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.
翻译:计数对象是一个根本性但具有挑战性的问题。 在本文中, 我们提出基于扩散的、 无几何的和不学习的方法来计算图像中对象的数量。 主要的想法是用一个独特的索引值来代表每个对象, 不论其强度或大小如何, 并简单计数索引值。 首先, 我们设置不同的矢量, 称为种子矢量, 在整个遮罩图像中统一使用。 遮罩图像含有对象的边界信息 。 第二, 种子在每一个对象中使用边比的调和变色优化模型来传播。 我们提出一个有效的算法, 以操作员分离方法和交替方向最小化应用法为基础, 并给出了这个算法的理论分析。 当分布种子完全分散时, 将每个对象以一个独特的密度来代表每个对象, 我们称之为一个缩放矢量的矢量。 对于计算结果, 我们建议将这些分散的指数作为计数对象( CODI ) 。 我们用某些矢量值和多维度的计算方法来进行计算, 我们用高维的种子迁移法来探索这个模型, 。 使用高空的矢量计算, 将种子的计算方法作为我们使用高位的 。