CODEWITHSUNDEEP
c++algorithmmatrixpca
rwvaldivia
rwvaldivia

Reputation: 395

What exactly happens after run Algorithm PCA (Principal Component Analysis)

At this moment I'm working with an image processing project. But I have a conceptual questions regarding the PCA.

What exactly happens to the matrix after applying the PCA in the matrix of an image?

I does not understand reading the literature on this subject.

Given an M x N matrix, the result is an matrix M' x N' and where M'< M and N' < N and M' x N' is proporcional to M x N?

Upvotes: 1

Views: 307

Answers (2)

MSalters
MSalters

Reputation: 179779

The concept of PCA is closely related to linear algebra, which is the domain of mathematics to which matrices belong. A common way to view a matrix is as a set of vectors, a MxN matrix are just M vectors in an N-dimensional space.

Now a general concept in linear algebra is that the choice of basis vectors is pretty arbitrary. If you choose another basis, you convert your matrix by multiplying it with the old basis expressed in the new basis (an NxN dimensional matrix itself).

PCA is a method to find a basis which isn't arbitrary, but specific to your matrix. In particular, it orders base vectors by the amount in which they're present in your set of vectors. If all your vectors point roughly in the same direction, that direction will be the first basis vector. If they're all roughly in the same plane, the major basis vectors for that plane will be your first two vectors. But remember: you'll generally a full MxN basis (unless your matrix is degenerate); it's up to you to decide how many of the Components are Principal.

Now here's the real question: what exactly is "the matrix of the image"? You generally can't treat a 1024 x 768 image as a set of 1024 vectors in 768-dimensional space. Sure, you can perform the PCA operation, and you will get a 1024x768 result matrix, but what does that even mean? They're the basis vectors of your input matrix, but that output doesn't have an image meaning exactly because your input is not a set of vectors.

Upvotes: 1

xyz
xyz

Reputation: 910

I'm no expert in PCA but I'll try to explain what I understand.

After apply PCA to the matrix of an image, you get the eigenvectors of that matrix, which represents a number invariant axis of the matrix. These vectors are all orthogonal to each other.

By measuring how disperse are your original data in matrix along these vectors, you can know how are they distributed. This can be useful, for example, if you wish to perform pattern categorization based on how the data are distributed along these "axis".

Although not accurate, you can imagine PCA helps you to draw "axis" along the blob of data that presence in your matrix, where the new "origin" of the axis are the center of your data.

The best part is that the data are distributed the most along the first eigenvector, follow by the second eigenvector and so on.

I hope I did not confused you.

There are a number of good reference about PCA in quora in addition to stackoverflow too. Here are a few examples: https://www.quora.com/What-is-an-intuitive-explanation-for-PCA http://www.quora.com/How-to-explain-PCA-in-laymans-terms

Again, I'm no expert and welcome others to correct/educate both rwvaldivia and me.

Upvotes: 1

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