Does Refreezing Ice Make it Clearer?

I am trying a series of experiments to find the best and most practical way to make clear ice in my home freezer. Future experiments include distilled vs. tap water, boiled vs. not boiled, hot vs. cold water, different shapes of containers, etc.

For this experiment, I started with plain tap water. I then froze it, took a picture, let it thaw back into liquid, and repeated the process. 

The test was to see if freezing and refreezing tap water would make it become clearer on successive freezings (due to the release of oxygen trapped in the ice) each time. 

So did it become clearer? The short answer is No. The long and visual answer is below.