Computer Vision Primer: beginner's guide to image analysis, data analysis, related mathematics (calculus, topology, linear algebra), image analysis software, and applications in sciences and engineering.

From Computer Vision Primer

Topology is usually defined as the science of the spacial properties that preserved under continuous transformations. So that you can bend, stretch and shrink etc but not tear or glue. This might be the only way to capture the essence in a single sentence but this sentence is meaningless to a person who knows nothing about topology. And the question "Why do we need to study topology?" still remains.

So, we start with examples instead. They come from three seemingly unrelated areas: computer vision, cosmology, and data analysis.

Computer vision

In industrial settings one might need to consider the integrity of objects being manufactured.

The first question may be: this is a bolt holding two things together, does it still or is there a crack in it?

Could be a bone too...

The second question may be: this material is supposed to hold liquid, is it water tight or is there leakage?

In other words: does it hold water?

The third question may be: to be strong this alloy is supposed to be solid, is it or are there air bubbles?

The opposite question is: does it hold air?

Observe that we consider here three different kinds of integrity as there may be a crack but no hole or vice versa etc.

We can describe these situations informally as:

• there are cuts in the object,
• there are tunnels,
• there are voids.

These three types of "damage" correspond to cycles of dimensions 0, 1, and 2 respectively. This is why:

• The simplest object with a "cut" is two points and points are 0-dimensional.
• The simplest object with a "tunnel" is a circle and curves are 1-dimensional.
• The simplest object with a "void" is a sphere and surfaces are 2-dimensional.

More on dimension. More on topology in computer vision: Topological features of images.

Cosmology

What is the shape of the universe? What is the topology? Does it have "cuts", "tunnels", or "voids"?

We are not living in the flat world of Euclid and Newton - we are living in the curved world of Riemann and Einstein. But if the universe curves and bends, does it close on itself? Maybe. Remember a similar discovery about the Earth that was made a long time ago?

So, it may be possible to travel on a straight line and arrive to the starting point from the opposite direction. Or the light of the sun may come from another direction as if from a distant star.

Such an event would provide us with important topological information. Observe now that there are no other ways to acquire such information. Unlike with objects we consider in the previous section, we can't step aside and take a view of the universe!

Since we can't see "cuts", "tunnels", or "voids" of the universe the only way we can discover its structure is by traveling. This suggests that using cycles as a tool is not just a matter of convenience but necessity.

In addition, as the universe is 3-dimensional, it might have 3-dimensional topological features. We can't even imagine what they look like. And we have to deal with even higher dimensions, below.

Data analysis

Datasets live outside of our tangible, physical, 3-dimensional world.

Suppose we have conducted 1000 experiments with a set of 100 various measurements in each. Then each experiment is a vector of dimension 100. The result is a point cloud of 1000 points in the 100-dimensional space.

It is impossible to visualize this data as any representation that one can see is limited to dimension 3 (by using colors one gets 6, time - 7). Yet we still need to answer the same questions about the object behind the point cloud: is it one piece or more ("clustering")? Is there a tunnel or a void? And what about possible 100-dimensional topological features?

Even more important than these "global" properties may be the local topology of the data. For example, in both of the images above the datasets are 3-dimensional but what's behind is 2-dimensional (surfaces). This is called dimensionality reduction.

More here: Topological data analysis.

Also Configuration spaces and robotics.

To summarize topology studies spacial properties that don't involve measuring.