# Math 21a Summer 2020

## Multivariable Calculus

# Math of the Pro Max

The Go Pro max is a**omni vision camera**. For more, see the Structure from Motion project from 2007 and the our Omnivis project from 2009 or some Slides from July 2007 from a talk at Colby. The math of 360 photography is pretty cool. What is nice about the Go Pro Max is that it gives the photographer tools to navigate the camera parameters. The math is pretty simple: Place the picture on a sphere, place a camera inside the sphere and give a sector to record. Now, you can move around inside the sphere and simultaneously also change the camera angle. This allows for pretty cool shots: Here were some examples recorded at the end of last week:

^{2}= Q. The later property is borrowed from projections in linear algebra. The point is however (and that happens in particular in 360 degree computer vision) that the map is non-linear. The image of Q is the

**retina**of the camera. In the case of the Go Pro max, the camera first projects three dimensional space orthogonally onto a sphere S=Q(N). This produces a picture on a sphere S. Now, you can place and orient yourself inside the sphere and record part of the retina. This is a second camera operation. In some sense, the camera has rebuild a 3D environment of the surrounding and then allows you to fly around in side that space. It is very simple but effective. Here are some shots from the two videos. (Click for a large 4000 pixel picture):