The lesser the distance, the more similar will be the distributions and vice-versa. Filling the holes result in converting one distribution to another. The least amount of work done to fill the holes completely gives us the EMD. This assumes one distribution to be a mass of earth or a pile of dirt and the other to be a collection of holes in that same space. Let’s say we have two distributions A and B whose distance we want to calculate. Let’s discuss the main concept behind this. This concept was first introduced by Gaspard Monge in 1781, in the context of transportation theory ( Wikipedia). So, in this blog, we will discuss the Earthmover’s distance also known as Wasserstein metric which is more suitable for finding distance or similarity between the distributions. Thus, the definition of distance becomes less apparent when we are dealing with distributions or sets of elements rather than single elements. Is it the distance between the centers of both the houses or the nearest distance between the two houses or any other form. For instance, if I say what is the distance from your house to the neighbor’s house? Most of you will come up with a number say x meters but where is this distance came from. In a formal sense, the notion of distance is more suitable to single elements as compared to distributions. In this blog, we will discuss a more robust method for comparing the distributions known as Wasserstein metric or Earthmover’s distance. Most of the methods we discussed were highly sensitive to blurring, local deformation or color shifts. In the previous blogs, we discussed various histogram comparison methods for image retrieval.
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