For comparisons, quantitative measures like what @Alex I explained is best. To do so, you need to define what is "correct" with a ground truth set and a way to consistently determine if a given image is correct or on a more granular level, how correct (some number like a percentage) it is. @Alex I gave a way to do that.
Another option that is often used in graphics research where there is no ground truth is user studies. Usually less desirable as they are time consuming and often more costly. However, if it is a qualitative improvement that you are after or if a quantitative measurement is just too hard to do, a user study is an appropriate solution.
When I mean user study I mean to poll people on how well a result is given the input image. You could give them a scale to rate things on and randomly give them samples from both your results and the results of another algorithm
And of course, if you still want more ideas, be sure to check out edge detection papers to see how they measured their results (I'd actually look here first as they've already gone through this same process and determined what was best for them: google scholar).