CVAug 25, 2023Code
Residual Denoising Diffusion ModelsJiawei Liu, Qiang Wang, Huijie Fan et al.
We propose residual denoising diffusion models (RDDM), a novel dual diffusion process that decouples the traditional single denoising diffusion process into residual diffusion and noise diffusion. This dual diffusion framework expands the denoising-based diffusion models, initially uninterpretable for image restoration, into a unified and interpretable model for both image generation and restoration by introducing residuals. Specifically, our residual diffusion represents directional diffusion from the target image to the degraded input image and explicitly guides the reverse generation process for image restoration, while noise diffusion represents random perturbations in the diffusion process. The residual prioritizes certainty, while the noise emphasizes diversity, enabling RDDM to effectively unify tasks with varying certainty or diversity requirements, such as image generation and restoration. We demonstrate that our sampling process is consistent with that of DDPM and DDIM through coefficient transformation, and propose a partially path-independent generation process to better understand the reverse process. Notably, our RDDM enables a generic UNet, trained with only an L1 loss and a batch size of 1, to compete with state-of-the-art image restoration methods. We provide code and pre-trained models to encourage further exploration, application, and development of our innovative framework (https://github.com/nachifur/RDDM).
CVSep 22, 2022Code
CCR: Facial Image Editing with Continuity, Consistency and ReversibilityNan Yang, Xin Luan, Huidi Jia et al.
Three problems exist in sequential facial image editing: incontinuous editing, inconsistent editing, and irreversible editing. Incontinuous editing is that the current editing can not retain the previously edited attributes. Inconsistent editing is that swapping the attribute editing orders can not yield the same results. Irreversible editing means that operating on a facial image is irreversible, especially in sequential facial image editing. In this work, we put forward three concepts and corresponding definitions: editing continuity, consistency, and reversibility. Then, we propose a novel model to achieve the goal of editing continuity, consistency, and reversibility. A sufficient criterion is defined to determine whether a model is continuous, consistent, and reversible. Extensive qualitative and quantitative experimental results validate our proposed model and show that a continuous, consistent and reversible editing model has a more flexible editing function while preserving facial identity. Furthermore, we think that our proposed definitions and model will have wide and promising applications in multimedia processing. Code and data are available at https://github.com/mickoluan/CCR.
CVSep 30, 2024Code
Domain Consistency Representation Learning for Lifelong Person Re-IdentificationShiben Liu, Huijie Fan, Qiang Wang et al.
Lifelong person re-identification (LReID) exhibits a contradictory relationship between intra-domain discrimination and inter-domain gaps when learning from continuous data. Intra-domain discrimination focuses on individual nuances (i.e., clothing type, accessories, etc.), while inter-domain gaps emphasize domain consistency. Achieving a trade-off between maximizing intra-domain discrimination and minimizing inter-domain gaps is a crucial challenge for improving LReID performance. Most existing methods strive to reduce inter-domain gaps through knowledge distillation to maintain domain consistency. However, they often ignore intra-domain discrimination. To address this challenge, we propose a novel domain consistency representation learning (DCR) model that explores global and attribute-wise representations as a bridge to balance intra-domain discrimination and inter-domain gaps. At the intra-domain level, we explore the complementary relationship between global and attribute-wise representations to improve discrimination among similar identities. Excessive learning intra-domain discrimination can lead to catastrophic forgetting. We further develop an attribute-oriented anti-forgetting (AF) strategy that explores attribute-wise representations to enhance inter-domain consistency, and propose a knowledge consolidation (KC) strategy to facilitate knowledge transfer. Extensive experiments show that our DCR achieves superior performance compared to state-of-the-art LReID methods. Our code is available at https://github.com/LiuShiBen/DCR.
CVJul 3, 2023
Review helps learn better: Temporal Supervised Knowledge DistillationDongwei Wang, Zhi Han, Yanmei Wang et al.
Reviewing plays an important role when learning knowledge. The knowledge acquisition at a certain time point may be strongly inspired with the help of previous experience. Thus the knowledge growing procedure should show strong relationship along the temporal dimension. In our research, we find that during the network training, the evolution of feature map follows temporal sequence property. A proper temporal supervision may further improve the network training performance. Inspired by this observation, we propose Temporal Supervised Knowledge Distillation (TSKD). Specifically, we extract the spatiotemporal features in the different training phases of student by convolutional Long Short-term memory network (Conv-LSTM). Then, we train the student net through a dynamic target, rather than static teacher network features. This process realizes the refinement of old knowledge in student network, and utilizes it to assist current learning. Extensive experiments verify the effectiveness and advantages of our method over existing knowledge distillation methods, including various network architectures and different tasks (image classification and object detection) .
LGMar 3
The power of small initialization in noisy low-tubal-rank tensor recoveryZHiyu Liu, Haobo Geng, Xudong Wang et al.
We study the problem of recovering a low-tubal-rank tensor $\mathcal{X}\_\star\in \mathbb{R}^{n \times n \times k}$ from noisy linear measurements under the t-product framework. A widely adopted strategy involves factorizing the optimization variable as $\mathcal{U} * \mathcal{U}^\top$, where $\mathcal{U} \in \mathbb{R}^{n \times R \times k}$, followed by applying factorized gradient descent (FGD) to solve the resulting optimization problem. Since the tubal-rank $r$ of the underlying tensor $\mathcal{X}_\star$ is typically unknown, this method often assumes $r < R \le n$, a regime known as over-parameterization. However, when the measurements are corrupted by some dense noise (e.g., Gaussian noise), FGD with the commonly used spectral initialization yields a recovery error that grows linearly with the over-estimated tubal-rank $R$. To address this issue, we show that using a small initialization enables FGD to achieve a nearly minimax optimal recovery error, even when the tubal-rank $R$ is significantly overestimated. Using a four-stage analytic framework, we analyze this phenomenon and establish the sharpest known error bound to date, which is independent of the overestimated tubal-rank $R$. Furthermore, we provide a theoretical guarantee showing that an easy-to-use early stopping strategy can achieve the best known result in practice. All these theoretical findings are validated through a series of simulations and real-data experiments.
LGDec 8, 2025
Efficient Low-Tubal-Rank Tensor Estimation via Alternating Preconditioned Gradient DescentZhiyu Liu, Zhi Han, Yandong Tang et al.
The problem of low-tubal-rank tensor estimation is a fundamental task with wide applications across high-dimensional signal processing, machine learning, and image science. Traditional approaches tackle such a problem by performing tensor singular value decomposition, which is computationally expensive and becomes infeasible for large-scale tensors. Recent approaches address this issue by factorizing the tensor into two smaller factor tensors and solving the resulting problem using gradient descent. However, this kind of approach requires an accurate estimate of the tensor rank, and when the rank is overestimated, the convergence of gradient descent and its variants slows down significantly or even diverges. To address this problem, we propose an Alternating Preconditioned Gradient Descent (APGD) algorithm, which accelerates convergence in the over-parameterized setting by adding a preconditioning term to the original gradient and updating these two factors alternately. Based on certain geometric assumptions on the objective function, we establish linear convergence guarantees for more general low-tubal-rank tensor estimation problems. Then we further analyze the specific cases of low-tubal-rank tensor factorization and low-tubal-rank tensor recovery. Our theoretical results show that APGD achieves linear convergence even under over-parameterization, and the convergence rate is independent of the tensor condition number. Extensive simulations on synthetic data are carried out to validate our theoretical assertions.
CVAug 5, 2025Code
Distribution-aware Knowledge Unification and Association for Non-exemplar Lifelong Person Re-identificationShiben Liu, Mingyue Xu, Huijie Fan et al.
Lifelong person re-identification (LReID) encounters a key challenge: balancing the preservation of old knowledge with adaptation to new information. Existing LReID methods typically employ knowledge distillation to enforce representation alignment. However, these approaches ignore two crucial aspects: specific distribution awareness and cross-domain unified knowledge learning, both of which are essential for addressing this challenge. To overcome these limitations, we propose a novel distribution-aware knowledge unification and association (DKUA) framework where domain-style modeling is performed for each instance to propagate domain-specific representations, enhancing anti-forgetting and generalization capacity. Specifically, we design a distribution-aware model to transfer instance-level representations of the current domain into the domain-specific representations with the different domain styles, preserving learned knowledge without storing old samples. Next, we propose adaptive knowledge consolidation (AKC) to dynamically generate the unified representation as a cross-domain representation center. To further mitigate forgetting, we develop a unified knowledge association (UKA) mechanism, which explores the unified representation as a bridge to explicitly model inter-domain associations, reducing inter-domain gaps. Finally, distribution-based knowledge transfer (DKT) is proposed to prevent the current domain distribution from deviating from the cross-domain distribution center, improving adaptation capacity. Experimental results show our DKUA outperforms the existing methods by 7.6%/5.3% average mAP/R@1 improvement on anti-forgetting and generalization capacity, respectively. Our code is available at https://github.com/LiuShiBen/DKUA.
CVApr 21, 2025Code
Distribution-aware Forgetting Compensation for Exemplar-Free Lifelong Person Re-identificationShiben Liu, Huijie Fan, Qiang Wang et al.
Lifelong Person Re-identification (LReID) suffers from a key challenge in preserving old knowledge while adapting to new information. The existing solutions include rehearsal-based and rehearsal-free methods to address this challenge. Rehearsal-based approaches rely on knowledge distillation, continuously accumulating forgetting during the distillation process. Rehearsal-free methods insufficiently learn the distribution of each domain, leading to forgetfulness over time. To solve these issues, we propose a novel Distribution-aware Forgetting Compensation (DAFC) model that explores cross-domain shared representation learning and domain-specific distribution integration without using old exemplars or knowledge distillation. We propose a Text-driven Prompt Aggregation (TPA) that utilizes text features to enrich prompt elements and guide the prompt model to learn fine-grained representations for each instance. This can enhance the differentiation of identity information and establish the foundation for domain distribution awareness. Then, Distribution-based Awareness and Integration (DAI) is designed to capture each domain-specific distribution by a dedicated expert network and adaptively consolidate them into a shared region in high-dimensional space. In this manner, DAI can consolidate and enhance cross-domain shared representation learning while alleviating catastrophic forgetting. Furthermore, we develop a Knowledge Consolidation Mechanism (KCM) that comprises instance-level discrimination and cross-domain consistency alignment strategies to facilitate model adaptive learning of new knowledge from the current domain and promote knowledge consolidation learning between acquired domain-specific distributions, respectively. Experimental results show that our DAFC outperforms state-of-the-art methods. Our code is available at https://github.com/LiuShiBen/DAFC.
CVMar 10, 2025Code
Unleashing the Potential of Large Language Models for Text-to-Image Generation through Autoregressive Representation AlignmentXing Xie, Jiawei Liu, Ziyue Lin et al.
We present Autoregressive Representation Alignment (ARRA), a new training framework that unlocks global-coherent text-to-image generation in autoregressive LLMs without architectural modifications. Different from prior works that require complex architectural redesigns, ARRA aligns LLM's hidden states with visual representations from external visual foundational models via a global visual alignment loss and a hybrid token, <HYBNEXT>. This token enforces dual constraints: local next-token prediction and global semantic distillation, enabling LLMs to implicitly learn spatial and contextual coherence while retaining their original autoregressive paradigm. Extensive experiments validate ARRA's plug-and-play versatility. When training T2I LLMs from scratch, ARRA reduces FID by 16.6% (ImageNet), 12.0% (LAION-COCO) for autoregressive LLMs like LlamaGen, without modifying original architecture and inference mechanism. For training from text-generation-only LLMs, ARRA reduces FID by 25.5% (MIMIC-CXR), 8.8% (DeepEyeNet) for advanced LLMs like Chameleon. For domain adaptation, ARRA aligns general-purpose LLMs with specialized models (e.g., BioMedCLIP), achieving an 18.6% FID reduction over direct fine-tuning on medical imaging (MIMIC-CXR). These results demonstrate that training objective redesign, rather than architectural modifications, can resolve cross-modal global coherence challenges. ARRA offers a complementary paradigm for advancing autoregressive models. The code is available at https://github.com/HKU-HealthAI/ARRA.
CVOct 5, 2020Code
Local Label Point Correction for Edge Detection of Overlapping Cervical CellsJiawei Liu, Huijie Fan, Qiang Wang et al.
Accurate labeling is essential for supervised deep learning methods. However, it is almost impossible to accurately and manually annotate thousands of images, which results in many labeling errors for most datasets. We proposes a local label point correction (LLPC) method to improve annotation quality for edge detection and image segmentation tasks. Our algorithm contains three steps: gradient-guided point correction, point interpolation and local point smoothing. We correct the labels of object contours by moving the annotated points to the pixel gradient peaks. This can improve the edge localization accuracy, but it also causes unsmooth contours due to the interference of image noise. Therefore, we design a point smoothing method based on local linear fitting to smooth the corrected edge. To verify the effectiveness of our LLPC, we construct a largest overlapping cervical cell edge detection dataset (CCEDD) with higher precision label corrected by our label correction method. Our LLPC only needs to set three parameters, but yields 30-40$\%$ average precision improvement on multiple networks. The qualitative and quantitative experimental results show that our LLPC can improve the quality of manual labels and the accuracy of overlapping cell edge detection. We hope that our study will give a strong boost to the development of the label correction for edge detection and image segmentation. We will release the dataset and code at https://github.com/nachifur/LLPC.
CVMar 24, 2024
Diverse Representation Embedding for Lifelong Person Re-IdentificationShiben Liu, Huijie Fan, Qiang Wang et al.
Lifelong Person Re-Identification (LReID) aims to continuously learn from successive data streams, matching individuals across multiple cameras. The key challenge for LReID is how to effectively preserve old knowledge while incrementally learning new information, which is caused by task-level domain gaps and limited old task datasets. Existing methods based on CNN backbone are insufficient to explore the representation of each instance from different perspectives, limiting model performance on limited old task datasets and new task datasets. Unlike these methods, we propose a Diverse Representations Embedding (DRE) framework that first explores a pure transformer for LReID. The proposed DRE preserves old knowledge while adapting to new information based on instance-level and task-level layout. Concretely, an Adaptive Constraint Module (ACM) is proposed to implement integration and push away operations between multiple overlapping representations generated by transformer-based backbone, obtaining rich and discriminative representations for each instance to improve adaptive ability of LReID. Based on the processed diverse representations, we propose Knowledge Update (KU) and Knowledge Preservation (KP) strategies at the task-level layout by introducing the adjustment model and the learner model. KU strategy enhances the adaptive learning ability of learner models for new information under the adjustment model prior, and KP strategy preserves old knowledge operated by representation-level alignment and logit-level supervision in limited old task datasets while guaranteeing the adaptive learning information capacity of the LReID model. Compared to state-of-the-art methods, our method achieves significantly improved performance in holistic, large-scale, and occluded datasets.
LGJan 22, 2024
Low-Tubal-Rank Tensor Recovery via Factorized Gradient DescentZhiyu Liu, Zhi Han, Yandong Tang et al.
This paper considers the problem of recovering a tensor with an underlying low-tubal-rank structure from a small number of corrupted linear measurements. Traditional approaches tackling such a problem require the computation of tensor Singular Value Decomposition (t-SVD), that is a computationally intensive process, rendering them impractical for dealing with large-scale tensors. Aim to address this challenge, we propose an efficient and effective low-tubal-rank tensor recovery method based on a factorization procedure akin to the Burer-Monteiro (BM) method. Precisely, our fundamental approach involves decomposing a large tensor into two smaller factor tensors, followed by solving the problem through factorized gradient descent (FGD). This strategy eliminates the need for t-SVD computation, thereby reducing computational costs and storage requirements. We provide rigorous theoretical analysis to ensure the convergence of FGD under both noise-free and noisy situations. Additionally, it is worth noting that our method does not require the precise estimation of the tensor tubal-rank. Even in cases where the tubal-rank is slightly overestimated, our approach continues to demonstrate robust performance. A series of experiments have been carried out to demonstrate that, as compared to other popular ones, our approach exhibits superior performance in multiple scenarios, in terms of the faster computational speed and the smaller convergence error.
LGFeb 1, 2025
Efficient Over-parameterized Matrix Sensing from Noisy Measurements via Alternating Preconditioned Gradient DescentZhiyu Liu, Zhi Han, Yandong Tang et al.
We consider the noisy matrix sensing problem in the over-parameterization setting, where the estimated rank $r$ is larger than the true rank $r_\star$ of the target matrix $X_\star$. Specifically, our main objective is to recover a matrix $ X_\star \in \mathbb{R}^{n_1 \times n_2} $ with rank $ r_\star $ from noisy measurements using an over-parameterized factorization $ LR^\top $, where $ L \in \mathbb{R}^{n_1 \times r}, \, R \in \mathbb{R}^{n_2 \times r} $ and $ \min\{n_1, n_2\} \ge r > r_\star $, with $ r_\star $ being unknown. Recently, preconditioning methods have been proposed to accelerate the convergence of matrix sensing problem compared to vanilla gradient descent, incorporating preconditioning terms $ (L^\top L + λI)^{-1} $ and $ (R^\top R + λI)^{-1} $ into the original gradient. However, these methods require careful tuning of the damping parameter $λ$ and are sensitive to step size. To address these limitations, we propose the alternating preconditioned gradient descent (APGD) algorithm, which alternately updates the two factor matrices, eliminating the need for the damping parameter $λ$ and enabling faster convergence with larger step sizes. We theoretically prove that APGD convergences to a near-optimal error at a linear rate. We further show that APGD can be extended to deal with other low-rank matrix estimation tasks, also with a theoretical guarantee of linear convergence. To validate the effectiveness and scalability of the proposed APGD, we conduct simulated and real-world experiments on a wide range of low-rank estimation problems, including noisy matrix sensing, weighted PCA, 1-bit matrix completion, and matrix completion. The extensive results demonstrate that APGD consistently achieves the fastest convergence and the lowest computation time compared to the existing alternatives.
IVMar 22, 2025
DVG-Diffusion: Dual-View Guided Diffusion Model for CT Reconstruction from X-RaysXing Xie, Jiawei Liu, Huijie Fan et al.
Directly reconstructing 3D CT volume from few-view 2D X-rays using an end-to-end deep learning network is a challenging task, as X-ray images are merely projection views of the 3D CT volume. In this work, we facilitate complex 2D X-ray image to 3D CT mapping by incorporating new view synthesis, and reduce the learning difficulty through view-guided feature alignment. Specifically, we propose a dual-view guided diffusion model (DVG-Diffusion), which couples a real input X-ray view and a synthesized new X-ray view to jointly guide CT reconstruction. First, a novel view parameter-guided encoder captures features from X-rays that are spatially aligned with CT. Next, we concatenate the extracted dual-view features as conditions for the latent diffusion model to learn and refine the CT latent representation. Finally, the CT latent representation is decoded into a CT volume in pixel space. By incorporating view parameter guided encoding and dual-view guided CT reconstruction, our DVG-Diffusion can achieve an effective balance between high fidelity and perceptual quality for CT reconstruction. Experimental results demonstrate our method outperforms state-of-the-art methods. Based on experiments, the comprehensive analysis and discussions for views and reconstruction are also presented.
CVSep 13, 2021
Effective Tensor Completion via Element-wise Weighted Low-rank Tensor Train with Overlapping Ket AugmentationYang Zhang, Yao Wang, Zhi Han et al.
In recent years, there have been an increasing number of applications of tensor completion based on the tensor train (TT) format because of its efficiency and effectiveness in dealing with higher-order tensor data. However, existing tensor completion methods using TT decomposition have two obvious drawbacks. One is that they only consider mode weights according to the degree of mode balance, even though some elements are recovered better in an unbalanced mode. The other is that serious blocking artifacts appear when the missing element rate is relatively large. To remedy such two issues, in this work, we propose a novel tensor completion approach via the element-wise weighted technique. Accordingly, a novel formulation for tensor completion and an effective optimization algorithm, called as tensor completion by parallel weighted matrix factorization via tensor train (TWMac-TT), is proposed. In addition, we specifically consider the recovery quality of edge elements from adjacent blocks. Different from traditional reshaping and ket augmentation, we utilize a new tensor augmentation technique called overlapping ket augmentation, which can further avoid blocking artifacts. We then conduct extensive performance evaluations on synthetic data and several real image data sets. Our experimental results demonstrate that the proposed algorithm TWMac-TT outperforms several other competing tensor completion methods.
CVJul 18, 2020
Tracking-by-Counting: Using Network Flows on Crowd Density Maps for Tracking Multiple TargetsWeihong Ren, Xinchao Wang, Jiandong Tian et al.
State-of-the-art multi-object tracking~(MOT) methods follow the tracking-by-detection paradigm, where object trajectories are obtained by associating per-frame outputs of object detectors. In crowded scenes, however, detectors often fail to obtain accurate detections due to heavy occlusions and high crowd density. In this paper, we propose a new MOT paradigm, tracking-by-counting, tailored for crowded scenes. Using crowd density maps, we jointly model detection, counting, and tracking of multiple targets as a network flow program, which simultaneously finds the global optimal detections and trajectories of multiple targets over the whole video. This is in contrast to prior MOT methods that either ignore the crowd density and thus are prone to errors in crowded scenes, or rely on a suboptimal two-step process using heuristic density-aware point-tracks for matching targets.Our approach yields promising results on public benchmarks of various domains including people tracking, cell tracking, and fish tracking.
CVDec 5, 2018
Dynamic Spatio-temporal Graph-based CNNs for Traffic PredictionKen Chen, Fei Chen, Baisheng Lai et al.
Forecasting future traffic flows from previous ones is a challenging problem because of their complex and dynamic nature of spatio-temporal structures. Most existing graph-based CNNs attempt to capture the static relations while largely neglecting the dynamics underlying sequential data. In this paper, we present dynamic spatio-temporal graph-based CNNs (DST-GCNNs) by learning expressive features to represent spatio-temporal structures and predict future traffic flows from surveillance video data. In particular, DST-GCNN is a two stream network. In the flow prediction stream, we present a novel graph-based spatio-temporal convolutional layer to extract features from a graph representation of traffic flows. Then several such layers are stacked together to predict future flows over time. Meanwhile, the relations between traffic flows in the graph are often time variant as the traffic condition changes over time. To capture the graph dynamics, we use the graph prediction stream to predict the dynamic graph structures, and the predicted structures are fed into the flow prediction stream. Experiments on real datasets demonstrate that the proposed model achieves competitive performances compared with the other state-of-the-art methods.
CVMay 18, 2017
A General Model for Robust Tensor Factorization with Unknown NoiseXi'ai Chen, Zhi Han, Yao Wang et al.
Because of the limitations of matrix factorization, such as losing spatial structure information, the concept of low-rank tensor factorization (LRTF) has been applied for the recovery of a low dimensional subspace from high dimensional visual data. The low-rank tensor recovery is generally achieved by minimizing the loss function between the observed data and the factorization representation. The loss function is designed in various forms under different noise distribution assumptions, like $L_1$ norm for Laplacian distribution and $L_2$ norm for Gaussian distribution. However, they often fail to tackle the real data which are corrupted by the noise with unknown distribution. In this paper, we propose a generalized weighted low-rank tensor factorization method (GWLRTF) integrated with the idea of noise modelling. This procedure treats the target data as high-order tensor directly and models the noise by a Mixture of Gaussians, which is called MoG GWLRTF. The parameters in the model are estimated under the EM framework and through a new developed algorithm of weighted low-rank tensor factorization. We provide two versions of the algorithm with different tensor factorization operations, i.e., CP factorization and Tucker factorization. Extensive experiments indicate the respective advantages of this two versions in different applications and also demonstrate the effectiveness of MoG GWLRTF compared with other competing methods.
CVJul 12, 2016
RGBD Salient Object Detection via Deep FusionLiangqiong Qu, Shengfeng He, Jiawei Zhang et al.
Numerous efforts have been made to design different low level saliency cues for the RGBD saliency detection, such as color or depth contrast features, background and color compactness priors. However, how these saliency cues interact with each other and how to incorporate these low level saliency cues effectively to generate a master saliency map remain a challenging problem. In this paper, we design a new convolutional neural network (CNN) to fuse different low level saliency cues into hierarchical features for automatically detecting salient objects in RGBD images. In contrast to the existing works that directly feed raw image pixels to the CNN, the proposed method takes advantage of the knowledge in traditional saliency detection by adopting various meaningful and well-designed saliency feature vectors as input. This can guide the training of CNN towards detecting salient object more effectively due to the reduced learning ambiguity. We then integrate a Laplacian propagation framework with the learned CNN to extract a spatially consistent saliency map by exploiting the intrinsic structure of the input image. Extensive quantitative and qualitative experimental evaluations on three datasets demonstrate that the proposed method consistently outperforms state-of-the-art methods.
CVJun 30, 2014
Pixel-wise Orthogonal Decomposition for Color Illumination Invariant and Shadow-free ImageLiangqiong Qu, Jiandong Tian, Zhi Han et al.
In this paper, we propose a novel, effective and fast method to obtain a color illumination invariant and shadow-free image from a single outdoor image. Different from state-of-the-art methods for shadow-free image that either need shadow detection or statistical learning, we set up a linear equation set for each pixel value vector based on physically-based shadow invariants, deduce a pixel-wise orthogonal decomposition for its solutions, and then get an illumination invariant vector for each pixel value vector on an image. The illumination invariant vector is the unique particular solution of the linear equation set, which is orthogonal to its free solutions. With this illumination invariant vector and Lab color space, we propose an algorithm to generate a shadow-free image which well preserves the texture and color information of the original image. A series of experiments on a diverse set of outdoor images and the comparisons with the state-of-the-art methods validate our method.