Contrast Enhancement For An Image Using A Gaussian Function
Convolution is a relatively simple operation to perform, but to make it fast involves a little extra work (Wikipedia article on convolution theorem). In general, if you convolve a discretized function F by a different discretized function G, you are simply performing the following operation:

$$(F\otimes G)(x) = \displaystyle\sum_{i=0}^NF(x-i)*G(i)$$

Where N is the number of discrete values of G. Notice this is effectively just summing the product of F and G "centered" around $x$.

In 2D (which is what an image is), this looks like:

$$(F\otimes G)(x,y) = \displaystyle\sum_{i=0}^N\displaystyle\sum_{j=0}^MF(x-i,y-j)*G(i,j)$$

The paper you're reading tells you that F is the V channel of the image in HSV space, and G is a 2D gaussian function. Code-wise, the above form should translate very simply into a double-nested for loop.

The paper should also tell you what the mean/variance of the gaussian function is as well as the number of samples it contains, but sometimes that information is omitted.