Noises on the I2C BUS
There are 2 different problems:
1. noise
2. you are getting errors in communication  

I try to answer the second one:
Try to use resistors with lower values, like 2.2k instead of 4.7k. Do not use capacitors for SDA or SCL lines. Use lower I2C frequency than 1MHz, you are probably getting errors not because of noise, but because too high frequency (according to first photo the rising edge is too slow to use 1MHz).  

The noise could be caused by many things. Most likely long wires (parasitic inductance), not very good grounding, etc.