  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.
