We address the problem of decomposing a single image into reflectance and shading. The difficulty comes from the fact that the components of image---the surface albedo, the direct illumination, and the ambient illumination---are coupled heavily in observed image. We propose to infer the shading by ordering pixels by their relative brightness, without knowing the absolute values of the image components beforehand. The pairwise shading orders are estimated in two ways: brightness order and low-order fittings of local shading field. The brightness order is a non-local measure, which can be applied to any pair of pixels including those whose reflectance and shading are both different. The low-order fittings are used for pixel pairs within local regions of smooth shading. Together, they can capture both global order structure and local variations of the shading. We propose a Consistency-aware Selective Fusion (CSF) to integrate the pairwise orders into a globally consistent order. The iterative selection process solves the conflicts between the pairwise orders obtained by different estimation methods. Inconsistent or unreliable pairwise orders will be automatically excluded from the fusion to avoid polluting the global order. Experiments on the MIT Intrinsic Image dataset show that the proposed model is effective at recovering the shading including deep shadows. Our model also works well on natural images from the IIW dataset, the UIUC Shadow dataset and the NYU-Depth dataset, where the colors of direct lights and ambient lights are quite different.
Ranking elements from their pairwise comparisons has been extensively studied in many fields @cite_33 @cite_8 @cite_45 . Angular Embedding @cite_23 adopts a cosine error function, which is proven to be more robust to outliers than the traditional @math or @math errors used by Least Squares Embedding @cite_33 . Angular Synchronization (AS) also uses the angular space @cite_2 , but it does not consider the confidences of pairwise measures.
@cite_33 @cite_8 @cite_45 @cite_23 @cite_2
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Given the size and confidence of pairwise local orderings, angular embedding (AE) finds a global ordering with a near-global optimal eigensolution. As a quadratic criterion in the complex domain, AE is remarkably robust to outliers, unlike its real domain counterpart LS, the least squares embedding. Our comparative study of LS and AE reveals that AE's robustness is due not to the particular choice of the criterion, but to the choice of representation in the complex domain. When the embedding is encoded in the angular space, we not only have a nonconvex error function that delivers robustness, but also have a Hermitian graph Laplacian that completely determines the optimum and delivers efficiency. The high quality of embedding by AE in the presence of outliers can hardly be matched by LS, its corresponding L1 norm formulation, or their bounded versions. These results suggest that the key to overcoming outliers lies not with additionally imposing constraints on the embedding solution, but with adaptively penalizing inconsistency between measurements themselves. AE thus significantly advances statistical ranking methods by removing the impact of outliers directly without explicit inconsistency characterization, and advances spectral clustering methods by covering the entire size-confidence measurement space and providing an ordered cluster organization. This paper examines the problem of ranking a collection of objects using pairwise comparisons (rankings of two objects). In general, the ranking of @math objects can be identified by standard sorting methods using @math pairwise comparisons. We are interested in natural situations in which relationships among the objects may allow for ranking using far fewer pairwise comparisons. Specifically, we assume that the objects can be embedded into a @math -dimensional Euclidean space and that the rankings reflect their relative distances from a common reference point in @math . We show that under this assumption the number of possible rankings grows like @math and demonstrate an algorithm that can identify a randomly selected ranking using just slightly more than @math adaptively selected pairwise comparisons, on average. If instead the comparisons are chosen at random, then almost all pairwise comparisons must be made in order to identify any ranking. In addition, we propose a robust, error-tolerant algorithm that only requires that the pairwise comparisons are probably correct. Experimental studies with synthetic and real datasets support the conclusions of our theoretical analysis. This paper aims to conduct a study on the listwise approach to learning to rank. The listwise approach learns a ranking function by taking individual lists as instances and minimizing a loss function defined on the predicted list and the ground-truth list. Existing work on the approach mainly focused on the development of new algorithms; methods such as RankCosine and ListNet have been proposed and good performances by them have been observed. Unfortunately, the underlying theory was not sufficiently studied so far. To amend the problem, this paper proposes conducting theoretical analysis of learning to rank algorithms through investigations on the properties of the loss functions, including consistency, soundness, continuity, differentiability, convexity, and efficiency. A sufficient condition on consistency for ranking is given, which seems to be the first such result obtained in related research. The paper then conducts analysis on three loss functions: likelihood loss, cosine loss, and cross entropy loss. The latter two were used in RankCosine and ListNet. The use of the likelihood loss leads to the development of a new listwise method called ListMLE, whose loss function offers better properties, and also leads to better experimental results. In this paper, we propose a rank minimization method to fuse the predicted confidence scores of multiple models, each of which is obtained based on a certain kind of feature. Specifically, we convert each confidence score vector obtained from one model into a pairwise relationship matrix, in which each entry characterizes the comparative relationship of scores of two test samples. Our hypothesis is that the relative score relations are consistent among component models up to certain sparse deviations, despite the large variations that may exist in the absolute values of the raw scores. Then we formulate the score fusion problem as seeking a shared rank-2 pairwise relationship matrix based on which each original score matrix from individual model can be decomposed into the common rank-2 matrix and sparse deviation errors. A robust score vector is then extracted to fit the recovered low rank score relation matrix. We formulate the problem as a nuclear norm and l 1 norm optimization objective function and employ the Augmented Lagrange Multiplier (ALM) method for the optimization. Our method is isotonic (i.e., scale invariant) to the numeric scales of the scores originated from different models. We experimentally show that the proposed method achieves significant performance gains on various tasks including object categorization and video event detection.