Topological Data Analysis is Superior to Pixel Based Methods i.e. Deep Learning

Aditya Yadav
Oct 30, 2019 · 4 min read

Lets consider TWO Scenarios

(1) Multi-Dimensional Data

(2) Vision/Speech/Touch

Computers See Points…

Humans See Shapes…

How do we go from Points to Shapes, And then to Reasoning about Shapes? Hopefully intelligently.

The Brain

The Brain is a Huge Huge Pattern Matching System. This is somewhat how our senses i.e. vision, hearing etc. works

And this is how Deep Learning based Vision works…

Which does help us solve some problems but NOT that of Intelligence or Intelligent Vision, Hearing etc.

It is Evident that Intelligence/Intelligent Vision is…

1.Translation Invariant

2.Scale Invariant

3.Rotation Invariant

4.Deformation Invariant

5.Color Invariant

6.Texture Invariant

7.Luminosity Invariant…

in 2D and 3D space atleast. And in N-Dim space in general (Mathematically)

Traditional Image Processing & Recognition Attempts

Many shape descriptors have been created so far. Some of them are very generic, like moment invariants or Fourier descriptors, while the others relate to specific shape characteristics, e.g. rectilinearity, tortuosity, orientability, convexity, etc.

Moment Invariants

Invariant Moments

Invariant Moments were Scale + Translation + Rotation Invariants and were used in early methods of Pattern Recognition since 1990’s or even earlier. Used with “Rigid” objects of “Fixed” Shapes.

Fourier Descriptors

Fourier descriptors are an interesting method for modeling 2D shapes that are described as closed contours. Unlike polylines or splines, which are explicit and local descriptions of the contour, Fourier descriptors are global shape representations, i. e., each component stands for a particular characteristic of the entire shape. If one component is changed, the whole shape will change. The advantage is that it is possible to capture coarse shape properties with only a few numeric values, and the level of detail can be increased (or decreased) by adding (or removing) descriptor elements. For example “cartesian” (or “elliptical”) Fourier descriptors, can be used to model the shape of closed 2D contours and they can be adapted to compare shapes in a translation-, scale- and rotation-invariant fashion.

Projective Invariants

Our discussion so far has focused on comparing 2-dimensional shapes. One area of significant research involves comparing 2-dimensional projections of 3-dimensional shapes. In other words, the question isn’t “do these 2-dimensional shapes match?” but “could these 2-dimensional shapes be different projections of the same 3-dimensional shape?” The key to this research is the notion of projective invariants — measurable properties that don’t change when projected.

Nearly all forms of invariants, whether projective or not, involve ratios. By constructing ratios, even if one quantity changes under a tranformation, as long as another quantity changes proportionally under the same transformation, their ratio stays the same.

What is Topological Data Analysis?

Topological Data Analysis (TDA) is an approach to the analysis of datasets using techniques from topology. … TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise.

What is Topology?

Topology is the study of shape of objects subject to smooth operations such as stretching and twisting. However, topology is not concerned with the exact geometric description of an object: it does not bother itself with how many edges an object has, or whether an object is round or oval in shape. Topology tries to describe an object in more general terms — how many distinct components does an object have, how many holes on its surface, or how many cavities.

Topology is independent of coordinate systems, coordinates or distances.

What is Algebraic Topology?

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

To Be Continued…

In the future articles we are going to see applications of Topology in

  • [Topological] Data Analysis using Python.
  • [Topological] Quantum Computing
  • Topology + Deep/Machine Learning

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