If you have the ground truth (the ideal binary image that is desired after the binarization process), you can just check MSE (Mean Squared Error). The process that yields smaller MSE is better. Comparing without ground truth can be a little tricky.  

Well, I assume there is no ground truth. Then basically there is no way you can evaluate the performance of binarization process. However, I can thin of one idea, and that is if the image has some well defined edges, the method should keep the edges connected. SO the more disconnected edges you have, the worse will be the binarization method. How to determine the connectivity of the edges, well, you may count the number of connected components left after the binarization process. A smaller total number of connected component could be a good measure.

Mr.  @Haider Adnan Khan, thank you for your suggestions. Indeed, if the correct form of a binary image we have seen earlier, the MSE can be applied. But for my case the correct binary image was not there before, because I studied is the image of a manuscript that has been previously written two centuries. I agree with the idea of Mr. @Mansour Jamzad . For my case, I will first calculate how many objects from the original manuscript (although this will take a long time and it seems pointless). Then the binary image would I count the number of objects remaining are the same or not. Are there any other suggestions on how this?

I assume the manuscript is all text and has no drawings, then are basically trying to measure how accurately text is separated from background.

Of course you can perform an analysis of the connected objects in the image and determine character objects detected, and characters missed. However, in handwritten, texts, as characters usually are connected, maybe this is not a valid solution. Also quality of segmented text is not so easy to determine, since to compare shapes of handwritten text is not so easy.

An option in handwritten connected texts is to analyse the connectivity of the segmented stroke.

Of course, you can try measure the readability of the segmented text, but is also not so easy to determine in an objective way.

One interesting way of testing in a practical way the quality of the binarization is the accuracy of OCR processes on the binary image.

Still continue talking about binerisasi. I have a case where there is a part of the document image is very high levels of darkness. When done binarization other parts dark levels of average or normal, would be well separated between objects with background. But in the dark it appears a lot of the black part that almost cover the object. For this case, what is the binarization method that suitable in view of the levels of darkness not the same in all parts of the document.

You should use local thresholding (different thresholds for different regions)  rather than global thresholding (one single threshold for the whole image). Another possible approach could be applying Contrast-limited Adaptive Histogram Equalization (CLAHE) prior to binarization (thresholding).  In MATLAB this can be done using: adapthisteq function.

he simple answer is to use a local technique, that is using an algorithm which uses different threshold values in different image regions. The other option is to process the image with a  technique to normalize the bright, or improve contrast. You can use also a segmentation or filtering technique invariant to the average gray level as a part of your algorithm  prior to the binarization.

For effective thresholding that covering lots of dark in the document. i may suggest you go for adaptive thresholding and while implementing that divide the image into 4 parts and apply adaptive thresholding algoithm on them and combine those divided images. like this you may got good binarized image.....