How to connect broken lines, but no enlarge line width?

I need to fill rectangle(run fill holes), but the broken lines in rectangle stop me to do this. due to the high density rectangles in figure, so I want to connect broken lines, but no enlarge line width. Could you help me?

I have tryed to do dilate/erode/open/close mask operations, but they all can't meet my request.

Before After

Answer 1

[埃博拉酱]

function Image=FixBroken(Image,GapSize)
try
	Image=imread(Image);
catch ME
	if ME.identifier~="MATLAB:imagesci:imread:badImageSourceDatatype"
		ME.rethrow;
	end
end
Image=imbinarize(rgb2gray(Image));
Image=imclose(Image, strel('line', GapSize, 0)) | imclose(Image, strel('line', GapSize, 90));

Answer 2

[Image Analyst]

Try this to completely fill the rectangle as you asked (untested - post before image separately if you want me to test it):

after = bwconvhull(before, 'union');