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(1) is just a minmax normalization

(2) is pow()

(3) is exp() (with a "funny" normalization term)

all in all, we get:

Mat im = imread("veins.png",0);
Ptr<CLAHE> cl = createCLAHE();
Mat draw, clout,g,final;
cl->apply(im,clout);

normalize(clout, g, 0, 1, NORM_MINMAX, CV_32F);  // (1) , and we need float values further on
draw = g.clone();                     // visualization
cv::pow(g,.4,g);                      // (2)  with ẞ=0.4 (guessed)
hconcat(draw,g,draw);                 // visualization
cv::exp(-g,final);                    // (3)
hconcat(draw,final,draw);             // visualization
float c = (255.0)/(exp(-1.) - 1.);    // (3)
(final - 1.) *= c;                    // (3)
hconcat(draw,final,draw);             // visualization
imshow("fuzzy",draw);
waitKey();