The Catenary Curve – On the way to the Calculus of Variations

In the beginning of the last decade of the 17th century, Jacob Bernoulli proposed the problem of the catenary; in 1696, Johann Bernoulli proposed the problem of the brachistochrone; some years later, Daniel Bernoulli would suggest to Euler the right functional to solve the elastica problem. The Bernoulli’s family had a great influence in the development of the Calculus of Variations, which was formalized by Euler in his book of 1744. Here the catenary problem and its solution are described in detail.

Definition and first model.

In 1690, Jacob Bernoulli stated the following problem:

Assume you have a perfectly elastic wire (it can be deformed by the action of some force, but its shape is recovered immediately after the forces are ceased) made of a material with uniform density \lambda.  The two extremities of the wire are attached at the top of two columns of the same height. Moreover, assume that the only forces acting on the wire are tension and gravity. What is the equation described by the wire?

Continue reading

Modeling Using Priors – A guide on the conception of an image denoising model

A model that works perfectly fine in every single case. Unfortunately, models that get closer to this tend to be very complex and, as a consequence, unpractical. Sometimes the best thing is to focus on a particular case and use all the assumptions that come with it in your favor and then devise a model to solve this particular class. In the following, we are going to work through image restoration models. The concepts introduced here will be motivated using the problem of Image Denoising.

An example from politics

Try to answer the question: Who is going to be the next president of Brazil?

It could be very hard to answer, even if you are a Brazilian. Nonetheless, you can be more confident on your answer if additional pieces of information are given. For example, your guess will be different in each of the scenarios below.

Continue reading