Aliasing

Hi,

I'm starting a serie of posts about the different forms of aliasing. Geometry aliasing is a well known subject, however aliasing is much more than that.

Aliasing takes its evilness in one of the most fundamental notion of realtime rendering : rasterization. Let's see why.

Rasterization

When rendering an image we do it pixel by pixel, usually doing the shading only once for every pixel, and using the center of the pixel as the point to render to:

The trouble is that pixels are not points at all. Would they be infinitely small that would be a perfectly fine approach. However even in 1080p, pixels are quite big. But quite big against what ? 

* Aliasing is certainly not worst on larger TV.

* On the other hand aliasing is worst if we reduce the resolution.

Let's view the scene as a collection of varying information that we can sample at any pixel position :

* Z position

* albedo

* normal

* etc

And let's now target one of these values (albedo for example) and see it how it is sampled :

Aliasing from texture sampling

Let's say our shading will simply set the pixel's value to the color from the texture :

Seems great ! 

But what happens if the red channel changes faster ? 

Something strange is happening here ! A low frequency pattern is appearing on our pixels while the source was actually very high frequency ! Bad bad !!!!

In other words : When sampling a signal, if the sampling frequency is too low (called undersampling), we can't reconstruct it properly.. 

Here is another example (the well known moiré pattern) :

High enough sampling

Undersampled

In our case the sampling frequency is driven by our resolution, and the signal frequency is driven by the input data -> geometry, texture, texel ratio and sampling method (from point to anisotropic sampling).

That seems a complex problem, however the Nyquist-Shannon sampling theorem is here to help us: 

In short it says that we can properly reconstruct a signal if the sampling frequency is at least twice the sampled signal maximum frequency (for proper intensity reconstruction target more like 4).

Full details here : Wikipedia Nyquist-Shannon sampling theorem

To simplify, let's agree that :

* triangle will always be far bigger than a pixel

* texel ratio (texel/pixel) will neither exceed one

Then our remaining worst case scenario is a "screenspace oriented" triangle with a texel ratio of one.

If we go back to our red shading case the maximum frequency in the texture is thus less than 2 pixels. Meaning any pattern less or equal to 2 pixels can cause problem. 

In real life we will probably have cases with texel ratio > 1, triangle subpixel, and/or a complex shading. 

However all is not lost : texture filtering  will help a lot as mipmapping will soften the input signal. Finaly this form of aliasing is mostly visible on repeating patterns (for example tiled details maps).

Additional food for thought (Thanks Jean-Michel!): To be sure to reconstruct the input signal properly (without losing it's intensity) one should sample at 3 to 4 times the input signal frequency. Furthermore using nearest sampling can have the effect of undersampling even if a lot of pixels are shaded. The texture being already a discrete information itself.

Geometry undersampling

For the sake of sanity we have agreed that :

* triangle will always be far bigger than a pixel

* texel ratio will never exceed one

Unfortunatly this is not always the case.

We have seen that texture texel ratio and content (frequency) can create aliasing. However this is not limited to texture. Every input signal can create aliasing if its undersampled.

Geometry is a very good candidate, Especially now that triangle are becoming smaller and smaller with the new generation ! That's gonna be the subject of next post :)