+ × T −→ R+ and Θ : R+ × T −→ T be
continuous on R+×T and C2 in R+∗×T that satisfy: R(·, ·) is surjective and R(0, ·)
is a constant function. Let f , g and h be C1 functions from R+∗ to R such that f
does not vanish. Then the following differential equalities hold
∀(r, θ) ∈ R+∗ × T, f(r) = R∂rR∂θΘ−R∂θR∂rΘ(2.1)
g(r) = (∂rR)
2 + (R∂rΘ)
2(2.2)
h(r) = (∂θR)
2 + (R∂θΘ)
2(2.3)
if and only if there exist ε1 and ε2 in {−1, 1} and Θ0 in T such that
(i) h is strictly increasing and continuous on R+ with h(0) = 0;
(ii) for all r > 0, f(r) = ε1
h′(r)
2
and g(r)h(r) > f 2(r);
(iii) the functions R and Θ are given by
∀(r, θ) ∈ R+ × T, R(r, θ) =
√
h(r)
Θ(r, θ) = ε1θ + Θ0 + ε2
∫ r
0
√
h(r∗)g(r∗)− f 2(r∗)
h(r∗)
dr∗.
Of important note is the fact that the assumptions made on R and Θ before the
differential system are here to ensure that they indeed describe a polar representant
of a planar deformation F . Also note that the solutions obtained above are indeed
spiral deformations (0.3).
The rest of this section is devoted to the proof of the theorem above. We first find
an equivalent version of (2.1)− (2.2)− (2.3) that is not quadratic. Second, we prove
that this equivalent problem can be seen as a specific case of a hyperbolic system
of equations solely constraint to (2.3). Finally we show that these two constraints
necessarily imply Theorem 2.1.
4 MARC BRIANT AND JULIE FOURNIER
2.1. A non quadratic equivalent. Here we prove the following proposition that
gives the shape of the derivatives of R and Θ.
Proposition 2.2. Let R, Θ, f , g and h be functions as described by Theorem 2.1.
Then, they satisfy the system (2.1) − (2.2) − (2.3) if and only if there exist p in N
and a continuous function Φ : R+ × T −→ T such that
∀(r, θ) ∈ R+ × T, ∂rR =
√
g(r) cos (Φ(r, θ))
R∂rΘ =
√
g(r) sin (Φ(r, θ))
∂θR = (−1)p
√
gh− f 2
g
(r) cos (Φ(r, θ))− f√
g
(r) sin (Φ(r, θ))
R∂θΘ = (−1)p
√
gh− f 2
g
(r) sin (Φ(r, θ)) +
f√
g
(r) cos (Φ(r, θ)) .
(2.4)
Proof of Proposition 2.2. First, functions satisfying (2.4) are solutions to our original
system (2.1)− (2.2)− (2.3) by mere computation.
Now assume that the functions are solutions of (2.1)− (2.2)− (2.3). The key is to
see the quantities involved as complex numbers functions: Z1 = ∂rR + iR∂rΘ and
Z2(r, θ) = R∂θΘ− i∂θR. Then the system (2.1)− (2.2)− (2.3) translates into
∀(r, θ) ∈ R+ × T, |Z1(r, θ)|2 = g(r) and |Z2(r, θ)|2 = h(r)
Re
(
Z1(r, θ)Z2(r, θ)
)
= f(r).
We first note that f(r)2 6 g(r)h(r) and recalling that f never vanishes on R+ it
follows that neither g nor h can be null on R+. Therefore since
g + h± 2f = (∂rR±R∂θΘ)2 + (∂θR∓R∂rΘ)2 > 0
it follows 2 |f | 6 g(r) + h(r) and therefore we must in fact have
(2.5) ∀r ∈ R+, f(r)2 < g(r)h(r).
We can thus define the complex numbers
W1(r, θ) =
Z1(r, θ)√
g(r)
, W2(r, θ) =
√
f(r)2
g(r) (g(r)h(r)− f(r)2)
[
Z1(r, θ)− g(r)
f(r)
Z2(r, θ)
]
.
which are of prime importance since they satisfy the following orthonormality prop-
erty :
∀(r, θ) ∈ R+ × T, |W1(r, θ)|2 = |W2(r, θ)|2 = 1 and Re
(
W1(r, θ)W2(r, θ)
)
= 0.
We deduce that there exist a continuous function Φ : R+ × T −→ T and an integer
p > 0 such that
∀(r, θ) ∈ R+ × T, W1(r, θ) = eiΦ(r,θ) and W2(r, θ) = e−iΦ(r,θ)+(2p+1)pi2 .
Coming back to the original Z1, Z2 and then to R and Θ concludes the proof. 
ISOTROPIC DIFFEOMORPHISMS FOR DEFORMED RANDOM FIELDS 5
2.2. A hyperbolic system under constraint. We now find a more general system
of equations satisfied by the functions we are looking for as well as a restrictive
property that defines them.
Lemma 2.3. Let R, Θ, f , g and h be functions as described by Theorem 2.1. Then,
they satisfy the system (2.1)− (2.2)− (2.3) if and only if they satisfy (2.3) and there
exist α, β : R+ −→ R continuous with β(r) > 0 such that
∀(r, θ) ∈ R+∗ × T, ∂θR = α(r)∂rR− β(r)R∂rΘ
R∂θΘ = α(r)R∂rΘ + β(r)∂rR.
(2.6)
Remark 2.4. Even if this set of equations still seems non-linear, it actually is linear
in X = (ln(R),Θ). It indeed satisfies a vectorial transport equation ∂θX+A(r)∂rX =
0 with A(r) being skew-symmetric and invertible. This equation is however non
trivial as even in the case α(r) = 0 we are left to solve ∂θ [ln(R)] = −β(r)∂θΘ and
∂θΘ = β(r)∂θ [ln(R)]. And so ln(R) and Θ are both solutions to ∂θθf+β(r)
2∂rrf = 0.
For more on this subject we refer the reader to [2].
Proof of Lemma 2.3. The necessary condition follows directly from the set of equa-
tions (2.4) given in Proposition 2.2, denoting α(r) = (−1)p
√
gh−f2
g
(r) and β(r) =
f(r)/g(r) and dividing by R > 0.
The sufficient condition follows by direct computation from (2.6) and (2.3), defin-
ing g(r) = h(r)
α2(r)+β2(r)
and f(r) = β(r)g(r) > 0. 
We now show that solutions to the hyperbolic system that are constraint by (2.3)
must satisfy that R is radially symmetric.
Proposition 2.5. Let R, Θ be solution to (2.6) with R and Θ verifying the assump-
tions of Theorem 2.1. Suppose that (R,Θ) also satisfies (2.3); then R is isotropic:
for all (r, θ) in R+ × T, R(r, θ) = R(r) with moreover R(0) = 0 and R′(r) > 0 for
all r > 0.
Proof of Proposition 2.5. Let us first prove that R(r, θ) = 0 if and only if r = 0.
The surjectivity of R implies that there exists (r0, θ0) such that R(r0, θ0) = 0. If
r0 6= 0 then by Lemma 2.3 R and Θ satisfy (2.1) at (r0, θ0) and thus f(r0) = 0. This
contradicts the fact that f does not vanish. Therefore r0 = 0 and since R(0, ·) is
constant we get that
(2.7) ∀θ ∈ T, R(0, θ) = 0.
Recall that R is positive and it follows
(2.8) ∃r0 > 0, ∀r ∈ (0, r0], ∀θ ∈ T, ∂rR(r, θ) > 0.
We now turn to the study of local extrema ofR(r, ·). For a fixed r > 0, if φ is a local
extremum of R(r, ·) then ∂θR(r, φ) = 0 and also, thanks to (2.6), α(r)∂rR(r, φ) =
6 MARC BRIANT AND JULIE FOURNIER
β(r)R∂rΘ(r, φ). Plugging these equalities inside the equation satisfied by R∂θΘ in
(2.6) and since β(r) 6= 0 we get
(R∂θΘ) (r, φ) =
α2(r) + β2(r)
β(r)
∂rR(r, φ).
Then we can apply the constraint on the angular derivatives (2.3) to conclude
(2.9) ∀(r, φ) such that ∂θR(r, φ) = 0, (∂rR(r, φ))2 = β
2(r)
(α2(r) + β2(r))2
h(r).
Finally, for any r > 0, R(r, ·) is 2pi-periodic and thus has a global maximum and
a global minimum on T. Let us choose r1 in (0, r0) where r0 has been defined in
(2.8). Functions R and Θ are C2 in R+∗ × T and for any θ, the determinant of
the Jacobian of (R, θ) is equal to f(r1)/R(r1), by (2.1). This determinant does not
vanish so by the local inverse function theorem there exist [r1 − δ1, r1 + δ1] ⊂ [0, r0]
and two functions φm, φM : [r1 − δ1, r1 + δ1] −→ T that are C1 and such that
∀r > 0, R(r, φm(r)) = min
θ∈[−pi,pi]
R(r, θ) and R(r, φM(r)) = max
θ∈[−pi,pi]
R(r, θ).
By definition, for all r in [r1− δ1, r1 + δ1], ∂θR(r, φm/M(r)) = 0 and because φm and
φM are C
1 this implies:
(2.10) ∀r ∈ [r1 − δ1, r1 + δ1], ∂r
(
R(r, φm/M(r))
)
= ∂rR(r, φm/M(r)).
Thanks to (2.10), (2.9) and (2.8) we thus obtain ∂r (R(r, φm(r))) = ∂r (R(r, φM(r)))
on [r1 − δ1, r1 + δ1]. This implies for all r ∈ [r1 − δ1, r1 + δ1]:
min
θ∈[−pi,pi]
R(r, θ) = max
θ∈[−pi,pi]
R(r, θ) +
[
min
θ∈[−pi,pi]
R(r1 − δ1, θ)− max
θ∈[−pi,pi]
R(r1 − δ1, θ)
]
.
To conclude we iterate: either r1 − δ1 = 0 and we define r2 = 0 or we can start
our argument again with r2 = r1− δ1. Iterating the process we construct a sequence
(rn)n∈N∗ either strictly decreasing or reaching 0 at a certain step and such that
∀r ∈ [rn+1, r1], min
θ∈[−pi,pi]
R(r, θ) = max
θ∈[−pi,pi]
R(r, θ) + min
θ∈[−pi,pi]
R(rn+1, θ)− max
θ∈[−pi,pi]
R(rn+1, θ).
(2.11)
This sequence thus converges to r∞ > 0. If r∞ 6= 0 then we could start our process
again at r∞ and construct another decreasing sequence still satisfying (2.11). In the
end we will construct a sequence converging to 0 so without loss of generality we
assume that r∞ = 0.
Hence, since R is continuous on R+ ×T and (2.7) holds true, it follows by taking
the limit as n tends to ∞ in (2.11):
∀r ∈ [0, r1], min
θ∈[−pi,pi]
R(r, θ) = max
θ∈[−pi,pi]
R(r, θ).
The equality above implies that θ 7→ R(r, θ) is constant for any r 6 r1.
The rotational invariance of R(r, ·) holds for any r1 < r0 where r0 is such that
(2.8) holds true. Therefore, denoting rM = sup {r > 0 : ∀θ ∈ T, ∂rR(r, θ) > 0} it
follows that for any r 6 rM , θ 7→ R(r, θ) is constant. Since R(·, ·) is C2 in R+∗ × T
we infer
∀r ∈ (0, rM), ∀θ ∈ T, ∂θR(r, θ) = 0.
ISOTROPIC DIFFEOMORPHISMS FOR DEFORMED RANDOM FIELDS 7
Suppose that rM < +∞ then by continuity of ∂θR we get ∂θR(rM , θ) = 0 for all θ.
But the definition of rM implies the existence of θM such that ∂rR(rM , θM) = 0. It
follows that
∂θR(rM , θM) = 0 and ∂rR(rM , θM) = 0.
Plugging the above inside the constraint (2.3) yields f(rM) = 0 which is a contra-
diction since rM > 0.
We thus conclude that rM = +∞ and that θ 7→ R(r, θ) is invariant for any r > 0.

2.3. Isotropic solutions and proof of Theorem 2.1.
Proof of Theorem 2.1. Let us consider R, Θ, f , g and h as in the statement of
Theorem 2.1 and satisfying the system (2.1)− (2.2)− (2.3). Thanks to Proposition
2.5, there exists R : R+ −→ R+ continuous on R+ and C2 on R+∗ such that
R(0) = 0, R′(r) > 0 and R(r, θ) = R(r) for any (r, θ) in R+ × T.
First, we recall (2.5): h(r)g(r) > f(r)2 for r > 0 and since f does not vanish it
follows that h(r) > 0 and g(r) > 0 for any r > 0.
Then, thanks to (2.3) we infer |∂θΘ(r, θ)| =
√
h(r)/R(r). The right-hand side
does not vanish so neither does ∂θΘ. By continuity it keeps a fixed sign s in {−1, 1}.
∀(r, θ) ∈ R+∗ × T, ∂θΘ(r, θ) = s
√
h(r)
R(r) .
We now recall that Θ maps R+ × T to T and since the right-hand side does not
depend on θ such an equality implies that
∃ε ∈ {−1, 0, 1} , ∀r > 0,
√
h(r)
R(r) = ε.
Recalling that R(0) > 0 we deduce that ε = 1 and thus
(2.12) ∀r ∈ R+, R(r) =
√
h(r) and Θ(r, θ) = ±θ + Θ(r)
where Θ is a function from R+ to T.
At last, we use (2.1) to obtain f(r) = ±h′(r)/2 for any r > 0. The hyperbolic
system (2.6), with α(r) = (−1)p
√
gh−f2
g
(r) (p an integer defined in Proposition 2.2))
and β(r) = f(r)/g(r), yields
∀(r, θ) ∈ R+∗ × T, ∂rΘ(r, θ) = α(r)R(r, θ)∂θΘ(r, θ)− β(r)∂θR(r, θ)
(α2(r) + β2(r))R(r, θ)
which implies
(2.13) ∀r > 0, Θ(r) = ±(−1)p
∫ r
0
√
g(r∗)h(r∗)− f 2(r∗)
h(r∗)
dr∗ + Θ(0).
This concludes the proof because equations (2.5), (2.12) and (2.13) are exactly the
conditions stated in Theorem 2.1. The sufficient condition is checked by direct
computations. 
8 MARC BRIANT AND JULIE FOURNIER
References
[1] Fournier, J. Identification and isotropy characterization of deformed random fields through
excursion sets. Preprint 2017 (submitted).
[2] Serre, D. Systems of conservation laws. 1. Cambridge University Press, Cambridge, 1999.
Hyperbolicity, entropies, shock waves, Translated from the 1996 French original by I. N. Sned-
don.
Marc Briant
Laboratoire MAP5, Universite´ Paris Descartes
45, rue des Saint-Pres,
75006, Paris, France
e-mail: briant.maths@gmail.com
Julie Fournier
Laboratoire MAP5, Universite´ Paris Descartes
45, rue des Saint-Pres,
75006, Paris, France
e-mail: julie.fournier@parisdescartes.fr


ISOTROPIC DIFFEOMORPHISMS: SOLUTIONS TO A DIFFERENTIAL SYSTEM FOR A DEFORMED RANDOM FIELDS STUDY

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ISOTROPIC DIFFEOMORPHISMS: SOLUTIONS TO A DIFFERENTIAL SYSTEM FOR A DEFORMED RANDOM FIELDS STUDY

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