How to detect the fuzzy edge in this image?
bacteria segmentation is a broad field with many approaches. here's an example. For your spesific image, I think you can do the following tricks:

d=double(imread('bac.png'));
d=d./max(d(:));

ws=7; % window size 
mean_d=imfilter(d,fspecial('average',ws),'replicate'); % this is a local mean operation 
d2=d-mean_d;

d3 = conv2(d2.*(d2&gt;0), fspecial('laplacian',0),'same');
subplot(3,1,1);imagesc(d);colorbar
subplot(3,1,2);imagesc(d2);colorbar
subplot(3,1,3);imagesc(d3);colorbar


you can take it from there...