This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

Dimension

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From a discussion board: "Why is it that you Americans are so two dimensional? Every possible idea for social reform must either be the republican or democrat position. If it can't be pigeon-holed into one of those viewpoints, it's not worth discussion. If it can, it quickly devolves into some form of partisan debate. Your two party system is killing you in every area, and you don't even know it."

This should become a mathematical discussion at some point...

1 1D

The Roomba has vision however rudimentary. It does not detect vertical changes, so it is fair to say that its vision is 1-dimensional. Observe that the parameter corresponding to this dimension is binary - it touches an obstacle or it does not. Another 1D vision system is radar (with continuous parameter).

2 2D

Photography provides 2D vision.

Here's a way to define 3D: “I can see the object AND I know how far it is”. The object is a 2D picture and the distance (depth) is the 3rd dimension. Without that, it’s not 3D. It does not matter whether the picture is curved as in panoramic shots.

3 Conversion of 2D to 3D

Generally, conversion of a single 2D image into 3D does not work. The simple reason is the lack of information.

To capture a 3D image you need a 3D camera.

What is a 3D camera? Any camera takes 2D pictures, so all you need to add is the 3rd dimension.

Time could be that, so a video camera is a 3D camera.

Or you could combine several cameras in a row - that row is the 3rd dimension (in fact just two cameras will do).

In either case, you can find the distance via stereo vision.

Or you could simply add to the camera a distance measuring device such as radar, etc.

Calibration can also solve the problem of 3rd dimension. The solution however is only as complete as your collection of objects of known size. Imagine walking through a forest...

Of course, here we refer to spatial dimensions of images. Time, color, the spectrum may be understood as image parameters.

4 2D analysis of 3D scenes

It is easy to assume that you can analyze a given image as 2D.

Spheres.jpg
Canc.JPG
Human red bc.jpg

Often 3D is crucial. For example, images may contain:

  • occluded objects,
  • objects well lit on one side and very dark on the other.

Better are: