Wavelets and Multiresolution Analysis 

Many thanks!

Wonderful!

Glad it was helpful!

Biologist here, I needed to understand image processing, your videos are very clear, thank you.

Yes, corrected @8:50

Thanks! Yes, this is a unique and invertible mapping.

for the interactive visuel contents

well, I though it should be a=2, and b=1 :D

It's incredible that you could understand these concepts without formal education, you are privileged. Definitely you should consider going to college, you would find mentorship, and you would be exposed to a great amount of ideas. With those tools, you could make a great contribution, for sure.

I would love to cover this, because then I would have a good reason to learn it better. On the list.

@@Eigensteve I would love to see this as well. I come across an article explaining that the SVD of a Hankel matrix has direct interpretable physical meanings related to space-time POD, but I find the jargons too difficult for someone who have no exposure to fluid dynamics to understand.

I read an article from University of Washington that Steve Brunton was left-handed. In the video, it appears as though he's writing with his right hand. He could have trained himself to write in reverse with his right hand but I think it's also possible for the video to be digitally altered (mirrored) after recording. Imagine you were standing behind him as he drew a straight line from left to right (i.e., the normal direction English is written in) onto a transparent (glass) surface. You'd observe that the line was drawn left to right. Now if you were to stand behind the glass surface, you'd observe the line drawn right to left. Imagine you magically turned into a camera and recorded what you saw behind the glass surface. With the video recorded, you're then able to 'mirror' every frame and end up with this recording. Steve would still be writing comfortably with his left hand. You can verify this with an experiment. Use a transparent plastic container, sharpie, and a smartphone (with a front-facing camera). Place the camera inside the plastic container and have it record video (as you would if you were taking a video selfie). Write normally on the outside wall of container that's being recorded. If you look at the video, you'd see that the "world" looks normal but the text you wrote looks reversed. Now, the i-phone allows you to play back your video with each frame flipped across a vertical. If you do that, the *text* appears normal but the "world" looks reversed (which is fine -- we don't care that Steve is left handed!).

@@MinhTran-wn1ri you are totally right

Dr Brunton's hair parts right in real life (the Clark Kent look). But in these videos, his hair parts left (like Superman). Does this answer the question?

I think this is a complicated question. It's probably similar to why it's good to find a basis where the basis vectors are orthogonal to one another in linear algebra. A non-orthogonal set can still span the space, but there are advantages in being able to interpret the coordinates if they are independent of one of another. In spectral techniques this may lead to functions that express independent information about the frequency content of whatever data you're looking at. For the fourier transform the cosines and sines at a specific frequency are orthogonal to those at another frequency and that means the information about the frequency content at each frequency is independent of the frequency content at any other frequency. I don't know if that's a great answer, but that's my understanding of it.

The sines and cosines are an orthogonal basis under an inner product defined by integrals. This is great because it allows us to use Graham Schmidt to find the coefficients of basis vectors! If they weren’t orthogonal, we wouldn’t be able to do this

if it was mirrored you would see him writing from left to right, so I think he actually writes backwards, which is crazy *o*