Basic Anti-aliasing Filter (Intro to Signals & Systems)

I am trying to create a function

zaa = antialias(z);

which inputs a hi-res image z and outputs high-res anti-aliased image zaa. The system needs to compute each point of zaa[x,y]:

.

The given "high-res" image z is:

z = round(exp(-1/w.^2*((y.'-30).^2+(x-40).^2)));

I am confused on how to implement the Zaa equation into my function. Here's what I have for my function thus far:

function zaa = antialias(z)
    
zaa = zeros(size(z,1), size(z,2)); 

for nn = 1:size(z,1) % grabs the number of rows
	for mm = 1:size(z,2) % grabs number of columns
		if nn > 1 && nn < size(z,1) && mm > 1 && mm < size(z,2) 
		
            %zaa(nn,mm) = 1/4 * (z(nn-1,mm)+z(nn+1,mm)+z(nn,mm-1)+z(nn,mm+1));
            %zaa(nn,mm) = 1/4 * z(nn-1,nn+1,mm-1,mm+1);
            %zaa(nn,mm) = 1/4 * z(nn,mm);
            
            %xx = [nn-1 nn+1 nn nn];
            %yy = [mm mm mm-1 mm+1];
            %zaa(nn,mm) = 1/4 * z(xx,yy);
            
		end
	end
end
end

The commented out section includes my 4 attempts at coding in the difference equation. I'm sure you can tell I am a novice with arrays and MATLAB. I am hoping for some guidence on how to write the function appropriately. Any help would be appreciated.

Thanks!

Answer 1

[Image Analyst]

Try this:

function zaa = antialias(z)
kernel = 1 + [0,1,0; 1,0,1; 0,1,0]/4;
zaa = conv2(z, kernel, 'same');

Note: Because your kernel is not normalized, and sums to 2, your output image zaa will be, on average, twice as bright as z. To avoid that, do this

function zaa = antialias(z)
kernel = 1 + [0,1,0; 1,0,1; 0,1,0]/4;
kernel = kernel / sum(kernel(:));
zaa = conv2(z, kernel, 'same');

Then the output image will have the same average brightness as the input image.