Two Nonlocal Variational Models for Retinex Image Decomposition

Frank W. Hammond

1 a

, Catalina Sbert

2 b

and Joan Duran

2 c

Higher Polytechnic School, University of the Balearic Islands, Spain

Department of Mathematics and Computer Science & IAC3, University of the Balearic Islands,

Cra. de Valldemossa, km. 7.5, E-07122 Palma, Illes Balears, Spain

Keywords:

Retinex Theory, Illumination, Reﬂectance, Image Decomposition, Low-Light Enhancement, Variational

Method, Total Variation, Nonlocal Regularization.

Abstract:

Retinex theory assumes that an image can be decomposed into illumination and reﬂectance components. In this

work, we introduce two variational models to solve the ill-posed inverse problem of estimating illumination

and reﬂectance from a given observation. Nonlocal regularization exploiting image self-similarities is used

to estimate the reﬂectance, since it is assumed to contain ﬁne details and texture. The difference between the

proposed models comes from the selected prior for the illumination. Speciﬁcally, Tychonoff regularization,

which promots smooth solutions, and the total variation, which favours piecewise constant solutions, are

independently proposed. A comprehensive theoretical analysis of the resulting functionals is presented within

appropriate functional spaces, complemented by an experimental validation for thorough examination.

1 INTRODUCTION

The Retinex theory (Land and McCann, 1971) aims

to explain and simulate how the human visual system

perceives color independently of global illumination

changes. Accordingly, an image can be decomposed

into luminance and reﬂectance components.

Many implementations of Retinex have been pro-

posed in the literature. Based on the center/surround

alternative algorithm (Land, 1986), Jobson et al. (Job-

son et al., 1996; Jobson et al., 1997) introduced a

method that ﬁlters the input image with Gaussian ker-

nels, taking the low-frequency result as the illumi-

nation and the residual image as the reﬂectance. In

(Horn, 1974; Morel et al., 2010), the Poisson equa-

tion is used to perform the decomposition.

Decomposing and image into illumination and

reﬂectance is mathematically ill-posed, thus prior

knowledge on the solution needs to be assumed. The

regularization theory assumes that the image which

is to be reconstructed is sufﬁciently smooth. In the

variational framework, this is formulated through the

minimization of functionals that induce a high energy

when the priors are not fulﬁlled.

Kimmel et al. (Kimmel et al., 2001) pioneered a

https://orcid.org/0009-0005-6890-0202

https://orcid.org/0000-0003-1219-4474

https://orcid.org/0000-0003-0043-1663

variational model to estimate the illumination, which

is assumed to be spatially smooth, in a multiscale set-

ting. The reﬂectance is not considered and needs to be

computed in post-processing. Guo et al. (Guo et al.,

2017) infer the illumination as the minimizer of a sim-

ple energy functional that incorporates the total vari-

ation (TV) seminorm as regularization term (Rudin

et al., 1992). The reﬂectance is obtained by pixelwise

divison between the input image and the estimated il-

lumination. However, this approach tends to amplify

the noise, especially in dark regions. To overcome this

issue, the use of denoising techniques becomes essen-

tial. Ng and Wang (Ng and Wang, 2011) estimate

illumination and reﬂectance simultaneously, penaliz-

ing gradient oscillations in the illumination through

norm and using TV for the reﬂectance, which is

thus assumed to be piecewise constant.

Many other variational methods perform the de-

composition in the logarithmic domain. In this set-

ting, Fu et al. (Fu et al., 2016) proposed a weighted

gradient-based variational model to avoid issues when

either the reﬂectance or the illumination is small.

Recently, an increasing number of deep learning

methods with different architectures have been pro-

posed (Chen et al., 2018; Wu et al., 2022). However,

these approaches are less ﬂexible and interpretable

than model-based methods, and the training and test-

ing of the networks require high computational costs.

Hammond, F., Sbert, C. and Duran, J.

Two Nonlocal Variational Models for Retinex Image Decomposition.

DOI: 10.5220/0012396800003660

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 19th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2024) - Volume 3: VISAPP, pages

551-558

ISBN: 978-989-758-679-8; ISSN: 2184-4321

551

In this paper, we propose two variational models

to simulatenously estimate the illumination and re-

ﬂectante components of an image. In both cases, non-

local regularization exploiting image self-similarities

is used to estimate the reﬂectance, since it is assumed

to contain ﬁne details and texture. For the illumina-

tion, Tychonoff regularization, favouring smooth so-

lutions, and TV, favouring piecewise constant solu-

tions, are independently proposed. A comprehensive

study of the existence and uniqueness of minimizers

in suitable functional spaces is provided.

The rest of the paper is organized as follows. In

Section 2, we present the nonlocal spaces and display

some tools from nonlocal vector calculus. Section 3

is devoted to the proposed models, where we prove

the existence and uniqueness of minimizer and intro-

duce the saddle-point formulations that will be used to

compute the solutions through the ﬁrst-order primal-

dual algorithm by Chambolle and Pock (Chambolle

and Pock, 2011). Section 4 evaluates the performance

of the proposed methods and compares them with

some state-of-the-art techniques. Finally, conclusions

are drawn in Section 5.

2 NONLOCAL THEORY

TV assumes that images consist of connected smooth

regions (objects) surrounded by sharp contours. Ac-

cordingly, it is optimal to reduce noise and reconstruct

the main geometry, but it fails to preserve ﬁne details.

On the contrary, nonlocal regularization (Gilboa and

Osher, 2009; Duran et al., 2014) allows any point to

interact directly with any other point in the domain.

The resemblance between them is usually evaluated

by comparing a patch around each point. Thus, the

underlying assumption is that images are self-similar,

thereby preserving ﬁne details and texture.

In this section, we introduce weighted L

spaces,

for which nonlocal regularization is well deﬁned, and

formalize a systematic and coherent framework for

nonlocal operators. Let Ω be a ﬁnite-measure sub-

set of R

, with n ≥2, and w: Ω →[0,+∞) a bounded

measurable function that is nonzero a.e. in Ω.

2.1 Weighted L

Spaces

Let p ∈ [1,+∞]. We deﬁne the weighted L

space as

(Ω) =



f : Ω → R :

Ω

(|f |

w)dx < +∞



which is endowed with the norm ∥f ∥

p,w

= ∥f w

1/p

∥

Note that, in order ∥·∥

p,w

to be a norm, w must be

nonzero a.e. in Ω, otherwise it will be a seminorm.

It is easy to show that L

(Ω) is a Banach space

containing L

(Ω), for p ∈ [1,+∞]. We are interested

in L

(Ω), which is a Hilbert space equipped with the

scalar product

⟨

f ,g

⟩

2,w

⟨

√

w,g

√

⟩

Ω

f gw dx.

Deﬁnition 2.1. We deﬁne the difference function of

u ∈ L

(Ω) as

u(x,y) = u(y) −u(x).

Note that ∥

u∥

≤2|Ω|

1/p

∥u∥

, where the norm on

the left is the L

(Ω×Ω)-norm and the one on the right

is the L

(Ω)-norm. Therefore,

u ∈ L

(Ω ×Ω).

Proposition 2.1. Let {u

}

n∈N

be a sequence in L

(Ω)

converging weakly to u in L

(Ω). Then, {

}

n∈N

con-

verges weakly to

u in L

(Ω).

Proof. Since L

(Ω ×Ω) is a Hilbert space, we can

identify f

∗

with some f ∈ L

(Ω × Ω) such that

∗

(x) =

⟨

f ,x

⟩

. We have

⟨

f ,

⟩

2,w

Ω

(y) −u

(x)) f (x,y)w(x, y)dxdy.

Let us deﬁne the functions F(x) =

Ω

f (x, y)w(x,y)dy

and G(y) =

Ω

f (x, y)w(x,y)dx. Then,

(F(x))

≤

Ω

f (x, y)

w(x,y)dy,

where M is an upper bound for w, so that

Ω

(F(x))

dx ≤ M|Ω| · ∥f ∥

2,w

< +∞. Thus, F ∈

(Ω). Similarly, it can be proved that G ∈ L

(Ω). It

follows that f

∗

(

) →

⟨

u,G

⟩

−

⟨

u,F

⟩

= f

∗

(

u).

Corollary 2.2. Let {u

}

n∈N

be a sequence in L

(Ω)

converging weakly to u in L

(Ω). Then,

liminf

∥

2,w

≥ ∥

u∥

2,w

Proof. It follows from previous result and weak lower

semicontinuity of norms in a Banach spaces.

2.2 Basic Nonlocal Vector Calculus

The notion of directional derivative extends to the

nonlocal case as ∂

u(x) =

u(x,y)

w(x,y). The non-

local gradient is then deﬁned as the vector of nonlocal

derivatives, i.e., ∇

u(x,y) = ∂

u(x).

The nonlocal divergence is deﬁned to satisfy the

adjoint relation

⟨

∇

u,v

⟩

2,Ω×Ω

⟨

u,−div

(v)

⟩

2,Ω

. A

sufﬁcient condition for this is deﬁning

div

(v)(x) =

Ω

v(y, x)

w(y, x) −v(x, y)

w(x,y)dy.

3 PROPOSED MODELS

Let us assume that the observed image S deﬁned on Ω

is the product of the illumination L ∈ (0,+∞) and the

VISAPP 2024 - 19th International Conference on Computer Vision Theory and Applications

552

reﬂectance R ∈ (0,1). By transforming S = R ·L into

the logarithmic domain, we get

l = s + r, (1)

where l = log(L), s = log(S) and r = −log(R). Based

on (1), we introduce two nonlocal variational models

to estimate l and r simultaneously, using nonlocal reg-

ularization for the reﬂectance, assumed to contain ﬁne

details and texture, and testing Tychonoff and TV for

the luminance, assumed to be smooth.

To ﬁnd a global optimal solution to the proposed

minimization problems, we use the ﬁrst-order primal-

dual algorithm introduced in (Chambolle and Pock,

2011). Therefore, we rewrite each problem in a

saddle-point formulation by introducing dual vari-

ables. The algorithm consists of alternating a gra-

dient ascent in the dual variable, a gradient descent

in the primal variable, and an over-relaxation for

convergence purposes. The gradient steps are given

in terms of proximity operators, which are deﬁned

for any proper convex function ϕ as prox

ϕ(x) =

argmin

ϕ(y) +

2τ

∥x −y∥

. For all details on convex

and functional analysis omitted in this section, we re-

fer to (Br

ezis, 2011; Chambolle and Pock, 2016).

3.1 Nonlocal and Tychonoff Terms

We propose to estimate r and l as the minimizers of

(r, l) = ∥

r∥

2,w

∥|∇l|∥

∥l −r −s∥

∥l∥

(2)

where α, β, γ > 0 are trade-off parameters. The last

term is added for purely technical reasons and has no

actual practical signiﬁcance.

Theorem 3.1. Let Λ = L

(Ω)×W

1,2

(Ω) be the space

of admissible functions and s ∈L

(Ω). There exists a

unique (r

∗

) ∈ Λ s.t. E

∗

) = inf

(r,l)∈Λ

(r, l).

Proof. Existence. We follow the direct method and a

proof similar to that in (Ng and Wang, 2011).

It is easy to see that E

is proper and bounded be-

low, thus b = inf

(r,l)∈Λ

(r, l) < ∞. Let {(r

)}⊆Λ

be such that E

) → b.

Since E

) is uniformly bounded, so are

Ω

|∇l

and ∥l

∥

, implying that {l

} is uniformly

bounded in W

1,2

(Ω). By the Rellich-Kondrachov the-

orem, there exists l

∗

∈ L

(Ω) such that l

→ l

∗

(Ω), and, up to a subsequence, l

⇀ l

∗

in W

1,2

(Ω)

due to the space being reﬂexive.

Furthermore, {r

} is uniformly bounded in L

(Ω)

as ∥r

∥

≤ ∥l

−r

−s∥

+ ∥s∥

+ ∥l

∥

. Thus, there

exists r

∗

∈L

(Ω) s.t., up to a subsequence, r

⇀ r

∗

(Ω). By Proposition 2.1,

⇀

∗

in L

(Ω ×Ω).

Finally, due to the weak lower semicontinuity of

the norms, b = lim inf

) ≥ E

∗

) ≥ b,

from which we deduce E

∗

) = b.

Uniqueness. It is a direct consequence of E

being

strictly convex.

3.1.1 Saddle-Point Optimization

Let Λ

′

= L

(Ω ×Ω) ×



(Ω)



and K : Λ → Λ

′

the linear operator K(r,l) = (∇

r, ∇l). We also con-

sider G : Λ → [0,+∞] and F : Λ

′

→ [0,+∞], respec-

tively deﬁned as G(r,l) =

∥r −l −s∥

∥l∥

and

F(a, b) = ∥a∥

∥

The minimization of (2) can be rewritten in a

saddle-point formulation as

min

(r,l)∈Λ

sup

(a,b)∈Λ

′

⟨

(a,b),K(r,l)

⟩

−F

∗

(a,b) + G(r,l),

where F

∗

denotes the convex conjugate of F.

In practice, the initial values of a sequence gen-

erated by an iterative algorithm may have errors due

to arbitrary initialization, which can accumulate and

lead to undesired results in the ﬁnal image. To miti-

gate this, an additional constraint is introduced. Since

l = s + r, with r ≥ 0, then l ≥ s. Therefore, it makes

sense to impose both r ≥ 0 and l ≥ s.

Finally, the luminance l and the reﬂectance r

are computed through the following Chambolle-Pock

primal-dual iterates, that is, initialize over-relaxiation

variables ˜r

= r

= 0,

= l

= s, and update the dual

variables as follows:

 

n+1



= prox

σF

∗



+ σ∇

(˜r

)

+ σ∇







Then, update the primal variables, impose the con-

straints r ≥ 0, l ≥ s and update the over-relaxation

variables.













n+1/2



= prox

τG



+ τdiv

n+1

)

+ τdiv(b

n+1

)



n+1

= max(r

n+1/2

,0), l

n+1

= max(l

n+1/2

,s)

˜r

n+1

= 2r

r+1

−r

n+1

= 2l

n+1

−l

The proximity operators involved are

prox

τG





(βτ+γτ+1)a+βτb−βτ(γτ+1)s

βγτ

+2βτ+γτ+1

βτa+(βτ+1)b+βτs

βγτ

+2βτ+γτ+1

prox

σF

∗





1+σ/2

1+σ/α

3.2 Nonlocal and TV Terms

We take s ∈ L

(Ω) and α, β,γ positive numbers,

and consider the functional E

over Λ = L

(Ω) ×

Two Nonlocal Variational Models for Retinex Image Decomposition

553

Building Lamp Bookcase

Horses Papiervert

Figure 1: Dataset used for the experiments.



BV(Ω) ∩L

(Ω)



(r, l) = ∥

r∥

2,w

+ αTV(l) +

∥l −r −s∥

∥l∥

(3)

Theorem 3.2. There exists a unique (r

∗

) ∈ Λ such

that

∗

) = inf

r,l

(r, l)

Proof. Existence. This proof is very similar to that

of Theorem 3.1. E

is clearly a proper functional,

and thus we can consider a minimizing sequence

) → b = inf E

. Again, each of the additive

terms of E

) will be uniformly bounded.

From Holder inequality it follows that

∥

≤

Ω

1/2

∥

is uniformly bounded.

Therefore, {TV(l

)}

n∈N

and {

∥

}

n∈N

are uni-

formly bounded, whence {l

}

n∈N

is bounded in

BV(Ω). Therefore, there exists some l

∗

∈ BV(Ω)

such that, up to a subsequence,

(Ω)

→ l

∗

and l

(Ω)

⇀ l

∗

∈ L

(Ω).

Because of the lower semicontinuity of norms in

BV(Ω) and L

(Ω),

liminf



αTV(l

) +

∥



≥ αTV(l

∗

) +

∥

∗

∥

This, as in 3.1, implies that E

∗

) = b.

Uniqueness. Again, a direct consequence of the

functional being strictly convex.

3.2.1 Saddle-Point Optimization

The same primal-dual algorithm as in the previous

section is used. The problem obtained is

min

r,l

T (K(r,l)) + G(r,l),

with K(r,l) = (∇

r, ∇l), T (a,b) =

∥

+ α

∥

|b|

∥

and G(r,l) =

∥

l −r −s

∥

is the same as in

the Tychonoff functional. The algorithm used is very

similar, the only difference being that the dual vari-

ables are updated via the proximity operator of T

∗

 

n+1



= prox

σT

∗



+ σ∇

(˜r

)

+ σ∇







The proximity operator is computed as follows:

prox

σT

∗





1+σ/2

max(1,

/α)

3.3 Proposed Weights

For the nonlocal regularization term in both mod-

els, we need to select an appropriate weight func-

tion w : Ω ×Ω → [0,+∞). We propose to use bilat-

eral weights that consider both the spatial closeness

between points and the similarity in the input image

S : Ω → R

. This similarity is computed by consid-

ering a whole patch around each point and using the

Euclidean distance across the color channels:

(S(x),S (y)) =

Ω

(z)

S(x + z) −S(y + z)

dz,

where |·| denotes the Euclidian norm in R

and G

a Gaussian kernel of standard deviation a ≥ 0.

The weights are deﬁned as

w(x,y) =

Γ(x)

exp

−

|x −y|

spt

−

(S(x),S (y))

sim

(4)

where h

spt

sim

> 0 are ﬁltering parameters that con-

trol how fast the weights decay with increasing spa-

tial distance or dissimilarity between patches, respec-

tively, and Γ(x) is the normalization factor

Γ(x) =

Ω

exp

−

|x −y|

spt

−

(S(x),S (y))

sim

dy.

Note that 0 < w(x,y) ≤ 1 and

Ω

w(x,y)dy = 1, but

the normalization factor breaks down the symmetry

of w. In the end, the average made between very sim-

ilar regions preserves the integrity of the image but

reduces its small oscillations, which contain noise.

For computational purposes, the nonlocal regular-

ization is limited to interact only between points at a

certain distance. Accordingly, the weight distribution

is in general sparse since a few nonzero values are

considered. In this setting, let N (x) denote a neigh-

bourhood around each x ∈Ω. Then, w(x,y) is deﬁned

as in (4) if y ∈ N (x), and zero otherwise. The nor-

malization factor is ﬁnally given by

Γ(x) =

N (x)

exp

−

|x −y|

spt

−

(L(x),L(y))

sim

dy.

VISAPP 2024 - 19th International Conference on Computer Vision Theory and Applications

554

α = 0.1,β = 0.1 α = 0.1,β = 5 α = 5,β = 0.1 α = 5,β = 5

Figure 2: Visual impact of the trade-off parameters α and β in the proposed models for the Papiervert image. The estimated

illumination maps are displayed in the ﬁrst and third rows for (2) and (3), respectively. The corresponding reﬂectance compo-

nents are displayed in the second and fourth rows for (2) and (3), respectively. Larger values of the regularization parameter

α provide smoother illumination components, while the geometry and colors remain in the reﬂectance maps. We observe that

the shadows due to lighting conditions are retained in the illumination components. Furthermore, the difference between the

two proposed models is evident in the estimated illumination maps. The results from (2) tend to be isotropically smooth due

to the use of a Tychonoff prior, while the results from (3) are piecewise constant, as expected from the TV.

In practice, the weight of the reference point is

set to the maximum of the weights in the neihbour-

hood, w(x,x) = max{w(x, y) : y ∈ N (x)}. This set-

ting avoids the excessive weighting of the reference

point. Furthermore, G

is not considered as it is only

necessary when the size of N (x) is large.

4 ANALYSIS AND EXPERIMENTS

In this section, we analyze the performance of the

proposed method for illumination and reﬂectance de-

composition. Figure 1 displays the images we used

in all experiments: Building (Petro et al., 2014),

Lamp (Guo et al., 2017), Bookcase (Wei et al., 2018),

Horses (Petro et al., 2014) and Papiervert (Morel

et al., 2010).

For our variational methods, we ﬁx the follow-

ing parameters throughout the experimental section:

γ = 10

−5

, h

spt

= 1.25, h

sim

= 2.5 and the number of

iterations is set to 2000. We revert the logarithmic

transformation by applying the exponential function

to the outputs of the primal-dual algorithm.

In Figure 2 we display the decomposition results

Two Nonlocal Variational Models for Retinex Image Decomposition

555

Illumination Reﬂectance Enhanced image

Figure 3: Resulting decomposition (α = 1, β = 5) of Lamp by the proposed Tychonoff (2) (ﬁrst row) and TV (3) (second

row) models, and the respective enhanced images after applying a gamma correction with parameter 0.4 to the illumination

component. We observe that the geometry, texture, color information, and noise are retained in the reﬂectance maps. Further-

more, the enhanced image by a simple gamma correction is able to discount the effect of the ilumination in the scene. We also

observe that the ilumination map produced by Tychonoff is isotropically smooth while that of TV is piecewise constant.

for different α and β combinations in both models on

Papiervert. Values were chosen taking into account

that the nonlocal regularization term has a coefﬁcient

of 1 in both models. Thus, the smaller the values α,β

are in relation to 1, the more importance is given to

nonlocal regularization. In both models, bigger val-

ues for α tend to yield smoother results for illumina-

tion, which was to be expected since we are imposing

that |∇l| be small in the functional. Smaller values

for α tend to yield illumination approximations that

are very close to the original image, and reﬂectance

approximations which are very close to R = 1. Big-

ger values for β tend to produce better results, since

this parameter corresponds to our ﬁdelity term. How-

ever, values too big (>> 20, as experience indicates),

tend to produce decompositions with the illumination

approximation too close to the original image. We

observe that the shadows due to lighting conditions

are retained in the illumination components. Further-

more, the difference between the two proposed mod-

els is evident in the estimated illumination maps. The

results from (2) tend to be isotropically smooth due

to the use of a Tychonoff prior, while the results from

(3) are piecewise constant, as expected from the TV.

Experience shows that best results are often ob-

tained for values of α and β smaller than 20, bigger

than 1 and β ≥ α. We propose α = 1, β = 5 as default

parameters in both models.

In Figure 3, we show the decomposition of Lamp

for our default combination α = 1 and β = 5. A

gamma correction of 0.4 is applied to illumination (al-

though the corrected version is not displayed) before

computing the resulting image.

In Figure 4, we compare the performance of our

method with state-of-the-art techniques Multiscale

Retinex (MSR) (Jobson et al., 1997), Kimmel et al.

VISAPP 2024 - 19th International Conference on Computer Vision Theory and Applications

556

α = 1, β = 10, Correction 0.5 α = 2, β = 10, Correction 0.35 α = 1, β = 5, Correction 0.6

α = 1, β = 10, Correction 0.5 α = 1, β = 1, Correction 0.35 α = 1, β = 5, Correction 0.6

Figure 4: Comparison between state-of-the-art techniques for low-light image enhancement and our models combined with a

gamma correction to the illumination component. Each row contains the results by Kimmel et al., LIME, MSR, our Tychonoff

model (2), and our TV model (3). Compared to our proposals, the method by Kimmel et al. produces similar results in terms of

illumination, but is more sensitive to noise, LIME produces oversaturated colors, while MSR is not robust to noise and exhibits

color issues. Our methods provide a good compromise between discounting the illumination, avoiding the ampliﬁcation of

noise, and preserving color and geometry. Note also that the results by Tychonoff are slightly more blurred than those of TV,

as seen in the clouds and grass of Horses and on trees on Building.

Two Nonlocal Variational Models for Retinex Image Decomposition

557

(Kimmel et al., 2001) and LIME (Guo et al., 2017)

on Horses, Bookcase and Building. The most suitable

parameters for all methods have been chosen based

on visual evaluation. We observe that neither MSR

nor the method proposed by Kimmel et al. are robust

to noise, LIME oversaturates color and MSR yields

greyish images. In contrast, both our methods cor-

rectly enhance illumination, respect color and are rel-

atively robust to noise. The Tychonoff model pre-

serves color slightly better than TV, but contrasts are

clearer in the latter.

5 CONCLUSION

In this paper, we proposed two variational models

to simultaneously estimate the luminance and re-

ﬂectance components from an observed image. Non-

local regularization has been employed as a prior for

the reﬂectance to help preserve colors and texture. Ty-

chonoff and TV regularizations have been tested for

the illumination component. We utilized this decom-

position for low-light image enhancement.

In future work, it may be interesting to explore

more sophisticated methods to enhance illumination

and experiment with different mechanisms to reduce

noise in our estimation of reﬂectance.

ACKNOWLEDGEMENTS

This work is part of the MaLiSat project TED2021-

132644B-I00, funded by MCIN/AEI/10.13039/

501100011033/ and by the European Union

NextGenerationEU/PRTR, and also of the Mo-

LaLIP project PID2021-125711OB-I00, ﬁnanced by

MCIN/AEI/10.13039/501100011033/FEDER, EU.

In memoriam Frank G. Hammond Figueroa.

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