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Still Needs:
The original image.
The image after sigmoidal contrast. Notice that the histogram shown in the screenshot includes the boundary around the image.
The raw image after its histogram has been matched to that of the sigmoidal contrast image. Notice that it is darker than the original and the histogram looks different.
Here is the histogram of just inner part of the "histogram matching" image (without the boundary); that image inherited its histogram from the entire "sigmoidal contrast" image (including the boundary, not just the inner part). The histogram of the inner part of the "histogram matching" image agrees nicely with the histogram of the entire "sigmoidal contrast" image (which is the histogram that we were trying to match).