johncostella

The kernels themselves should be applied in linear space, and their properties are independent of the signals to which they are (assumed linearly) applied. If the practical examples I applied them to should have had gamma compression removed before being transformed, then that's an oversight in the code that could be easily corrected. The general results won't be affected.

Yes, and if you're comfortable pulling down the code and running 'make' on any *nix system, there is an executable 'resize_image' that lets you test it out for yourself. I'm always looking out for corner cases that stress-test the differences between the algorithms, so let me know if you find any good ones.

Thanks. It was unexpected to me as well. The original "magic" kernel was really along the lines of interpolation (particularly by large factors — there were all sorts of algorithms out there, e.g. repeatedly upsizing by small factors, to avoid aliasing ands pixelation artifacts) and wasn't seeking to approximate the properties of the sinc function / rectangular low-pass filter at all. Only after really understanding in practical terms (i.e. on millions of test images) the blurring nature of the continuum generalization of that kernel did I apply sharpening, and that's when it starts to resemble the Lanczos approximations. It's really the properties around harmonics of the sampling frequency that make the big difference — and that's where the high-order zeros come into play, that the Lanczos approximations don't possess.

Ah nice, I will correct that statement, which I was sure was bound to have a counterargument. :)

Nice — it did look to be a more general class of interpolating functions, and that does indeed look to be the case. Interesting!

They are actually pretty good — I believe much better than truncated sinc (likely because of the discontinuity in first derivative of the latter), but I haven't done that analysis myself.

Granted, it's a bad example for a Security Engineer to not offer HTTPS. Unfortunately I use a provider that doesn't make it easy at all. I will get there eventually.

The paper includes examples of Magic Kernel Sharp 3 (the original), 4, 5, and 6. The C code allows you to use any value 'a' of these "generations," and any version 'v' of the Sharp kernel listed in the paper (up to a = 6, v = 7), for any case you want to try. I may add more versions later.

The Sharp+ is just extra sharpening. It's included in the executable, but isn't fundamentally anything more than extra sharpening over what tries to be as-close-to-ideal resizing as possible.

I generated bicubic and Lanczos versions in 2006. It was stated that the Lanczos one evidently came from a version of GIMP that was buggy, so I removed it a long time back. The bicubic is still there from that era. If it's wrong, I will have to redo that part of the page.

The Lanczos images in the paper should be correct.

"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.

Right. Fundamentally, Magic Kernel Sharp seeks to be an efficient but good approximation to a rectangular low-pass filter; it doesn't add information, and ideally doesn't destroy or distort it either.

Yes, I have seen other methods for guessing or dropping in extra detail. (Indeed, my old "UnBlur" method of deblurring effectively makes a best guess as to what an image looked like before it was blurred, which in the presence of noise is always a selection of one particular possible solution.) There's nothing wrong with that — it's just a different operation.

I put the Magic Kernel into the public domain in 2011, long before starting at Facebook. Yes, the Sharp step is nothing more than a three-tap sharpening filter, not at all remarkable in itself, and yes the "+" is just extra sharpening with the same filter to match existing properties; it was convenient in practical terms in that it did not require extra processing, but is not special in itself.