These images illustrate three different methods for reducing the effect of histogram equalization (HE). The original image (CA), 512x512 pixels, was equalized using a 41x41 window and 30 terms of the Fourier series to create image AD (repeated as images BD and CD). Histogram equalization was used to generate images AA, AB and AC. In each case the local histograms were smoothed using Gaussians with widths 90, 58 and 22 respectively.
Image BA was generated by subtracting the local mean, that is the mean in the window surrounding each pixel. Images BB and BC were created using the basic equalization process, but replacing the cumulation function (a sign function in full HE) with a signed power-law function. The power (alpha) was 0.65 for BB and 0.3 for BC. In fact, local-mean subtraction for image BA was implemented by using alpha=1. These results are very similar to Gaussian smoothing, and the Gaussian widths were chosen to illustrate this. Differences in detail are perhaps most clear in the petals.
One can recover the original image CA by replacing the local mean in BA. Images CB and CC were created by replacing scaled proportions (beta) of the local mean in BB and BC. The proportions were beta=0.65 for CB and 0.3 for CC. As can be seen, the degree of contrast equalization can be varied between nothing (that is retaining the original image) and full HE by varying alpha and setting beta to a similar value.
These widths for images AA through AC correspond to the standard deviation of a Gaussian in grey levels (0-255), and the process was implemented using the (very close) approximation of modifying the Fourier series coefficients.
The arrangement of the images matches that in the paper. However, on your system you may prefer the images arranged horizontally.