Question

I’ve been reading up a bit on anti-aliasing and it seems to make sense, but there is one thing I’m not too sure of. How exactly do you find the maximum frequency of a signal (in the context of graphics).

I realize there’s more than one case so I assume there is more than one answer. But first let me state a simple algorithm that I think would represent maximum frequency so someone can tell me if I’m conceptualizing it the wrong way.

Let’s say it’s for a 1 dimensional,finite, and greyscale image (in pixels). Am I correct in assuming you could simply scan the entire pixel line (in the spatial domain) looking for a for the minimum oscillation and the inverse of that smallest oscillation would be the maximum frequency?

Ex values {23,26,28,22,48,49,51,49}

Frequency:Pertaining to Set {}

(1/2) = .5 : {28,22}

(1/4) = .25 : {22,48,49,51}

So would .5 be the maximum frequency?

And what would be the ideal way to calculate this for a similar pixel line as the one above?

And on a more theoretical note, what if your sampling input was infinite (more like the real world)? Would a valid process be sort of like:

Predetermine a decent interval for point sampling Determine max frequency from point sampling while(2*maxFrequency >  pointSamplingInterval) { pointSamplingInterval*=2 Redetermine maxFrequency from point sampling (with new interval) } 

I know these algorithms are fraught with inefficiencies, so what are some of the preferred ways? (Not looking for something crazy-optimized, just fundamentally better concepts)

Answer 1

The proper way to approach this is using a Fourier Transform (in practice, an FFT,or fast fourier transform)

The theory works as follows: if you have an set of pixels with color/grayscale, then we can say that the image is represented by pixels in the ‘spatial domain’; that is, each individual number specifies the image at a particular spatial location.

However, what we really want is a representation of the image in the ‘frequency domain’. Instead of each individual number specifying each pixel, each number represents the amplitude of a particular frequency in the image as a whole.

The tool which converts from the ‘spatial domain’ to the ‘frequency domain’ is the Fourier Transform. The output of the FT will be a sequence of numbers specifying the relative contribution of different frequencies.

In order to find the maximum frequency, you perform the FT, and look at the amplitudes that you get for the high frequencies – then it is just a matter of searching from the highest frequency down until you hit your ‘minimum significant amplitude’ threshold.

You can code your own FFT, but it is much easier in practice to use a pre-packaged library such as FFTW