Continuous Exposure Learning for Low-light Image Enhancement using Neural ODEs
Authors: Donggoo Jung
Daehyun Kim
Tae Hyun Kim
Venue: ICLR 2025 Spotlight (2025)
Keywords: NeuralODE
Low-light Enhancement
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: This is a low-light image enhancement method using NeuralODE.

Abstract: Low-light image enhancement poses a significant challenge due to the limited information captured by image sensors in low-light environments. 
  Despite recent improvements in deep learning models, the lack of paired training datasets remains a significant obstacle. 
  Therefore, unsupervised methods have emerged as a promising solution. 
  In this work, we focus on the strength of curve-adjustment-based approaches to tackle unsupervised methods. 
  The majority of existing unsupervised curve-adjustment approaches iteratively estimate higher order curve parameters to enhance the exposure of images while efficiently preserving the details of the images. 
  However, the convergence of the enhancement procedure cannot be guaranteed, leading to sensitivity to the number of iterations and limited performance.
  To address this problem, we consider the iterative curve-adjustment update process as a dynamic system and formulate it as a Neural Ordinary Differential Equations (NODE) for the first time, and this allows us to learn a continuous dynamics of the latent image. 
  The strategy of utilizing NODE to leverage continuous dynamics in iterative methods enhances unsupervised learning and aids in achieving better convergence compared to discrete-space approaches. Consequently, we achieve state-of-the-art performance in unsupervised low-light image enhancement across various benchmark datasets.

Published as a conference paper at ICLR 2025
CONTINUOUS EXPOSURE LEARNING FOR LOW-LIGHT IMAGE ENHANCEMENT USING NEURAL ODES
Donggoo Jung"!, Daehyun Kim*!, Tae Hyun Kim
Dept. of Artificial Intelligence', Dept. of Computer Science?, Hanyang University {dgjung, daehyun, taehyunkim @hanyang ac.
ABSTRACT
Low-light image enhancement poses a significant challenge due to the limited information captured by image sensors in low-light environments. Despite recent improvements in deep learning models, the lack of paired training datasets remains a significant obstacle. Therefore, unsupervised methods have emerged as a promis- ing solution. In this work, we focus on the strength of curve adjustennttmentt-toe based approaches to tackle unsupervised methods. The majority of existing unsupervised curve-adjustment approaches iteratively estimate higher order curve parameters to enhance the exposure of images while efficiently preserving the details of the images. However, the convergence of the enhancement procedure cannot be guar- anteed, leading to sensitivity to the number of iterations and limited performance. consider the iterative curve-adjustment update process
To address this problem, as a dynamic system and formulate it as a Neural Ordinary Differential Equations (NODE) for the first time, and this allows us learn continuous dynamics of the latent image. The strategy of utilizing NODE to leverage continuous dynamics in it- erative methods enhances unsupervised learning and aids in achieving better conver- gence compared to discrete-space approaches. Consequently, we achieve state-of- the-art performance unsupervised low-light image enhancement across various benchmark datasets. Code is available at https//gthub.com/gjjng000000
INTRODUCTION
Images taken in various low-light environments suffer from insufficient light, leading to the capture of limited information by the camera's image sensor. Therefore, many studies have been conducted to improve the quality of the low-light images and achieve images with optimal exposure levels. In particular, recent supervision-based deep learning approaches (Wang al., 2022b; Cai et al., 2023; Hou et al., 2023) have shown remarkable performance in enhancing low-light images. However, the process of collecting pairs of low-light scenes and their corresponding ground-truth images for supervised learning is time consuming and resource intensive. As a result, unsupervised approaches that rely solely on low-light images have been proposed address this problem.
Among many unsupervised low-light image enhancement approaches, aujve-adjustment- baasstteettt-tdedddddde meth- ods, conventionally used in photo editing software (e.g.. Photoshop), have received much attention.
After the introduction of first learning- based curve-adjustment work by Yuan and Sun (Yuan & Sun, 2012), iterative surve-adjustment based methods have been explored various subsequent studies. These unsupervised methods achieve enhancement without using the ground-truth images by fitting the brightness values of pixels in the input image to specific curves. In addition, is advantageous to preserve local structural information adaptively by allowing efficient pixel-by-pixe computations. For example, ZeroDCE (Guo et al., 2020; Li ct al., 2021) 1) introduced fast and lightweight neural network to predict pixel-wise curve parameter maps within fixed iteration step. In addition, ReLLIE (Zhang et al., 2021b) produced more accurate image enhancement results by using reinforcement learning to predict the curve parameter map at each iteration step, with users able to adjust the number of iterations.
Equal contribution.
Corresponding author.

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+0.00E
TountLiC bUILE
Rethsssee MSSCC
Avonte
Input Image
ZerODCE
RetineFFomer
CLODE (Ours)
Number Recamtters
(a)
(b)
Figure 1: (a) Quantitative Evaluation: The average PSNR values on the LSRW (Hai et al., 2023) and LOL (Chen Wei, 2018), together with the respective parameter numbers for each model. (b) Visual Comparisons with ZeroDCE (Guo et al., 2020) (unsupervised), RetinexFormer (Cai et al., 2023) (supervised) and proposed CLODE (unsupervised).
In general, these sevre-ajuuttment-basse methods, which have fewer parameters, offer the advantage of fast and efficient training and also demonstrate the effectiveness of using higher-order curves for low-light image adjustment. However, conventional iterative approaches in discrete-space with fixed update steps do not arrive at the optimal solution and cannot guarantee convergence of the optimization. Therefore, we alleviate this problem in the discrete-spp space updating process of existing methods. In doing we bring out the strengths of curve fitting methods by reformulating the iterative update formula into ordinary differential equations. This allows the iterative approach to be transformed from discrete-space to continuous-space and to find input-specific higher-order curves until convergence within a specified tolerance. To be specific, we present the Neural Ordinary Differential Equations (NODE) model for the low-light enhancement task for the first time. By solving the NODE problem using conventional ODE solvers, we obtain better approximate solutions to the curve aunee-ddjsttmen problem. The proposing method therefore produces more accurate results than conventional results from iterative updates in discrete-space by exploring the continuous exposure dynamics. In this work, introduce Continuous exposure learning for Low-light image enhancement using neural Ordinary Differential Equations (CLODE), which is the first dynamic system for low-light image enhancement. Our main contributions can be summarized as follows:
CLODE is the first approach formulate the higher-order curve estimation problem NODE problem, enabling effective and accurate solutions with standard ODE solvers.
By transforming the discrete update formula into NODE, which is solvable continuous- space, we significantly enhance the unsupervised low-light image enhancement results across various benchmark datasets as shown in Fig. 1. This effectively bridges the performance gap between supervised and unsupervised approaches.
CLODE also offers user controllability without altering the network architecture, enabling users manually adjust the desired level of exposure as needed.
RELATED WORKS
2.1 UNSUPERVISED LOW-LIGHT IMAGE ENHANCEMENT
Obtaining well-exposed ground-truth images paired with corresponding low-light images is inherently challenging, which limits the use of supervised learning in low-light image enhancement. To address this limitation, many unsupervised methods have been developed to tackle the problem. First, there are some approaches (Liu al., 2021; Ma al., 2022; Zhao al., 2021; Fu et al., 2023) that utilize the principles of retinex-theory Among them. PairLIE (Fu et al., 2023) utilizes
teinex-theory, to identify the reflectance and illumination, and employs gamma correction with user-defined gamma values to enhance the illumination. In addition, UDCN (Jiang et al., 2022) and HEP (Zhang et al., 2021a) use histogram equalization results as a reference for exposure enhancement. Moreover, recent approaches using GANs have shown remarkable improvements by additionally utilizing unpaired images of normal exposed (Jiang et al., b; Jin et al., 2022). Lastly, there are curve-adjustment based methods (Guo et al., 2020; Li et al., 2021; 2022; Zhang et al., 2021b) 1b) that transform images through tone mapping. These methods have advanced the curve-fitting techniques from traditional editing tools into deep learning-based approaches, enhancing images by predicting the fitting curves
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pixel-by-pixel. By repeating the pixel-wise curve fitting and exposure enhancement for fixed number of iterations in discrete-space, these approaches aim to handle locally varying exposure levels (t.e., single image with both underexposed and overexposed areas) in an unsupervised manner. Our CLODE also follows this unsupervised curve-adjustment-haar based method and reformulates the curve- fitring problem into a neural ordinary differential equation (NODE). By solving the NODE problem using conventional ODE solvers, we increase the accuracy of curve fitting and thus significantly improve the performance of low-light image enhancement.
2.2 NEURAL ORDINARY DIFFERENTIAL EQUATIONS
An ordinary differential equation (ODE) a fundamental concept in mathematics that describes how function changes with respect to single variable. It captures the relationship between a function and its derivatives, providing a powerful tool for modeling dynamic systems, such as Newton's s Second Law of Motion. To effectively apply the strength of ordinary differential equations to the deep learning model, the concept of neural ordinary differential equations (NODE) is introduced in (Chen al., 2018). The of NODE facilitates model definition and evaluation, highlighting its effectiveness in parameter efficiency, adaptive computation, and modeling continuous data. In order to effectively capture more complicated functions, the Augmented Neural ODE (ANODE) (Dupont et al., 2019) has been introduced. Furthermore, for seamless continuous time-series modeling, Latent ODE (Rubanova al., 2019) proposed and recently, ClimODE (Verma al., 2023) proposed a continuous- time NODE models for numerical weather prediction. To be specific, in the field of computer vision, the Vid-ODE approach (Park et al., 2021) has been introduced to generate continuous-time videos. NODEO (Wu et al., 2022b) has presented a versatile architecture tailored for deformable image registration, and a temporal deformation model using the capabilities of NODE has been developed in (Jiang et al., 2021a) to address the challenges associated with future prediction tasks in the context of 4D reconstruction. With advantages like continuous-spat modeling. adaptive computation, and memory efficiency, NODE (Chen et al., 2018) is utilized in various deep learning tasks. However, it has not been extensively explored in the field of image restoration. While NODE-SR (Park & Kim, 2022) has been introduced address the arbitrary scale super-resolution problem, our methodology marks the first application in image exposure enhancement, In contrast to NODE-SR (Park& & Kim, 2022), which learns the continuous variation of the scaling factor for the arbitrary scale super-resolution problem, our CLODE learns the continuous variation of image exposure through curve-adjustment.
3 PROPOSED METHOD
3.1 PRELIMINARY
In photo editing applications, the curve-adjustment method is often used to adjust the tone of input images and provides effective exposure control. While this method is useful for pixel-wise manipulation, it is not well suited for images that contain areas of extreme over- or under-exposure. Additionally, a notable drawback of this approach is its reliance on manual adjustments (e.g., the number of updates) by the user for each input image. This can be time-consuming and potentially less accurate in certain scenarios. To address this problem, Yuan and Sun (Yuan & Sun, 2012) have proposed a solution that aims to mitigate the limitations of manual adjustments. They introduced an automated approach that involves estimating an image-specific S-shaped nonlinear tone curve (referred to as an S-curve) tailored to each input image. Specifically, for given low-light image lo. where each pixel value is in the range [0, 1]. the S-curve formula for the enhanced image 1o can be represented as follows:
T = Tn + dx (T)) - Φb PA(1 - To);
(1)
where de and dh represent parameters for the amount of shadow and highlight. respectively. The function Pa serves as an increasing function for the adjustment that manipulates the intensity of individual pixels within the input of the function.
While E allows for adjusting the brightness of an entire image using a single global curve parameter, existing iterative curve-adjustee adjusttntss approaches (Guo al., 2020; et al., 2021; Zhang et al., 202 lb; Huang et al., 2023) operate on a pixel-wise basis of the input images. Furthermore, they introduce the necessity of higher-order curves, which enhances images by fitting higher-order curves for fixed
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(a) Continuous update procedure of CLODE
(b) ODElunc
ODE (0.
1:
Co.
Trajenory Inperseneent
fa
fe
Fe. 008lunc
A
d4
Noise Removal Module (g)
Curve Parameter Estimation (h)
3 -I
Figure 2: (a) Illustration of continuous update procedure of CLODE. Optimal iterative update can be achieved through the ODE equation. (b) Illustration of our ODEfunc fa. ODEfunc contains the Noise Removal (g), Curve Parameter Estimation (h) module, and Eq. 9 to obtain the derivative value.
iteration steps while using a deep learning model to predict curve parameters on pixel-by-pixe basis. Specifically the update formula enhances an image I, the n-th step an image Ia+1 the next step as follows:
(2) where A. E CC represents pixel-wise varying curve parameter map and C, H, and W represent the number of channels, height, and width of the image I, and & operation denotes dlement-wise multiplication. Note that, the elements of A, corresponding to the curve parameters at each pixel location are in the range [-1, 1] and determine the quadratic curve for the pixel-wise exposure adjustment during the enhancement process. Conventional curve-adjustment methods (Guo et al., 2020; Li et al., 2021; Zhang et al., 202 Li et al., 2022) iteratively follow this process for N times, fitting an appropriate higher-order curve to produce the final well-exposed output image. On the contrary, our CLODE performs curve adjustment for image enhancement by reformulating Eq. 2 as an ordinary differential equation. This approach facilitates memory-effiee training and yields more accurate results through adaptive computation using modern ODE solvers.
T_ I, An I & (1 I,.,
3.2 CONTINUOUS EXPOSURE LEARNING FOR LOW-LIGHT IMAGE ENHANCEMENT USING NEURAL ODES
Although conventional curve-adjustment--aseed iterative methods offer advantages in terms of lightweight network architecture and local robustness, these approaches cannot guarantee con- vergence the update process. ZeroDCE (Guo et al., 2020) empirically determines the iteration number N and enhances low-light images by iterating the curve-adjustment formula 8 (=N) times. While ReLLIE (Zhang et al., 2021b) provides users with optional flexibility, it requires manual selection of the value of N for each input image to further improve image quality. To tackle this challenge in optimization, we reformulate the curve-adjustmenttbase Neural Ordinary Differential Equations (NODE). Then, we can solve the NODE with conventional ODE solvers (e.g., Euler, RK4, dopri5) which guarantees the convergence of loss within tolerances. Specifically, reformulate the original surve-adjustment-base based formula by introducing continuous state instead using the discrete state n as follows:
formula outlined in Eq. 2 as
I++1 I + fo(
where fo is a neural network with trainable parameters 0 that satisfies fo(T,, = A 8 1 It). Then, parameterize the derivative of the enhanced image during the update using the network fe if the continuous update step is very small, and it is given by.
dI, dt = fo (I,,, t).
(4)
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By transforming the original curve fitting problem into NODE problem with an initial condition Io- we can estimate not only the derivative value of each state but also recover the enhanced image by solving the problem, and the initial value problem is given by,
fo(1. tydt, IT 1 lo /
(5)
where Iy denotes the well-exposed image at the final state T. Finally, the low-light image enhance- ment process to output /r is accomplished by using the ODE solver as:
IT ODE Solver(fo. [0. fo.
(6)
where ODE_Solve denotes a conventional algorithm for solving the ordinary differential equations. In our experiments, CLODE adopts the well-known dopri5 (Dormand Prince 5th order Runge-Kute as an adaptive ODE solver, that determines an input-specific number of iterations for each imput and dynamically adjusts the step size. Using the adaptive solver, we can adaptively compute the optimal state for different exposure levels, thereby enabling a more accurate approximation of the solution. This is in contrast to conventional methods, which use the same fixed number of iterations for all input images and cannot guarantee optimality and convergence. To the best of our knowledge, our approach is the first to define the low-light image enhancement problem novel NODE problem with an initial condition.
3.2.1 ODE FUNCTION (ODEFUNC)
We can solve the NODE problem in Eq. 5 by integrating fo over the time interval [0, with the given initial value to (e.g.. low-light image). In practice, conventional ODE solvers are used to address this problem, iteratively enhancing the low-light images using Eq. 3. In Fig. 2 (a). we illustrate the continuous update procedure of our CLODE approach. Notably, the ODE function (ODEfunc) fo computes continuous dynamics of the latent image and is key element in the update procedure. The detailed configuration of our ODEfune fa is shown in Fig. 2 (b). To be specific, our ODEfunc includes Noise Removal (g) and the Curve Parameter Estimation (f) modules with trainable parameters, and die the continuous dynamics of I, Please refer to Appendix A. I. .,2 for more details.
outputs di
Noise Removal In the ODEfunc, we first employ a pre- rre-proeessing step to eliminate the artifacts from I, and generate the denoised image I, in order to produce more accurate curve adjustment parameters A.. To minimize computational costs within the Sa, we employ a simple and lightweight three-layer convolutional neural network our Noise Removal module, expressed as follows:
g(t))
(7)
The refined image I is then used as the input to the subsequent Curve Parameter Estimation stage.
Curve Parameter Estimation Inspired by (Yuan & Sun, 2012; Wang et al., 2022a), enhance both under- and over- exposed areas, we not only use the denoised image and its inverted version (1 I<) as inputs to the Curve Parameter Estimation module. The formulation is given by:
A = h(i,, 1- I.),
(8)
where A, represents the curve parameter map at I, and h represents the Curve Parameter Estimation module. For efficacy, this module is also lightweight convolutional neural network. In particular, we apply layer normalization (Ba et al., 2016) to all intermediate features. Notably, the use of layer normalization enables CLODE handle the diverse exposure ranges of input images. Furthermore, all convolutional layers within the Curve Parameter Estimation module take the continuous state t as a conditional input, allowing for time-varying outputs during the integration interval [o, T'] as in (Chen et al., 2018).
Continuous Dynamies Lastly, the derivative value of the one-step state / is computed in our ODEfunc, and it is expressed as follows:
dl,
=
4
2 4
4t)
dt
Notably, unlike conventional adjue-adj based update formulas that discretize update steps, continuous dynamics allows the desired level of accuracy and produces more accurate solutions.
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1-77-00
Figure 3: Illustration of User Controllable Design. By manually changing the integration interval from -(T' + At) +T' At), ours can produce results with different exposure levels.
3.3 INFERENCE PROCESS OF CLODE
Inference Process Given a low-light input image 1o. CLODE undergoes successive image enhance- ment through fo until convergence within the specified tolerance of the ODE solvers, resulting in well-exposed image Ir. Note that, the output image IT may contain some noise that is amplified during the image enhancement process. Therefore, we use the noise-free free-freeeeee image Iy as our final outcome by applying the Noise Removal module g.
User Controllable Design CLODE learns the low-light exposure adjustment mechanism in the continuous-space, and trained output IT by integrating the states from 0 to 7 in Eq. 5 using fixed T. However, as shown in Fig. 3, users can manually adjust the integration interval by changing the final state value T the test stage, allowing them output images with the preferred exposure level and even produce images darker than the input. In practice, by controlling the final state from -TT At) (T + At), the exposure level of the output image can be easily controlled provide more user-friendly exposure level.
3.4 REFFRREFEEENCE Loss FUNCTIONS
To address the challenge posed by the lack of ground truth, we use five zero-reference loss functions for unsupervised training.
Spatial Consistency Loss While the given low-light input image To is enhanced during the update procedure, maintaining spatial consistency in pixel brightness order crucial for preserving image details. Specifically, we measure the difference in spatial consistency between the input image Io and our prediction IT by comparing the differences neighboring pixel values. Similar to (Guo et al., 2020), compute the spatial consistency after applying 4-by-4 average pooling to both To and Ir, and the spatial consistency loss Lspa is expressed as:
Cepa
(Im4(Ir): m4(4r)a (mm)))llmmmmoo)) m,((o),1)². JER(()
Σ
(10)
The 4-by-4 average pooling operation denoted as (+) and (ii)) includes neighboring pixels in four directions (left, right, top, bottom) centered at position The normalization factor K denotes the number of pixels in the reduced image after the pooling operation, and K is given by H x C.
To enforce consistent exposure level across pixels, conventional unsupervised methods incorporate exposure guidance into the loss function (Guo et al., 2020). Similarly, we
Exposure Loss
introduce a desired exposure level parameter E and define the exposure loss Leap as:
Leep [ ElI2
(11)
In our experiments, we the exposure level E 0.6, which corresponds to the gray level in the RGB color space. To maintain the overall exposure level in the results, we minimize the difference between the pixel values of the predicted image Ir and the desired exposure level E. after performing a 16-by-16 average pooling operation m=s(-) on the output image Tr.
Color Constancy Loss In conventional zero-reference methods, two main approaches are used to enforce spatial color constancy: one based on the retinex-theory. and the other based on Gray-War World hypothesis in (Buchsbaum, 1980). In this work, the color constancy loss Lost is based on the Gray-World World hypothesis as (Guo et al., 2020; Zhang et al., 2021 a), and the formulation given by.
Lcsl
R B) [R G + (G - B)",
(12)
where R, G. and B are the mean pixel values of the red, green, and blue channels in the predicted image Irs respectively. We minimize the color constancy loss Lool to correct the potential color deviations in the enhanced image.

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Parameter Regularization Loss
To prevent rapid changes of pixel values in nearby regions, employ the spatial regularization to enforce smoothness among neighboring curve parameter values in A,, and the formulation is given by.
Cparam = (oV + v.,,' +... (IV.Ar-1l + | yr)r11)"
(13)
where the linear operations Vx and V, compute the horizontal and vertical gradients from the parameter map A4, respectively. For better understanding, represent I - 1 the stage before the final enhancement. We employ the parameter regularization loss at each update step (e.g., red points in Fig. (a)) and accumulate the loss while solving the NODE problem.
Noise Removal Loss To estimate a spatially smooth A regardless of the noise in the image 4, we use the Noise Removal module (g) to remove the noise. To train the Noise Removal module, utilize a seff-superrission-bale
loss Lnoise that follows the Noise2Noise approaches (Lehtinen et al., 2018; Huang et al., 2021; Mansour & Heckel, 2023). Specifically, we employ the loss introduced in Zeroshot-N2N (Mansour & Heckel, 2023). Our Lnoise has two components at state t: the residual loss fres and the consistency loss Coooss We minimize these losses using two different down-
0.5 0
samplers: D1 and D2. Notably, D1 and D2 represent fixed 2D convolutional kernels: 0
0.5]
0 0.5
and
0.5 0 respectively. These kernels used for downsampling through convolutions with
stride of two. First, our Ctes fits the noise within 4e through symmetric loss function similar to the approach in (Chen & He, 2021) and yields:
L 2 (1) ((Di(I)) - 1 +ID=(TT) - --op(((DD)))))) - (T)
(14)
Next, in (Mansour & Heckel, 2023), foons ensures spatial consistency by maintaining similarity in noise distributions, even if the order of denoising and downsampling is altered. Specifically, C also adopts symmetric loss and defined as each update step (e.g., red points Fig. 2 (a)):
C C
(())) (ID) Dz - ( + (IP)(I)) ((D((I)) D2 g (1)))
(15)
Therefore, our final noise removal loss Lnoise can be represented accumulating during the update
procedure
Lnoise (Ches + Coons) + (CTe- - As with Eq. 13, represent T - as the stage before the final enhancement. A more detailed description of the noise removal loss is provided in Appendix A.5.
(16)
Final Objective Function The final objective function to optimize is given as follows:
Liotal = Wapa Lepa + Weep"" Leep + Wool Leo! + Wparam Cporam + Wnoise" Enoiser (17)
where Wspa. Wespp Weol, Wparams and Wnoise are hyper-parametee used control the relative significance of each associated loss during the training process.
4 EXPERIMENTS
4.1
IMPLEMENTATION DETAILS
Please refer Appendix A.I for more implementation details and training scheme.
4.2
EXPERIMENTAL SETUP
In this work, we use the LOL (Chen Wei, 2018) and SICE (Cai et al., 2018) Partl datasets for training. The results of low-light image enhancement are evaluated on the LOL and LSRW (Hai al., 2023) benchmark datasets. In addition, the SICE (Cai et al., 2018) Part2 dataset is used as benchmark dataset for evaluation under various exposure conditions. SICE Part2 contains 229 image sequences with different exposure levels, and we use the entire sequences as the evaluation dataset. By default, each comparison model uses its official network weights. In cases where the official code is available but weights not provided, the models are retrained using the official code and settings, except for ReLLIE (Zhang et al., 2021b). We present the performance of ReLLIE on the LOL dataset as reported in their original manuscript.
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Input
[11
ZaroDCE (6)
PurLIE [15]
Eighn-Ehhnnee(111
ILFlow [1]
RetinasForme [3]
CLODE (Ouns)
Ground-Truth
Figure 4: Visual comparisons. From top to bottom: LOL (Chen Wei, 2018), under- and over-exposed image of the SICE (Cai et al., 2018) Part2. For more visual results, please refer to Fig. 13 in the Appendix.
4.3 QUANTITATIVE COMPARISONS
First, we quantitatively compare the performance low-light image enhancement on different datasets. Notably, in the experimental results, CLODE represents proposed method without requiring additional user input (by default), while CLODEt represents the result of adjusting the final state T to the user's preferred level, introduced in Sec. 3.3.
In Table 1, we compare the low-light image enhancement performance on the LSRW (Hai et al., 2023) and LOL (Chen Wei, 2018) benchmark datasets in terms of peak signal-to-noise ratio (PSNR) and structural similarity (SSIM). The term "GT Mean" refers to the evaluation method used by KinD (Zhang al., 2019) and LLFlow (Wang et al., 2022b), which matches the average value of the output pixels to that of the ground truth pixels. CLODE and CLODEi outperform other unsupervised learning methods. Notably, CLODE even surpasses the PSNR of state-of-the-arr supervised learning methods by 0.73 dB, when averaging the results from the LSRW and LOL datasets in the rightmost columns, without using GT Mean. Moreover, two notable points can be highlighted in Table 1. First, the effectiveness of using NODE to compute accurate higher order curves is evident, as demonstrated by its superiority over aurve-ajjustment-bae based methods; ZeroDCE (Guo et al., 2020) and ReLLIE (Zhang et al., 2021b). Second, unlike other models trained on the same training dataset (LOL), our model shows robust performance on both the LSRW and LOL test datasets, indicating that our model generalizes better than conventional approaches.
In Table 2, we demonstrate the robustness under various exposure conditions including both under- and over- exposures, and evaluate the performance SICE Part2 (Cai et al., 2018). The results show that CLODE exhibits robust performance compared to other models, even under various exposure conditions. It outperforms other unsupervised learning methods, and even when compared to supervised learning methods, CLODEi and CLODE achieve the best and second best results, respectively. Despite being an unsupervised method, CLODE narrows the performance gap with state-of-the offte--f-the-rr supervised methods. Additionally, it operates robustly under challenging conditions such as various exposure conditions in SICE Part2, surpassing supervised approaches. These strengths distinguish CLODE from other unsupervised learning methods.
4.4 PERCEPTUAL AND VISUAL COMPARISONS
In Table 2, we also provide a perceptual comparison of the results with other methods. The evalua- tion is conducted on SICE Part2, which includes a combination of underexposed and overexposed images. To measure the perceptual quality, we adopt Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al., 2018), and non-reference metrics; natural image quality evaluator (NIQE) (Mit- tal al., 2012b), blind/refferenceless image spatial quality evaluator (BRISQUE) (Mittal et al., 2012a), perception index (PI) (Blau et al., 2018), and Entropy (Chen et al., 2019). In these four aspects, both CLODE and CLODEt show outstanding performance compared to existing unsupervised methods. The visual results compared in Fig. 4. CLODE shows robust and natural image enhancement results compared to other comparison methods, regardless of the exposure conditions of the input image.
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Table 1: Quantitative results on LSRW (Hai al., 2023) and LOL (Chen Wei, 2018) datasets. For a fair comparison, we re-trained some models on LOL and marked them with * Among the unsupervised approaches, the best score displayed in red, the second best in blue, and the third best in black. For more comparison results in terms of non-reference metrics, please refer to Appendix A.5.4.
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DARKFaw
15.16
0.833
18.0
0.434
13.01
0.536
3964
0.500
14.48
0.667
1886
0.467
Inaqenttdd
KLAS (1. 2021)
ZexOCEY (Glank cl 2000
03804
0.0944
1OL
14.23
14.59
04233
0.403
37.00
18.57
02.00
0.867
16.19
0.991
0.923
30.99
18.00
0.50
0.566
1550
0.653
1338
1955
13334
0.532
PeilUEE (T% 2033;
0.3417
LOL SICE
16.97
0.88
1882
0.533
[9.5]
673
23.10
0.73
1824
D617
23.66
0,33
Kigh Tifenomeen tiee ot 2002)
67.011
LOL
14.34
0.47
19.09
0.564
31.53
6763
34.55
0.81
17:8
8688
31.72
0.66
CLODE!
0.3177
LOL
21.77
0.562
38.04
0.56
23.58
6754
34.47
0.799
33.15
665
33.71
0.664
Table 2: Quantitative results on SICE (Cai et al., 2018) Part2. For a fair comparison, we re-trained some models on SICE Part and marked them with * Within the unsupervised approaches, the best score is displayed in red, the second in blue and the third in black.
Training
Method
Train darasct
PSNK-
SSIMT
LPIPS]
NIQEL
Nomna
BRISQUEL
PL
Emimps
PSNRT
G7 Myan
SSIMI
UReiineeNe (Wa 20226)
LOE, SICE
12:15
0,708
0.303
4.250
15.633
3372
5.926
17.81
0.686
LLFlou "oruu*"Vanng (Wang el. 202203
SICE
14,34
0.503
1.279
3.643
17.011
3.481
5.565
19.59
0.65%
(CLuet ata y
SICE
13.99
0.562
0.290
4.279
24,570
3.520
5.919
16:56
0.690
Sapervisd
FECNa (Hunng ct al, 2022a)
SICE
14.25
0.500
0.291
3.786
17:454
3,225
7.635
16.47
01639
RatineFFamma (Clr 2025)
SICE
19:12
0.570
1.369
4,452
24.768
4.573
7.025
20.50
0.578
RetinexPommer (Cai at. 20.3)
MIT-SK
13.25
0.564
0.263
3.348
17.350
2.863
5.881
16.35
0.609
(Ma 2022)
MIT-3K
9.87
4.276
3.226
16.44
0.62
medium 2022)
LOL LSKW
9.77
0.510
0.454
5.727
33.200
4.392
52212
15.83
0.374
SCI-diflicatt 2012y
DukFFaee
11.13
3.107
16.85
0.647
SCT (Ma
SICE
10.67
11.331
4.289
3,570
17.99
0.75
Unsuervised
RUAS*
2021)
SOCE
9.12
11539
8.097
6.004
5,101
15.52
0.331
ZemFXCFF (Cain 2020)
12.67
3,886
18.85
0.68
(Fu al., 2023)
LAI. SACE
13.39
5.268
19.22
0.663
Night Enhannement" 200.2)
13:18
4.728
0.660
15.01
3,005
0.706
CLODE!
16:18
4.026
2.970
7.045
21.55
0.113
4.5 COMPARISON ON COLOR CASTS
CLODE enhances the image in an unsupervised manner based on the color statistics of the input image, which can lead to color casts. Nevertheless, CLODE exhibits less color casts compared to previous unsupervised methods.
In Table 3, we measure the degree of color casts between the output image and the ground -rrth image using the color-matching histogram loss (Afif et al., 2021a), which is designed to control the color of the input image by matching color histogram with the target image. Additionally, we present the results of color correction performance in the LAB color space using A F2000 and Á Fab. well-established color metrics. For all metrics used in the evaluation, lower values indicate closer match to the colors of the ground-truth image. As shown in Table 3, CLODE using the NODE scenario demonstrates superior performance in terms of naturalness, image quality metrics, and color-matching histogram loss compared to existing methods. Moreover, the results of A E2000 and A Fab confirm that CLODDi achieves the best performance, and CLODE ranks second, implying that our method leads to less color cast problem and offers superior enhancement effects.
Table 3: Quantitative comparisons on LOL (Chen Wei, 201 1088/SICH (Cai et al., 2018) dataset.
Method
NIQEL
BRISQUE,
color-natching bistogram loss]
A Paggl- (Shomta et al, 2005)
& (Shamta & Rala, 2017)
SCI-enay (Ma et ol,, 2022)
7.1544.28
12.42/2185 85
O1 0,466ND-4788
31 11.99/7 00
39 20/51/33111 13
SCI CC-mmdium (Md al., 2022)
7,86/5.73
25.87/33.20
0,453000 1911
19,60/27 28
27 27.28/35.99 5,96
SCI-Hifficlt
2022)
8.06/4.64
26. 26.62/23.62
0.3854/0 4872
21.06/24 02
03/11
RLAS (Lin al., 2021)
6.30/8.10
11.98/52.92
0.4471/0.5100
16.80/29.J5
18/28,83
Zero DCB (Gon al., 2020)
7.7K3388
27.30/2163
0.4855K 1617
21,93/21 26
26.60127.10
CLODE
4.52/4,05
8.22/18.66
0.4381 10.466
12.73/17.04
15.71/22.39
CLODEt
4.25/4.03
8.81/18.21
0.3848/0.4462
9.21/14.46
11.86/19.32
4.6 ABLATION STUDY
Effeettvvnness of NODE To validate the impact of NODE, we compare curve adjustment in discrete
(w/o NODE) and continuous (w/ NODE) spaces using CLODE'S architecture, as shown in Table 4.
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Table 4: Comparative experiments according using NODE on LSRW (Hai et al., 2023)/LOL (Chen Wei, 2018).The The "Discrete" refers to performing curve ad- justment in discrete steps, similar to the conventional methods (Guo al., 2020; Zhang et al., 2021b), and
"Continuous" refers to the reformulation of NODE.
Method
Case
Step (N)
PSNRT
SSIM
BRISQUEL
(a))
11.19/9.236
0.2770.362
41.137/11 169
Discrete
(cl)
(b1)
16.12/17447
1399416.18
0.41900.716
0.3950.520
31. 42143/333
12.262/32.245
Figure 5: Trajectories of improvement (el)
(dl)
12,95/14,44
0,373/0.00
33.537/34.911
(el)
12,87/14.97
0.375/0.509
33.537/35.342
and (fI) in Table 4. PCA dimension reduction is used to visualize the trajectories.
Contasous
(81)
S. 30 (adapeive)
17.28/19.61
0.533/0.118
18.426/8.200
Table 5: Impact of the modules in fo- Noise Removal and the layer normalization (LN) significantly improve performance.
Table 6: Execution time and performance.
Training
Method
PSNR/SSIM
ePicans (M)
Time (5)
KeturesNer (Chon 201 2015
15-1501:55
01140
0.337
Supervisal
(Wang el al. 20226
17.52/0500 509
38.859
Case
Noise Removal
LN
PSNRT
SSIM
Retine-Fumer (Cai 2003)
17.7600517
1,4057
SCI medium et
15.240.424
(x2)
14.72
0.538
RUAS (Lu 2021)
14.2761470
000334
(c2)
(b2)
15.19
18.67
0.489
0.577
Unsupeviied
ZcxDCE (Guss al, 2020)
PaiLIE (Fu ct al., 2023)
CLODE
16.97/0.498
17.260UNES
15.8100440
0.3167
0.0754
0,5417
0.088
0.064
0.066
(d2)
19.61
0.718
CLODESS
16.9301457
0.0044
0.065
In the discrete setting, similar to (Guo et al., 2020), curve parameters are estimated in parallel for fixed steps [1, 5, 10, 20, 30J (a))) while in the continuous setting, parameters are sequentially estimated for adaptive steps, up to 30 ((fl)). Table 4 shows that the sequential continuous updates produce more accurate parameters, demonstrating superior performance over the conventional discrete approach. Additionally, Fig. 5 visualizes latent image trajectories during updates using PCA, revealing that continuous adjustments ((f1)) converge more accurately than discrete updates ((el)), highlighting NODE's contribution to image enhancement. For visualization result on each and further detailed explanation on NODE, please refer to Sec. A.1.1 and Fig. 6 of the Appendix.
Effect of the Modules In Table 5, we conduct ablation experiments the modules used in ODEfunc fo- We verify the effects of the Noise Removal module g and the layer normalization (L.) in the Curve Parameter Estimation module h. Each module shows performance improvements compared to the baseline (a2). In particular, our final model (d2) achieves the largest performance gain in terms of PSNR/SSIM. Furthermore, case (c2), which includes layer normalization, has about 4dB gain in PSNR compared to (a2), which does not include layer normalization. This shows essential to use layer normalization to
that during the image enhancement process in NODE, normalize each state. The visual results can be seen in Fig. 9 of the Appendix.
5 LIMITATIONS
Table 6 presents the PSNR/SSIM performance, parameter count, and execution time measured on LSRW (Hai et al., 2023) using an NVIDIA RTX 4090. CLODE demonstrates a size advantage over supervised methods. While its iterative ODE solving takes longer than lightweight unsupervised models, achieves comparable speed and performance to supervised approaches. Additionally, a smaller variant, CLODE-S (Appendix 1.2.2), shows promising enhancement with inference times similar to unsupervised models.
CONCLUSIONS
In this work, we address the unsupervised low-light image enhancement problem by reframing discrete iterative curve-adjustment methods into a continuous space using Neural Ordinary Differ- ential Equations (NODE), Our proposed CLODE method effectively overcomes the limitations of existing approaches, demonstrating superior convergence and adaptability in diverse low-light and multi-exposure scenarios. Additionally, CLODE introduces user- controllability. leveraging NODE's inherent flexibility deliver customizable brightness adjustments. By incorporating novel curve- adjustment framework and offering enhanced interpretability, our method bridges the gap between unsupervised and supervised approaches, representing meaningful contribution the field.
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ACKNOWLEDGMENTS
This work was supported by Samsung Electronics Co., Ltd, and Samsung Research Funding Center of Samsung Electronics under Project Number SRFCIT1901-06, and Institute of Information & commu- nications Technology Planning Evaluation (ITPP) grant funded by the Korea government (MSIT) (No.2022- 0-00156, Fundamental research on continual meta- lettallarnnng for quality enhancement of casual videos and their 3D metaverse transformation).
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A APPENDIX
A.1 IMPLEMENT DETAILS
The training set of images is resized 128x128, we employ the Pytorch framework on NVIDIA A6000 GPU with batch size of 8. The ADAM optimizer used with default parameters and fixed learning rate of le to optimize the parameters our network. The weights for the loss function Weols Wporams Wapa. Wenp and Whosse set 20, 200, 1, 10 and respectively. to balance the scale of losses. Furthermore, we adopt torchdiffeq (Chen, 2018) for Neural ODEs implementation. The training process is conducted for 100 epochs.
A.1.1 IMPLEMENTATION DETAILS OF NODE
Piacrate metheds: discrete space
613
CLODE dynamicy. contemmuus spuce
Balr
CMt 23RE EI segrreeeeatttton
Peverrrnee sotee
Figure 6: Further analysis of CLODE and previous discrete methods as described in Table 4 of main manuscript.
As mentioned in NODE (Chen et al., 2018), ODE reformulation provides benefits such as continuous space estimation, memory efficiency, and accurate problem-solving with ODE solvers. In Fig. 6, we describe additional analysis CLODE. The top of Fig. shows discrete trajectories of models (al) to (el) from Table 4, while the bottom shows CLODE trajectories with Euler and dopri5 (corresponding to Table 4. (fl)) solvers. (Top) Discrete methods (al) l) to (cl) enhance images but don' achieve optimal exposure. (Bottom) CLODE (dopri5) provides more realistic image enhancement in continuous space.
Additionally, the early stop mechanism of the adaptive solver is explained at the bottom of Fig. 6. CLODE (dopri5) uses an early stop mechanism. tracks error at each state, terminating when the error is within allowable error rate. For dopri5, k-order solutions (k=5) used calculate error (T,) as follows:
T atol + rtol norm(lok - -KK--1);
(18)
where the k-order solution at time is denoted as OK and the (k- 1)-order solution denoted as O atol is absolute tolerance, and rtol is relative tolerate, and the norm being used is mixed L-infinity/RMS norm.
If 1OOKK -K H2 the step size is re-adjusted, or it's within Fr, the solution is deemed optimal, and the process terminates. ODE solvers are designed find optimal solutions through iterative steps. The top part of Fig. 6 shows that discrete methods can't guarantee optimal solutions, which led us to develop the NODE method for continuous ODE problems. Thus, improvements are due more to NODE reformulation than to iteration count. Table 4 shows that NODE outperforms simple discrete repetition. For example, using the Euler method in 30 steps achieves better performance than method (The visual result in Fig. 6 (el) is inferior to CLODE with Euler applied.). We chose dopri5 (Dormand- Prmnand-Prince Runge-Kutta across platforms like MATLAB. The maximum allowed step for the adaptive solver is set 30. In
Kutta of Order 5) as CLODE's default solver for its stability and reliability
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Eq. 18, the relative and absolute tolerances for the error rate calculation uniformly to le" set both atol and rtol to le-5.
A.1.2 DETAILS OF THE CLODE ARCHITECTURE
This section presents the architectural details of the CLODE network architecture, with particular focus on the ODEfune module. The Noise Removal module g employs simple and lightweight three-layer convolutional network. In Curve Parameter Estimation module h, a shallow network with two branches utilized, wherein filters of varying sizes are employed at each branch to capture image features across different filter scales. We also provide architectural details of CLODE-S as mentioned in Sec. of the main manuscript. This version omits the Noise Removal module for speed and uses a 2- 2-layer network with 1xl convolutions.
Noiue Samoool ls)
Curve Parameter Extmntion (N
U.1 +
000ucc
41
(a) Illustration modules in CLODE
(b) 00Efunc of CLODE-S
Figure 7: Illustration of architecture details of (a) modules of ODEfunc in CLODE and (b) ODEfunc of CLODE-S.
A.2 IMPACT OF EACH LOss FUNCTIONS
Input
(o3) w/o Eexp
(b3) w/o
Apara
Cyp
(e3) Koep Kew
Input
(e3) w/o Lnotse
(f3) CLODE
Figure 8: Visual results for the ablation study of each loss function. CLODE combines five reference loss functions in training for producing optimal enhancement results.
CLODE combines five non-reference loss functions train NODE, producing optimal improvements. We present ablation experiments for each loss function, and the results are presented in Table 7 and Fig. 8. The results of each image ablation experiment demonstrate that appropriate improvement results can only be obtained when using CLODE with all loss functions. The characteristics of the loss function observed in each ablation follows: ((a3) w/o Lexp): Brightness improvement
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Table 7: Ablation study each non-reference losses. The experiment is evaluated LOL (Chen Wei,
2018).
Casc
Capa
Leep
Leat
Lparam
Lnoise
PSNR
SSIM
(a3)
/
8.84
0.323
(b3)
/
/
14.72
0,566
(c3)
/
/
1
14.76
0.535
(d3)
/
/
18.76
0,580
(e3)
/ /
18.92
0.582
(f3)
/
19.61
0.718
not achieved in low-exposure enhancement, ((b)) w/o Loot): Severe color distortion occurs in over-exposure enhancement, damaging structural details. ((c3) w/o pprram): Structural distortion occurs, creating artifacts. ((d3) w/o Sspa): While showing better results than other experiments, occurs loss of structural details compared to (e3). ((e3) w/o Nooiee): Compared to the proposed version (f3), it produces improved results with noise present.
A.3 VISUALIZATION OF CURVE PARAMETER MAP .A
We provide visual comparison results for the module ablation experiments in Sec. 4.6 of the main manuscript. In the visual results without Noise Removal module (c2), we observe the noise in A. The enhanced result of (c2) using A with noise shows overall color discrepancy compared to the ground-truth. in contrast to the enhanced result of (d2) where the Noise Removal module applicd. The enhanced result of (d2) shows robust color similarity with the ground-true -tuth image. We can confirm that removing noise for A is important for surve-adjusteent-basse method.
Input
enhanced
(d2)
(d2) enhanced
Ground-Truth
Figure 9: A visual comparison of the results for (c2) and (d2) from Table 5 in the main manuscript. The enhanced result (d2) using A with noise removal module demonstrates improvement more similar to the ground-truth.
A.4 CLODE STEP STATISTICS ACROSS EXPOSURE CONDITIONS.
In CLODE, the maximum allowable step for the ODE solver empirically set to 30, considering speed. The ODE solver terminates early if finds the optimal solution within the maximum steps. Furthermore, in order to provide the statistical analysis, the average number of steps for the ODE solver was calculated across the SICE, BSD100, DIV2K, and L.OL. datasets. The SICE dataset comprises five to seven images per sample, with exposure levels ranging from under-exposed to over-exposed. Additionally, BSD100 and DIV2K were used to provide additional statistics for normal-exposed conditions. The results based on the exposure conditions are Table
Table 8: Step Statistics across exposure conditions
Condition
SICE
BSD100, DIV2K
LOL
Under-exposed 13.716
21.120
Normal-exposed
20.754
28.210
26.400
Over-exposed 18.906
The number of calculation steps increases in the order of under-exposed, over-exposed, and normal- exposed images. This is because improving normal-exposed image is considered stiff problem.
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For normal-exposed images, where minimal improvement is required, the dynamics more stiff than for other images. Specifically, for dynamically stiff ODE problems, the step size taken by the solver is forced down to small level even in region where the solution curve is smooth, and these decreased step sizes may require more evaluation steps.
For clearer comprehension, time, CLODE aims to find the optimal solution by minimizing the loss functions, therefore Fig. 10 (Left), the y-axis represents the non-reference loss value, while the x-axis represents time, with each point indicating a step. The dopri5 solver, which we primarily use, is non-stiff solver. Although stiff solver (e.g., ode15s) could potentially reduce the number inference steps for normally exposed images, it proves inefficient for improving under- or over-exposed images, which are closer non-stiff problems. Future research may explore dynamically adaptive ODE solver algorithms that adjust based on input image conditions by this observation.
present additional graph results in Fig. 10 (Left). In the inference
under-exposed
rormal-exposed
0.15
Lossss
0.10
0.05
0.0
0.2
0.4
0.6
0.8
1,0
Input
Zero-DCE
CLODE
Figure 10: (Left) The trajectory of loss changes over time. (Right) Comparison of enhancement results for normal-exposed inputs.
A.5 BACKGROUND NOISE REMOVAL Loss
In Sec.3.4 we provide information about the zero-reference loss functions that we used. Unlike the others, the Noise Removal Loss (Cnaise) requires more explanation due to its unfamiliarity in the field of low-light enhancement, so we provide additional explanation for it.
A.5.1 NOISE2NOISE BACKGROUND
In supervised denoising studies, neural networks are aimed at denoising the noisy image to the clean image x. Since the noisy y an addition of the clean image and the noise e, the network trained to map the noise which is called Noise2Clean (N2C) method. If the network parameter is d22c, the object function the supervised denoising method with the network gø can be written as:
N2 arg min E (1)
(19)
Denoising networks can also be trained to output the noisy image from the noisy input image y that comes from the same clean image x. This noise-to-noise manner can achieved by assuming that the noise has mean of zero as introduced in Noise2Noise (N2N) (Lehtinen et al., 2018). This is the objective function for the N2N network parameter ON2N:
Φ2NN = arg min E (1)
-y 1
(20)
The N2N manner shows close performance compare to N2C manner with sufficient training data since the objective functions of N2C and N2N are aimed on the same network parameter. If y, = x + e,,
yo cb and the mean value of e, and e, zero, the proof is as follows:
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DN2c arg min E [190(8)) x11]
arg min (y2)l 2x gx + [|x
arg min IE (y)) 2x 2x
N2N arg min [195 11
(21)
arg min IF [190 (x2) (x + e e:)
arg min E (y2) 2x (x22)(y)) (x) ||x+ e1l2
arg min |9g 22)) 2x (x22) (0
arg min E |1 (y2)) (x))
By Eq. we can confirm that object of ΦN2C and ΦNNN is the identical one.
A.5.2 ZEROSHOT NOISE2NOISE METHOD
In spite of N2N approaches, it is hard obtain two different noisy images from the same clean scene. To address this hurdle, the Neighbor2Neighbor (Huang et al., 2021) method is proposed. This allows pair of noisy images be augmented from a single noisy image coming from the same clean image. In Zeroshot-N2N (Mansour & Heckel, 2023), which is adopted in our proposed method, Neighbor2Neighbor is achieved by using two different 2D convolutional kernels (D₁ and D2) on noisy images. If the noisy image is pair of down-sampled images Y1. V2 can be represented as:
y1 Di(y),y2 D2(y).
(22)
For noisy image y with size of и W C. the size of y1 and Y2 is H downsampled images yı and y2, the loss optimizes to fit the noise as:
C. With
arg min (y,) -22l
(23)
Zeroshot-N2N (Mansour & Heckel, 2023) emphasizes that residual learning, symmetry loss, and an additional coherence- enhanccng term are critical for good performance. Zeroshot-N2N proposes two different loss functions, the residual loss frss and the consistency loss Leons- First, the residual loss optimizes the network dø to fit the noise instead of image. The loss then becomes as:
arg ly, (x,)) 12113.
(24)
To fit the noise in y, and y2 both, symmetric loss (Chen & He, 2021) is applied as:
1
Crea ()) (Ily, (o(r)) y21 +ly 96 (v2) 11))
(25)
Second, the method constrain consistency by making denoised output of the downsampled image and downsampled result of the denoised image like:
arg min Ily, -go(Y)) -D(y, - -
(26)
Same Eq. 25, with the adoption of symmetric manner, the consistency loss is represented as:
Lcons (0) = 2
(1ly, - (x)) )- Di(y, -
(y ()1)) llyz (o - D2(y2 - (o()))) (27)
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The noise removal loss function Enoise in Zeroshot-N2N becomes the sum of Eq. 25 and Eq. 27, expressed as:
Euoise Lres Leons.
(28)
A.5.3 MORE STUDIES ON THE EFFECTIVENESS OF DENOISER
Since NODEs rely on inuulation-baseed training, as the denoiser (Noise Removal module) becomes more complex, requires more time for training. To mitigate this, we utilize a lightweight 3-layer network (0.085MB) as the denoiser in CLODE. Although the Noise Removal module has fewer parameters, performs critical task. It learns to denoise the image at each step (Eq. 16), integrating with the image enhancement process and assisting in predicting fine-gratd tinegrainedd curve parameter maps every step.
We train the denoiser concurrently with the image cahancement process to maximize the module's effectiveness. To demonstrate this, we compared three different scenarios: Pre-denoising, CLODE, and Post-denoising. For clarity, 'Pre-denoising' refers to training the denoiser only on the input image, while 'Post-denoising' involves training the denoiser solely on the enhanced image. CLODE outperforms the other approaches, suggesting that it the optimal scenario among the three (Table 9).
Table 9: Results of denoising ablations.
Method
PSNR
SSIM
Pre denoising 19.38
0.661
CLODE
19.61
0.718
Post denoising
19.16
0.659
The reason for these results is that low-light images have low pixel values, which provide insufficient information for effective denoising. After enhancement, the original noise becomes entangled with the image content. Therefore, we believe that continuous denoising is crucial for effective low-light correction, as noise tends to be amplified with successive exposure enhancements.
Using existing methods We observe performance improvements by utilizing existing denoisers to obtain Tr. For our experiments, we employed DrCNN (Zhang et al., 2017) and Restormer (Zamir al., 2022) as denoisers. Quantitatively, as shown in Table 10, Restormer provided 0.66 gain in SSIM, while DnCNN achieved 0. gain in PSNR. We also obtained visually superior results, as illustrated in Fig.
Table 10: Quantitative Results of existing denoising methods.
Method
#params (M)
PSNR
SSIM
LPIPS
CLODE CLODE DNCNN
0.2167
19.61
0.718
0.263
0.8629
19.71
0.774
0.199
CLODE Restormer
26.306
19.69
0.784
0.228
Input
CLODE
CLODE + DnCNN
CLODE + Restormer
Figure 11: Visual results with conventional denoising methods.
A.5.4 MORE QUANTITATIVE RESULTS
We present the comparison results for non-reference metrics, which we did not include in Table 1. Table 11 demonstrates that CLODE outperforms other unsupervised methods in terms of perceptual
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quality. Notably, it demonstrates competitive results in terms of BRISQUE and PI, even when compared to state-of-the-ar supervised methods. Additionally, in Table 12, we provide LPIPS (Learned Perceptual Image Patch Similarity) (Zhang et al., 2018) performance results. CLODE exhibits superior average LPIPS performance compared to other methods. Moreover, performances on non reference metric for unpaired low-light datasets (DICM (Lee et al., 2013), MEF (Ma et al., 2015), LIME (Guo et al., 2016), NPE (Wang et al., 2013), and VV (Vonikakis et al., 2018)) are provided in Table 13.
Table 11: Comparison results LSRW (Hai al., 2023) and LOL (Chen Wei, 2018) in terms of NIQE (Mittal et al., 2012b), BRISQUE (Mittal et al., 2012a), PI (Blau et al., 2018) and Entropy (Chen al., 2019). Within the unsupervised approaches. the best score is displayed in Red. CLODE performs better than all other methods, including supervised methods, in terms of PI (Perceptual Index).
Training
Method
NIQE,
BRISQUEL
LSRW
PIL
Entropy
NIQUL
BRISQUT!
LOL
Entropy
Añfi (Afif et al., 2021b)
6.470
RetinesNet (Cheo Woi, 2018)
8.871
51.813
4,955
6.835
Supervised
URetinexNet (Wa et 2022s)
4.154
23.614
3.495
6.762
4,250
15,633
3.372
6.926
LLFlow (Wang al, 30226)
3,756
26.671
3.176
7,369
5,709
35,022
4.530
7.141
RetinexFormer (Cai al., 2023)
3,549
15.951
3.208
7.131
3,478
17.101
3.102
7.074
SCI-easy (Ma ol., 2022)
25,859
3,259
6.388
7.153
12.424
5,437
5.825
SCI-medium (Ma al., 2022)
3.917
22.416
3.159
6.494
7,861
25,870
4.583
6.842
SCI-difieult (Ma ct. 2022)
4,368
20,692
3,851
5,975
8.060
26.823
4,664
6.675
Unsupervised
RUAS (Lio al., 2021)
Zer (Guo ol., 2020)
3.776
23.867
3.156
6.526
7.777
27.301
4,459
6.608
Night- Enhhncement (Jin al,, 2022)
7,208
$1,356
6.801
6.544
4.491
27.122
4,436
7.1.99
PairLIE (Po al. 2023)
CLODE
3.827
18.426
3.115
7.025
4,516
8.220
2.914
7.053
Table 12: Quantitative results terms of LPIPS (Zhang et al., 2018). The best average result displays
in Red.
Dataset
URetinexNet
RetinexFormeer
SC1
RUAS
ZeroDCE
NightEnhanccement
PairLIE
CLODE
LSRW
0.308
0.315
0.398
0.469
0.317
0.583
0.342
0.331
LOL
0.121
0.131
0.358
0.270
0.335
0.241
0.248
0.263
SICR
0.264
0.263
0.486
0.608
0.239
0.360
0.305
0.235
MSEC
0.393
0,362
0,396
0.668
0.329
0.462
0.431
0.223
Average
0.272
0.268
0.410
0.504
0.305
0.412
0.332
0.263
Table 13: Comparison results on DICM, MEF, LIME, NPE, and VV terms of NIQE (Mittal et al.,
201 201),), BRISQUE (Mittal et al., 2012a), NIMA (Talebi & Milanfar, 2018) and Entropy (Chen et al., 2019).
ESCC
RTC.
Latogy:
8815QU-1)
KIMA1
$40000
N33
Suarpy
Seary?
3.90
1933
0.735
1,015
11:52
<.10
X.71
0.11%
14:90
4.50
1.33
6212
11.31
4.27
1221
1.50
10.5
1090
BL.4
1150
(741
-11%
4413
5465
34.30
4.12
6338
5430
25.50
4.30
6.95
113
48.03
4111
4.00
1100
4063
S.111
0.30
2033
0.380
01.0
1200
2.31
4.30
1.31
0.30
23 35
1.84
183
1330
$3.06
0
1.36
1.20
(180
2430
A.5.5 COMPARISON WITH OTHER ITERATIVE METHODS
Fig. 12 shows the changes in performance over steps of each surve-ajjustment-eee method. Each comparison method is retrained for 10 steps in the official code provided by the author. To fix the number of steps in CLODE to 10, we replace CLODE's ODE solver with the Euler method, and referred as CLODE-Euler. The results show that even within the same number steps, CLODE- Euler performs better than other curve adjustment-basee methods. Furthermore, the proposed version, CLODE, demonstrates higher performance compared to other methods in most iterative steps. In case of ReLLIE (Zhang al., 2021b), it exhibits decline in performance after 7 steps, emphasizing the need for careful selection of the number of iterative steps itself to achieve optimal result, this makes the method impractical to use.
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Comparison with curve-adjistmenn--aseed methods
ZeroDCE (10 steps)
ZeroDCE+ [10 steps)
staps)
CLODE-Buler (10 steps)
CLODE
PSNR (dB)
Step (Scaled to 0-1)
Figure 12: Changes in PSNR (Peak Signal-to-Noiss Ratio) over steps of CLODE, CLODE-Euler ReLLIE (Zhang al., 2021b), ZeroDCE++ (Li et al., 2021), and ZeroDCE (Guo al., 2020). As CLODE employs continuous adaptive step according to the input image, we represent the steps by scaling them from 0 to 1. CLODE demonstrates superior performance compared to other methods at almost every step.
A.6 MORE VISUAL RESULTS
We show additional results for CLODE enhancement that we did not show in the main manuscript due to lack of space. We present additional visual comparison results for PairLIE (Fu et al., 2023) and Night-Enne Eicht-Enhancement (Jin et al., 2022), which demonstrated the best quantitative performance among the unsupervised methods Table the main manuscript, except for our proposed method (CLODE), in Fig. 13. CLODE shows the most robust enhancement results across various image exposure conditions.
Fig. 14, Fig. 15, Fig. 16 and Fig. 17 show the results for CLODE and CLODEt on LOL (Chen Wei, 2018) and SICE (Cai et al., 2018) validation dataset. Additionally, Fig. 18 shows the visual results with different exposures for photos extracted from MSEC (Afifi et al., 2021b) and the internet (Filckr: CC BY-NC 2.0).
Input
(a)
(b)
CLODE
Figure 13: Comparative visualization results with (a) PairLIE (Fu et al., 2023) and (b) night- enhancement (Jin al., 2022) on LOL (Chen Wei, 2018) and SICE (Cai al., 2018). Images are taken from LSRW (Hai et al., 2023) and SICE (Cai et al., 2018) Part2.
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Input
CLODE
CLODE
Ground-Tree Truth
Figure 14: Visualization results on LOL (Chen Wei, 2018). While CLODE demonstrates superior enhancement results, control with CLODEt produces images that more closely resemble the ground-truth image.
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Input
CLODE
CLODE
Ground- Ground-Truth
Figure 15: Visualization results LOL (Chen Wei, 2018) and SICE (Cai et al., 2018). While CLODE demonstrates superior enhancement results, user control with CLODEt produces images that more closely resemble the ground-truth image.
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Input
CLODE
CLODE
Ground-Truth
X
Figure 16: Visualization results on SICE (Cai et al., 2018). While CLODE demonstrates superior enhancement results, user control with CLODE produces images that more closely resemble the ground-truth image.
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Input
CLODE
CLODEt
Ground-Trud Truth
Figure 17: Visualization results SICE (Cai al., 2018). While CLODE demonstrates superior enhancement results, user control with CLODEt produces images that more closely resemble the ground-truth image.
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Input
CLODE
Input
CLODE
Jolochaa CC BYINC
Figure 18: Visualization results on MSEC (Afifi et al., 2021b) and extracted from internet (Flickr by julochka). Even with diverse inputs of various exposures, CLODE show robust result in unsupervised manner.
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Reviews:
[{"review_id": "PkZz1tPO1F", "reviewer": "Reviewer_b2uw", "summary": "This paper introduces a novel approach for enhancing low-light images by formulating the problem as a Neural Ordinary Differential Equations (NODE) problem. The authors propose CLODE, a dynamic system that leverages continuous dynamics of latent images to improve exposure levels in images. The method is unsupervised, which is particularly useful given the difficulty of obtaining paired low-light and well-exposed images for supervised learning.", "questions": "Please refer to weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper presents a creative application of NODEs to the problem of low-light image enhancement, which is a contribution to the field.\n\n2. Addressing the challenge of lacking paired datasets, the unsupervised nature of CLODE makes it highly applicable to real-world scenarios where ground truth data is scarce.\n\n3. The paper provides a thorough explanation of the CLODE model, including its theoretical foundations and implementation details.\n\n4. The method achieves competitive results across multiple benchmarks, which is a strong point of its potential impact.", "weaknesses": "1. The paper does not explicitly address the computational cost of solving NODEs compared to other methods, which is an important consideration for practical applications. Authors should report this results compared with other similar methods.\n\n2. The paper does not discuss the potential for overfitting, especially since the model is tailored to low-light images, which may have unique characteristics.\n\n3. In my opinion, the author's comparative experiment is still not sufficient. It should be compared on more real benchmarks to highlight the superiority of the method. The current comparison seems to be insufficient, including the following:\n\n[1] Contrast enhancement based on layered difference representation.\n\n[2] Perceptual quality assessment for multi-exposure image fusion.\n\n[3] Structure-revealing low-light image enhancement via robust retinex model.\n\n[4] Naturalness preserved enhancement algorithm for non-uniform illumination images.\n\n[5] On the evaluation of illumination compensation algorithms.\n\n4. There are many different paradigms for unsupervised methods. Compared with this type of method, is the author's method more scalable? If it can be improved on other methods, I think this article will have higher value.\n\nIn general, I will make further adjustments based on the author's rebuttal.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mrvWYNxIBK", "reviewer": "Reviewer_Asgg", "summary": "In this article, the author discovers that the traditional curve-based iterative methods cannot guarantee the convergence of the enhancement process, leading to sensitivity to the number of iterations and limited performance. The author regards the iterative curve adjustment process as a dynamic system and, for the first time, formulates it as a neural ordinary differential equation (NODE) to learn the continuous dynamics of underlying images. Specifically, this method leverages NODE to harness the continuous dynamics within iterative methods, achieving better convergence than discrete space methods. Finally, the article demonstrates superior performance over current unsupervised low-light image enhancement methods on various benchmark datasets.", "questions": "1. Since this article addresses issues in low-light scenarios, presenting more objective results in complex scenarios would enhance the persuasiveness of the article.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. In this article, for the first time, the iterative curve adjustment update process is regarded as a dynamic system and formulated as a neural ordinary differential equation (NODE) to learn the continuous dynamics of underlying images. This approach offers an alternative perspective for addressing low-light issues；\n2. The article employs a substantial amount of mathematical language to demonstrate how to transform the iterative curve adjustment update into a continuous process, which enhances the logical strength of the paper；\n3. The language of the article is quite fluent and adheres to English writing standards；", "weaknesses": "1. Since this article addresses issues in low-light scenarios, presenting more objective results in complex scenarios would enhance the persuasiveness of the article.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1sf9Sd82wZ", "reviewer": "Reviewer_NMzG", "summary": "This manuscript introduces a CLODE method that enhances low-light images by framing exposure adjustment as a Neural Ordinary Differential Equation (NODE) problem.  It improves on traditional curve-adjustment techniques by automatically adjusting exposure without manual input and converges to an optimal exposure using adaptive ODE solvers. CLODE also includes noise removal and curve parameter estimation, and users can control exposure levels by adjusting integration intervals.\n===================================================================\n\nThe authors have raised many of my concerns during the rebuttal phase. I am now ready to increase the rating.", "questions": "1. Adaptive ODE solvers increase computation time and memory usage. Any idea regarding the number of pixel operations taking place here?\n\n2. The curve parameter estimation module might not generalize well to diverse lighting conditions. Can you provide some generalization test regarding this.\n\n3. Manual adjustment of exposure intervals introduces subjectivity. An ablation can give us more insights.\n\n4. NODE-based adjustments may struggle with high-dynamic-range images, risking instability and suboptimal exposure adjustments. Can you provide the separate loss curves for better understanding of the training dynamics.\n==========================================================================", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Strengths of the CLODE method:\n\n1. CLODE’s use of Neural ODEs allows it to dynamically adjust exposure levels based on image-specific needs.\n2. The approach is really interesting. Mainly the noise removal block is a very good idea for this type of tasks.\n3. Ablation study is really exhaustive.\n4. Results are evaluated well.\n5. Manuscript is well written and organized. Very easy to follow.\n6. Technical content is really well written with proper citations.\n\nGood Paper!!!", "weaknesses": "Weaknesses:\n\n1. Novelty is somewhat limited.\n2. Comparisons with recent-most SOTA methods should be included.\n3. Flowcharts should be improved like Figure 2.\n4. Conclusion should be improved although I can understand there is a space limitation. Please make other parts smaller and add some key findings in conclusions.\n5. Some more recent references from 2024 is required.\n\nBy framing the enhancement process as a Neural ODE, CLODE relies on numerical approximation methods (e.g., dopri5) that inherently carry approximation errors. This can provide solvers with adaptive step sizes but they cannot guarantee analytical precision or stability across all image conditions, particularly in complex cases with high dynamic ranges. This approximation could lead to unpredictable or suboptimal results, especially when the integration steps or the neural network's learned parameters are insufficiently fine-tuned for certain low-light conditions. Can the authors provide some low-light images from real-world dark conditions and demonstrate the method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)