1. Check whether Noise Reduction is on or not.
2. Check your ISO setting whether too high or not. Recommended to be 200 and below.
3. If even NR is on, do a pixel mapping after taking 2-3 very long shutter shots with your lens cap on.

Hope these will solve your problem. Best is to post a sample with the EXIF intact.

Those blue and red dots you got from the picture are noise. Since the C-720UZ doesn't have a NR feature, there is not much you can do. However, do try the pixel mapping that I've mentioned in my previous post. It should help.

You can remove the noise using software such as Photoshop Elements.

That's a very good job done. Try to avoid the shortcomings of your camera when using it and you will take great pictures!