Inference-time computation techniques, analogous to human System 2 Thinking, have recently become popular for improving model performances. However, most existing approaches suffer from several limitations: they are modality-specific (e.g., working only in text), problem-specific (e.g., verifiable domains like math and coding), or require additional supervision/training on top of unsupervised pretraining (e.g., verifiers or verifiable rewards). In this paper, we ask the question "Is it possible to generalize these System 2 Thinking approaches, and develop models that learn to think solely from unsupervised learning?" Interestingly, we find the answer is yes, by learning to explicitly verify the compatibility between inputs and candidate-predictions, and then re-framing prediction problems as optimization with respect to this verifier. Specifically, we train Energy-Based Transformers (EBTs) -- a new class of Energy-Based Models (EBMs) -- to assign an energy value to every input and candidate-prediction pair, enabling predictions through gradient descent-based energy minimization until convergence. Across both discrete (text) and continuous (visual) modalities, we find EBTs scale faster than the dominant Transformer++ approach during training, achieving an up to 35% higher scaling rate with respect to data, batch size, parameters, FLOPs, and depth. During inference, EBTs improve performance with System 2 Thinking by 29% more than the Transformer++ on language tasks, and EBTs outperform Diffusion Transformers on image denoising while using fewer forward passes. Further, we find that EBTs achieve better results than existing models on most downstream tasks given the same or worse pretraining performance, suggesting that EBTs generalize better than existing approaches. Consequently, EBTs are a promising new paradigm for scaling both the learning and thinking capabilities of models.
We introduce a novel weighted convolution operator that enhances traditional convolutional neural networks (CNNs) by integrating a spatial density function into the convolution operator. This extension enables the network to differentially weight neighbouring pixels based on their relative position to the reference pixel, improving spatial characterisation and feature extraction. The proposed operator maintains the same number of trainable parameters and is fully compatible with existing CNN architectures. Although developed for 2D image data, the framework is generalisable to signals on regular grids of arbitrary dimensions, such as 3D volumetric data or 1D time series. We propose an efficient implementation of the weighted convolution by pre-computing the density function and achieving execution times comparable to standard convolution layers. We evaluate our method on two deep learning tasks: image classification using the CIFAR-100 dataset [KH+09] and image denoising using the DIV2K dataset [AT17]. Experimental results with state-of-the-art classification (e.g., VGG [SZ15], ResNet [HZRS16]) and denoising (e.g., DnCNN [ZZC+17], NAFNet [CCZS22]) methods show that the weighted convolution improves performance with respect to standard convolution across different quantitative metrics. For example, VGG achieves an accuracy of 66.94% with weighted convolution versus 56.89% with standard convolution on the classification problem, while DnCNN improves the PSNR value from 20.17 to 22.63 on the denoising problem. All models were trained on the CINECA Leonardo cluster to reduce the execution time and improve the tuning of the density function values. The PyTorch implementation of the weighted convolution is publicly available at: https://github.com/cammarasana123/weightedConvolution2.0.
The paper introduces the weighted convolution, a novel approach to the convolution for signals defined on regular grids (e.g., 2D images) through the application of an optimal density function to scale the contribution of neighbouring pixels based on their distance from the central pixel. This choice differs from the traditional uniform convolution, which treats all neighbouring pixels equally. Our weighted convolution can be applied to convolutional neural network problems to improve the approximation accuracy. Given a convolutional network, we define a framework to compute the optimal density function through a minimisation model. The framework separates the optimisation of the convolutional kernel weights (using stochastic gradient descent) from the optimisation of the density function (using DIRECT-L). Experimental results on a learning model for an image-to-image task (e.g., image denoising) show that the weighted convolution significantly reduces the loss (up to 53% improvement) and increases the test accuracy compared to standard convolution. While this method increases execution time by 11%, it is robust across several hyperparameters of the learning model. Future work will apply the weighted convolution to real-case 2D and 3D image convolutional learning problems.
Remote sensing image (RSI) denoising is an important topic in the field of remote sensing. Despite the impressive denoising performance of RSI denoising methods, most current deep learning-based approaches function as black boxes and lack integration with physical information models, leading to limited interpretability. Additionally, many methods may struggle with insufficient attention to non-local self-similarity in RSI and require tedious tuning of regularization parameters to achieve optimal performance, particularly in conventional iterative optimization approaches. In this paper, we first propose a novel RSI denoising method named sparse tensor-aided representation network (STAR-Net), which leverages a low-rank prior to effectively capture the non-local self-similarity within RSI. Furthermore, we extend STAR-Net to a sparse variant called STAR-Net-S to deal with the interference caused by non-Gaussian noise in original RSI for the purpose of improving robustness. Different from conventional iterative optimization, we develop an alternating direction method of multipliers (ADMM)-guided deep unrolling network, in which all regularization parameters can be automatically learned, thus inheriting the advantages of both model-based and deep learning-based approaches and successfully addressing the above-mentioned shortcomings. Comprehensive experiments on synthetic and real-world datasets demonstrate that STAR-Net and STAR-Net-S outperform state-of-the-art RSI denoising methods.
Self-supervised image denoising methods have garnered significant research attention in recent years, for this kind of method reduces the requirement of large training datasets. Compared to supervised methods, self-supervised methods rely more on the prior embedded in deep networks themselves. As a result, most of the self-supervised methods are designed with Convolution Neural Networks (CNNs) architectures, which well capture one of the most important image prior, translation equivariant prior. Inspired by the great success achieved by the introduction of translational equivariance, in this paper, we explore the way to further incorporate another important image prior. Specifically, we first apply high-accuracy rotation equivariant convolution to self-supervised image denoising. Through rigorous theoretical analysis, we have proved that simply replacing all the convolution layers with rotation equivariant convolution layers would modify the network into its rotation equivariant version. To the best of our knowledge, this is the first time that rotation equivariant image prior is introduced to self-supervised image denoising at the network architecture level with a comprehensive theoretical analysis of equivariance errors, which offers a new perspective to the field of self-supervised image denoising. Moreover, to further improve the performance, we design a new mask mechanism to fusion the output of rotation equivariant network and vanilla CNN-based network, and construct an adaptive rotation equivariant framework. Through extensive experiments on three typical methods, we have demonstrated the effectiveness of the proposed method.
All-in-one image restoration aims to recover clear images from various degradation types and levels with a unified model. Nonetheless, the significant variations among degradation types present challenges for training a universal model, often resulting in task interference, where the gradient update directions of different tasks may diverge due to shared parameters. To address this issue, motivated by the routing strategy, we propose DFPIR, a novel all-in-one image restorer that introduces Degradation-aware Feature Perturbations(DFP) to adjust the feature space to align with the unified parameter space. In this paper, the feature perturbations primarily include channel-wise perturbations and attention-wise perturbations. Specifically, channel-wise perturbations are implemented by shuffling the channels in high-dimensional space guided by degradation types, while attention-wise perturbations are achieved through selective masking in the attention space. To achieve these goals, we propose a Degradation-Guided Perturbation Block (DGPB) to implement these two functions, positioned between the encoding and decoding stages of the encoder-decoder architecture. Extensive experimental results demonstrate that DFPIR achieves state-of-the-art performance on several all-in-one image restoration tasks including image denoising, image dehazing, image deraining, motion deblurring, and low-light image enhancement. Our codes are available at https://github.com/TxpHome/DFPIR.
In imaging inverse problems, we would like to know how close the recovered image is to the true image in terms of full-reference image quality (FRIQ) metrics like PSNR, SSIM, LPIPS, etc. This is especially important in safety-critical applications like medical imaging, where knowing that, say, the SSIM was poor could potentially avoid a costly misdiagnosis. But since we don't know the true image, computing FRIQ is non-trivial. In this work, we combine conformal prediction with approximate posterior sampling to construct bounds on FRIQ that are guaranteed to hold up to a user-specified error probability. We demonstrate our approach on image denoising and accelerated magnetic resonance imaging (MRI) problems. Code is available at https://github.com/jwen307/quality_uq.
Breast cancer is the most frequently diagnosed human cancer in the United States at present. Early detection is crucial for its successful treatment. X-ray mammography and digital breast tomosynthesis are currently the main methods for breast cancer screening. However, both have known limitations in terms of their sensitivity and specificity to breast cancers, while also frequently causing patient discomfort due to the requirement for breast compression. Breast computed tomography is a promising alternative, however, to obtain high-quality images, the X-ray dose needs to be sufficiently high. As the breast is highly radiosensitive, dose reduction is particularly important. Phase-contrast computed tomography (PCT) has been shown to produce higher-quality images at lower doses and has no need for breast compression. It is demonstrated in the present study that, when imaging full fresh mastectomy samples with PCT, deep learning-based image denoising can further reduce the radiation dose by a factor of 16 or more, without any loss of image quality. The image quality has been assessed both in terms of objective metrics, such as spatial resolution and contrast-to-noise ratio, as well as in an observer study by experienced medical imaging specialists and radiologists. This work was carried out in preparation for live patient PCT breast cancer imaging, initially at specialized synchrotron facilities.
Echocardiography (echo), or cardiac ultrasound, is the most widely used imaging modality for cardiac form and function due to its relatively low cost, rapid acquisition time, and non-invasive nature. However, ultrasound acquisitions are often limited by artifacts and noise that hinder diagnostic interpretation in clinical settings. Existing methodologies for denoising echos consist solely of traditional filtering-based algorithms or deep learning methods developed on radio-frequency (RF) signals which prevents clinical applicability and scalability. To address these limitations, we introduce the first deep generative model capable of simulating ultrasound noise developed on B-mode data. Using this generative model, we develop a synthetic dataset of paired clean and noisy echo images to train a downstream model for real-world image denoising and demonstrate state-of-the-art performance in both internal and external experiments. In both held-out test sets, our method results in echo images with higher gCNR in comparison to noisy image counterparts and images derived from a comparable method which is consistent with provided visual comparisons. Our experiments showcase the potential of our method for future clinical use to improve the quality of echo acquisitions. To encourage further research into the field, we release our source code and model weights at https://github.com/echonet/image_quality.
Noise synthesis is a promising solution for addressing the data shortage problem in data-driven low-light RAW image denoising. However, accurate noise synthesis methods often necessitate labor-intensive calibration and profiling procedures during preparation, preventing them from landing to practice at scale. This work introduces a practically simple noise synthesis pipeline based on detailed analyses of noise properties and extensive justification of widespread techniques. Compared to other approaches, our proposed pipeline eliminates the cumbersome system gain calibration and signal-independent noise profiling steps, reducing the preparation time for noise synthesis from days to hours. Meanwhile, our method exhibits strong denoising performance, showing an up to 0.54dB PSNR improvement over the current state-of-the-art noise synthesis technique. Code is released at https://github.com/SonyResearch/raw_image_denoising
Image denoising is essential in low-level vision applications such as photography and automated driving. Existing methods struggle with distinguishing complex noise patterns in real-world scenes and consume significant computational resources due to reliance on Transformer-based models. In this work, the Context-guided Receptance Weighted Key-Value (\M) model is proposed, combining enhanced multi-view feature integration with efficient sequence modeling. Our approach introduces the Context-guided Token Shift (CTS) paradigm, which effectively captures local spatial dependencies and enhance the model's ability to model real-world noise distributions. Additionally, the Frequency Mix (FMix) module extracting frequency-domain features is designed to isolate noise in high-frequency spectra, and is integrated with spatial representations through a multi-view learning process. To improve computational efficiency, the Bidirectional WKV (BiWKV) mechanism is adopted, enabling full pixel-sequence interaction with linear complexity while overcoming the causal selection constraints. The model is validated on multiple real-world image denoising datasets, outperforming the existing state-of-the-art methods quantitatively and reducing inference time up to 40\%. Qualitative results further demonstrate the ability of our model to restore fine details in various scenes.
This paper presents an overview of the NTIRE 2025 Image Denoising Challenge ({\sigma} = 50), highlighting the proposed methodologies and corresponding results. The primary objective is to develop a network architecture capable of achieving high-quality denoising performance, quantitatively evaluated using PSNR, without constraints on computational complexity or model size. The task assumes independent additive white Gaussian noise (AWGN) with a fixed noise level of 50. A total of 290 participants registered for the challenge, with 20 teams successfully submitting valid results, providing insights into the current state-of-the-art in image denoising.
Effective denoising is crucial in low-dose CT to enhance subtle structures and low-contrast lesions while preventing diagnostic errors. Supervised methods struggle with limited paired datasets, and self-supervised approaches often require multiple noisy images and rely on deep networks like U-Net, offering little insight into the denoising mechanism. To address these challenges, we propose an interpretable self-supervised single-image denoising framework -- Filter2Noise (F2N). Our approach introduces an Attention-Guided Bilateral Filter that adapted to each noisy input through a lightweight module that predicts spatially varying filter parameters, which can be visualized and adjusted post-training for user-controlled denoising in specific regions of interest. To enable single-image training, we introduce a novel downsampling shuffle strategy with a new self-supervised loss function that extends the concept of Noise2Noise to a single image and addresses spatially correlated noise. On the Mayo Clinic 2016 low-dose CT dataset, F2N outperforms the leading self-supervised single-image method (ZS-N2N) by 4.59 dB PSNR while improving transparency, user control, and parametric efficiency. These features provide key advantages for medical applications that require precise and interpretable noise reduction. Our code is demonstrated at https://github.com/sypsyp97/Filter2Noise.git .
Even though Deep Neural Networks are extremely powerful for image restoration tasks, they have several limitations. They are poorly understood and suffer from strong biases inherited from the training sets. One way to address these shortcomings is to have a better control over the training sets, in particular by using synthetic sets. In this paper, we propose a synthetic image generator relying on a few simple principles. In particular, we focus on geometric modeling, textures, and a simple modeling of image acquisition. These properties, integrated in a classical Dead Leaves model, enable the creation of efficient training sets. Standard image denoising and super-resolution networks can be trained on such datasets, reaching performance almost on par with training on natural image datasets. As a first step towards explainability, we provide a careful analysis of the considered principles, identifying which image properties are necessary to obtain good performances. Besides, such training also yields better robustness to various geometric and radiometric perturbations of the test sets.
The indicator matrix plays an important role in machine learning, but optimizing it is an NP-hard problem. We propose a new relaxation of the indicator matrix and prove that this relaxation forms a manifold, which we call the Relaxed Indicator Matrix Manifold (RIM manifold). Based on Riemannian geometry, we develop a Riemannian toolbox for optimization on the RIM manifold. Specifically, we provide several methods of Retraction, including a fast Retraction method to obtain geodesics. We point out that the RIM manifold is a generalization of the double stochastic manifold, and it is much faster than existing methods on the double stochastic manifold, which has a complexity of \( \mathcal{O}(n^3) \), while RIM manifold optimization is \( \mathcal{O}(n) \) and often yields better results. We conducted extensive experiments, including image denoising, with millions of variables to support our conclusion, and applied the RIM manifold to Ratio Cut, we provide a rigorous convergence proof and achieve clustering results that outperform the state-of-the-art methods. Our Code in \href{https://github.com/Yuan-Jinghui/Riemannian-Optimization-on-Relaxed-Indicator-Matrix-Manifold}{here}.
While deep neural networks have revolutionized image denoising capabilities, their deployment on edge devices remains challenging due to substantial computational and memory requirements. To this end, we present DnLUT, an ultra-efficient lookup table-based framework that achieves high-quality color image denoising with minimal resource consumption. Our key innovation lies in two complementary components: a Pairwise Channel Mixer (PCM) that effectively captures inter-channel correlations and spatial dependencies in parallel, and a novel L-shaped convolution design that maximizes receptive field coverage while minimizing storage overhead. By converting these components into optimized lookup tables post-training, DnLUT achieves remarkable efficiency - requiring only 500KB storage and 0.1% energy consumption compared to its CNN contestant DnCNN, while delivering 20X faster inference. Extensive experiments demonstrate that DnLUT outperforms all existing LUT-based methods by over 1dB in PSNR, establishing a new state-of-the-art in resource-efficient color image denoising. The project is available at https://github.com/Stephen0808/DnLUT.
Low-light and underwater videos suffer from poor visibility, low contrast, and high noise, necessitating enhancements in visual quality. However, existing approaches typically rely on paired ground truth, which limits their practicality and often fails to maintain temporal consistency. To overcome these obstacles, this paper introduces a novel zero-shot learning approach named Zero-TIG, leveraging the Retinex theory and optical flow techniques. The proposed network consists of an enhancement module and a temporal feedback module. The enhancement module comprises three subnetworks: low-light image denoising, illumination estimation, and reflection denoising. The temporal enhancement module ensures temporal consistency by incorporating histogram equalization, optical flow computation, and image warping to align the enhanced previous frame with the current frame, thereby maintaining continuity. Additionally, we address color distortion in underwater data by adaptively balancing RGB channels. The experimental results demonstrate that our method achieves low-light video enhancement without the need for paired training data, making it a promising and applicable method for real-world scenario enhancement.
Many studies have concentrated on constructing supervised models utilizing paired datasets for image denoising, which proves to be expensive and time-consuming. Current self-supervised and unsupervised approaches typically rely on blind-spot networks or sub-image pairs sampling, resulting in pixel information loss and destruction of detailed structural information, thereby significantly constraining the efficacy of such methods. In this paper, we introduce Prompt-SID, a prompt-learning-based single image denoising framework that emphasizes preserving of structural details. This approach is trained in a self-supervised manner using downsampled image pairs. It captures original-scale image information through structural encoding and integrates this prompt into the denoiser. To achieve this, we propose a structural representation generation model based on the latent diffusion process and design a structural attention module within the transformer-based denoiser architecture to decode the prompt. Additionally, we introduce a scale replay training mechanism, which effectively mitigates the scale gap from images of different resolutions. We conduct comprehensive experiments on synthetic, real-world, and fluorescence imaging datasets, showcasing the remarkable effectiveness of Prompt-SID. Our code will be released at https://github.com/huaqlili/Prompt-SID.
Diffusion models transform an unknown data distribution into a Gaussian prior by progressively adding noise until the data become indistinguishable from pure noise. This stochastic process traces a path in probability space, evolving from the original data distribution (considered as a Gaussian with near-zero variance) to an isotropic Gaussian. The denoiser then learns to reverse this process, generating high-quality samples from random Gaussian noise. However, standard diffusion models, such as the Denoising Diffusion Probabilistic Model (DDPM), do not ensure a geodesic (i.e., shortest) path in probability space. This inefficiency necessitates the use of many intermediate time steps, leading to high computational costs in training and sampling. To address this limitation, we propose the Geodesic Diffusion Model (GDM), which defines a geodesic path under the Fisher-Rao metric with a variance-exploding noise scheduler. This formulation transforms the data distribution into a Gaussian prior with minimal energy, significantly improving the efficiency of diffusion models. We trained GDM by continuously sampling time steps from 0 to 1 and using as few as 15 evenly spaced time steps for model sampling. We evaluated GDM on two medical image-to-image generation tasks: CT image denoising and MRI image super-resolution. Experimental results show that GDM achieved state-of-the-art performance while reducing training time by a 50-fold compared to DDPM and 10-fold compared to Fast-DDPM, with 66 times faster sampling than DDPM and a similar sampling speed to Fast-DDPM. These efficiency gains enable rapid model exploration and real-time clinical applications. Our code is publicly available at: https://github.com/mirthAI/GDM-VE.
Modern foundation models often undergo iterative ``bootstrapping'' in their post-training phase: a model generates synthetic data, an external verifier filters out low-quality samples, and the high-quality subset is used for further fine-tuning. Over multiple iterations, the model's performance improves--raising a crucial question: how should the total budget on generation and training be allocated across iterations to maximize final performance? In this work, we develop a theoretical framework to analyze budget allocation strategies. Specifically, we show that constant policies fail to converge with high probability, while increasing policies--particularly exponential growth policies--exhibit significant theoretical advantages. Experiments on image denoising with diffusion probabilistic models and math reasoning with large language models show that both exponential and polynomial growth policies consistently outperform constant policies, with exponential policies often providing more stable performance.