The claim that the best interpolation kernel for 2d images is just a product of two Lanczos kernels in the X and Y directions seems wrong to me. Rather, there is a proper 2D analog to the sinc function known as the jinc, or "Sombrero" function https://en.wikipedia.org/wiki/Sombrero_function . (This function is mathematically related to the "Airy" ringing patterns found in the optics of lenses.) Resampling with a jinc-based circular-window kernel should show a far lower intensity of ringing artifacts compared to the simple composition of two 1D Lanczos filters.
On a different point, when approximating a "brickwall" frequency spectrum an 'equiripple' pattern is generally considered desirable, and this will lead to a different kernel than the Lanczos approach does.
I don't really claim that Lanczos interpolation as presented is the "best" 2D interpolation there is. It is definitely popular though, and I couldn't find a source explaining how it is derived, so I thought it'd be an interesting topic for a blog post.
One way to think of the sinc function is as the reproducing kernel (the point evaluation functional) for the space of bandlimited functions, thought of as a reproducing kernel Hilbert space. I guess the Shannon/Nyquist theorem just kind of falls out of this.
It's fascinating how the "ringing" around the edges looks so much like JPEG compression that it took me around 10 minutes to stop ignoring it and actually start looking at the images properly
I know bicubic is similar to lanczos with less ringing and less sharp but still felt like a glaring omission not comparing it with lanczos. That said, great article and great details, I learned a lot.
Cubic interpolation is not included in the showcase since it is a family of filters rather than a single filter — most cubic filters used in practice end up looking similar to Lánczos, although probably a bit less sharp but with less ringing.
I came to say just that. Why are there no pictures comparing the two? A competent high school student could write a method that beats linear and nearest.
An interesting article, especially given the links to the YouTube interview of Cornelius Lánczos. I've used the Lánczos algorithm for years for interpolation but until now I'd not put a face and voice to his work.
It's interesting to compare Lánczos (and other) resampling algorithms in digital imaging with what's known as K-factor (aka K-rating)—a measurement in analog television for rating image quality. There are interesting similarities between the two.
An image is an image whether it's generated digitally or by analog means, so it's only to be expected that ways of measuring image quality between these two systems would have some things in common. That's done by comparing the output signal with the original image but it's not as straightforward as it seems as human perception and subjectivity get in the way.
As per article we've seen Lánczos, (sync, (sin x)/x) resampling quality is better than say nearest neighbour, Mitchell, triangle, etc. but the problem of human subjectivity remains as it's often difficult to compare image quality visually and or consistently. Analog television has long had methods of objectively evaluating images without the human factor and again the solution is mathematical, and as I'll show it has some interesting parallels with Lánczos resampling.
To determine image quality/K-factor of a television transmission system an electronic test signal replaces the subjective image and it's measured for distortion products after it exits the system. The mathematical parameters of this test signal are carefully defined to detect distortions and artifacts that are most noticeable to the human eye.
The test signal consists of a sine-squared pulse of specified duration followed by bar (a square wave with a transient response the same as the pulse). The K-factor is determined by measuring the deviation in the pulse and bar risetimes together with generated artifacts such as ringing and under and or overshoot. As the Pulse & Bar is a precision test signal input/output comparisons aren't necessary, thus a single measurement simplifies testing.
Unfortunately, this ref. is behind a firewall: Macdiarmid, I.F. and Phillips, B. 'A Pulse & Bar Waveform Generator for Testing Television Links.' Proc I.E.E. Vol. 105, Part B, p.) 440. 1958.
That was a very nice and detailed description of the Láczos filter. It gave some new perspectives that were not covered by the sources I've read earlier.
Difference with Gabors kernels (that are in our visual cortex and what the first level CNN kernels look to converge to) is the scaling factor 1/t instead of e^(-t)
The claim that the best interpolation kernel for 2d images is just a product of two Lanczos kernels in the X and Y directions seems wrong to me. Rather, there is a proper 2D analog to the sinc function known as the jinc, or "Sombrero" function https://en.wikipedia.org/wiki/Sombrero_function . (This function is mathematically related to the "Airy" ringing patterns found in the optics of lenses.) Resampling with a jinc-based circular-window kernel should show a far lower intensity of ringing artifacts compared to the simple composition of two 1D Lanczos filters.
On a different point, when approximating a "brickwall" frequency spectrum an 'equiripple' pattern is generally considered desirable, and this will lead to a different kernel than the Lanczos approach does.
(I'm the author)
I don't really claim that Lanczos interpolation as presented is the "best" 2D interpolation there is. It is definitely popular though, and I couldn't find a source explaining how it is derived, so I thought it'd be an interesting topic for a blog post.
Thank you for this article. Are the graphs images? and how were they generated?
Kind of related: https://johncostella.com/magic/ and its discussion https://news.ycombinator.com/item?id=41756205
Not only a very nice explanation, but it's also presented in such an aesthetically pleasing way! The layout, the typography, the colors..
The blog is a thing of beauty.
One way to think of the sinc function is as the reproducing kernel (the point evaluation functional) for the space of bandlimited functions, thought of as a reproducing kernel Hilbert space. I guess the Shannon/Nyquist theorem just kind of falls out of this.
It's fascinating how the "ringing" around the edges looks so much like JPEG compression that it took me around 10 minutes to stop ignoring it and actually start looking at the images properly
It's the exact same phenomenon. :-) The high frequencies are abruptly cut off in both cases.
I know bicubic is similar to lanczos with less ringing and less sharp but still felt like a glaring omission not comparing it with lanczos. That said, great article and great details, I learned a lot.
Author does explain why it in a footnote:
Cubic interpolation is not included in the showcase since it is a family of filters rather than a single filter — most cubic filters used in practice end up looking similar to Lánczos, although probably a bit less sharp but with less ringing.
So just one method with common paramters and display those.
I came to say just that. Why are there no pictures comparing the two? A competent high school student could write a method that beats linear and nearest.
An interesting article, especially given the links to the YouTube interview of Cornelius Lánczos. I've used the Lánczos algorithm for years for interpolation but until now I'd not put a face and voice to his work.
It's interesting to compare Lánczos (and other) resampling algorithms in digital imaging with what's known as K-factor (aka K-rating)—a measurement in analog television for rating image quality. There are interesting similarities between the two.
An image is an image whether it's generated digitally or by analog means, so it's only to be expected that ways of measuring image quality between these two systems would have some things in common. That's done by comparing the output signal with the original image but it's not as straightforward as it seems as human perception and subjectivity get in the way.
As per article we've seen Lánczos, (sync, (sin x)/x) resampling quality is better than say nearest neighbour, Mitchell, triangle, etc. but the problem of human subjectivity remains as it's often difficult to compare image quality visually and or consistently. Analog television has long had methods of objectively evaluating images without the human factor and again the solution is mathematical, and as I'll show it has some interesting parallels with Lánczos resampling.
To determine image quality/K-factor of a television transmission system an electronic test signal replaces the subjective image and it's measured for distortion products after it exits the system. The mathematical parameters of this test signal are carefully defined to detect distortions and artifacts that are most noticeable to the human eye.
The test signal consists of a sine-squared pulse of specified duration followed by bar (a square wave with a transient response the same as the pulse). The K-factor is determined by measuring the deviation in the pulse and bar risetimes together with generated artifacts such as ringing and under and or overshoot. As the Pulse & Bar is a precision test signal input/output comparisons aren't necessary, thus a single measurement simplifies testing.
For those interested see BBC Monograph 58 'Sine-squared Pulse and Bar Testing in Colour Television'. 1965. PDF https://www.bbc.co.uk/rd/publications/bbc_monograph_58
Unfortunately, this ref. is behind a firewall: Macdiarmid, I.F. and Phillips, B. 'A Pulse & Bar Waveform Generator for Testing Television Links.' Proc I.E.E. Vol. 105, Part B, p.) 440. 1958.
That was a very nice and detailed description of the Láczos filter. It gave some new perspectives that were not covered by the sources I've read earlier.
Difference with Gabors kernels (that are in our visual cortex and what the first level CNN kernels look to converge to) is the scaling factor 1/t instead of e^(-t)