If you are expecting the Arduino to display the video and more importantly blur the video, stop right there. An Arduino does not have the processing power to do video on its own, let alone blur the images.
A Rasberry Pi probably has enough power to display the video, but perhaps not enough power to blur the video. There I would store the video in several versions with different amounts of blurryness that you preprocessed ahead of time, and just use the distance sensor to select which video to output. http://www.raspberrypi.org/