Edit: If you want to do even better than the tensor product Lanczos filtering, you can do a filter based on Euclidean distance. Make sure that you work in linear (non-gamma-adjusted) color space.

A couple more resources to read: http://www.imagemagick.org/Usage/filter/#cylindrical http://www.imagemagick.org/Usage/filter/nicolas/

Edit 2: really weird that this is getting downvoted. There’s not really anything to dispute here. It is straight-forward to show that the Lanczos method has objectively better output. Moreover, Alvy Ray Smith’s paper is a classic which anyone interested in image processing should read.

Pixels are often not representative of little squares (rectangles). However, they are more likely to be representative of little rectangles now than they were in 1995, now that most photographs are captured on Bayer arrays of square sensors, most computer-generated images are rendered as an average of point samples weighted according to estimated coverage of a square pixel, and everything is displayed on LCD or OLED panels with square pixels comprised of rectangular subpixels. If you take a random JPEG or PNG off the Internet, interpreting the pixel data as point samples will usually be less accurate than interpreting them as integrated over a rectangle. Interpreting the data as integrated over a gaussian distribution is also less realistic than the rectangle interpretation. Doing image processing in a point sample or gaussian context is certainly useful, but it's definitely not more fundamentally right than the little square model. Historically, that context was at best a neutral choice that was equally unrealistic no matter what kind of hardware you were working with, but mathematically convenient.

The paper's arguments about coordinate systems (whether the y-coordinate should grow toward the top of the screen or the bottom, and whether pixels should be centered on half-integer points) are also a waste of time for the modern reader.

Whether pixels are on the half-integer points is a completely arbitrary decision, unless you're trying to mix raster and vector graphics. Then the correct solution will become obvious.

[1] Most sensors are bayer patterns so some extra considerations apply to color resolution

If you want to do the correct thing from a signal processing perspective, you should upsample your images with a square-pixel filter until the Nyquist frequency is below the limit of human vision first. Then you can do your operations on the pixels as point samples before downsampling again with a square-pixel filter.

In practice most images go through multiple layers of processing, physical sensor and display pixels come in all kinds of wacky shapes (and as you point out have channels offset by different amounts, etc.). It’s usually better to treat pixels as point samples for generic intermediate processing because you have no idea what kind of source an image comes from, and you have no idea what someone down the line is going to do with your image before sending a final signal to the display hardware. Creating synthetic images by integrating over little rectangles produces markedly inferior results to applying some real resampling filter to the samples, but gets done by computer games etc. because it’s computationally cheap, with the hope that there are enough pixels moving fast enough that someone won’t notice the artifacts.

e.g. http://hhoppe.com/filtering.pdf

This may be true in terms of the sheer amount of GPU rasterized imagery.

I wrote the pixel filtering code currently used in one of the major production-quality film renderers. I can tell you that it uses the classical approach of treating pixels as point samples. Notionally, it convolves irregularly placed camera samples with a reconstruction filter to create a continuous function which is then point sampled uniformly along the half-integer grid to produce the rendered image.

Regarding the squarish pixels on the shown screen, I like to view them as just an analog convolution of those discrete samples so that the photoreceptors in our eyes can resample them.

You don’t have an email address in your HN profile. Is there one folks trying to contact you should prefer (before I hunt around the internet looking for one)?

And no, intentionally not patented.

The real point of the paper is that a pixel is a sample in a band-limited signal, not something that covers area. That's still just as true today, no matter what camera or display, no matter what pixel shape you're using. The point behind the paper still stands, even if the shape turns out to be a square, so we shouldn't get too hung up on the title and language railing against a square specifically.

While true that display pixels are more square today than when it was written, that's only one minor piece of the puzzle. Because we're talking about image resizing, there are multiple separate filters to consider, and for resizing it would be bad to treat pixels as squares even if you can.

If a camera's pixels are little squares, and we want to sample and then resize that image, our choice of resize filter needs to account for the little squares. We can't use a lanczos filter at all, we'd have to use something else entirely.

The big problem you have is that a sampled signal is band limited, and we treat them as perfectly band-limited. We have a body of knowledge about how to use and reason around perfectly band-limited signals, we don't have a strong image resizing theory for sampled data that is made of high-frequency samples.

If you don't convert to an ideal band-limited signal during initial sampling, then you'd have to keep the kernel shape with the image as some kind of meta data, and you'd have to use that during image resizes. If we don't have a perfectly band-limited signal, then our resize filter will always be larger than the ideal resize filter, and resizes with square pixels will take longer than resizes with band-limited point samples.

> The paper's arguments about coordinate systems are also a waste of time for the modern reader.

I'm curious why? These are still issues if you write a ray tracer, or if you mix DOM and WebGL in the same app. The paper was written for what was the SIGGRAPH going audience -- professors and Ph.D. Students -- at the time, which were all graphics researchers just learning about signal processing theory for the first time. Graphics textbooks today still cover Y-up vs Y-down for images and 0.5 offsets for pixels.

Or, in the case of fonts, they're rendered by computing the exact area coverage of the polygon over…a square representing each pixel.

If you're downsampling a line-art image, for example, you may actually be better off with some variation of the box filter. The negative lobes on the Lanczos filter can induce objectionable ringing because line-art tends to be full of what are basically step functions. It's a similar issue as with strong mosquito artifacts on JPEG compressed line-art.

A filter function that is everywhere non-negative, such as the box filter or a Gaussian can't suffer from this problem. Of the two, the box filter will give you sharper results but of course it doesn't antialias as well, either.

You have reasonable behavior along the orthogonal dimensions, but you're introducing a sqrt(2) stretch factor along diagonals.

The stretch factor along diagonals will be the same for any separable filter will it not?

My own pipe-dream is that we would use a triangular grid (if you like, the Voronoi cells here are hexagonal pixels) for intermediate image representations. This is more spatially efficient and has nicer 2-dimensionnal frequency response than a square grid, and displays are so heterogeneous nowadays that we need to do some amount of resampling for output pretty much all the time anyway, and our GPUs are getting fast enough that resampling at high quality from a hexagon grid to a square grid for output should add only relatively cheap overhead.

Faster depends on implementation details, but from the looks of it, to implement pixel mixing, you either have to generate your convolution kernel dynamically, or chop some source pixels into potentially 4 separate pieces? I would guess that it's easier to optimize a static kernel than to make a dynamic kernel faster than a static one. But I'm not entirely sure how fast pixel mixing could be made.

> It can be confused with a box filter or with linear interpolation, but it is not the same as either of them... [pixel mixing] Treats pixels as if they were little squares, which gets on some experts’ nerves.

This is a funny way of putting it. Pixel mixing is definitely using a box filter, just slightly differently than what people normally call box filter resizing. The author even says that later: "Another way to think of pixel mixing is as the integral of a nearest-neighbor function." Pixel mixing as described here is clipping the source image under the box filter rather than using a static kernel. The reason that a box filter isn't ideal is well understood. It's because the filter itself has high frequencies in it. This is the reason that the quality of pixel mixing resizes is low.

One of the benefits of a box filter is that applied multiple times, it becomes a better filter, and approximates a Gaussian after several times. I'm not sure but I would bet that clipping to the exact filter boundary actually prevents you from being able to do that with pixel mixing.

Did you call it pixel mixing in the 80s? I've been doing image resizing for decades and never heard the term before tonight. I have implemented a couple of very similar algorithms to pixel mixing for CG films, once in a shader and once for antialiasing in a particle renderer.

I hadn't either. I just searched for a name for the algorithm I also came up with independently. The imagemagick docs had a particularly comprehensive discussion on resizing options and linked this page with this naming for the algorithm.

I worked on a major application years later that was calling it bilinear, until someone pointed out that bilinear had a completely different definition. I think we renamed it Weighted Average.

Obviously, your implementation could be further optimized, but quality drawbacks will remain the same.

On the quality drawbacks I'd have to do some more checking. This algorithm is closest to what the image would be if you had a camera with that native sensor size. The standard filtering approach may very well have plenty of cases where it produces better output but it can also cause issues so I wonder if this isn't a conservative solution.

I've implemented it generally by weighing the pixels by the actual area overlapping so border pixels get weights <1.0 for non power of two reductions. Haven't seen any artifacts from it.