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Fixman's work in 1974 and the follow-up studies have developed a method that can factorize the inverse of mass matrix into an arithmetic combination of three sparse matrices—one of them is positive definite and needs to be further factorized by using the Cholesky decomposition or similar methods. When the molecule subjected to study is of serial chain structure, this method can achieve O ( n ) time complexity. However, for molecules with long branches, Cholesky decomposition about the corresponding positive definite matrix will introduce massive fill-in due to its nonzero structure. Although there are several methods can be used to reduce the number of fill-in, none of them could strictly guarantee for zero fill-in for all molecules according to our test, and thus cannot obtain O ( n ) time complexity by using these traditional methods. In this paper we present a new method that can guarantee for no fill-in in doing the Cholesky decomposition, which was developed based on the correlations between the mass matrix and the geometrical structure of molecules. As a result, the inverting of mass matrix will remain the O ( n ) time complexity, no matter the molecule structure has long branches or not. [ABSTRACT FROM AUTHOR]

This paper introduces a new algorithm for Boolean operations on rasterized geometric shapes that are represented with chain codes. The algorithm works in three steps. Firstly, the chain code symbols are transformed in the Hilbert space, where the overlaid chain code symbols are recognised. After that, a suitable starting cell is determined. Finally, the walk-about through the sequence of the initial chain code symbols is performed to obtain the sequence of chain code symbols representing the shape of the required Boolean operation. The algorithm is demonstrated on Freeman chain code in four directions. The time and space complexity of the proposed algorithm is linear, which was proven theoretically and confirmed by experiments. [ABSTRACT FROM AUTHOR]

In this paper, a novel crowd descriptor, termed as bilinear CD (Curl and Divergence) descriptor, is proposed based on the bilinear interaction of curl and divergence. Specifically, the curl and divergence activation maps are computed from the normalized average flow. A local curl patch and the corresponding divergence patch are cropped respectively from the activation maps. The outer product of the two local patches is defined as the bilinear CD vector. Through sliding a window on the activation maps, we can get hundreds to thousands local bilinear CD vectors. To encode them into a compact representation, fisher vector pooling and PCA algorithms are applied on the local descriptors. Experiments on the CUHK crowd dataset show that the proposed bilinear dynamics can improve the performance of video classification and retrieval by a noticeable margin when compared with the existing crowd features. [ABSTRACT FROM AUTHOR]

It is well known that sparse approximation problem is NP-hard under general dictionaries. Several algorithms have been devised and analyzed in the past decade under various assumptions on the coherence μ of the dictionary represented by an M × N matrix from which a subset of k column vectors is selected. All these results assume μ = O ( k − 1 ) . This article is an attempt to bridge the big gap between the negative result of NP-hardness under general dictionaries and the positive results under this restrictive assumption. In particular, it suggests that the aforementioned assumption might be asymptotically the best one can make to arrive at any efficient algorithmic result under well-known conjectures of complexity theory. In establishing the results, we make use of a new simple multilayered PCP which is tailored to give a matrix with small coherence combined with our reduction. [ABSTRACT FROM AUTHOR]

Counting objects is a fundamental but challenging problem. In this paper, we propose diffusion-based, geometry-free, and learning-free methodologies to count the number of objects in images. The main idea is to represent each object by a unique index value regardless of its intensity or size, and to simply count the number of index values. First, we place different vectors, refer to as seed vectors, uniformly throughout the mask image. The mask image has boundary information of the objects to be counted. Secondly, the seeds are diffused using an edge-weighted harmonic variational optimization model within each object. We propose an efficient algorithm based on an operator splitting approach and alternating direction minimization method, and theoretical analysis of this algorithm is given. An optimal solution of the model is obtained when the distributed seeds are completely diffused such that there is a unique intensity within each object, which we refer to as an index. For computational efficiency, we stop the diffusion process before a full convergence, and propose to cluster these diffused index values. We refer to this approach as Counting Objects by Diffused Index (CODI). We explore scalar and multi-dimensional seed vectors. For Scalar seeds, we use Gaussian fitting in histogram to count, while for vector seeds, we exploit a high-dimensional clustering method for the final step of counting via clustering. The proposed method is flexible even if the boundary of the object is not clear nor fully enclosed. We present counting results in various applications such as biological cells, agriculture, concert crowd, and transportation. Some comparisons with existing methods are presented. [ABSTRACT FROM AUTHOR]