How to solve lighting differences on stitching images?

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Laplacian blending will indeed make stitching seamless but it can't solve illumination problems. You will have to solve it yourself prior to stitching, i.e. you will have to multiply intensity of one of the images by some factor prior to stitching. The tricky part here is: how to determine this factor (while unknown parts of image are changed due to presence of new objects)? It can be solved in various ways. Personally I like the following one:

Step 0: Perform the registration between images. We are only interested in region that present in both images.

Step 1: Divide ROI into some reasonable amount of sub-region (10x10 for example).

Step 2: For each sub-region calculate intensity in both image, and calculate ratio between those intensities.

Step 3: Take median (M) of those ratios.

Step 4: Multiply relevant image by factor M.

Step 5: Perform stitching.

It works when most of the image area differs only because of illumination change, not because new objects are present.