Revisiting the Estimation of Fractal Dimension for Image Classification

Classification is a well-established use case for Machine Learning. Though textbook examples abound, standard examples include the classification of email into ham versus spam, or images of cats versus dogs.

Circa 1994, I was unaware of Machine Learning, but I did have a use case for quantitative image classification. I expect you’re familiar with those brave souls known as The Hurricane Hunters – brave because they explicitly seek to locate the eyes of hurricanes using an appropriately tricked out, military-grade aircraft. Well, these hunters aren’t the only brave souls when it comes to chasing down storms in the pursuit of atmospheric science. In an effort to better understand Atlantic storms (i.e., East Coast, North America), a few observational campaigns featured aircraft flying through blizzards at various times during Canadian winters.

In addition to standard instrumentation for atmospheric and navigational observables, these planes were tricked out in an exceptional way:

For about two-and-a-half decades, Knollenberg-type [ref 4] optical array probes have been used to render in-situ digital images of hydrometeors. Such hydrometeors are represented as a two-dimensional matrix, whose individual elements depend on the intensity of transmitted light, as these hydrometeors pass across a linear optical array of photodiodes. [ref 5]

In other words, the planes were equipped with underwing optical sensors that had the capacity to obtain in-flight images of

hydrometeor type, e.g. plates, stellar crystals, columns, spatial dendrites, capped columns, graupel, and raindrops. [refs 1,7]

(Please see the original paper for the references alluded to here.)

Even though this is hardly a problem in Big Data, a single flight might produce tens to hundreds to thousands of hydrometeor images that needed to be manually classified by atmospheric scientists. Working for a boutique consultancy focused on atmospheric science, and having excellent relationships with Environment Canada scientists who make Cloud Physics their express passion, an opportunity to automate the classification of hydrometeors presented itself.

Around this same time, I became aware of fractal geometrya visually arresting and quantitative description of nature popularized by proponents such as Benoit Mandlebrot. Whereas simple objects (e.g., lines, planes, cubes) can be associated with an integer dimension (e.g., 1, 2 and 3, respectively), objects in nature (e.g., a coastline, a cloud outline) can be better characterized by a fractional dimension – a real-valued fractal dimension that lies between the integer value for a line (i.e., 1) and the two-dimensional (i.e., 2) value for a plane.

Armed with an approach for estimating fractal dimension then, my colleagues and I sought to classify hydrometeors based on their subtle to significant geometrical expressions. Although the idea was appealing in principle, the outcome on a per-hydrometeor basis was a single, scalar result that attempted to capture geometrical uniqueness. In isolation, this approach was simply not enough to deliver an automated scheme for quantitatively classifying hydrometeors.

I well recall some of the friendly conversations I had with my scientific and engineering peers who attended the conference at Montreal’s Ecole Polytechnique. Essentially, the advice I was given, was to regard the work I’d done as a single dimension of the hydrometeor classification problem. What I really needed to do was develop additional dimensions for classifying hydrometeors. With enough dimensions then, the resulting multidimensional classification scheme would be likely to have a much-better chance of delivering the automated solution sought by the atmospheric scientists.

In my research, fractal dimensions were estimated using various algorithms; they were not learned. However, they could be – as is clear from the efforts of others (e.g., the prediction of fractal dimension via Machine Learning). And though my pursuit of such a suggestion will have to wait for a subsequent research effort, a learned approach might allow for the introduction of much more of a multidimensional scheme for quantitative classification of hydrometeors via Machine Learning. Of course, from the hindsight of 2018, there are a number possibilities for quantitative classification via Machine Learning – possibilities that I fully expect would result in more useful outcomes.

Whereas fractals don’t receive as much attention these days as they once did, and certainly not anything close to the deserved hype that seems to pervade most discussions of Machine Learning, there may still be some value in incorporating their ability to quantify geometry into algorithms for Machine Learning. From a very different perspective, it might be interesting to see if the architecture of deep neural networks can be characterized through an estimation of their fractal dimension – if only to tease out geometrical similarities that might be otherwise completely obscured.

While I, or (hopefully) others, ponder such thoughts, there is no denying the stunning expression of the fractal geometry of nature that fractals have rendered visual.

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