You can use many of the available morphological operations to acheive your taget image. In your case, I think image erosion will help in removing the noise. You can try the following:
% Read the image
imageData = imread('image1.png');
BW = bwareaopen(rgb2gray(imageData),2);
se = strel('disk',2);
I = imerode(BW,se);
figure,imshow(I);