c# - How do I achieve supersampling / anti-aliasing in pixel shaders?

I am trying to write a couple pixel shaders to apply to images similar to Photoshop effects. For example this effect:

But I noticed there is aliasing in my resulting image (the above image is a sample from the linked website).

What can I do to prevent this? I couldn't find any article that explains how to solve this. I read some techniques like rendering in a higher resolution and then downsizing the image, but I need to solve this in a pixel shader capacity.