of the Magic Kernel Sharp algorithm, and discovered that it possesses high-order zeros at multiples of the sampling frequency, which explains why it produces such amazingly clear results: these high-order zeros greatly suppress any potential aliasing artifacts that might otherwise appear at low frequencies..."
[...]
>"Following that work, I analytically derived the
Fourier transform
of the Magic Kernel in closed form, and found, incredulously, that
it is simply the cube of the sinc function.
This implies that the Magic Kernel is just the
rectangular window function convolved with itself twice
-which, in retrospect, is completely obvious. This observation, together with a precise definition of the requirement of the Sharp kernel, allowed me to obtain an analytical expression for the exact Sharp kernel, and hence also for the exact Magic Kernel Sharp kernel, which I recognized is just the third in a sequence of fundamental resizing kernels. These findings allowed me to explicitly show why Magic Kernel Sharp is superior to any of the Lanczos kernels. It also allowed me to derive further members of this fundamental sequence of kernels, in particular the sixth member, which has the same computational efficiency as Lanczos-3, but has far superior properties."
PDS: My thoughts: Absolutely fascinating! (Also, that the the "Magic Kernel is just the rectangular window function convolved with itself twice" -- was not obvious to me, so it's good you included that!)
"Not obvious": yes, I was originally writing up the paper just to contain the derivation and results of the Fourier transform calculation — whatever it turned out to be — and was dumbstruck when it turned out to be sinc^3 x. The original title was just "The Fourier Transform of Magic Kernel Sharp."
It all turned out to be more interesting than a boring calculation. Some of it also doesn't quite flow properly because the story kept changing as the mystery unfolded, but I think it's close enough to summarize it all.
I also realized while writing it that the original doubling "magic" kernel (the 1/4, 3/4) was just the linear interpolation kernel for a factor of 2 with "tiled" positions, which I added to the page after writing the paper. I have since seen that others had previously made that comment (which I missed, but they are completely right). It's interesting that the infinitely repeated application of this "second" order kernel, i.e. what I now call m_2(x), originally led me (i.e. in 2011) to the "next" one, i.e. m_3(x). I still haven't fully sorted through how that works, but it's lucky that it did.
I'm always curious when/where/how we see Fourier Transforms of any sort -- because we see them in so many diverse fields, and in so much diverse Mathematics to consider them almost ubiquitous and fundamental for any meaningful understanding in many areas of Physics...
Fourier properties
of the Magic Kernel Sharp algorithm, and discovered that it possesses high-order zeros at multiples of the sampling frequency, which explains why it produces such amazingly clear results: these high-order zeros greatly suppress any potential aliasing artifacts that might otherwise appear at low frequencies..."
[...]
>"Following that work, I analytically derived the
Fourier transform
of the Magic Kernel in closed form, and found, incredulously, that
it is simply the cube of the sinc function.
This implies that the Magic Kernel is just the
rectangular window function convolved with itself twice
-which, in retrospect, is completely obvious. This observation, together with a precise definition of the requirement of the Sharp kernel, allowed me to obtain an analytical expression for the exact Sharp kernel, and hence also for the exact Magic Kernel Sharp kernel, which I recognized is just the third in a sequence of fundamental resizing kernels. These findings allowed me to explicitly show why Magic Kernel Sharp is superior to any of the Lanczos kernels. It also allowed me to derive further members of this fundamental sequence of kernels, in particular the sixth member, which has the same computational efficiency as Lanczos-3, but has far superior properties."
Paper: http://www.johncostella.com/magic/mks.pdf
PDS: My thoughts: Absolutely fascinating! (Also, that the the "Magic Kernel is just the rectangular window function convolved with itself twice" -- was not obvious to me, so it's good you included that!)