Use PSNR, SSIM,... for the edge. The edge can be extracted by using the norm of gradient of image. Something looks like this:

1) Image u (clean), v (noisy) => Their edge maps D(u) and D(v), D=norm(Grad(.))

2) PSNR(D(u), D(v)) or SSIM(D(u), D(v))

You also use other edge maps detection by following http://www.owlnet.rice.edu/~elec539/Projects97/morphjrks/moredge.html

Check the following link

https://in.mathworks.com/matlabcentral/fileexchange/65245-edge-preservation-indx

Use PSNR, SSIM,... for the edge. The edge can be extracted by using the norm of gradient of image. Something looks like this:

1) Image u (clean), v (noisy) => Their edge maps D(u) and D(v), D=norm(Grad(.))

2) PSNR(D(u), D(v)) or SSIM(D(u), D(v))

You also use other edge maps detection by following http://www.owlnet.rice.edu/~elec539/Projects97/morphjrks/moredge.html