# The Helmholtz Principle

Chapter
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 34)

The Helmholtz principle can be formulated two ways. The first way is commonsensical. It simply states that we do not perceive any structure in a uniform random image. In this form, the principle was first stated by Attneave [Att54]. This gestaltist was to the best of our knowledge the first scientist to publish a random noise digital image. This image was actually drawn by hand by U.S. Army privates using a random number table. In its stronger form, of which we will make great use, the Helmholtz principle states that whenever some large deviation from randomness occurs, a structure is perceived. As a commonsense statement, it states that “we immediately perceive whatever could not happen by chance”. Our aim in this chapter is to discuss several intuitive and sometimes classical examples of exceptional events and their perception. We will see how hard it can be to calculate some rather simple events. This difficulty is solved by introducing a universal variable adaptable to many detection problems, the Number of False Alarms (NFA). The NFA of an event is the expectation of the number of occurrences of this event. Expectations are much easier to compute than probabilities because they add. After we have treated three toy examples in Section 3.1, we will define in Section 3.2 what we call ε-meaningful events, namely events whose NFA is less than ε. This notion is then applied to a first realistic problem: the dot alignment detection in an image.

## Keywords

Black Pixel Universal Variable Markov Inequality Bibliographic Note Exceptional Event
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