The voxel rendering algorithm already collected depth information. A while ago I added some code that collects this information and stores it in a depth buffer. Hence it can be used in post-processing steps, such as the screen space ambient occlusion (SSAO) lighting technique. I implemented the SSAO filter using this tutorial. However, my rendering algorithm does not produce normal information, so I had to change it slightly.
I also added some code to allow super sampling anti aliasing (SSAA), which basically renders the image at a much higher resolution and then downscales this to the output resolution. This a prohibitively expensive anti-aliasing technique, but still, my renderer takes only 2-4 seconds to render a 7680×4320 image and downscale it to 1920×1080, which results in 16x SSAA. Combined with the SSAO this resulted in the pictures attached below. To emphasize the effect of SSAO, I exaggerated the resulting shadows.