Combining Total Variation and Nonlocal Variational Models for

Low-Light Image Enhancement

Daniel Torres

, Catalina Sbert

and Joan Duran

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:

Low-Light Image Enhancement, Illumination Estimation, Reﬂectance Estimation, Retinex, Variational

Method, Total Variation, Nonlocal Regularization.

Abstract:

Images captured under low-light conditions impose signiﬁcant limitations on the performance of computer

vision applications. Therefore, improving their quality by discounting the effects of the illumination is crucial.

In this paper, we present a low-light image enhancement method based on the Retinex theory. Our approach

estimates illumination and reﬂectance in two steps. First, the illumination is obtained as the minimizer of an

energy functional involving total variation regularization, which favours piecewise smooth solutions. Next,

the reﬂectance component is computed as the minimizer of an energy functional involving contrast-invariant

nonlocal regularization and a ﬁdelity term preserving the largest gradients of the input image.

1 INTRODUCTION

Enhancing images captured under low-light condi-

tions is crucial for many applications in computer vi-

sion. Various strategies have been proposed to tackle

this problem (Wang et al., 2020), broadly classiﬁed

into histogram equalization (Thepade et al., 2021;

Paul et al., 2022), Retinex-based methods, fusion ap-

proaches (Fu et al., 2016a; Buades et al., 2020), and

deep-learning techniques.

The Retinex theory (Land and McCann, 1971),

which aims to explain and simulate how the human

visual system perceives color independently of global

illumination changes, has been an important basis for

addressing low-light image enhancement. One of the

most used models assumes that the observed image L

is the product of the illumination T , which depicts the

light intensity on the objects, and the reﬂectance R,

which represents their physical characteristics:

L = R ◦T, (1)

where ◦ denotes pixel-wise multiplication. The illu-

mination map is assumed to be smooth, while the re-

ﬂectance component contains ﬁne details and texture

(Ng and Wang, 2011; Li et al., 2018).

https://orcid.org/0009-0000-5829-8557

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

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

Decomposing an image into illumination and re-

ﬂectance is mathematically ill-posed. To address this

issue, patch-based (Land and McCann, 1971), par-

tial differential equations (Morel et al., 2010), cen-

ter/surround (Jobson et al., 1996) and variational

(Kimmel et al., 2001; Ng and Wang, 2011; Ma and

Osher, 2012; Fu et al., 2016b; Guo et al., 2017; Gu

et al., 2019) methods have been proposed.

Recently, the growing popularity of deep learning

has lead to an increase in enhancement methods (Wei

et al., 2018; Lv et al., 2021). However, the structure

of these architectures is often non-intuitive, and their

training and testing require high computational costs.

In this paper, we present a low-light image en-

hancement method taking into account the decompo-

sition model given in (1). We propose to estimate illu-

mination and reﬂectance separately. In the ﬁrst step,

the illumination component is obtained as the mini-

mizer of an energy functional involving total varia-

tion (TV) regularization (Rudin et al., 1992), which

favours piecewise smooth solutions. We assume that

the channels of color images share the same illumina-

tion map. In a second step, the reﬂectance component

is computed as the minimizer of an energy involving

nonlocal regularization, which exploits image self-

similarities, and a ﬁdelity term preserving the large

gradients of the input low-light image. Importantly,

the nonlocal regularization depends on a weight func-

tion that is contrast-invariant.

508

Torres, D., Sbert, C. and Duran, J.

Combining Total Variation and Nonlocal Variational Models for Low-Light Image Enhancement.

DOI: 10.5220/0012386300003660

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

508-515

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

2 RELATED WORK

In the variational framework, the enhanced image is

computed as the minimizer of an energy functional

that comprises data-ﬁdelity and regularization terms.

The latter quantiﬁes the smoothness of the solution

by usually prescribing priors on the gradient of the

illumination or reﬂectance components.

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

a variational model to estimate the illumination in a

multiscale setting. The illumination is assumed to be

spatially smooth, thus gradient oscillations are penal-

ized through L

norm. Ma et al. (Ma and Osher, 2012)

introduced TV to directly compute the reﬂectance.

On the contrary, Ng et al. (Ng and Wang, 2011) es-

timate illumination and reﬂectance simultaneously.

One of the most celebrated works is LIME (Guo

et al., 2017). The authors propose a simple TV vari-

ational model to estimate T . Then, the reﬂectance is

computed as R =

+ε

, where ε > 0 is a small con-

stant and T

is the γ-corrected illumination. However,

this strategy tends to amplify the noise, especially in

dark regions. To overcome this issue, the authors pro-

pose as enhanced image R ◦T

+ R

◦(1 −T

), with

being the denoised reﬂectance. Note that any dis-

crepancy in the illumination will impact the result.

Several other variational methods use logarithms

to linearize (1). However, this transformation ampli-

ﬁes errors in gradient terms. In (Fu et al., 2016b),

weights are introduced to address issues arising from

large gradients when either R or T is small.

3 PROPOSED MODELS

In this section, we introduce our low-light image en-

hancement method. We estimate R and T separately

using two different variational models. Let Ω be an

open and bounded domain in R

, with n ≥ 2.

3.1 TV Model for Luminance

Based on the classical ROF denoising model (Rudin

et al., 1992), T is obtained as the minimizer of

Ω

+ λ

Ω

|T (x) −

T (x)|

dx, (2)

where

Ω

is the TV semi-norm,

λ > 0 is a trade-off parameter, and

T is an

initial illumination estimate computed as

T (x) =

max

c∈{R,G,B}

(x). The assumption underlying TV is

that images consist of connected smooth regions (ob-

jects) surrounded by sharp contours. Accordingly, TV

is a good prior for the luminance since it is optimal

Bookcase Clothes rack Plush toy

Figure 1: Pairs of low-light and ground-truth images from

the LOL dataset (Wei et al., 2018) used for the experiments.

to reduce noise and reconstruct the main geometrical

shape.

The energy in (2) is derived from LIME (Guo

et al., 2017). However, we calculate the minimizer

for the exact functional instead of an approximation.

3.2 Nonlocal Model for Reﬂectance

To estimate the reﬂectance component, we use non-

local regularization (Gilboa and Osher, 2009; Duran

et al., 2014), which assumes that images are self-

similar, thereby preserving ﬁne details and texture.

3.2.1 Basic Deﬁnitions and Notations

Let u = (u

,...,u

) : Ω →R

be a color image, where

C denotes the number of channels. We also consider

nonlocal functions p = (p

,. .., p

) : Ω ×Ω → R

Let ω : Ω×Ω →R

≥0

be a weight function, commonly

deﬁned in terms of differences between patches in u.

Deﬁnition 3.1. Given u : Ω → R

, its nonlocal gra-

dient ∇

u : Ω ×Ω → R

is, for each k ∈ {1,...,C},

(∇

)(x, y) =

ω(x,y) (u

(y) −u

(x)).

Given p : Ω × Ω → R

, its nonlocal divergence

div

p : Ω → R

is, for each k ∈ {1,...,C},

(div

)(x)

Ω



(x,y)

ω(x,y) − p

(y,x)

ω(y,x)



dy.

Deﬁnition 3.2. The nonlocal vectorial total variation

(NLVTV) of u ∈ L

(Ω;R

) is deﬁned as

Ω

u| = sup

p∈Y



−

Ω

⟨u(x),(div

p)(x)⟩dx



with Y = {p ∈C

∞

(Ω×Ω; R

) : |p|

(x) ≤1,∀x ∈Ω}

and |p|

(x) =

∑

k=1

Ω

(x,y))

dy.

3.2.2 Nonlocal Energy Functional

We propose to estimate the reﬂectance R : Ω → R

usually C = 3, as the minimizer of the functional

Combining Total Variation and Nonlocal Variational Models for Low-Light Image Enhancement

509

λ = 0.005 λ = 0.05 λ = 0.5

Figure 2: Visual impact of the trade-off parameter λ in (2). Each row contains the illumination, reﬂectance and enhanced

image, respectively. We observe a piecewise smooth illumination in all cases. However, the differences are more noticeable

in the reﬂectances and the enhanced images. Indeed, the smaller λ is, the brighter the result.

Ω

R|+

Ω

|∇R −G|

Ω

|R −R

, (3)

where α,β > 0 are trade-off parameters,

Ω

R| is

the NLVTV, and R

is an initial estimate of the re-

ﬂectance computed as R

(x) =

L(x)

T (x)+ε

, with T being

the minimizer of (2) and ε > 0 a small constant. The

weight function ω will be deﬁned in terms of differ-

ences between patches in L. Therefore, the underlying

assumption behind NLVTV is that images are self-

similar, making it a good prior for the reﬂectance.

The second term in (3) enforces closeness be-

tween the gradients of the reﬂectance and those of an

adjusted version of the low-light image, strengthening

the structural information. Following (Li et al., 2018),

we deﬁne G =



1 + λ

−|∇

L|/σ



◦∇

L, with

∇

L =



0 if |∇L| < ε

∇L otherwise,

where σ

controls the ampliﬁcation rate of different

gradients, λ

controls the degree of the ampliﬁcation,

and ε

is the threshold that ﬁlters small gradients.

3.2.3 Contrast-Invariant Weights

For the function ω : Ω × Ω → R

≥0

involved in

NLVTV, we propose to use bilateral weights that con-

sider both the spatial closeness between points and the

similarity in L : Ω → R

. This similarity is computed

by considering a whole patch around each point and

using the Euclidean distance across color channels:

d (L(x), L(y))

Ω

(L(x + z) −µ(x)) −(L(y + z) −µ(y))

dz,

(4)

where µ(x) denotes the mean value of the patch cen-

tered at x. This allows the distance (4) to be contrast

invariant between the selected patches.

The weights are deﬁned as

ω(x,y) =

Γ(x)

exp

−

|x −y|

spt

−

d(L(x),L(y))

sim

(5)

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. Note that

0 < ω(x,y) ≤1 and

Ω

ω(x,y) dy = 1, but Γ(x) breaks

down the symmetry of ω. The average between very

similar regions preserves the integrity of the image

but reduces small oscillations, which contain noise.

For computational purposes, NLVTV is limited to

interact only between points at a certain distance. Let

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

510

α = 0.0001, β = 0.5 α = 0.01, β = 0.5 α = 1, β = 0.5

α = 0.01, β = 0.05 α = 0.01, β = 0.5 α = 0.01, β = 1

Figure 3: Visual impact of the trade-off parameters α and β in (3) on the ﬁnal enhanced image. Larger values of α or β result

in an over-smoothed image, as can be appreciated in the third column. On the other hand, when the role of the regularization

term is less relevant, as it is the case for β = 0.05, the noise is enhanced in the ﬁnal result.

N (x) denote a neighbourhood around x ∈ Ω. Then,

ω(x,y) is deﬁned as in (5) if y ∈ N (x), and zero oth-

erwise. The normalization factor is ﬁnally given by

Γ(x) =

N (x)

exp

−

|x −y|

spt

−

d (L(x), L(y))

sim

dy.

3.3 Enhanced Image

Once we obtain the illumination and the reﬂectance,

we need to adjust the lighting by applying a gamma

correction with parameter γ > 0 to T , helping us ad-

dress the over-saturation problem. This transforma-

tion involves the following operation at each pixel:

′

(x) = S



T (x)



, (6)

where S is a constant typically set to 1 to ensure that

the inputs and outputs are within the same range. Fi-

nally, the enhanced image is computed as L

′

= R ◦T

′

4 PRIMAL-DUAL

OPTIMIZATION

In the discrete setting, L,R ∈ R

N×C

and T ∈ R

where N is the number of pixels and C is the num-

ber of channels. Both ∇T ∈ R

N×2

and ∇R ∈ R

N×C×2

are computed via forward differences and Neumann

boundary conditions, and denoted for each pixel i

and channel k by (∇T)

= ((∇T )

i,1

,(∇T )

i,2

) and

(∇R)

i,k

= ((∇R)

i,k,1

,(∇R)

i,k,2

), respectively.

Let M be the size of the neighborhood around

each pixel where the weights of the NLVTV regu-

larization term are nonzero. The nonlocal gradient

∇

R ∈R

N×C×M

, denoted for each pixel i and channel

k by (∇

i,k

= ((∇

i,k,1

,. ..,(∇

i,k,M

), is de-

ﬁned as (∇

i,k, j

√

i, j



j,k

−R

i,k



, where {ω

i, j

}

contains the discretization of the weights (5). In prac-

tice, the weight of the reference pixel is set to the

maximum of the weights in the neihbourhood, i.e.,

i,i

= max

1≤j≤M

i, j

. This setting avoids the exces-

sive weighting of the reference pixel.

Both minimization problems (2) and (3) are con-

vex but non-smooth. To ﬁnd a global optimal so-

lution, we use the ﬁrst-order primal-dual algorithm

introduced in (Chambolle and Pock, 2011). To do

so, we rewrite each problem in a saddle-point for-

mulation by introducing dual variables. The algo-

rithm consists of alternating a gradient 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 the proximity op-

erator, which is deﬁned for any proper convex func-

tion ϕ as prox

ϕ(x) = arg min

{ϕ(y) +

2τ

∥x −y∥

The efﬁciency of the algorithm is based on the as-

sumption that proximity operators have closed-form

representations or can be efﬁciently solved. For all

details on convex analysis omitted in this section, we

refer to (Chambolle and Pock, 2016).

4.1 Estimation of the Illumination

The discrete variational model related to (2) is

min

T ∈R

∥∇T ∥

+ λ∥T −

T ∥

, (7)

Combining Total Variation and Nonlocal Variational Models for Low-Light Image Enhancement

511

γ = 0.25 γ = 0.4 γ = 0.55

Figure 4: Study of the effect of the gamma correction (6) with parameter γ. In the ﬁrst row, we display the γ-corrected

illuminations and the second row contains the resulting enhanced images. We observe that an excessive gamma correction

(γ = 0.25) leads to a saturated image, while a less signiﬁcant transformation (γ = 0.55) results in an overly dark image.

where ∥∇T ∥

∑

i=1

|(∇T )

|. The saddle-point for-

mulation of (7) is given by

min

T ∈R

max

p∈P

⟨∇T, p⟩−δ

(p) + λ∥T −

T ∥

where P = {p ∈ R

N×2

: |p

i,:

| ≤ 1,∀i} and δ

is the

indicator function of P .

Therefore, T is computed through the following

primal-dual iterates:











n+1

i, j

+ σ(∇T

)

i, j

max



1,|p

+ σ(∇T

)



n+1

+ τ divp

n+1

+ τλ

1 + τλ

n+1

= 2T

n+1

−T

4.2 Estimation of the Reﬂectance

The discretization of the variational model (3) is

β∥∇

R∥

∥∇R −G∥

∥R −R

∥

, (8)

where ∥∇

R∥

∑

i=1

∑

k=1

|(∇

i,k

and ∥·∥

applies across channel and pixel dimensions.

The NLVTV term is treated analogously to the

TV case. We also need to dualize the α-term since

its proximity operator has no closed-form. By tak-

ing into account that

∥x∥

= sup

⟨x,y⟩−

2α

∥y∥

, the

saddle-point formulation of (8) is

min

R∈R

N×C

max

p∈R

N×C×2

,q∈Q

⟨∇R, p⟩+ ⟨∇

R,q⟩

−

2α

∥p −G∥

−δ

(q) +

∥R −R

∥

with Q = {q ∈ R

N×C×M

∑

k=1

∑

j=1

i,k, j

≤ β

,∀i}.

Therefore, R is computed through the following

primal-dual iterates:











n+1



+ σ∇R



+ σG

α + σ

n+1

i,k, j



i,k, j

+ σ(∇

)

i,k, j



max



β,|q

+ σ(∇

)



n+1

+ τdivp + τdiv

q + τR

1 + τ

n+1

= 2R

n+1

−R

5 ANALYSIS AND EXPERIMENTS

In this section, we analyze the performance of the pro-

posed method for low-light image enhancement. Fig-

ure 1 displays the pairs of low-light and ground-truth

images from the LOL dataset (Wei et al., 2018) used

in the experiments. We have also used Lamp from

(Guo et al., 2017), a natural low-light image.

5.1 Ablation Study

Figure 2 illustrates the impact of the trade-off param-

eter λ in the variational model (2). We observe that a

piecewise smooth illumination is obtained in all cases.

The slight variation in the illumination is ampliﬁed

in the reﬂectance maps and the enhanced images, re-

sulting in darker or brighter images, depending on

whether more or less weight is given to the ﬁdelity

term compared to the regularization term.

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

512

Bookcase MSR Kimmel et al. SRIE

LIME RetinexNet AGLLNet Ours

Plush toy MSR Kimmel et al. SRIE

LIME RetinexNet AGLLNet Ours

Clothes rack MSR Kimmel et al. SRIE

LIME RetinexNet AGLLNet Ours

Figure 5: Comparison between state-of-the-art techniques and our method on dataset in Figure 1. We observe that MSR, SRIE

and RetinexNet are not robust to noise and exhibit color issues in all experiments. The method by Kimmel et al. is not able

to correctly remove the effect of the illumination, while LIME produces oversaturated results. AGLLNet produces greyish

images and introduces color artifacts as seen for instance in the blue bottle and the red ball of yarn. Our method provides the

best compromise between discounting the illumination effect, avoiding the ampliﬁcation of noise, and preserving the color

and geometry structure of the scene.

Combining Total Variation and Nonlocal Variational Models for Low-Light Image Enhancement

513

Lamp MSR Kimmel et al. SRIE

LIME RetinexNet AGLLNet Ours

Figure 6: Comparison between state-of-the-art methods and our proposal on a natural low-light image. We observe that only

LIME, RetinexNet and our method are able to remove the effect of the illumination. However, LIME and RetinexNet are

affected by noise and also produce enhanced images with artifacts, like the halo surrounding the lamp.

In Figure 3, we discuss about the inﬂuence of the

trade-off parameters α and β in (3) on the enhanced

image produced by the proposed method. We observe

that larger values of α or β result in an over-smoothed

image. On the other hand, when the role of the reg-

ularization term is less relevant, as it is the case for

β = 0.05, the noise is ampliﬁed.

Gamma correction is an essential tool for adjust-

ing the lighting, as shown in Figure 4. Indeed, we ob-

serve how varying γ signiﬁcantly affects the ﬁnal im-

age: an excessive gamma correction (γ = 0.25) leads

to a saturated image, while a less signiﬁcant transfor-

mation (γ = 0.55) results in an overly dark image.

5.2 Comparison with the State of the

Art

We compare our method with Multiscale Retinex

(MSR) (Jobson et al., 1997), Kimmel et al. (Kim-

mel et al., 2001), SRIE (Fu et al., 2016b), LIME

(Guo et al., 2017) and the deep learning techniques

RetinexNet (Wei et al., 2018) and AGLLNet (Lv

et al., 2021). Kimmel et al. and LIME have been im-

plemented by ourselves, while SRIE is sourced from

(Ying et al., 2017) and MSR from (Petro et al., 2014).

The trained networks of RetinexNet and AGLLNet

are provided by the authors on their webpages. The

most suitable parameters for state-of-the-art and our

method have been chosen based on visual evaluation.

Table 1: Quantitative evaluation on results in Figure 5.

PSNR ↑ SSIM ↑

MSR 12.41 0.7229

Kimmel et al. 14.27 0.8287

SRIE 17.27 0.8483

LIME 18.09 0.8592

RetinexNet 17.56 0.7819

AGLLNet 16.82 0.8661

Ours 20.08 0.8868

Figure 5 displays a comparison among all meth-

ods on Bookcase, Clothes rack and Plush toy. We

observe that MSR, SRIE and RetinexNet are not ro-

bust to noise and exhibit color issues in all cases. The

method by Kimmel et al. is not able to correctly re-

move the effect of the illumination, while LIME tends

to oversaturate the scene. AGLLNet produces grey-

ish images, introduces color artifacts, as seen in the

blue bottle and the red ball of yarn in Plush toy, and

modiﬁes the texture of the reference image, as seen in

the wood of Bookcase. Our method provides the best

compromise between discounting the illumination ef-

fect, avoiding the ampliﬁcation of noise, and preserv-

ing the color and geometry structure of the scene.

We also evaluate the performance of all methods

in terms of PSNR and SSIM. Table 1 displays the av-

erage values on the results in Figure 5. Our method

outperforms the others in terms of both metrics.

An additional analysis has been performed with

the natural low-light image Lamp in Figure 6. Only

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

514

LIME, RetinexNet and our method are able to remove

the effect of the illumination. However, LIME and

RetinexNet are affected by noise and also produce ar-

tifacts, like the halo surrounding the lamp.

6 CONCLUSION

In this paper, we have proposed a low-light image en-

hancement method that estimates illumination and re-

ﬂectance separately using variational models. In par-

ticular, we have introduced a contrast-invariant non-

local regularization term for recovering ﬁne details

in the reﬂectance component. The experiments have

shown that our method obtains state-of-the-art results

and performs well in terms of noise reduction, color

recovery, geometry and texture preservation.

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.

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