In this paper, we propose a path following replicator dynamic, and investigate its potentials in uncovering the underlying cluster structure of a graph. The proposed dynamic is a generalization of the discrete replicator dynamic. The replicator dynamic has been successfully used to extract dense clusters of graphs; however, it is often sensitive to the degree distribution of a graph, and usually biased by vertices with large degrees, thus may fail to detect the densest cluster. To overcome this problem, we introduce a dynamic parameter, called path parameter, into the evolution process. The path parameter can be interpreted as the maximal possible probability of a current cluster containing a vertex, and it monotonically increases as evolution process proceeds. By limiting the maximal probability, the phenomenon of some vertices dominating the early stage of evolution process is suppressed, thus making evolution process more robust. To solve the optimization problem with a fixed path parameter, we propose an efficient fixed point algorithm. The time complexity of the path following replicator dynamic is only linear in the number of edges of a graph, thus it can analyze graphs with millions of vertices and tens of millions of edges on a common PC in a few minutes. Besides, it can be naturally generalized to hypergraph and graph with edges of different orders. We apply it to four important problems: maximum clique problem, densest k-subgraph problem, structure fitting, and discovery of high-density regions. The extensive experimental results clearly demonstrate its advantages, in terms of robustness, scalability and flexility.
Since dense subgraphs correspond to high-density regions in the data, the evolution process of path following replicator dynamic can be considered as a shrink process of high-density regions. The task of estimating high-density regions from data samples is a fundamental problem in a number of works, such as outlier detection and cluster analysis @cite_30 @cite_34 . The advantage of our method for this task is that our method can gradually reveal the landscape of multiple high-density regions of various shape at different scales. High density regions usually represent modes of data, and in this sense, our method is also closely related to mode-finding methods, such as mean shift @cite_27 .
@cite_30 @cite_27 @cite_34
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For crowded scenes, the accuracy of object-based computer vision methods declines when the images are low-resolution and objects have severe occlusions. Taking counting methods for example, almost all the recent state-of-the-art counting methods bypass explicit detection and adopt regression-based methods to directly count the objects of interest. Among regression-based methods, density map estimation, where the number of objects inside a subregion is the integral of the density map over that subregion, is especially promising because it preserves spatial information, which makes it useful for both counting and localization (detection and tracking). With the power of deep convolutional neural networks (CNNs) the counting performance has improved steadily. The goal of this paper is to evaluate density maps generated by density estimation methods on a variety of crowd analysis tasks, including counting, detection, and tracking. Most existing CNN methods produce density maps with resolution that is smaller than the original images, due to the downsample strides in the convolution pooling operations. To produce an original-resolution density map, we also evaluate a classical CNN that uses a sliding window regressor to predict the density for every pixel in the image. We also consider a fully convolutional (FCNN) adaptation, with skip connections from lower convolutional layers to compensate for loss in spatial information during upsampling. In our experiments, we found that the lower-resolution density maps sometimes have better counting performance. In contrast, the original-resolution density maps improved localization tasks, such as detection and tracking, compared to bilinear upsampling the lower-resolution density maps. Finally, we also propose several metrics for measuring the quality of a density map, and relate them to experiment results on counting and localization. For crowded scenes, the accuracy of object-based computer vision methods declines when the images are low-resolution and objects have severe occlusions. Taking counting methods for example, almost all the recent state-of-the-art counting methods bypass explicit detection and adopt regression-based methods to directly count the objects of interest. Among regression-based methods, density map estimation, where the number of objects inside a subregion is the integral of the density map over that subregion, is especially promising because it preserves spatial information, which makes it useful for both counting and localization (detection and tracking). With the power of deep convolutional neural networks (CNNs) the counting performance has improved steadily. The goal of this paper is to evaluate density maps generated by density estimation methods on a variety of crowd analysis tasks, including counting, detection, and tracking. Most existing CNN methods produce density maps with resolution that is smaller than the original images, due to the downsample strides in the convolution pooling operations. To produce an original-resolution density map, we also evaluate a classical CNN that uses a sliding window regressor to predict the density for every pixel in the image. We also consider a fully convolutional adaptation, with skip connections from lower convolutional layers to compensate for loss in spatial information during upsampling. In our experiments, we found that the lower-resolution density maps sometimes have better counting performance. In contrast, the original-resolution density maps improved localization tasks, such as detection and tracking, compared with bilinear upsampling the lower-resolution density maps. Finally, we also propose several metrics for measuring the quality of a density map, and relate them to experiment results on counting and localization. Abstract In many microscopy applications the images may contain both regions of low and high cell densities corresponding to different tissues or colonies at different stages of growth. This poses a challenge to most previously developed automated cell detection and counting methods, which are designed to handle either the low-density scenario (through cell detection) or the high-density scenario (through density estimation or texture analysis). The objective of this work is to detect all the instances of an object of interest in microscopy images. The instances may be partially overlapping and clustered. To this end we introduce a tree-structured discrete graphical model that is used to select and label a set of non-overlapping regions in the image by a global optimization of a classification score. Each region is labeled with the number of instances it contains – for example regions can be selected that contain two or three object instances, by defining separate classes for tuples of objects in the detection process. We show that this formulation can be learned within the structured output SVM framework and that the inference in such a model can be accomplished using dynamic programming on a tree structured region graph. Furthermore, the learning only requires weak annotations – a dot on each instance. The candidate regions for the selection are obtained as extremal region of a surface computed from the microscopy image, and we show that the performance of the model can be improved by considering a proxy problem for learning the surface that allows better selection of the extremal regions. Furthermore, we consider a number of variations for the loss function used in the structured output learning. The model is applied and evaluated over six quite disparate data sets of images covering: fluorescence microscopy, weak-fluorescence molecular images, phase contrast microscopy and histopathology images, and is shown to exceed the state of the art in performance. Purpose To present and validate a new method that formalizes a direct link between k-space and wavelet domains to apply separate undersampling and reconstruction for high- and low-spatial-frequency k-space data. Theory and Methods High- and low-spatial-frequency regions are defined in k-space based on the separation of wavelet subbands, and the conventional compressed sensing problem is transformed into one of localized k-space estimation. To better exploit wavelet-domain sparsity, compressed sensing can be used for high-spatial-frequency regions, whereas parallel imaging can be used for low-spatial-frequency regions. Fourier undersampling is also customized to better accommodate each reconstruction method: random undersampling for compressed sensing and regular undersampling for parallel imaging. Results Examples using the proposed method demonstrate successful reconstruction of both low-spatial-frequency content and fine structures in high-resolution three-dimensional breast imaging with a net acceleration of 11–12. Conclusion The proposed method improves the reconstruction accuracy of high-spatial-frequency signal content and avoids incoherent artifacts in low-spatial-frequency regions. This new formulation also reduces the reconstruction time due to the smaller problem size. Magn Reson Med 70:1306–1318, 2013. © 2012 Wiley Periodicals, Inc.