Use something like bwlabel() to label the image. Then use regionprops to find the Euler number for each of the regions. For contours that have no loops, the number will be 1 (one object minus zero holes), and for open holes the number will be 0 (one object minus one hole).
Note that the long contour near your bright spot includes a closed loop and so would be characteristic 0 just like the open circles. To distinguish that, you could look at the regionprops Eccentricity, which will be close to 0 for the circles.
